tag:blogger.com,1999:blog-26881382909719158302024-03-14T10:39:03.131-07:00Interdependent ScienceJimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.comBlogger133125tag:blogger.com,1999:blog-2688138290971915830.post-23272807419071369332024-03-13T19:07:00.000-07:002024-03-14T10:38:29.470-07:00Finding the Phase Transition <div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwcrMzflQTi4GOCGQD8FEOLwHdr5sGDoqUEkpXFSgRHPrvivceDCTbftZdRvt_V1s7reOTnTvQZZG4rKpQ6UbN9krgBoILaOi1kPuRL33Fk_UMl2u3uiKG3tFeHzYqAT5bV68CSDrfAxKksADulPd92tG6UnEiQ1vi3b2DzrYpdOiUhpoPau6L0j1eLUQ/s910/31edo%20peak.jpg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="660" data-original-width="910" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwcrMzflQTi4GOCGQD8FEOLwHdr5sGDoqUEkpXFSgRHPrvivceDCTbftZdRvt_V1s7reOTnTvQZZG4rKpQ6UbN9krgBoILaOi1kPuRL33Fk_UMl2u3uiKG3tFeHzYqAT5bV68CSDrfAxKksADulPd92tG6UnEiQ1vi3b2DzrYpdOiUhpoPau6L0j1eLUQ/s400/31edo%20peak.jpg"/></a></div>
<p>
I've been using an algorithmic approach to music composition, based on thermodynamic simulation. One advantage of an algorithmic approach is its generality. One can use the same software to generate music in a wide variety of tuning systems. Other parameters can be adjusted easily, too. Of course, no algorithm is likely to generate music of the quality that a skilled human composer could produce. On the other hand, there can be some value to music that is outside the usual patterns.
<p>
These days a different algorithmic approach to composition has gained some traction, an approach that works with a large number of existing compositions, extracts some patterns, and then follows these patterns to generate a new composition that sounds much like the existing compositions. Just to be clear, the approach I am using does not use existing compositions in the execution of the software. I listen to a lot of music, and I listen to the output of my software; I tweak my software in an effort to coax it into producing something as music-like as I can. But this is a long way from the Deep Learning methods of the predominant Artificial Intelligence software.
<p>
Thermodynamic simulation is a randomized process that repeatedly adjusts the values of a large number of inter-related variables. The variables are connected in some kind of network, that defines the interactions between the variables.
In my music composition software, the variables are the pitches of what is to be played at each particular time by each particular voice. When a voice is to sound a pitch at one time and then a second pitch at a succeeding time, these pitches should be nicely related if the music is to sound good. The pitches should not be too far apart, and should be reasonably consonant. Similarly, if one voice is sounding one pitch, and another voice is sounding another pitch at the same time, these two pitches need to be consonant if the music is to sound good.
<p>
If a piece is ten minutes long, each voice might involve a thousand pitch choices. During the simulation, all of these choices have provisional pitch values. Again and again, one or another of these choices is selected at random, and then the pitch selection for that specific time is reevaluated, in the context of the provisional choices in place for what that voice is to sound before and after, etc. The software will choose a new pitch for that voice at that time, preferring pitches that are consonant with the other pitch choices nearby in space and time. Then some other voice and time will have its pitch reevaluated. Over the course of the composition process, each pitch will change hundreds of times. Other related pitches will have changed between one evaluation and the next, so which pitch is most consonant may well change over the course of the simulation.
<p>
Thermodynamic simulation is driven by a key parameter, the temperature. At high temperature, the preference for consonant pitches is not very strong. At low temperature, only the most consonant pitches will be assigned. At very high temperature, the simulation will assign pitches essentially at random, so the music will be pure noise. At very low temperature the simulation will strive to maximize consonance. But if the pitches are initially very random and then the simulation is run at very low temperature, very often it will happen that in evaluating the best pitch for a particular voice at a particular time, the related pitches don't pull in a consistent direction. One pitch choice will be consonant with some neighbors but dissonant with others. There will often be no choice that is consonant with all the nearby pitches.
<p>
The way to generate pitch assignments that are mutually consonant throughout the network is to start the simulation at a high temperature and then to slowly lower the temperature. Each pitch selection provides some communication between more remote regions of the composition. The entire system can eventually negotiate mutually agreeable pitch choices in this way.
<p>
Thermodynamic simulation thus has the capability of generating pure noise at high temperature and pure order at low temperature. Neither of these makes interesting music: either extreme is quite dull! Interesting music happens in the region between total noise and total order.
<p>
The fascinating thing about this kind of thermodynamic system, whether simulated or in real physical systems, is that the transition between order and disorder is often not smooth and gradual, but can be quite abrupt. Right at the boundary between the ordered phase and the disordered phase, the system can exhibit fractal fluctuations as it wavers between the behaviors of the different phases. Fractal fluctuations are a characteristic of interesting music. So the approach I generally use for generating interesting music with thermodynamic simulation is to set the temperature to where the phase transition happens and make the pitch choices at that temperature, where consonant choices are preferred but not too strongly.
<p>
One challenge with this approach is that the temperature at which the phase transition happens is not something one can calculate in any simple way. One is basically stuck with simulating the system at different temperatures, observing its behaviors, and identifying an abupt shift. That's what the graph at the top of this post illustrates. At each temperature, the system will settle into an overall level of consonance, which corresponds to energy in thermodynamics. A highly consonant system has very low energy; a highly dissonant system has very high energy.
<p>
The graph above has a clear enough abrupt shift at a temperature of around 380. There is a sudden drop in the energy with a small change in the temperature. Locating this sudden drop is a bit tricky though, because of the random nature of the simulation. The energy is always bouncing around even at a fixed temperature. What I do to filter out this randomness is to fit a smooth curve through each small family of temperature and energy measurements. Then I look for which such small curve shows the most abrupt change in energy over a small change in temperature. That tells me the temperature of the phase transition.
<p>
Once the temperature of the phase transition has been determined, I can set the simulation temperature to that value and let the simulation run so all the pitch choices come to reflect that boundary behavior, to exhibit fractal fluctuations.
<p>
Here are two pieces generated using this approach:
<ul>
<li><a href="https://app.box.com/s/wj77440ubj0f1quezb1me5myxc05wi1u">50edo</a>
<li><a href="https://app.box.com/s/x0okfj7ktbh48msja07itqn0k0gct6q4">31edo</a>
</ul>
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com0tag:blogger.com,1999:blog-2688138290971915830.post-30664590616602022612024-02-29T21:32:00.000-08:002024-03-01T07:53:34.592-08:00A Tale of Two Unconventional TuningsNowadays the notes available on a conventional piano, seven white keys and five black keys per octave, form the building blocks for almost all the music in circulation. And of course these building blocks have been very effective at enabling the crafting of a vast treasure chest of music, diverse, profound, and beautiful. And yet, there is value in exploring other tuning systems.
<ul>
<li>Around the world, there are still many different traditional tuning systems in use.
<li>Tuning systems evolved over the centuries in Europe, only settling on the present convention some two centuries ago.
<li>Different tuning systems enable different musical structures; they are a rich compositional resource.
<li>Conventional tuning can be better understood in perspective, as being one alternative among many.
</ul>
<p>
One could spend a lifetime learning about different tuning systems, their histories and features etc. But sometimes when encountering a large building it can be difficult to find an entrance! Recently I have been exploring the tuning system that divides octaves into fifty equal intervals, rather than the conventional twelve. Dividing octaves into fifty three equal intervals is another useful tuning. The sizes of the intervals in these two tunings is not very different, yet the tunings have quite different strengths. Comparing these two systems could work as a doorway into the world of alternate tunings.
<p>
A general foundation for tuning theory is the observation that two pitches sound consonant when the ratio between their frequencies is a simple rational fraction. For example, the A above middle C is conventionally tuned to 440 Hertz. The next higher A, an octave higher, is at 880 Hz, a ratio of 2:1. If one tunes the E in between to 660 Hz, it will sound very nicely consonant with either of the As, with ratios of a perfect fifth, 3:2, or a perfect fourth, 4:3. Tuning the C# to 550 Hz will complete a consonant major triad. The interval from the A of 440 Hz and the C# of 550 Hz is a major third, with a frequency ratio of 5:4. The interval from the C# of 550 Hz to the E of 660 Hz is a minor third, with frequency ratio 6:5.
<p>
The pitches involved in a piece of music form a network. Each pitch is related to several other pitches, and these related pitchs then relate to yet other pitches. Pitches are thus related by chains of simple intervals. The whole pitch space forms a kind of network. If the simple relationships are built from the consonant relationships of fifths and thirds described above, the network of pitches will look something like this:
<p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFuYX-kAoTMLLO4U6fZ1OeQcmoKdn5jMweaiPp31LiYWlL4hmkvHXGr5SQMyurpijNTuy3BqS5OAL13s5c9z-8P1vBGbfBdclQwApFwI6lZBZgIuG9NHLiaHdwqfR2i977psHEkx7dpJ2siI4lBttR64k2LpqzYkY7nNXQFLTrlYzfQqHJaEryfgdy-VQ/s513/tale%20just.jpg" style="display: block; padding: 1em 0; text-align: center; clear: left; float: left;"><img alt="" border="0" width="400" data-original-height="101" data-original-width="513" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFuYX-kAoTMLLO4U6fZ1OeQcmoKdn5jMweaiPp31LiYWlL4hmkvHXGr5SQMyurpijNTuy3BqS5OAL13s5c9z-8P1vBGbfBdclQwApFwI6lZBZgIuG9NHLiaHdwqfR2i977psHEkx7dpJ2siI4lBttR64k2LpqzYkY7nNXQFLTrlYzfQqHJaEryfgdy-VQ/s400/tale%20just.jpg"/></a></div>
<p>
These pitches are all inside a single octave range - the network could be replicated in as many octaves as needed. The network can also be extended arbitrarily in any and all directions.
<p>
While one can make perfectly good music with a tuning system like this, with very precisely consonant frequency ratios, it does run into difficulties. As the network is extended, each octave gets broken up more and more finely, without limit. It's hard to build instruments that can play so many notes, hard for players to hit the right notes, and hard for listeners to distinguish among so many notes. Over the centuries, musicians, composers, and instrument builders have developed simpler tuning systems that approximate these ideal intervals while avoiding the infinite division problem. And then music has evolved to take advantage of opportunities these simpler tuning systems provide for harmonic movement. A tuning system is a network of pitches with a particular shape. Music is then a kind of dance that moves around through that shape.
<p>
The fine divisions brought about by precise consonance first arise with the 81:80 pitch on the right column of the tuning network above. It is very difficult to distinguish that from the 1:1 in the center. So the first tuning simplification is to adjust the pitches in the network somehow so that 81:80 is changed back to 1:1. This changes the shape of the tuning network from a flat plane to a cylinder. If one travels in a suitable constant direction on the surface of a cylinder, one can end up back where one started.
<p>
There are many ways to adjust the pitches in the network so the 81:80 is flattened slightly to become 1:1, but in general this tuning system is known as meantone. The way pitches are named in European music is a reflection of the meantone system:
<p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjV8nkuXn34tW8BPvk8D82cisbpLMj5w9WwYUGJC4TMxCV-3JXCu62tHKuFkTmMU64WsE41KTY9720MbWe0EfO-H8iiUK-DoQVB0Taur-HVFND4c2MoO92tbtlEYoDpjPvOZQriX3hLH-41tsRa8Pe6clP8k2KSvmGVfrKl1YlpWrXBpPOJOJC5TaXIYrQ/s761/tale%20names.jpg" style="display: block; padding: 1em 0; text-align: center; clear: left; float: left;"><img alt="" border="0" width="400" data-original-height="181" data-original-width="761" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjV8nkuXn34tW8BPvk8D82cisbpLMj5w9WwYUGJC4TMxCV-3JXCu62tHKuFkTmMU64WsE41KTY9720MbWe0EfO-H8iiUK-DoQVB0Taur-HVFND4c2MoO92tbtlEYoDpjPvOZQriX3hLH-41tsRa8Pe6clP8k2KSvmGVfrKl1YlpWrXBpPOJOJC5TaXIYrQ/s400/tale%20names.jpg"/></a></div>
<p>
While this system does allow unbounded movement, that movement needs to flow around the cylinder, along the diagonal strip where the sharps and flats don't get too wild. Old keyboard instruments sometimes have extra black keys to accommodate a wider range of movement, but still, it can be challenging to dance freely when there is an abrupt edge that one must steer away from. So the next step of evolution is to wrap the cylinder into a torus:
<p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgv7MtDnBV4_2vFbm7ZwRFPbR9x0jHcBr4sXtWbuKoK0GM5Pl4yzdWWfxzGQTkXTznCKVeZMfKZevmYGlJRF4zcxhbVgu5cl5O0YNNCE9PMKKQ9UfwDpSLT5eFYzOHdaakV2ulRwfe-dIGOIlS7VkSpRTW9zZKJqVq1Q7YRBST0acoy7mmH0LR9eBQNCIU/s761/tale%20well.jpg" style="display: block; padding: 1em 0; text-align: center; clear: left; float: left;"><img alt="" border="0" width="400" data-original-height="181" data-original-width="761" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgv7MtDnBV4_2vFbm7ZwRFPbR9x0jHcBr4sXtWbuKoK0GM5Pl4yzdWWfxzGQTkXTznCKVeZMfKZevmYGlJRF4zcxhbVgu5cl5O0YNNCE9PMKKQ9UfwDpSLT5eFYzOHdaakV2ulRwfe-dIGOIlS7VkSpRTW9zZKJqVq1Q7YRBST0acoy7mmH0LR9eBQNCIU/s400/tale%20well.jpg"/></a></div>
<p>
If one moves a perfect fifth from G#, one arrives at Eb. The network of pitches has been tweaked somehow so that D# and Eb are the same pitch. There are various ways to do this, but the simplest way is to adjust all the fiths and thirds in the same way, so the system is totally uniform. This is our conventional tuning of today.
<p>
To review the development so far:
A network of precise consonances splinters the pitch space to an impractical unbounded extent. Adjusting, or tempering, the intervals allow the network to wrap back on itself, so the number of pitches required can be limited.
<p>
Fundamentally, a tuning system is a compromise between simplicity and precision. But tuning must serve music. The shape of the tuning network enables some kinds of harmonic movement but prevents other sorts. Music and tuning evolve in response to each other, meeting each other's demands and taking advantage of each other's opportunities.
<p>
One can build a tuning system by dividing octaves into equal intervals of any number. A good tuning system will provide intervals that are close approximations of the precise consonances of 3:2 and 5:4. Dividing octaves into 50 or 53 equal parts will provide reasonable approximations:
<p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2pr_lMWS0yLyqWsZFmaV_qQS0M1Th6WzYPiYpKDPhEJ16yptSaltHEYHkJT55k9ghoPgtjMKy5xH5FTP5smYKQcmOLuMNg4nAioETCVi5A0FcbDLY3QrUPMQfUT78pk-BSL287rXb1YmPd3fslUTSizVNajsVvmZFbVbildIGDD5vhw_IZD49Bf7hCNA/s257/tale%20table.jpg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="109" data-original-width="257" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2pr_lMWS0yLyqWsZFmaV_qQS0M1Th6WzYPiYpKDPhEJ16yptSaltHEYHkJT55k9ghoPgtjMKy5xH5FTP5smYKQcmOLuMNg4nAioETCVi5A0FcbDLY3QrUPMQfUT78pk-BSL287rXb1YmPd3fslUTSizVNajsVvmZFbVbildIGDD5vhw_IZD49Bf7hCNA/s320/tale%20table.jpg"/></a></div>
<p>
This table gives the error, in cents, for each tuning system for each consonant interval. One can see that the conventional tuning system has somewhat large errors for several intervals, though it comes quite close for 3:2. The 53 steps per octave system is quite accurate for all the intervals. The 50 step system is not so good for 3:2, but it is at least better than conventional tuning for the thirds 5:4 and 6:5.
<p>
It might seem that, since 53 steps per octave is only slightly more than 50 steps per octave, and provides a significant improvement in precision, that the 50 step per octave system is not very useful. But beyond simplicity and precision, one must look at the shape of the tuning network:
<p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi11eHCS8Q74ZImNNZ8q2aP18upH_xECrZByh2ymRSkq3O0d0CpGQK0qqcwOex310aYRTIPoZGzR_XBmVUUQ-oGKptkwQ41Qru57_oVRGnH5anLo3wBBA4qS7Qp6YXRQ8VWFxd8_Eofvdr8Pf3ovVu8IKe4o98TjNPQKYheoDn2NfCKkHX1l-4P9CdOqO8/s430/50edo%20scale%203%20of%205.jpg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="381" data-original-width="430" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi11eHCS8Q74ZImNNZ8q2aP18upH_xECrZByh2ymRSkq3O0d0CpGQK0qqcwOex310aYRTIPoZGzR_XBmVUUQ-oGKptkwQ41Qru57_oVRGnH5anLo3wBBA4qS7Qp6YXRQ8VWFxd8_Eofvdr8Pf3ovVu8IKe4o98TjNPQKYheoDn2NfCKkHX1l-4P9CdOqO8/s320/50edo%20scale%203%20of%205.jpg"/></a></div>
<p>
The bright blue highlighted cells marked "0" show the way the torus is wrapped back on itself. Those closely spaced "0" cells along a line sloping slightly down to the right, those cells are wrapped in exactly the way that the meantone tuning system is wrapped. What this means is that most any music written for the meantone system will be playable in the 50 step per octave system. The 50 step system will support even triple sharps and triple flats. It would be a rare piece of music that requires more sharps and flats than that!
<p>
The 53 step per octave system has a very different shape:
<p>
<div class="separator" style="clear: both;"> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEGhFbXibRYZjLOvmMDTsK2ubocOgr1fhnDkrSo4ExawTDy3YSzsYrAgv-8qM00trnBqs9iJeyZOqD_Kf7Rt5ztiIcy42tTmc-tz7rrNFeeKUnIW9fwDwBiOq7DZNIMpApEFwY37c5ehtlN8dUFXESl5r6lBIWYGvMBJ-BxsKXdRP0xWvfEBLp3_e2GcQ/s417/53edo%20scale%2031%20at%2012.jpg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="401" data-original-width="417" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEGhFbXibRYZjLOvmMDTsK2ubocOgr1fhnDkrSo4ExawTDy3YSzsYrAgv-8qM00trnBqs9iJeyZOqD_Kf7Rt5ztiIcy42tTmc-tz7rrNFeeKUnIW9fwDwBiOq7DZNIMpApEFwY37c5ehtlN8dUFXESl5r6lBIWYGvMBJ-BxsKXdRP0xWvfEBLp3_e2GcQ/s320/53edo%20scale%2031%20at%2012.jpg"/></a></div>
<p>
The pattern of repeated cells, the way the tuning torus is wrapped, does not match the meantone system at all. Music written in the meantone system will fail to return or connect back properly if one tries to play it in the 53 step system.
<p>
I have been exploring some of the unconventional musical possibilities of these two tuning systems. For each, I picked a subset of the available pitches to work as a scale. In both systems I built the scale to form a path from lower left to upper right, which is, roughly speaking, a chromatic scale. I then used my algorithmic composition software to generate some music that would flow with the shapes of the scales:
<ul>
<li><a href="https://app.box.com/s/t2pjiyu0xbqii5ywncgc2yi1q8tft7dv" target="_blank">50 step</a>
<li><a href="https://app.box.com/s/e564gmttmg9mxkj6gcl1e9i3ogvc7b2w" target="_blank">53 step</a>
</ul>
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com0tag:blogger.com,1999:blog-2688138290971915830.post-11106925315327155732023-04-23T20:49:00.004-07:002023-11-05T10:33:53.274-08:00The Disintegration of ScienceIn the 1990s, the science wars were fought between advocates of science and folks who saw flaws in science. Nowadays, the science wars underway are between folks claiming scientific support for wildly differing claims. Does our global use of fossil fuels for energy have significant impact on the climate? Are covid vaccines safe and effective? Of course scientific progress is driven by debate, so perhaps these disagreements are healthy.
<p>
A healthy organism is constantly fighting off infections and other disturbances. The integrity of an organism is constantly under threat. For a while, various homeostatic processes manage to preserve that integrity, but eventually those processes are overwhelmed, and the organism loses its integrity. Sometimes this lack of integrity means the death of the organism, but it can also mean division into multiple separate organisms. Presevation of integrity and subsequent loss of integrity can happen at many scales, from single cells to cell colonies to insect colonies to human societies.
<p>
Science nowadays, for the most part, maintains a very healthy level of integrity. A key component of this integrity is the vision of scientific knowledge as a coherent whole. All the bits and pieces of our scientific knowledge fit together somehow, or eventually will. We're always discovering inconsistencies, but our processes of research and mutual critique keep these inconsistencies under sufficient control that the overall integrity of the system is not under threat. The loud arguments over e.g. climate change are a sore point, but they are certainly at a small enough scale not to threaten the entire system.
<p>
And yet... these superficial rashes could be symptoms of a larger systemic problem. Is the rough coherence of scientific knowledge something inevitable? What processes maintain this coherence? What could threaten this coherence?
<p>
The coherence of science is maintained by a kind of circulatory system. Information circulates: researchers publish papers but also exchange preliminary results, critiques of draft versions of papers, and also text books and other coordinated summaries of scientific knowledge. People circulate: researchers meet to discuss their work, but also visit each other's laboratories to collaborate on research. Students are trained in one research organization and then get hired to work in other research organizations. Equipment and materials circulate: measuring devices can be calibrated to common standards. Experimental samples are exchanged between laboratories.
<p>
What would precipitate the disintegration of science would be the breakdown of this circulatory system. Circulation is supported by the larger social context. Freedom of the press allows research results to be published. Freedom of travel allows people to collaborate. Free trade enables the exchange of equipment and materials.
<p>
These freedoms are the hallmarks of liberal society. Science and liberal society have emerged together since early modern times. A free market of ideas allows the best ideas to emerge. Basing policy on effective ideas leads to success and growth, to progress. This progress provides a platform for further exploration, leading to better ideas, more effective policies, and further growth. We have been riding this feedback loop for four hundred years. It's not just science that is coherent, but our global society.
<p>
The general pattern in biological systems is that growth is followed by decline. Perhaps this time it will be different, but that is a position that requires a lot of faith! Just as science, liberalism, and progress supported each other in a feedback loop of expansion, there are signs that the same feedback loop may be picking up momentum in the direction of decline.
<p>
Of course one can pick a measure of prosperity to support whatever argument one wishes to advance. But it really seems like the financial crash of 2008 is one we have not really recovered from. The rise of vehement anti-liberalism is largely driven by the failure of liberalism. We were promised progress but that is not what we are experiencing. The underlying cause for the lack of progress is probably our reaching various ecological limits, but that's not a message that sells. Science and liberalism have built their castles on progress. As progress falters, so will liberalism, and so will science. Liberalism maintained the circulatory system on which scientific coherence depended.
<p>
Of course change is the nature of things. How science might best maintain itself in a new dark age, that is one worthy puzzle. It is valuable to step back a bit, to try to think strategically. How things will unfold in the coming decades and centuries, it is impossible to foresee with any accuracy. What is more feasible is to consider a range of possible trajectories, and to prepare responses across some plausible range. Insurance policies, diversified portfolios, hedged bets: these are effective approaches to dealing with uncertainty. We need to bring these approaches into our investments in scientific research programs.
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com0tag:blogger.com,1999:blog-2688138290971915830.post-52862180698200670322023-03-30T10:56:00.002-07:002024-02-29T22:09:35.665-08:00Heat Pump EfficiencyThermodynamics is a fundamental branch of physics. It gets a bit subtle: I find myself getting tripped up often enough!
<p>
The cornerstone of thermodynamics is the Carnot cycle, an ideal process for converting heat to work. It's a model for what steam engines do, for example. The Carnot cycle sets a limit on how efficient an engine can be: it is not possible to convert all the energy from heat to mechanical work.
<p>
A heat pump is simply an engine running backwards. An engine has heat flowing from a hot reservoir to a cold reservoir, converting some of that heat to mechanical work. A heat pump uses mechanical work to push heat from a cold reservoir to a hot reservoir. The amount of heat added to the hot reservoir will be the sum of the energy from the work and the heat energy removed from the cold reservoir.
<p>
To heat a home, one can use a natural gas furnace, or one can use a heat pump. The heat pump runs off electricity, much of which is generated from an engine running off natural gas. Energy is lost when the natural gas heat energy is converted to electricity, but then energy is gained when the electricity is used to heat the home. Since the heat pump is just an engine running backwards, these losses and gains are in some sense reflections of each other, and might seem to cancel out. But they don't!
<p>
The missing detail is that there are three heat reservoirs involved. The engine at the utility power generation plant has energy flowing from a furnace to the environment, converting some of that to electrical energy. The heat pump has energy flowing from the environment to the interior living space, driving that with electrical energy:
<p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiFRACyVhNDDRuf1yJFWO_1XPi4_NguGAZYXG_LiEJId3mCfgTgSsE9GTI1soK2fYiZztoD1NyHNMHMv4HV5sniRB16C4QYQFNZSdq0_2vhvwnnc4Ko6IjTI08kjO-5FOO32JIEcFXk1jp5NReeT7w1WaL-ptdrTM5a7pFWbj4J4YE2SP-cZsY-D1U/s960/heat%20pumps.jpg" style="display: block; padding: 1em 0; text-align: center; clear: left; float: left;"><img alt="" border="0" width="400" data-original-height="720" data-original-width="960" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiFRACyVhNDDRuf1yJFWO_1XPi4_NguGAZYXG_LiEJId3mCfgTgSsE9GTI1soK2fYiZztoD1NyHNMHMv4HV5sniRB16C4QYQFNZSdq0_2vhvwnnc4Ko6IjTI08kjO-5FOO32JIEcFXk1jp5NReeT7w1WaL-ptdrTM5a7pFWbj4J4YE2SP-cZsY-D1U/s400/heat%20pumps.jpg"/></a></div>
<p>
The two efficiency factors have inverse forms, but the numbers involved are different, so they don't cancel each other.
<p>
Plugging in some roughly plausible numbers, a graph can be generated for maximum effiency of the overall system as a function of the outside temperature. As the outside temperature warms to near the interior living space temperature, the round trip efficiency increases without bound. At cold temperatures, the utility's power generation engine can run more efficiently, but the reduction in effectiveness of the heat pump is more dramatic, so the overall effiency is reduced.
<p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUmVBcVPyvzrwGSUfZA7h1IsaSnhyvsz0reVYGVLcXOxAfL1tD9EC5MeQXPx_uWZn3v7pBoqnlfJ5chWfMTBlsuJw8lnkAoIVHZCTvxX5WaiJGe9NC5q_FwAqVmviv6Ziw4zdVoXXt13yBwGKhW3eJTCmT1L5XHJaZccy12Bm7nEBGfz7wzZs2163l/s911/heat%20pump%20graph.jpg" style="display: block; padding: 1em 0; text-align: center; clear: left; float: left;"><img alt="" border="0" width="400" data-original-height="663" data-original-width="911" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUmVBcVPyvzrwGSUfZA7h1IsaSnhyvsz0reVYGVLcXOxAfL1tD9EC5MeQXPx_uWZn3v7pBoqnlfJ5chWfMTBlsuJw8lnkAoIVHZCTvxX5WaiJGe9NC5q_FwAqVmviv6Ziw4zdVoXXt13yBwGKhW3eJTCmT1L5XHJaZccy12Bm7nEBGfz7wzZs2163l/s400/heat%20pump%20graph.jpg"/></a></div>JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com0tag:blogger.com,1999:blog-2688138290971915830.post-60265396714691347632023-03-24T14:25:00.002-07:002024-02-29T22:09:46.961-08:00Aperiodic TilingI've been seeing <a href="https://www.newscientist.com/article/2365363-mathematicians-discover-shape-that-can-tile-a-wall-and-never-repeat/" target="_blank">reports of an aperiodic tiling</a>. At first, I couldn't imagine how a tiling could be aperiodic. Now the pendulum has swung to the other extreme, where it seems trivial:
<p><div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAaWdDpqCq7elbE-dW42B8bx7GAYpPqHCBIQOTjLV7304ct1fMSpasIhLgfJ5hULPsI53lBaJn3PdwkgPrGa5D50YJq8Y41fAyZXLUjiXGJMzXeoz17BizbzF4xE5-hkyMP7USOQJBJDDsSY6D_wJH6VLxsVvOmBHXgzvB4G491aBhIShuJ5OTvDe7/s960/aperiodic.jpg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="720" data-original-width="960" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAaWdDpqCq7elbE-dW42B8bx7GAYpPqHCBIQOTjLV7304ct1fMSpasIhLgfJ5hULPsI53lBaJn3PdwkgPrGa5D50YJq8Y41fAyZXLUjiXGJMzXeoz17BizbzF4xE5-hkyMP7USOQJBJDDsSY6D_wJH6VLxsVvOmBHXgzvB4G491aBhIShuJ5OTvDe7/s400/aperiodic.jpg"/></a></div>
<p>
The tile is just a 1x2 rectangle. Mostly they are all placed vertically, but there is a line along which horizontal tiles are placed. One could interpret the pattern of absence or presence of a horizontal tile in the sequence of columns as expressing a fraction in base 2. If the fraction is irrational, the pattern will be aperiodic. Hmmm, even if there was just one horizontal tile in the middle, the pattern would be aperiodic!
<p>
There must be some trickier definition in play, of what aperiodic means. But anyway, now it doesn't seem so impossible!
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com1tag:blogger.com,1999:blog-2688138290971915830.post-17157434569921143302023-03-17T19:30:00.008-07:002023-11-05T10:34:01.574-08:00Scientific Equipment <div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKvkTRX_hBDbbJZqxB1MlNtzgykRBp2Wkt3cQWmJXnwFP-k5yA7_mo4CX5yjesOUNfaV_h1ldpgUDvgHycVRF6BG9XEPzR98D_GaNWx5RDzlsYo5MRmMGLdQGGG3kMuoRx3slo02nnFtqa7DU0dICHm5Zcx4meT-7O1qZ7JfF6PvBlYS1I-UMjgbyS/s958/Science%20System.jpg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="300" data-original-width="958" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKvkTRX_hBDbbJZqxB1MlNtzgykRBp2Wkt3cQWmJXnwFP-k5yA7_mo4CX5yjesOUNfaV_h1ldpgUDvgHycVRF6BG9XEPzR98D_GaNWx5RDzlsYo5MRmMGLdQGGG3kMuoRx3slo02nnFtqa7DU0dICHm5Zcx4meT-7O1qZ7JfF6PvBlYS1I-UMjgbyS/s400/Science%20System.jpg"/></a></div>
<p>
Galileo did not invent the telescope. Galileo looked at the night sky with a telescope that someone else built. Scientists do sometimes invent and build the equipment they need, but in general scientists take advantage of existing equipment to enable them to do science. Science is not a free-standing activity; it is an integral part of a much larger world. Science uses its connections with the world, just as the world uses science.
<p>
This relationship of mutual use creates a self-reinforcing feedback loop. Scientific discoveries enable new equipment to be constructed, and new equipment enables yet further scientific discoveries. The astounding technological capabilities of our time are the fruit of this system. However, the system is more complex. Our global-scale industry has global-scale impact on the environment. Climate change may be the most immediate concern, but we are seeing many other effects too. It is of course difficult to predict exactly how environmental limits will impace the availability of scientific equipment. But a starting point would be a reflection on the variety of ways that science uses what the world makes available.
<p>
Instruments with which to observe and measure natural phenomena are surely at the head of the list. Nowadays we have telescopes in orbit, detecting a wide range of electromagnetic frequencies: not just in orbit around earth, but around other planets too. And we have robots on the surface of Mars, observing at close range. At the tiniest scale we have particle accelerators and electron microscopes. Chromotography, spectroscopy, magnetic resonance imagery... a catalog of today's observation and measurement equipment would fill an encyclopedia.
<p>
Another way that science uses what industry provides is the acquisition of raw materials. All kinds of very pure simple and complex substances are available. There is also a rich variety of materials processing equipment by which raw materials can be processed to form both experimental samples and also custom observational devices. Vacuum pumps are a curious sort of equipment, since their function is to remove material rather than to supply it. But vacuum pumps are fundamental to preparing a suitable environment for observations, back to the time of Boyle at the birth of modern science.
<p>
Recording experimental observations can rely on little more than paper and pencil, though nowadays all sorts of automated recording devices make continuous accurate measurement and recording possible. A variety of automated analysis can be performed by computers, so the scientist need only attend to a summary report.
<p>
Science is a communal enterprise. Scientists compare results, critique each other's methods, exchange tools and materials, hire each other's students, etc. The worldwide transportation and communication networks make these exchanges possible. Scientists travel, too, to observe phenomena that occur at special locations, such as biological species in their native habitat, or geological phenomena in place.
<p>
Another sort of equipment that science needs is social. At the most basic level, there need to be scientists, people with the capability and freedom to pursue scientific research. The various physical equipment necessary must not only exist but be made available for use by scientists. For the self-amplifying feedback loop of scientific advancement to work, industry must be confident in the validity of scientific results so that the know-how produced by science will be applied to produce the next generation of more capable scientific equipment.
<p>
The reader is invited to augment this list. But a further exercise is to consider what impact environmental limits might have on any of these sorts of equipment. There could be other potential feedback loops that get excited as we enter some new regime of system behavior. It is not impossible that environmental limits push industry into less efficient processes, which accelerate the impact of those limits.
<p>
It seems clear enough that science has a large share of responsibility for creating our modern world, with all its miraculous technological capabilities. That is another facet of the self-amplifying feedback loop: powerful people understand how science has enhanced their power, and so they promote scientific research. We certainly seems to be at very real risk of entering a new regime, where our miraculous technological capabilities are seen instead as driving us ever more violently against environmental limits. Just has science earned support by taking credit, science may well lose support by taking blame.
<p>
Science is not a free-standing activity, but is embedded in a multi-faceted world. This relationship has been at the heart of modern industrial civilization, which is about 200 years old. We seem to be headed for a major shift. If science is to survive the shift in good health, the scientific community will need to find ways to adapt to the new patterns.
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com0tag:blogger.com,1999:blog-2688138290971915830.post-82608979011457613362023-03-10T11:54:00.002-08:002024-02-29T22:10:43.792-08:00ConsequencesOur actions have consequences. When we're being careful, we choose our actions so they'll have the best possible consquences. Most commonly what this means is that we try to change the world so it satisfies our desires more. But our actions don't just change the world, they change our selves. We often divide our activities into two phases, e.g. rehearsal and performance. The purpose of rehearsal is to refine our capabilities. Performance is when apply those capabilities to create an aesthetic experience for an audience, for example. But this division is just a rough cut. All of our actions change who we are at the same time that they change the world.
<p>
This division of experience into self and world is problematic. An athlete might consider their own body to be a component of the world. One's actual self might be perhaps the rational component of mind, something constant underlying even one's shifting mental capabilities. One of the essential insights of the Buddhist tradition is that the search for this constant underlying component of the self is futile. And yet this framework of thinking, e.g. "I will do this," seems practically unavoidable. If we want to use a conceptual framework of self and world, how can we think about this without getting distracted by illusions?
<p>
Organizational behavior is a doorway to a different perspective. It is not just individual human beings who act. All kinds of organizations act: political, military, industrial, academic, religious, etc. At a planetary scale, all of humanity acts. A basic principle of systems theory is that analysis starts with a clear definition of the system to be analyzed: what is part of the system, and what is not. A complementary axiom is in easy reach: the self is what is not in the system. The key point here is that the division of experience into self and world is like establishing a coordinate system or a frame of reference. It has no ontological foundation but is a practical step to allow conceptual elaboration for solving specific problems.
<p>
In organizational situations, it is commonly understood that actions both change the world and also change the self, i.e. change the organization engaged in the action. Teams develop cohesion by working together.
<p>
That what we are is a dynamic pattern that is constantly being shaped by our actions and experiences, that an important factor in choosing our actions is how those choices will reshape who we are... this perspective seems easier to achieve when we feel safe and secure. When things are good, we are happy to train ourselves to make them even better. When things are difficult, our entire focuse is on fixing problems with the world so we have no desire or opportunity to train ourselves. People do train themselves to be able to respond to difficult situations, though mostly that is to make themselves more capable of making whatever necessary changes to the world. But sometimes people do understand that shaping the world to meet their desires is not going to go very far, and they need to shape their own expectations. Aging gracefully can include such adjustments. What an older person can do is not the same as what a younger person can do. There is a lot less frustration in playing the hand you've been dealt.
<p>
At the planetary scale, the growing human population and the growing levels of consumption are driving us up against ecological limits, mostly prominently due to climate change but many other problems are accelerating too, such as aquifer depletion and ocean desertification. The reflex response is to demand that the world change in order to let us preserve our way of life. But of course our way of life is always changing and will continue to change as a consequence of our actions. However one chooses to partition the situation, it is always a dance between self and world. Our habits change, our understandings change, our values change. This dynamism is both a challenge and an opportunity. If our response to our discomfort is to become ever more stubborn and insensitive, we can certainly ramp up the level of mutual frustration to a catastrophic breaking point. But if we can respond to discomfort with care and flexibility, then we can discover tender joys in the most suprising places.
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com0tag:blogger.com,1999:blog-2688138290971915830.post-8605696559428101942023-02-10T19:42:00.004-08:002024-02-29T22:11:05.577-08:00What Is To Be Done?We are constantly faced with decisions about what to do, at every scale. Oranges are on sale: how many should I buy? Fossil fuels are creating climate havoc: should we switch to nuclear fission power?
<p>
In the simplest situation, one can foresee with sufficient accuracy the results of each alternative action, and choose the one with the most positive result. This formula outlines some major dimensions of a decision-making situation. One needs a set of alternative actions from which to choose; one needs to understand the results of each possible action; one needs to evaluate each of these possible results.
<p>
Commonly enough it is not possible to predict accurately the results of actions. We must decide in the face of uncertainty. We might have a pretty good idea about the probilities of the possible results of each possible action. For example, in a card game, we can calculate quite accurately the probabilities for each combination of cards we might draw from a well-shuffled deck. When developing a financial plan for living in retirement, actuarial tables can give reasonable estimates for survival to whatever age. Comparing the uncertain results of various possible actions is quite difficult. Given a choice between one action whose result is a certain $1, against another action whose result is $0 with probability 99% and $100 with probability 1%.... the expected value for each action is the same, $1. Whether to buy a raffle ticket for $1, that is a choice where expected return is not going to sufficient information to make a decision. <p>
Many real world situations do not permit probability calculations with any realistic promise of accuracy. Probabilities are applicable in situations that repeat. Of course situations never repeat exactly, but a large number of situations can be similar enough so that the outcomes of each possible action can be tabulated to provide guidance for what to do when the situation occurs yet again. But sometimes situations don't repeat with any reasonable similarity. What's the probability that Donald Trump will be elected President in 2024? Of course one can assign this whatever probability seems appropriate, but there is no way to check this number against the facts. In situations like this, one can look at the set of plausible outcomes of each possible action. An action might turn out well, or might turn out badly. How well? How badly? Comparing these sets of plausible outcomes is not simple or mechanical, but that's what's required for deciding on what action to take.
<p>
Sometimes a decision involves a significant action that takes place essentially at a single point of time. For example, if I am considering a major purchase, at some point I have to signal my decision to complete the transaction. But oftentimes what is called for is an ongoing series of actions. There is deciding what to cook for dinner tonight, and then there is deciding on my diet, on my pattern of meal selection. I don't have to plan out my meals for the rest of my life; I can decide on meals more or less on the spot, depending on my schedule, my activities, the availability and prices of various food items, etc. In a game like chess, there is no way to plan out the full sequence of moves one should make in order to win. Each move must take into account the preceding moves of one's opponent, which cannot be predicted with anything like sufficient accuracy. One can, however, potentially decide on a strategy. A plan is a sequence of actions. A strategy is like a table of possible situations that might arise in the future and what action to take in each situation. Market orders versus limit orders in the stock market would be an example. A market order is the decision to buy or sell some number of shares. A limit order is conditional: whether any shares are bought or sold depends on the market price. A market order is a plan, a limit order is a strategy.
<p>
Deciding on a strategy can be very difficult. It can be impractical or impossible to tabulate all the possible situations that can arise in the future. And when one encounters a situation in the future, one might choose a quite different action than whatever had seemed the wisest back when one was contemplating future possibilities. Our understanding of actions and outcomes evolves: we are always learning, or at least we can be learning. So an effective strategy for action is one that enhances the quality of one's future decisions, by providing opportunites for learning along the way, and leaving open as wide a range of possible actions in the future as possible.
<p>
We should not be planning to imprison ourselves; we should be planning to liberate ourselves.
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com0tag:blogger.com,1999:blog-2688138290971915830.post-41612132796605472252023-01-31T14:03:00.008-08:002023-01-31T21:26:19.632-08:00Fission PowerEvidence continues to mount that fossil fuel combustion is causing climate havoc. Floods and droughts, damage to cities and farms: it is becoming clear to more and more people that we need to wean ourselves off fossil fuels somehow. This is, however, an enormous challenge. We humans live very large on the earth nowadays, in our combinations of large populations and comfortable lifestyles. We consume energy globally at a rate of about 20 TeraWatts. We cook, heat our homes, drive our cars, run our factories... energy is fundamental to our modern way of life. Most of this energy comes from fossil fuels: coal, petroleum, and methane. To avoid enormous difficulties from any total change to our way of life, we need to substitute non-fossil sources to continue to provide energy at the required scale. Maybe in the future we will develop new sources, but in the next few decades at least we will need to rely on existing technology. Renewable sources such as solar, wind, and hydro are already in widespread use. Energy storage systems can help bridge the gap between fluctuating supply and fluctuating demand. But how to scale up renewable sources to meet the requirements of our modern society remains a daunting challenge. Nuclear fission is another existing technology that already provides steady reliable power at large scale. It is an very real option on the table for addressing climate change.
<p>
When your credit card bill is due and your checking account is empty, it is tempting to pay one credit card bill by borrowing from a different credit card. The general temptation is to solve short term problems by creating even larger long term problems. It's not an entirely invalid approach, but it's definitely smart to go down that road with eyes wide open. If we do choose to ramp nuclear power up by the factor of about 25x that would be needed to meet our energy needs, how might that move fit into a longer term strategy?
<p>
The long term strategy for modern society is rather cloudy but still worth considering. There is not going to be any kind of consensus possible, but that shouldn't stop a person from thinking about it. Some of the main options:
<ul>
<li>The world is due to end quite soon, so a long term strategy has no application.
<li>We cannot have any idea about the future. Long term planning is an absurd pretense.
<li>Technology will continue to advance at an ever more astounding pace. Any problems we create now will easily be fixed by the people
of the future with their capabilities that will be almost miraculous by our present standards.
<li>Maybe after a few thousand more years of expanding population and increasing comfort, humanity will start to bump up against actual planetary limits, but there is no point in worrying about that now.
<li>We are clearly hitting real planetary limits already. But it takes time for us to shift our various systems, such as agriculture, to more sustainable patterns. We cannot continue to consume energy at today's rate, but we need a few decades to shift. The immediate dangers of climate change mean that we need to shift to non-fossil sources sooner than we can reduce our energy consumption. Nuclear power can provide a bridge from today's unsustainable way of living to a future sustainability.
</ul>
It's worth thinking through what nuclear power would look like under these various scenarios. To ramp up nuclear power by 25x over the next decade or two is already a daunting prospect. If energy consumption continues to double every 50 years or so... what this would mean exactly in terms of uranium mining, waste management, fuel transport, etc. - I don't have answers, but it would be worth exploring such possibilities.
<p>
To flesh out such visions of how nuclear power could be scaled up in the future, perhaps the baseline assumption might be that everything goes according to plan. But effective engineering requires us to think about what might go wrong. If we are considering the option of walking down a tightrope to get to our destination, we'd be wise to understand how high off the ground that rope is!
<p>
Some of the unpleasant surprises worth considering:
<ul>
<li>Natural disasters such as earthquakes can cause radioactive material to escape containment.
<li>Safe management of nuclear material can require a somewhat advanced level of industrial capabilities to make available the necessary equipment and materials. Even with scaled up nuclear power, other factors could cause our industrial capabilities to be significantly reduced.
<li>All kinds of human bungling are not just possible but unavoidable. People are not perfect - not even close to perfect.
<li>It's not just that people make mistakes. People will quite deliberately act to benefit themselves at whatever cost to others. It may be possible to build a very safe reactor, but it will cheaper to build one that is less safe.
<li>People are always involved in conflicts at every scale. Nuclear technology can be weaponized in any number of ways. Of course we have very many nuclear explosive devices already built and ready for action. But the more we have fissile material circulating and the machinery for refining it etc., the easier it will be to build more explosive devices.
<p>Weaponization is not limited to nuclear explosives. Depleted uranium is already in widespread use in various types of bullets and other projectiles, just because of its metallurgical properties. Easy availability of radioactive materials will make them attractive for all sorts of uses. Various sorts of dirty bombs, conventional explosives coupled with radiactive shrapnel, are also straightforward possibilities. We have seen in the Ukraine where Russian troops occupied nuclear power facilities, because Ukrainian forces would not likely attack them there because of the risk of releasing radiative materials into the environment.
<li>Nuclear technology can be a source of conflict. A nation might be developing nuclear technology for entirely peaceful purposes, but this unavoidably also increases their ability to build nuclear weapons. Their enemies will be motivated to attack and destroy their nuclear facilities, to cut off that nuclear capability.
</ul>
It's also important to think about how we should evaluate consequences. We could just decide that it is too difficult to wean ourselves off fossil fuels, and just accept the ensuing climate change. We could cut our energy consumption dramatically to avoid climate change, and just accept the ensuing disruptions to our way of life. Or, if we decide to scale up nuclear power and some of the possible negative consequences arise, how bad could they be? Nowadays I see folks arguing that nuclear war wouldn't be so bad. Perhaps any cost short of human extinction should be considered acceptable. Even if ramping up nuclear power leads to human extinction... well, humans will surely go extinct sooner or later anyway, and if nuclear power improves our lives before that point, maybe it is a worthwhile bargain.
<p>
Understanding the various risks is very difficult. Many of the numbers involved are simply unknown, especially when the time scales involve many thousands of years. But there are also more complicated sources of uncertainty. Government inspectors will help prevent dangerous cost-cutting in nuclear facilities, but then government inspectors are themselves corruptible too. Nuclear advocates will point out that there have been no documented fatalities due to plutonium toxicity. But of course the people that handle plutonium employ many safety measures. Is plutonium safe because we know how dangerous it is? It's a bit like how the Mutually Assured Destruction provided by nuclear weapons has made the world a safer place, in some sense or other.
<p>
How can we decide what to do, in a game with such high stakes, with such high uncertainty, faced with such paradoxical logic? At least if we can get some common understanding of the predicament, that might be a start!
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com1tag:blogger.com,1999:blog-2688138290971915830.post-56392844563365205802023-01-24T20:30:00.005-08:002023-01-24T22:54:06.810-08:00Steady GrowthThere is a notion around that humanity requires steady growth to be healthy and happy. Steady growth clearly cannot continue for long on a finite planet. So there is another related notion around, that interplanetary colonization is required for humanity to be healthy and happy. Even the solar system is finite of course, so interstellar colonization is a natural next step. Why not intergalactic?!
<p>
But there are other physical limits that will constrain growth. Of course it could be that we will discover that our notions about physical limits are not accurate. But then our notions about the need for growth could be wrong, too. Any and all of our ideas could be wrong, but still, we're thinking beings; if we expect to succeed with interstellar colonization, we'd better hone the precision of our thinking!
<p>
One of the most fundamental physical limits in our theories today is the speed of light. Perhaps we'll find a way to colonize other galaxies, but it will take us a very long time to get to any of them!
<p>
Steady growth generally means exponential growth. Over a generation, the growth in whatever segment of the population will grow in proportion to the size of that segment. If health and happiness is to be equitably distributed, and if health and happiness requires growth, then growth will be exponential.
<p>
Physics comes in because humans, whatever else they might be, are also physical objects. The disciples of Ray Kurzweil might quibble: perhaps humans, in essence, are actually information. But even information requires some minimal physical substrate to be stored and processed! In any case, I am certainly not proposing that the specific numbers of my back-of-envelope calculations here should be taken with any seriousness. My point here is that steady growth will eventually bump up against the physical limit of the speed of light. I invite everyone to run the numbers as they see fit.
<p>
Suppose humanity's domain is some large sphere, centered on the earth presumably, and stretching out through interstellar space toward the distant galaxies. Since humanity is steadily growing, its domain is also growing. If humanity is growing exponentially, the volume of its domain will also be growing exponentially. Of course humanity can grow, to some extent, while in some fixed domain. That's what we've been doing on earth so far.
<p>
What exactly the carrying capacity of earth is, that's difficult to say. But, again, there are physical limits. The earth's mass is about 10^13 times the total mass of humanity. If the population grows at a steady 1% per year, then in about 3000 years, the total mass of humanity will exceed the total mass of the planet earth. Obviously we will run into serious trouble long before that; it is difficult to predict the exact course of our battle against limits to growth. The point of my quick calculations here is that they set some quite hard bounds. If humanity is to continue to grow at a steady 1%, certainly before 3000 years have gone by, we will need to be well down the road of interplanetary colonization.
<p>
It's easy to run similar numbers for the solar system. In less than 5000 years, the steadily growing mass of humanity will exceed the total mass of the solar system. Probably we will not find a way to digest the sun, so we will need to be colonizing distant stars well before then.
<p>
So let's say that we have spread out in the galaxy out to some radius R. If humanity is growing at 1% per year, the volume of its domain must also be growing at 1% per year, and then the radius will need to grow at 0.3% per year. Once that radius hits 300 light years, that steady growth will require the radius to grow more than one light year per year, i.e. faster than the speed of light!
<p>
So a reasonable bound on steady growth of 1% per year is that the domain of humanity will hit a hard physical limit at radius 300 light years. That's a volume of about 3 x 10^61 cm. Given the rough density of galactic matter, the total mass in that volume would be about 3 x 10^40 grams. A human weighs about 10^5 grams, so that would be a maximum population of about 3 x 10^35... assuming humans have incorporated all material into their bodies! Today's population is about 10^10, so that's a population growth of a factor of 3 x 10^25. At a steady 1% growth rate, we'll hit the speed of light in about 6000 years.
<p>
Of course these rough calculations involve many very unrealistic assumptions. There is no way that humanity will absorb into their bodies the entire mass of galactic matter inside a sphere of radius 300 light years. But even if they could, we'd hit the speed of light in 6000 years, given a steady 1% growth rate. 6000 years is already not an absurdly long time - it's roughly our historical horizon. Absurdly generous assumptions about the success of humanity's battle against the limits to growth already run into limits that are not absurdly far away.
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com1tag:blogger.com,1999:blog-2688138290971915830.post-52433524000440595722022-11-03T16:20:00.001-07:002022-11-03T16:20:35.703-07:00Non-Euclidean ScienceMaybe I should call it non-Aristotelean, or non-Platonic, but the exact name isn't the point. Euclid built geometry up from postulates; Aristotle explained motion as objects returning to their natural state; Plato portrayed experience as the shadow play of forms in an ideal realm. In each case, a complex field of phenomena is explained as the outgrowth of some simple essential foundation. Perhaps I should call my proposal non-foundational science. But I am not proposing any kind of freed-floating science. I am proposing a science that is founded on reality, on the vast tangled web of lived experience. Science is an extract, like resin extracted from the sap of a tree. There is a lot more to a tree than such resin. The tree itself is embedded in soil and climate, in an ecological web, flying pollinators and mycorrhizal fungi. The simple essence emerges from the whole, rather than the whole emerging from the simple essence.
<p>
Science as a quest for an inner key that explains everything - such science takes us on a quest into ever more remote realms. It distances us from experiencing what is right at hand. Of course, building and launching the James Webb infrared telescope surely involved considerable attention to experiences right at hand - precision torquing of many bolts, etc. Galaxies and quarks are not objects of direct experience, but neither are they disconnected from direct experience. What I am proposing is no neglect of any corner of the world. I am suggesting a shift in how we understand the way all the bits and pieces fit together.
<p>
Non-Euclidean geometry provides an excellent analogy. The surface of a sphere, such as the surface of the earth, is a perfect concrete instance. Euclidean geometry is plane geometry, the geometry of a flat surface. At the scale of a few square miles, the earth is extremely close to a flat surface, and can be mapped onto a flat sheet of paper with great precision. But as the area to be mapped increases to include a significant fraction of the earth's surface, inevitably distortions arise. There is no perfect flat map of the earth.
<p>
The impossibility of perfection does not mean that we just give up and produce fantastic maps that have lost any connection with the lived experience of moving around on the earth. The value of a map is exactly in how it relates to such lived experience. Whether a map is good or not, that depends on how the map is to be used. A map that is good for navigation will typically not be a good map for estimating agricultural productivity.
<p>
Pure science is science that neglects its relationship to its use. Applied science is science that orients itself to its use. The classical scientific attitude is that applied science grows out of pure science. I am proposing that a healthier approach to science is to see pure science growing out of applied science. Applied science connects to the vast complexity of lived experience. Refining our ideas requires chopping out local regions to be precisely mapped. This always involves distortion and omission: the inevitable price of precision. It's like taking a photograph: a fast shutter speed can reduce blurring from movement, but requires opening the aperture which increases blurring from less depth of field.
<p>
Our scientific quest for ultimate theories is like the old searches for the alchemical philosopher's stone or the healer's panacea, a medicine to cure all diseases. Good science requires following the clues wherever they lead, but it also requires a perspective on the actual situation so that one doesn't chase clues just for the sake of the chasing. Good science is science that is engaged with the lived reality of an actual situation.
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com0tag:blogger.com,1999:blog-2688138290971915830.post-42235052463967230462022-07-01T20:08:00.005-07:002022-07-01T21:15:00.702-07:00Double Helices<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEircdsGGCYR_mKKNn9fpgzpceweNbjU9yBsSWrbJEtlp-F9oFDqT-CtCrHPQytrsz5xLlfz22uHhpOu3SGtCSnOpDrbRwLPSuOjZ6AWoChQKGY9OKcFatQGKoSoZkL85VAf7-tf_hAq44hXLDTRknqiR55MqSpRwmSVsu9_MP5f4sU0NR8k4wDsYryD/s2100/torus289c.jpg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="1300" data-original-width="2100" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEircdsGGCYR_mKKNn9fpgzpceweNbjU9yBsSWrbJEtlp-F9oFDqT-CtCrHPQytrsz5xLlfz22uHhpOu3SGtCSnOpDrbRwLPSuOjZ6AWoChQKGY9OKcFatQGKoSoZkL85VAf7-tf_hAq44hXLDTRknqiR55MqSpRwmSVsu9_MP5f4sU0NR8k4wDsYryD/s320/torus289c.jpg"/></a></div>
<p>
Here is an image of a torus with a square lattice drawn on it. This lattice is formed by the intersections of two helices drawn on the torus. There are many ways to draw such helices on a torus, and the lattice pattern emerges from the combination of two such helices. This kind of double helix square lattice on a torus is a broad family of geometric shapes.
<p>
This kind of geometric shape can be used in music several ways. It can serve as a model for the time evolution of a piece of music. It can also serve as a model for the harmonic relationships between the pitches used in music. Since music is, in large part, a relationship between time and pitch, a piece of music can be modeled as a relationship between two different toruses, a torus of time and a torus of pitch. Of course most music won't fit this model very well or at all. But it can serve as a blueprint for creating music.
<p>
Musical time as a helical path on a torus... maybe it's because I have been thinking this way for decades, but it seems quite natural. Of course a piece of music often has more of an arc structure, a beginning, a middle, and an end. But often within that large arc, whole stretches are largely repetitive, where the end of each repetition joins smoothly with the beginning of the next. If the repetitions were exactly the same, this would simply be a circle. But perhaps the words of verses change or other details, so each repetition is slightly different than the last repetition. To bring the last repetition close to the first repetition is of course a more arbitrary choice, but not a very wild one. I hope this makes sense of the notion of musical time as a helical path on a torus.
<p>
The idea of pitches being related harmonically in a way similar to a helical lattice on a torus... this is hardly a new idea in the world of music theory. There are many ways to use this kind of geometric shape to represent harmonic relationships. The circle of fifths is the most basic. Major thirds are another fundamental relationship between pairs of pitches. These two intervals then create a mesh of relationships that can be laid out on a torus:
<p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1ZOk9T8uj3wrVTbtnmb__AYqi3WgHb-0lyxmmcPbojNQrRJLUiP8lLOqSBVO2OgjjrR8PIsU3OfpyZHwCvGeVSzH-SyYBnkX6bl7XojOOUoUe02048g_r4qUlwXvk1SD5DAEh1pzvuiAZJtB6pJGrol_rRhGzzYuVWWTsp530s2IEiMuqJOShjFnp/s960/torus12%20labeled.jpg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="720" data-original-width="960" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1ZOk9T8uj3wrVTbtnmb__AYqi3WgHb-0lyxmmcPbojNQrRJLUiP8lLOqSBVO2OgjjrR8PIsU3OfpyZHwCvGeVSzH-SyYBnkX6bl7XojOOUoUe02048g_r4qUlwXvk1SD5DAEh1pzvuiAZJtB6pJGrol_rRhGzzYuVWWTsp530s2IEiMuqJOShjFnp/s320/torus12%20labeled.jpg"/></a></div>
<p>
Here the green line traces the circle of fifths. There are four red loops, representing the circles of major thirds.
<p>
This torus of harmonic relationships can be drawn for alternate tunings. The different topologies generated display the different musical possibilities of these alternate tunings. One important alternate tuning divides octaves into 19 equal steps instead of the usual 12. The torus of harmonic relationships for 19edo looks like:
<p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcKb-Uy_WyYiy3sI66ZAKt8-opNpwXpuzpNc43Tiu6XiuGJDN0GUCONpC2SQKZHUZnX0kRLu6Tf9EkZHx89I73X4ABk6DT2887ux-HJKO2PTZC4snuBqmJ_sGnpagldhoSSbytgLISWoTuTN5cNVCtMPA5MIqLqZ4fRdCR9bZD5Mmo5LTawQX-Hs4u/s960/19edo%20torus%20labels.jpg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="720" data-original-width="960" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjcKb-Uy_WyYiy3sI66ZAKt8-opNpwXpuzpNc43Tiu6XiuGJDN0GUCONpC2SQKZHUZnX0kRLu6Tf9EkZHx89I73X4ABk6DT2887ux-HJKO2PTZC4snuBqmJ_sGnpagldhoSSbytgLISWoTuTN5cNVCtMPA5MIqLqZ4fRdCR9bZD5Mmo5LTawQX-Hs4u/s320/19edo%20torus%20labels.jpg"/></a></div>
<p>
The green loop here again represents the circle of fifths. What's more interesting is that major thirds no longer divide up the pitches into separate loops. Instead there is one large loop traversed by major thirds.
<p>
With these two toruses and their helical lattices,
the harmonic structure of a piece of music can now be mapped out. For the most part, one would expect phrases that are closely related in musical time to be closely related harmonically. There might be abrupt transitions, but they make sense in this approach in the context of surrounding smooth relationships.
<p>
The simplest non-trivial mapping uses a loop on the pitch torus, some path through the lattice that returns back to the starting point. This loop is then traversed in musical time. It could be that each repetition traverses the loop. Or perhaps the repetitions don't move much internally, but each repetition moves slightly relative to the previous repetition, so the harmonic loop is traversed over the course of the whole piece.
<p>
Algebraic topology is the mathematical discipline where these sorts of smooth mappings are enumerated. Mapping a torus onto a torus is a rather elementary problem in algebraic topology... but there is still a rich variety of possibilities to be explored musically!
<p>
Another feature of alternate tunings is that additional basic harmonic relationships can be introduced. Exactly what makes pitches sound harmonically close, that is an endless topic of study and debate. But one fundamental notion with a long history is that frequency ratios very close to a simple rational ratio, that's the basis of close harmonic relationships. An octave is a frequency ratio of 2:1. A perfect fifth is a frequency ratio of 3:2. A major third is a frequency ratio of 5:4. In conventional music, these basic intervals are the foundation of harmony.
<p>
One natural step in extending music into wider worlds is to introduce yet another basic interval, governed by the frequency ratio 7:4. This extra relationship makes the torus of harmonic relationships much more difficult to draw... it's not anything that could physically exist in our three dimensional world. But of course mathematically it is nothing very complicated to manage. A tuning that can represent this new interval quite accurately, along with the more conventional intervals, is 171edo, the tuning that divides octaves up into 171 equal steps.
<p>
An instrument with so many notes would be physically unwieldy. But with a software synthesizer and algorithmic composition, it is not so hard to build a piece of music based on a mapping of this more complex torus onto a torus of musical time:
<p>
<a href="https://app.box.com/s/tu9fjd6dugvhnbrvkovyjz9glmv5d0x5" target="_blank">Double Helices</a>
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com2tag:blogger.com,1999:blog-2688138290971915830.post-39388312709937793822022-03-22T19:52:00.013-07:002022-07-01T20:10:02.140-07:00Sliding SchismasFor some years now I have been exploring music and tuning, through algorithmic composition. I have a computer program that I tweak, to change tuning systems, scales, etc. How much of the tweaking that I do is actually reflected in the output, in any perceptible way? That's a question worth examining!
<p>
Here is a set of musical pieces. The only thing I changed in the software from one piece to the next is that I changed the seed for the random number generator. The random numbers it generates control very many choices in the execution of the program, so these pieces will vary quite a bit. But the primary choice in question is the harmonic movement involved, the key changes. Some of the pieces don't have any movement at all. Others have a progression that is six steps long. Some of the pieces move along the progression in the forward order, other move in the reverse order.
<p>
So the question here is: can you divide these pieces into three groups, one group with no key changes, another group that moves in one direction, and a final group that moves in the opposite direction. Can I tell the difference? (The names of the pieces are the seeds I used to initialize the random number generator for each piece.)
<ul>
<li><a href="https://app.box.com/s/9au8p4qdsdcp28b7gpupe6zh8igvw9db">7000</a>
<li><a href="https://app.box.com/s/t6749wz8s6dsny26rzsfqoeo4lo32hjm">7001</a>
<li><a href="https://app.box.com/s/qkci4i3jhjv1lkihetr02ysvs0mw2t32">7002</a>
<li><a href="https://app.box.com/s/4472lpszhjwiln62vu49u071iwfzk5aj">7003</a>
<li><a href="https://app.box.com/s/lsdpbeiw18gc7foxh2xmrfs2jg1w1oau">7004</a>
<li><a href="https://app.box.com/s/fdx1qjnsejrs79jko47qdl2zd18ok8hl">7005</a>
<li><a href="https://app.box.com/s/p8loyctnakclt8u0q1zzob5alget15ta">7006</a>
<li><a href="https://app.box.com/s/sozt7chg1sodcwdyqdnxouu93bgespmd">7007</a>
</ul>
<p>
These pieces all use the 53edo tuning system, where octaves are divided into 53 equal steps rather than the conventional 12. These pieces all work with a schisma[17] scale, where 17 notes are selected in each octave out of the full set of 53. In the pieces with no key changes, the scale is constant throughout the piece. In the other pieces, the key changes in a regular pattern, shifting every measure. With six key changes, the scale returns to the starting scale.
<p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOqKRe9pch4UTCb_cen5lNWjwtHuTp-OKkGihkL9kQfDwKRABP8XnXKc9p9f3xma4bN9jae2GG8yTFj_T9r8gqGfOkxbZFX9yc3nNIhyvHYJ2oHHFeDgcHHa1qLAtsvaxwJ3vB8ntWAOeeTKKTfe9qxgUIeeq-rNrl7OJEoWGI1jAfpCV6Y-xbiYCv/s1179/53edo%20scale%20stripe.jpg" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="400" data-original-height="361" data-original-width="1179" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOqKRe9pch4UTCb_cen5lNWjwtHuTp-OKkGihkL9kQfDwKRABP8XnXKc9p9f3xma4bN9jae2GG8yTFj_T9r8gqGfOkxbZFX9yc3nNIhyvHYJ2oHHFeDgcHHa1qLAtsvaxwJ3vB8ntWAOeeTKKTfe9qxgUIeeq-rNrl7OJEoWGI1jAfpCV6Y-xbiYCv/s400/53edo%20scale%20stripe.jpg"/></a></div>
<p>
Each row in this picture shows which notes are in the scale in one of the keys. In the pieces with key changes, from one measure to the next the scale will shift to the next row up or down in the diagram; in some pieces the key changes move up in the diagram, in other pieces the key changes move down. I repeated the sequence three times in the diagram, and also extended the scale a bit beyond an octave, just to make clear that the pattern continues smoothly through time and up and down the pitch space.
<p>
One could play the pieces with no key changes on a piano reasonably accurately. There are five pairs of notes that are very close togther, just one step apart of the 53 per octave. These would correpond to spit keys on a deluxe piano, a slightly sharp version of a note and a slightly flat version. Thinking of the split note as just two versions of a single note, then there are twelve coarse notes per octave, very close to a conventional piano.
<p>
The sequence of key changes in the other pieces involve two different shifts in the scale. Moving along the sequence in one direction, the scale shifts five times by a minor third, and once by a minor sixth. Moving in the other direction, the shifts are the inversions, i.e. five major sixths and one major third. In the 53edo tuning system, this combination of key changes brings the scale back to its starting point.
<p>
When the scale is shifted by a minor third, the new position of the scale include eight of the notes of the scale before the shift. The shift by a minor sixth has a similar amount of overlap. This overlap allows for good continuity of musical phrases across the shift.
<p>
Listen to the pieces above: some have key shifts, and some don't. Can you tell the difference?
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com0tag:blogger.com,1999:blog-2688138290971915830.post-89935979695215788962022-01-15T15:24:00.014-08:002022-01-15T20:28:28.411-08:00Tuning TangleThe appearance of orderly structure in the world is a fascinating puzzle. Mathematics studies the properties of orderly structures. Are mathematical objects features of the world, or features of our minds? Do the mathematical regularities we see in the world appear just because that's how our minds process sensory data? Aren't our minds part of the world, anyway?
<p>
The vision of the world as mathematically structured is traditionally credited to Pythagoras. One of the cornerstones of this vision is the notion of musical consonance as mathematically structured. Music is built from consonant intervals, the relationships between tones that sound good together. Musically consonant intervals correspond to mathematically simple integer frequency ratios. An "A" pitch with frequency 440 Hertz and the "A" pitch an octave higher, with frequency 880 Hertz, have the frequency ratio 2:1. The 440 A relates to the 660 E that is a perfect fifth above it, with a frequency ratio of 3:2.
<p>
Musically, a song is a pattern of notes that are related by a variety of such consonant intervals. Of course songs also involve rhythmic patterns etc., but here I am just focusing on harmonic patterns.
<p>
Patterns arise in many ways, but generally they are the outcome of some sort of process. For example, tree rings appear from the varying growth rate of the tree through the regular changing of the seasons. Another kind of pattern arises as liquids cool and solidify. A quick cooling will form finer grained crystals; slow cooling allows the crystals to grow larger. Thermodynamic phase transitions, such as freezing and melting, are a rich field for the study of how order can emerge spontaneously. Musical patterns can be generated by thermodynamic simulation; consonant clusters of notes, such as chords, are similar to crystals that emerge from the process of freezing.
<p>
The algorithmic composition method I describe here relies on thermodynamic simulation to choose the pitches to be played at each time. The simulation works with a matrix of points at which a pitch is to be played. This matrix defines connections between such points. Pitches to be played at the same time are connected; pitches played at successive times are connected. Musical patterns generally have a structure of repetition and variation. The matrix is constructed with a fixed repetition structure: connections are made between pitches played at the corresponding points in successive cycles of repetitions.
<p>
Thermodynamic simulation is driven by temperature as a key control parameter. Degrees of consonance correspond to energetic possibilities. At high temperatures, pitches are chosen relatively freely; only the most dissonant choices are discouraged. At low temperature, only the most consonant choices are allowed between connected points in the matrix. Initially the points in the matrix are assigned random pitches. The simulation begins at a very high temperature, and then gradually the temperature is reduced. The pitches in the matrix are randomly reassigned again and again. Gradually patterns of mutual consonance begin to emerge.
<p>
While the temperature is still quite high, very little orderly structure has emerged:
<a href="https://app.box.com/s/bx3yi6d9it4y3ijt80pyyxvre6relc69">118edo 3x3x3x3x3 1</a>.
<p>
A graphical score also shows a lack of structure:
<p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEhcJ_061kclj5Pc46SbpyWGpMxmOKFSfSeGuW3FUFP65C5VUgJAbTZGtBYOfRasulGlUUJeZPk-wMiP0ebUqOHjUHbwh3-Q-RWYpod0X4HFBV30Rc4KFfHUBLqxx3sjHaXQoHRa5_jRgX_E7nvi0jp9kkjEzBs5LbI1ddWPNgne-4u2clFbwjxWJMxk=s911" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="662" data-original-width="911" src="https://blogger.googleusercontent.com/img/a/AVvXsEhcJ_061kclj5Pc46SbpyWGpMxmOKFSfSeGuW3FUFP65C5VUgJAbTZGtBYOfRasulGlUUJeZPk-wMiP0ebUqOHjUHbwh3-Q-RWYpod0X4HFBV30Rc4KFfHUBLqxx3sjHaXQoHRa5_jRgX_E7nvi0jp9kkjEzBs5LbI1ddWPNgne-4u2clFbwjxWJMxk=s320"/></a></div>
<p>
Here the vertical axis is the pitch, and the horizontal axis is time.
<p>
A slow cooling process will allow long range order to emerge, so eventually the entire matrix becomes consonant:
<a href="https://app.box.com/s/k62wbbharsnfp6x3u49hyiz5t67grc98">118edo 3x3x3x3x3 22</a>
<p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEj0rivtuagrvAv9SRVhisuHOevRVytgGVyjJ8F8EPTvwCJtwH0oIXuKPyIJ3A451Sc2ivO0baw0-WaAJarwQw9vHjOXgTT-h5NJg7f1HYHsLTTVZFDj34wIkbGck2zx0pryeTfcUzHz7xFLqp6key0ChvVGRaiQzefIyKRsWy4wc4adzy8O-uZczYrW=s911" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="662" data-original-width="911" src="https://blogger.googleusercontent.com/img/a/AVvXsEj0rivtuagrvAv9SRVhisuHOevRVytgGVyjJ8F8EPTvwCJtwH0oIXuKPyIJ3A451Sc2ivO0baw0-WaAJarwQw9vHjOXgTT-h5NJg7f1HYHsLTTVZFDj34wIkbGck2zx0pryeTfcUzHz7xFLqp6key0ChvVGRaiQzefIyKRsWy4wc4adzy8O-uZczYrW=s320"/></a></div>
<p>
At an intermediate temperature, there can be fluctuations within an overall harmonic framework, a balance of order and variation that approaches musicality: <a href="https://app.box.com/s/pcdbyfxivhhc6rxghsdxcschitpri1v2">118edo 3x3x3x3x3 13</a>
<p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgWM6sryyrOIzta7JEmWiRnxABb8oiKhpownVZchpc8kTuM_LAfjbYl7uL-o0vUlpZKiocxgeFoziExHyFW6dOWbYZ5AsMDV1jnxcYNq8SEY_LU7NZQi5yduHYZDOXx2Zb6UwiOQ0_l8EG3cKr-AlvDWqmNm_7I_HFDa7IC7mItpUDIOxyavKWzxDFq=s911" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="662" data-original-width="911" src="https://blogger.googleusercontent.com/img/a/AVvXsEgWM6sryyrOIzta7JEmWiRnxABb8oiKhpownVZchpc8kTuM_LAfjbYl7uL-o0vUlpZKiocxgeFoziExHyFW6dOWbYZ5AsMDV1jnxcYNq8SEY_LU7NZQi5yduHYZDOXx2Zb6UwiOQ0_l8EG3cKr-AlvDWqmNm_7I_HFDa7IC7mItpUDIOxyavKWzxDFq=s320"/></a></div>
<p>
The harmonic movement here is quite limited. One avenue that can open up a richer harmonic landscape is the introduction of tempered tuning. The tuning used here divides octaves into 118 equal steps (118edo), instead of the conventional 12 equal steps (12edo) of a piano. Dividing octaves into some moderate number of equal steps is a practical way to organize the set of pitches used in a composition. If the pure rational intervals such as the perfect fifth 3:2 and the major third 5:4 are used, these can be combined in an infinite number of ways. If the number of equal steps per octave is chosen carefully, good approximations for these pure intervals are available: four steps of 12edo is 1.2599, quite close to the pure 1.25. 38 steps of 118edo is a frequency ratio of 1.2501, imperceptably close to the pure 1.25.
<p>
Another feature of these tempered tunings is that the infinite number of ways to combine the fundamental consonances will give only a finite number of results, within an overall pitch range. A given interval can be constructed from multiple combinations of fundamental consonances. For example, in 12edo, a major third can be reached by moving four perfect fifths up and then down two octaves. Each tuning has a different pattern of such combinational coincidences. A <i>Tonnetz</i> diagram provides a useful summary:
<p>
<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEgQfRm06zYY-PbpOjjsEI_Y9e0moBJ5wsESexh-r_hGVfFwuGwAR8PrGPu85qtq19OYrv6koLQTs208GpG7dCUI9QKthAmWo4iq-5lP-xFWHfbwtA3GqGSNIc4BiLpIUSHSq0DWtile2ofZXOkPOgGLaQvm7AfMQk8rDubPOwYuUPoBXzIkOZOJFQs-=s960" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="720" data-original-width="960" src="https://blogger.googleusercontent.com/img/a/AVvXsEgQfRm06zYY-PbpOjjsEI_Y9e0moBJ5wsESexh-r_hGVfFwuGwAR8PrGPu85qtq19OYrv6koLQTs208GpG7dCUI9QKthAmWo4iq-5lP-xFWHfbwtA3GqGSNIc4BiLpIUSHSq0DWtile2ofZXOkPOgGLaQvm7AfMQk8rDubPOwYuUPoBXzIkOZOJFQs-=s320"/></a></div>
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In this diagram, the octaves are omitted. E.g. all the ways to play a "C" note in various octaves are all represented as just "C". This diagram is for the 118edo tuning, so instead of the usual 12 note names like "C", "C#", etc., the numbers 0 to 117 are used.
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The repeating structure in this diagram, e.g. the multiple occurrances of the 0 pitch, are a result of the tempering of the tuning. E.g., moving by 8 perfect fifths and then a major third will result in the same pitch where one started (moving as many octaves as needed). This property of tempered tunings introduces the possibility of loops in a compositional structure. The Tonnetz diagram shows that loops in 118edo need to be quite long: there are no short paths from a 0 pitch to another 0 pitch in the diagram.
<p>
The compositional matrix used above was given a repetition/variation structure of a five dimensional torus with circumferences uniformly size 3. This created a large space but where no large loops will easily arise. Another large compositional space is a two dimensional torus with circumferences size 18. The compositional torus can easily accommodate tuning loops as long as 18 measures. This is long enough that several loops in the tuning space can fit.
<p>
Starting the thermodynamic simulation from a random pitch assignment and gradually cooling, these sorts of tuning loops will tend to get trapped in the matrix. When the system is cooled to a very low temperature, the tuning loops remain: <a href="https://app.box.com/s/t2qh69ilheyxlugy7jm4jaqsjjy85176">118edo 18x18 cold</a>.
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEh9oyLaLTRcABlG3WeabXWDJKgNxatyKFj_75Ym6QXfih7a9vnml2cOI2cxFowfIu8QTIGi1aMyhwdOCQP6_6-v5Ro6-aaoHpZafiYfbp1GK7seyP3VjFiXAqnUr12GnUpDkx8mw7bhd7o73A_g7b0-LcY3GJVS4b5yLFQ84B5j5MXNHnUJTmQ0xq_q=s911" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="662" data-original-width="911" src="https://blogger.googleusercontent.com/img/a/AVvXsEh9oyLaLTRcABlG3WeabXWDJKgNxatyKFj_75Ym6QXfih7a9vnml2cOI2cxFowfIu8QTIGi1aMyhwdOCQP6_6-v5Ro6-aaoHpZafiYfbp1GK7seyP3VjFiXAqnUr12GnUpDkx8mw7bhd7o73A_g7b0-LcY3GJVS4b5yLFQ84B5j5MXNHnUJTmQ0xq_q=s320"/></a></div>
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The harmonic movement makes even this very orderly pattern somewhat interesting. At a moderately higher temperature, there are short term fluctuations together with long range movement, producing a composition that is even more musical: <a href="https://app.box.com/s/h14dszfos4zy7d4po3mih2f9540jsznb">118edo 18x18 10</a>.
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/a/AVvXsEiPKoKm3ldzKWWWCW0I3vwXAf-mop8LPU-Rua17QqIlney8OeYuz6AoUuKgmY9lasvRz3krkpeTK-J1FqsysnE__T5ZGZrShVfPGFUTdcd_ITZzmTAhAg7AFckX_y1SMnsFIjOtr0RtVqd0NSQdd3yP9MGxSBKP0O53riDVbFxVCSAJ_y0qKf9mJWn-=s911" style="display: block; padding: 1em 0; text-align: center; "><img alt="" border="0" width="320" data-original-height="662" data-original-width="911" src="https://blogger.googleusercontent.com/img/a/AVvXsEiPKoKm3ldzKWWWCW0I3vwXAf-mop8LPU-Rua17QqIlney8OeYuz6AoUuKgmY9lasvRz3krkpeTK-J1FqsysnE__T5ZGZrShVfPGFUTdcd_ITZzmTAhAg7AFckX_y1SMnsFIjOtr0RtVqd0NSQdd3yP9MGxSBKP0O53riDVbFxVCSAJ_y0qKf9mJWn-=s320"/></a></div>
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com0tag:blogger.com,1999:blog-2688138290971915830.post-84571894793883483022022-01-07T20:46:00.001-08:002022-01-07T21:04:48.031-08:00Science without ProgressThere's a notion of science for which progress is essential to science. Science is a process of steadily broadening, deepening, and refining our knowledge about the world. It's a process of steady improvement. This year's science is better than last year's science, and next year's will be better yet. Whether this process converges on some ultimate theory that captures precisely the way things are, that's a bit beside the point. The sequence of integers 1, 2, 3, etc. steadily get bigger, without ever converging on some final largest integer.
<p>
For this kind of steady progress to be the way science works, two things must be true. First, we need a way to compare our scientific knowledge at one time to our scientific knowledge at another time. We need a way to tell which state of scientific knowledge is better. Once we have that measuring stick, then we can at least check empirically whether science is constantly improving. We can develop some kind of model of the evolution of scientific knowledge, and check whether at least the model guarantees continual progress into the future.
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It's easy to sketch out a model of the evolution of scientific knowledge that implies perpetual progress. Such a model may not be accurate, though! A major question in examining the dynamics of science is its coupling with the world outside science, with social, ecological, and geological systems. Science is a social institution, intimately connected with the rest of society. When sources of funding, materials, equipment, and personnel dry up, science cannot thrive.
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One measure of the state of scientific knowledge is the size of the total accumulation of scientific publications. As long as some library somewhere continues to accumulate the mass of literature, as long as scientific literature is not lost, then scientific knowledge will continue to advance, by this measure.
<p>
There are two problems with this logic. First, it is unreasonable to expect all scientific literature to be preserved in perpetuity. It's not even clear what exactly should count as scientific literature. Parapsychology, the study of phenomena such as telepathy, is an example of a discipline whose scientific status has been debated. Should raw data accumulated by scientific instruments count as scientific literure? As our boundary that defines scientific literature changes, our measuring stick to detect progress is being updated. We don't have a consistent measure by which to determine whether science progresses consistently.
<p>
Even if we maintained a constant definition of what should count as scientific literature, it is not reasonable to expect all such literature to be maintained in perpetuity. There is some expense involved in preserving information. There is additional expense involved in converting old literature to new formats. Not all printed literature is scanned to digital form. Digital formats are steadily changing, and obscure literature will generally be given a low priority for format conversion.
<p>
Even if a record of some coherent piece of scientific knowledge has been preserved in a library somewhere, it can easily happen than no one is alive any more who can make any sense of it. The papers involved may easily refer to scientific instruments that no longer exist, for example.
<p>
One can slog through endless such details to determine whether scientific progress is inevitable. In the face of impending climate catastrophe and the profound social upheavals that will bring, the idea that science will somehow weather the storm despite all the challenges... perhaps no amount of detailed argument will convince a true believer!
<p>
If progress is essential to science, but if progress is not a secure ground on which to build... must science then crumble, too? Can science survive and even thrive without progress? Is progess, after all, essential to science?
<p>
It is a vital project to develop a vision of science that does not depend on progress. We in that part of the world that supports science are at grave risk for a major decline in our general level of prosperity. Science will participate fully in the trajectory of decline and collapse. If we can maintain a thriving science despite that decline, our ability to cushion that decline will be significantly enhanced. We will be better able to respond to recurring crises in medicine, agriculture, etc. If the scientific community cannot find a way to dance with circumstances, we will all suffer from that failure.
<p>
An analogy should be useful in developing a vision for science that doesn't depend on progress. Darwin's theory of evolution shows how species are constantly adapting themselves to their circumstances. The steady extension and refinement of scientific knowledge is similar to biological evolution. But biological evolution does not imply any kind of progress. Species today are not more advanced or better adapted than were species ten million years ago. Species ten million years ago were reasonably well adapted to their circumstances back then, which were very different than the circumstances of species today. Some of these changes are surely geological, but they are largely due to the interdependence of species, the nature of the ecological web. When one species develops some new characteristic, that changes the circumstances of other species, pushing them to adapt in new ways. There is no fixed measuring stick by which to determine whether one species is more advanced that some other species.
<p>
When we dream of some ultimate scientific truth and view science as a path leading to that goal, progress seems to be essential to science. But if we understand science to be a practical approach to engaging with our experience, enabling us to respond more effectively to our circumstances, then it becomes natural that our scientific knowledge must shift and adapt as our circumstances change.
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com0tag:blogger.com,1999:blog-2688138290971915830.post-10984711508920749862021-12-12T15:05:00.003-08:002021-12-12T21:36:45.153-08:00The Paradox at the End of ModernityModernity can be defined as a culture of faith in progress: newer can be better, should be better, is better. The engine of progress is science. Science is a process of refining our understanding of the world. We are constantly learning about the world, correcting our misunderstandings and extending the frontiers of our knowledge. Science doesn't go backwards. Tomorrow's science is better than yesterday's science. We can use our constantly improving scientific understanding to improve conditions in the world around us, to cure diseases, increase crop yields, etc.
<p>
This vision of progress based on science was elaborated by Francis Bacon in the early 17th Century, at the beginning of the modern era. The road of progress we can see in front of us remains limitless. Colonizing Mars, autonomous robots, the extension of life expectancy to multiple centuries and beyond... what barrier can we not imagine transcending? And if we can imagine it, step by step we can use the scientific method to resolve whatever problems limit our ability to achieve it.
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On the other hand, as science refines our understanding of the world, it reveals some very challenging limits. Of course the way science understands limits on one day may be overturned the next day. Perhaps the rudest limit science has discovered is the speed of light. As the vastness of the universe has been revealed to us, so has its remoteness. Will we figure out some clever way to leap across distances of thousands of light years? This might be the most elementary form of the paradox we are caught in. An irresistable force is contending against an immovable object. What will happen?
<p>
Back down on earth, of course, the speed of light doesn't seem like anything worth much worry. We have plenty of technical problems with much more immediate impact. The general problem of global resources: climate change, water, biodiversity; this problem is foremost among our challenges. Pandemics such as covid-19 are to some degree a result of humanity running up against planetary limits, but there is also the problem of pathogens evolving to evade our countermeasures. Our problems can get worse more quickly than our pace of finding new solutions.
<p>
At this point, it is not too farfetched to observe that our progress in scientific understanding is revealing more about the limits to our technical progress than it is enabling further technical progress... "technical" meaning our ability to improve our world.
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Nowadays it is very easy to find literature championing each of the two poles of this paradox. There are books that show how things have always been improving and will continue to improve. There are books that map out the trajectory of the collapse of modern civilization that we are riding along.
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Will the irresistable force succeed in dislodging the immovable object, or will it be defeated?
<p>
Paradoxes, like the paradox of progress that we are caught in today, are not generally resolved by the victory of one pole over the other. In general some kind of deeper understanding of the apparent contradiction is required. A good starting point is simply acknowledging that we really are facing a paradox. It would be foolish to dismiss either the vitality of scientific progress or the reality of the planetary limits to growth. To develop a new understanding of our situation that can encompass these two aspects, that is the challenge we face.
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I found Kurasawa's film <i>Dreams</i> to be a wonderful vision of how we got here and where might might be heading. The final segment is "The Village of the Watermills," where we see a joyful celebration of a funeral. Death is a reality. However over the top the visions of the colonization of Mars might be, the visions of human immortality make those look very tame. We really do need to grow up and learn, not just to accept our limited situation, but to cherish it. A joyful funeral is one way to do this. But how we age, that is another vital dimension. What can it mean, to be healthy and old? To be healthy and dying? Such a vision might provide a model for our modern civilization as it runs up against planetary limits.
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com0tag:blogger.com,1999:blog-2688138290971915830.post-5117584591549427752021-10-17T21:39:00.003-07:002022-01-07T21:05:00.977-08:00The Need for GrowthI've heard many times people say that our economic system requires growth in order to function. Usually people explain this by saying that
the only way that interest can be paid on debt is if the money supply increases. This is not true, though. In a debt-based money system, the sum of money accounts is always zero. People who owe money need to be able to provide goods and services that people who have lent the money will purchase, but as long as that is true, there is exactly enough money floating around to pay any debts that are floating around. Understanding this, we can see that paying interest doesn't require a growing economy. So, is a growing economy actually required at all?
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Here's a different notion of how our economic system requires growth to function. Our system requires inflation to encourage people to spend and invest. In a deflationary environment, it's better just to hold onto money - in which case, the economy freezes up completely. Inflation has two different meanings that are of course related. One meaning is an overall increasing level of prices. Another meaning is an increasing money supply. If the supply of dollars goes up, then the value of each dollar goes down, which means prices go up.
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Our economic system needs prices to go steadily up in order to encourage people to spend and invest, which is what keeps the economy functioning. Prices will go up as the money supply goes up. The money supply goes up as debt goes up. This is what a growing economy is. I.e., our economy indeed requires growth in order to function.JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com0tag:blogger.com,1999:blog-2688138290971915830.post-44924885472528403542021-08-18T20:26:00.004-07:002022-01-07T21:05:12.950-08:00Digging Down to the FoundationsLately I'm reading Michael Millerman's book <i>Beginning with Heidegger: Strauss, Rorty, Derrida, Dugin and the Philosophical Constitution of the Political</i>. Dugin is the target of the book, and the main reason I'm reading it. I'm in the middle of the Rorty chapter at the moment. The overall notion seems to be that Dugin is the one who has picked up Heidegger's ball and is running with it. Strauss and Rorty have either misread Heidegger or anyway have refused to pick up his ball, for opposite reasons. Strauss is more fundamentalist than Heidegger, and Rorty is more historicist.
<p>
In a curious coincidence, my wife has been reading <i>The Great Bliss Queen</i> by Anne Klein. She tells me that Klein is discussing a debate within feminism between essentialists and constructivists. It sounds pretty much the same as the debate between Strauss and Rorty - or their followers, anyway. These debates are a bit like the conundrum, "Why not tolerate intolerance?" It's like a dog chasing its own tail.
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This brings to mind a simple analogy that I use to illustrate the potential for Buddhist thinking to provide a way to escape the deadend represented by these debates. We're trying to investigate the true reality underlying the diverse appearances that we experience in the world. We start digging down through the shifting sands of the surface, looking for the solid bedrock that holds everything up.
<p>
The fundamentalist essentialist vision is that indeed, we can cut through the fog and confusion, and whether we land on the Bible or the U. S. Constitution or Feynman's Lectures on Physics, we will find solid ground. The constructivist historicist vision is that we can dig our way straight through to empty space on the other side. The web of appearances is free floating. It might be a fair amount of work to move the whole mess, but it is quite possible, and perhaps a worthy project. We have that freedom.
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The middle way of Buddhism, or of the Madhyamaka school of Buddhism at least, is a third alternative. It's not that we find some third sort of thing once we have dug through appearances. The vision is that we can dig and we can keep digging and actually we can just keep on digging endlessly. The investigation of appearances never reaches any kind of point where further investigation isn't possible. Of course we might run out of the resources needed to keep investigating. But we can also relax our desperate search for foundations once we realize that every layer of appearance is supported by yet another layer of appearance. There is no bottom. It's not that the bottom is hollow - that's the constructivist historicist vision. There is no bottom.
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What are the practical consequences of this vision, that's hard to say. Mostly it's a matter of avoiding futile and destructive projects. The MAGA crew seems to want to scrape away the shifting sands to return society to whatever solid ground they put their faith in; once they've killed off all the liberals they can start killing each other over transubstantiation versus consubstantiation etc. The progressive crew seem to want to pick up the whole mess and move it to a less strife-filled place; maybe an annual cycle of presentations from the Human Relations department will do a lot, but the inertia of the entire system will assert itself long before we start knocking up against the constancy of the speed of light and the limits it imposes on interstellar colonization.
<p>
Buddhist practise seems to be mostly a matter of letting go of grasping. The subtle details come from a deepening perception of how we are grasping. The extremes of eternalism and nihilism are classical mirages at which we grasp. Fundamentalism and constructivism are modern manifestations of these philosophical extremes.
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I've become interested in Dugin because he seems to be a major philosophical inspiration behind the right wing movement. Recently I read an observation, that the right wing extreme in the USA is not really philosophically grounded. It's basically a gang of street thugs. There's this character in the movie <i>A Fish Named Wanda</i>, this thug who lies around reading Nietzsche and shooting his pistol. Perhaps this is a good model for someone like Steve Bannon.
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com1tag:blogger.com,1999:blog-2688138290971915830.post-64318458685478947282021-01-14T13:44:00.006-08:002021-01-15T11:42:02.709-08:00JusticeI'm no lawyer, but what we saw happening at the U. S. Capitol building on the 6th sure looked like criminal activity. What should we do about that?
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Generally speaking, the justice system is our collective means to respond to crime. We bring criminals to justice. But there are voices, mostly friendly to the criminals, warning us that justice will further division at a time when we especially need unity. Justice, division, unity: these are complicated ideas with many possible meanings.
<p>
What is the proper function of justice? Revenge? Punishment? Compensation? A crime has been committed: what should we do about it?
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A crime is a breaking of social bonds. The proper function of justice is to heal those bonds. Justice should be social therapy.
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Punishment and compensation can work as components of a therapeutic program. No matter what, a path must be provided by which criminals can be reintegrated into society. Justice must exercise discrimination but never promote division. The wisdom of Solomon is indeed required to judge what form justice should best take in any given situation. We mortals are stuck bumbling along the best we can. But if we at least understand what we're trying to accomplish, that ought to improve results.
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Unity: to the extent that it opposes diversity, it is a flawed goal. One image of unity is a watertight boundary surrounding perfect uniformity within. This is a kind of death. A vital society is a cohesive society. Cohesion means rich relationships among diverse components. Diversity without cohesion is merely plural unity, just as dead as singular unity. Society is a fabric, a network.
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The core work of justice is in repairing and strengthening social relationships.
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com0tag:blogger.com,1999:blog-2688138290971915830.post-90968023731854507662020-11-24T14:41:00.006-08:002020-11-24T16:12:00.720-08:00Diaschismic TuningI continue my wandering in the vast world of musical tuning...
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One of the first tuning calculations I performed was in 1976. I wrote a simple formula to give scores for possible tunings, to look for good ones. My idea was that a simple calculus-based optimization would yield an excellent result. I soon learned that this formula was not a matter of simple calculus. Now I have returned to evaluating tunings, with a simpler scoring forumula and simpler tunings. Still, a simple calculus optimization is not going to work with a graph like:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVXnT0hBMpUQF51vELmU0qdfBD2ofoOAxxOTPX-H5DjcsjKPyAnfK9-M2BhxKBvdMVKL_Jw2RgMGhVgbhqRwYGNdMCZxb0govsOsGLeNp8vJqnF-u9SUs8YdadtRBCR4QMarccDnEpoQw/s910/goodness+graph.jpg" style="display: block; padding: 1em 0; text-align: center; clear: left; float: left;"><img alt="" border="0" width="400" data-original-height="662" data-original-width="910" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVXnT0hBMpUQF51vELmU0qdfBD2ofoOAxxOTPX-H5DjcsjKPyAnfK9-M2BhxKBvdMVKL_Jw2RgMGhVgbhqRwYGNdMCZxb0govsOsGLeNp8vJqnF-u9SUs8YdadtRBCR4QMarccDnEpoQw/s400/goodness+graph.jpg"/></a></div>
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But here I am just looking at dividing octaves into some whole number of equal parts, so I can just sort the possibilities to see which are the best:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIFvw0gTeTFfW3NGL9WpY5WZuw37HuZLCNskn91-lnWJ7walHrfcEAMaTO9eRQJIUotkhpiqMGGe2LZVHZtpTGPtrd8RsHnf8bSaADMukwag8zAA0O6xY0YPVVGRKftehppVm7tqhpVgg/s513/edo+goodness.jpg" style="display: block; padding: 1em 0; text-align: center; clear: left; float: left;"><img alt="" border="0" width="400" data-original-height="301" data-original-width="513" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIFvw0gTeTFfW3NGL9WpY5WZuw37HuZLCNskn91-lnWJ7walHrfcEAMaTO9eRQJIUotkhpiqMGGe2LZVHZtpTGPtrd8RsHnf8bSaADMukwag8zAA0O6xY0YPVVGRKftehppVm7tqhpVgg/s400/edo+goodness.jpg"/></a></div>
<p>
Here the number of steps in an octave is the column on the left, and the score is on the right. The score is based on how well the tuning approximates the exact harmonics 3 and 5.
<p>
The tunings 118edo, 87edo, and 53edo are ones I had worked with already. I had never given thought to 34edo, but there it is on the table! So I thought I should give it a try!
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Another key feature of tunings like these is the commas that they temper. A <a href="https://en.wikipedia.org/wiki/Tonnetz">Tonnetz</a> diagram should help:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwbOKCLwztFTTnMLQXMqhQQ_bCukprHR8XsXSWRbJeecoiL980oapreJ44wiaV9mx_YcHEHJW1fClA3XcFfY488U5MsQIEOkIq08E5LMHdCzp7-VkkviRoSx-B_MuB6q693ZvJri4SheU/s959/34edo+commas.jpg" style="display: block; padding: 1em 0; text-align: center; clear: left; float: left;"><img alt="" border="0" width="400" data-original-height="492" data-original-width="959" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwbOKCLwztFTTnMLQXMqhQQ_bCukprHR8XsXSWRbJeecoiL980oapreJ44wiaV9mx_YcHEHJW1fClA3XcFfY488U5MsQIEOkIq08E5LMHdCzp7-VkkviRoSx-B_MuB6q693ZvJri4SheU/s400/34edo+commas.jpg"/></a></div>
<p>
Each square represents a pitch class, e.g. the pitch C in conventional tuning. C can exist in any octave, but if we just ignore the octave, it's plain old C. There are 12 such pitch classes in conventional tuning. Here there are 34, and instead of using letters I use numbers. Since we're ignoring octaves, one microstep up from pitch class 33 is 0.
<p>
A key feature of these tunings is that they only have a finite number of pitch classes. Musically one can move up and down by various intervals, such as octave, perfect fifths, major thirds, etc. in an infinite variety. But there are only a finite number of places to land on. So, for any pitch class, there are an infinite number of ways to get there!
<p>
The Tonnetz diagram helps show how this works. Moving from any square to the square above it, that is moving by a major third (plus any number of octaves). In 34edo, a major third is 11 microsteps. Similarly, moving to the right one square is moving a perfect fifth, which is 20 steps. The part of the diagram I show isn't enough to see directly that e.g. by continuing to move by perfect fifths one will eventually circle back to the original pitch class. 20 and 34 have a common divisor 2, so 34 has two seperate circles of fifths, each 17 steps long: one circle for the even-numbered pitch classes, and one circle for the odd-numbered pitch classes.
<p>
But the Tonnetz diagram makes it clear that that are many other ways to move in harmonic space and end up back where you started. Intervals in 34edo are near approximations to just-tuned intervals: that's what my scoring formula was measuring. If one were to wander in some direction in just-tuned space, one would never return to the same pitch class except by undoing all the steps one had taken, though perhaps in a different order. Just-tuning requires, or provides, an infinite number of pitch classes. But if the 34edo intervals are close to the just-tuned ones, when 34edo reaches the same 34edo pitch class, the matching just-tuned movement must have reached some just-tuned pitch class quite close to 1:1. Such a just-tuned pitch class is known as a <i>comma</i>. A tempered tuning like 34edo is said to temper a comma when the harmonic movement that would result in that comma in just tuning instead, in the tempered tuning, returns to the same pitch class. In the Tonnetz diagram I have given the common names for the main commas tempered by 34edo.
<p>
I use algorithmic composition to explore alternate tunings. I can coax the algorithm in various ways to loop through, or pump, one or more commas tempered by a tuning. Here's <a href="https://app.box.com/s/db3ktsrz29bu9v71ovm6bb9e2nrwbfcq">a diaschisma pump in 34edo</a>.
<p>
Studying the diaschisma a bit more, I realized that a nice 12 pitch class subset can be used to tune a conventional piano:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguq_Kc-VWBkyQ5npXpmT0LgZKD0wCZhjul5B9j_n83LV0C2-x-DETHcNfbnpU1da5xwGvSZHlo31AGwxVnlP8oknxxCcdis2oRdqnC3P5B_9qG2o2G6SUYI7NOT-odOXMnA9txBvDahXk/s897/diaschismic+tuning+in+34edo.jpg" style="display: block; padding: 1em 0; text-align: center; clear: left; float: left;"><img alt="" border="0" width="400" data-original-height="481" data-original-width="897" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguq_Kc-VWBkyQ5npXpmT0LgZKD0wCZhjul5B9j_n83LV0C2-x-DETHcNfbnpU1da5xwGvSZHlo31AGwxVnlP8oknxxCcdis2oRdqnC3P5B_9qG2o2G6SUYI7NOT-odOXMnA9txBvDahXk/s400/diaschismic+tuning+in+34edo.jpg"/></a></div>
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These pitches can be assigned to piano keys:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIo3sRBOgobPGVHvKYYmsN-F7YmPZODlex8mOPYC0OXC4h_N3Uu3WvXTF8_1dEBhVq84KN4-FhwfpHWApk_vgCV1t0FZvZk60wMxUumUjBm61w8XRYfHVh41kbCAyLiU1_bmBjMMWaelE/s715/diaschisma+piano.jpg" style="display: block; padding: 1em 0; text-align: center; clear: left; float: left;"><img alt="" border="0" width="400" data-original-height="438" data-original-width="715" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIo3sRBOgobPGVHvKYYmsN-F7YmPZODlex8mOPYC0OXC4h_N3Uu3WvXTF8_1dEBhVq84KN4-FhwfpHWApk_vgCV1t0FZvZk60wMxUumUjBm61w8XRYfHVh41kbCAyLiU1_bmBjMMWaelE/s400/diaschisma+piano.jpg"/></a></div>
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The main advantage of such a tuning is that some intervals are closer to their precise just tuning values than e.g. conventional 12edo. From the tuning goodness score table at the top of this post, one can see that 34edo has much better major thirds than does 12edo, while the perfect fifths are significantly worse. In that way, 34edo is similar to a meantone tuning like 31edo, also in the table. But then again, tempered commas are actually a compositional resource. Each tuning creates its own opportunities for making music.
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That said, it's interesting to hear what a conventional composition sounds like in an unconventional tuning. This is a minuet by Telemann that I had put into software some years back, so it was easy to convert the tuning. I should note: conventional 12edo was not conventional in Telemann's time. I have no idea what he would have used: in those days, folks were quite creative in finding fresh ways to manage the compromises involved in tuning. I'd like to think that Telemann wouldn't object: <a href="https://app.box.com/s/5x9ncenpchllwpq59uu4tvmuinaxy2h3">Telemann minuet in 34edo diaschismic tuning</a>.
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Many high-end digital keyboards allow individual control of the tuning of each pitch class. Here is a tuning table for anyone who'd like to try this:
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<div class="separator" style="clear: both;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZWDf7atilGH3TSG8lL82kHvDagSEmSYC3b_bT42WrVczdic_qKxUE9q18hO46h_JJfq8I1p6jTj5ZnEVthh_TNoPQ67BIBrsUuR2v3FIJvGA-zX72aONOYrUfHKpvA2hBf8SMnKu2fkg/s760/diaschisma+tuning+table.jpg" style="display: block; padding: 1em 0; text-align: center; clear: left; float: left;"><img alt="" border="0" width="400" data-original-height="321" data-original-width="760" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZWDf7atilGH3TSG8lL82kHvDagSEmSYC3b_bT42WrVczdic_qKxUE9q18hO46h_JJfq8I1p6jTj5ZnEVthh_TNoPQ67BIBrsUuR2v3FIJvGA-zX72aONOYrUfHKpvA2hBf8SMnKu2fkg/s400/diaschisma+tuning+table.jpg"/></a></div>JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com0tag:blogger.com,1999:blog-2688138290971915830.post-2640053721683222562020-11-01T18:56:00.003-08:002020-11-01T19:17:37.596-08:00The Conduct of EmptinessBuddhism and science share the view that much suffering is due to mistaken ideas about the way the world works, and that much suffering can be avoided by replacing those mistaken ideas with ones that better fit the world. The biggest difference between Buddhism and science is their general strategies for avoiding suffering. Shantideva’s analogy captures the difference. Walking barefoot over rocky terrain is painful. The scientific solution, or at least the technological solution, is to pave a smooth road. The Buddhist solution is to wear shoes. The focus of science is outward; the focus of Buddhism is inward. The more accurate understanding of the world provided by science allows engineers to change the world into a less painful configuration. The understanding of mind taught by the Buddha enables Buddhists to train their minds to respond to the world in less painful ways.
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The Buddhist view of the nature of reality is not monolithic. In general, the idea is that the coarse objects of our everyday perception and interpretation, the individual persons and things of our everyday world, do not exist in the cleanly defined discrete way that they appear to. For example, careful examination of a cart reveals that it is just an arrangement of parts such as wheels, an axle, etc. As such analysis is pursued ever deeper, is there some natural stopping point, where analysis reveals some elementary discrete objects that cannot be analyzed any further? Over the millennia of the development of Buddhism, a variety of doctrines have emerged. The Madhyamika school, pioneered by Nagarjuna but also followed by Shantideva, holds that no such elementary objects can exist. Whatever object appears in our investigations of the world, that object can be further analyzed to understand how it arises from the interplay of other objects. Emptiness is a term used to refer to the way objects are never discrete, elementary, and unanalyzable. Interdependent origination is a term used to refer to the way objects arise from the interplay of other objects.
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The distinction between the two ways to reduce suffering, changing the world versus changing the mind: this distinction is not so clear and discrete either. Effective engineering requires an understanding of the human mind. For example, designing safe and efficient roadways requires an understanding of how the human perceptual and interpretive systems will parse the various road markings. Mind and world are intimately coupled; indeed, each gives rise to the other: they co-emerge. The Buddhist path of mental transformation is not separate from actions in the world. Ethical and compassionate conduct is one of the cornerstones of the path, along with view and meditation. Cultivating the view of emptiness and interdependent origination is a classic meditation exercise. The view comes alive as it is reflected in one’s conduct.
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This perspective flows into conduct mainly through the channels of alertness, curiosity, and sensitivity. Our actions arise out of our perceptions and interpretations of a situation. When we know that further analysis would surely alter those perceptions and interpretations, we don’t commit 100% to the interpretation of the moment. We remain open and curious. The new experiences that unfold as we act can inform us and give us fresh understandings so we can adapt and improvise.
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There are classical teaching stories that illustrate the need for keeping our interpretations tentative. One story is about a farmer who finds a beautiful horse and the events that ensue. What first seems like good fortune then turns out to be misfortune; what first seems like misfortune turns out to be good fortune. Things are never quite what they seem.
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A further channel for living the view of emptiness and interdependent origination is through community, through collective exploration. So much of our common conversation revolves around who is right and who is wrong. More productive discussion can happen when we understand that no one is altogether right and no one is altogether wrong. Each interpretation of a situation provides another perspective. We may need to take action, and so need to resolve a coherent interpretation on which to base that action. But we can work to act in a way that keeps open opportunities to learn more, for our interpretation to evolve, rather than closing down our perceptions in order to stabilize our interpretation, our actions reinforcing our justifications.
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com1tag:blogger.com,1999:blog-2688138290971915830.post-34820162136907221192020-07-04T15:29:00.002-07:002020-07-04T15:34:24.941-07:00The Character of CharacterIt’s a simple easy idea that there are good people and bad people. Bad people should be locked up. Bad people should not be allowed on the police force. Police should get tough on bad people. Good people should be trusted. Good people should have opportunities to improve their circumstances.
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Perhaps it is the core idea of Buddhism that this notion of good people and bad people is dangerously simplistic. Certainly there is some validity to it. But the more one looks closely, the reality is not so simple. Buddhism is known as The Middle Way for a variety of reasons, but one reason is that it can encompass a simple idea along with an understanding of the limits of that idea, ways that an idea can blind us to important details.
How can we be blinded by the idea that some people are good and other people are bad?
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How somebody behaves is highly situational. A person whose children are starving and whose efforts to feed them by legitimate means have been constantly frustrated, such a person might be reduced to stealing to save the lives of their children. Another person might have a very active physiology and have a hard time sitting quietly for a long time. Put in a situation where they need to sit quietly, they may not be good at following the rules. In a different situation where constant movement is required, their behavior may well be very good.
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Whether a particular behavior is good or bad can be strongly dependent on the perspective of the observer. Someone strongly defending one side of a divisive issue can appear good or bad depending on which side of the issue the observer leans toward.
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The behavior of a person is generally very complex. In a day or a month or a year, even in an hour, a person can perform many actions, some of which may be very good and others not so good at all. A person experiences many different situations and also what’s going through their head is always shifting. One situation could make a person angry and they carry that anger into the next situation which can open the door to rather bad behavior. A person could be a hard worker on the factory floor but then go home and abuse their family, taking out their workplace resentments and frustrations.
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People do change over time. Some of that is just from getting older and slowing down. Age can bring experience and understanding. But age can also bring bitterness and frustration. There is a feedback loop where a person’s behavior leads them into situations which can then steer their behavior. For example, prisons have a reputation for converting petty criminals into hardened criminals. The possibility for such a feedback loop to lead in a positive direction is equally present.
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The most potent driver of change is a conscious intention. Parents, teachers, ministers, supervisors, therapists: each of these can work to guide a person to evolve in a more positive direction. It can certainly happen too that one person works to steer another person into some negative behavior pattern, to exploit that person one way or another. To help another person improve is one of the best behaviors possible; to lead another person astray is one of the worst.
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To realize that one’s own pattern of response and behavior is malleable and within one’s own power to shape, this is the essence of the Buddhist path. None of us are fixed quantities. We are each a process of constant becoming. To cherish our own positive potential, and that of everyone we meet, is the seed that yields the fruit that nourishes most deeply.
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The dangerously simplistic notion of good people and bad people, that seems to be at the foundation of much of our current political strife. Can we reform society effectively by putting all the bad people in prison? Can we reform the police effectively by firing all the bad police? Certainly we need to find ways to suppress bad behavior of all sorts. A good starting point would be to work to understand what sorts of situations promote bad behavior and to keep people out of those situations. Putting violent criminals in prison and firing abusive police officers, these are reasonable and necessary actions. But those are only superficial remedies.
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com0tag:blogger.com,1999:blog-2688138290971915830.post-37180404549900585712020-03-24T20:40:00.001-07:002020-06-24T18:35:59.553-07:00What is EssentialSaving lives or saving the economy? That is a truly bizarre dichotomy! More lives could be lost from the collapse of the economy than from the pandemic: that is a perspective that at least focuses on what matters. Merely staying alive, that is a very low bar indeed, but it is certainly a good place to start.
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The natural instinct in a crisis like this is to work to reestablish the routine by which we had been living adequately enough. Of course, to ignore a pandemic is not going to make it go away. Wishing doesn't make it so. There is no way to escape this unscarred, but facing the situation and realistically engaging with it, that is the approach that will minimize the damage. We do have to balance lives lost due to supply chain disruptions etc. against lives lost directly from the pandemic. People need food, clothing, shelter, and medicine. There is enough to go around. A modicum of ingenuity can keep people supplied.
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The shock of this pandemic reveals in a variety of ways that our routines have not served our needs so very adequately. This virus is the kind of thing that comes around every decade or so. A healthy economy, a healthy society, is one that can handle such challenges effectively and efficiently. So far, the United States appears to be stumbling.
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In the short run, we need extraordinary measures to make sure that people are fed and housed. Insuring adequate medical care seems, sadly, out of reach in this crisis. In the longer run, we need to restructure our routines so that the next pandemic doesn't knock us off our feet. Our way of life has been driving us against many ecological limits. Plague, drought, flood, famine: we can expect ever more frequent crises as long as we mistake recovering our routines as essential.
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com0tag:blogger.com,1999:blog-2688138290971915830.post-13165771696164348502020-02-26T14:15:00.000-08:002020-06-24T18:44:29.920-07:00Pride and FallI was discussing interplanetary colonization with an acquaintance recently. I don’t foresee that in any likely near future, while my interlocutor is convinced it’s a near certainty. Opinions do differ! The discussion did degenerate, sadly, into <i>ad hominem</i> characterizations. I was put into the “genteel-poverty crowd”; I called my acquaintance a “technocrat”. He accepted the label gracefully enough.
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Predicting the future involves some kind of model. The simplest common model is an exponential function. For example, the world population might be increasing at 2% per year. With that simple model we can forecast the population at any time in the future. Science is largely a matter of developing, testing, and refining models for various facets of the natural world, by comparing the various models’ predictions against real world experience.
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How this whole dialectic interplay between theory and experiment actually proceeds, or should proceed, is a topic of endless discussion among philosophers of science. Scientists have got the orbit of the moon around the earth figured out with remarkable precision. Philosophers of science continue to struggle to come to any basic understanding of how that precise figuring has come about.
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Immanuel Kant’s philosophy is a major waypoint in understanding how science progresses. His notion of synthetic <i>a priori</i> judgements accounts for the fact that models cannot be inferred directly from the data of experience. It seems clear by now that our thinking is more adaptable than Kant gave us credit for. But still, it often takes a succession of generations of scientists for a new conception to take root. Our ways of seeing the world are quite deeply rooted.
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In my recent discussion on interplanetary colonization, my acquaintance made a remarkable declaration: “Technocrats don’t have preconceptions.” I don’t think this is any kind of unique or even unusual attitude. Most scientists dismiss philosophy of science as being irrelevant. Generally they take for granted a kind of direct insight into the nature of phenomena. The task of science looks, from this perspective, a bit like that of a surveyor mapping out a new territory. Probably even real surveying is a bit trickier than this kind of naïve notion would portray it!
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Our modern civilization is built on science. Practically every facet of our lives has been explored scientifically. Our lives are, in turn, structured by these scientific models, be they mechanical, chemical, biological, geological, or whatever. It is quite easy to slip into thinking that we have pretty much figured it all out. The stable structures of our lives mesh neatly with the stable structures of our ideas.
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It may be just my own preconception, but things do change. The notion of environmental constraints has been talked about at least since Malthus, a couple hundred years ago, but clearly we humans have been very successful in circumventing whatever limits have appeared. How long this run of success will continue… that’s one of the core controversies of our time. What interests me here is not this or that model on which to build a forecast. What interests me is how the shape of the future can appear unquestionable to the cultish wing of the technocratic faith. More precisely, how might this kind of blind faith affect our ability to navigate any turbulence in the coming decades.
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There is a curious paradox here. The notions of Malthusian limits, of climate change, of turbulence on the horizon: these are all scientific ideas. Science forecasts change, but science is not really ready for change.
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Supposing that we actually do run out of miracles and Malthusian limits do finally get their teeth into us. What’s the impact going to be? Will it just be a matter of maybe replacing our air conditioners with more efficient units? Or are we facing famine and plague and the decimation of the population? These questions have been discussed extensively. A question that gets much less attention, though: what will happen to science? Certainly there are deep visionaries in the scientific community who can foresee the coming turbulence and begin to sketch what a post-collapse laboratory might look like. But a large fraction of the members of the greater scientific-technocratic subculture share my acquaintance’s blind faith, I fear, in the eternal stability of the present system: a steady growth in watts per capita and the rest of it. This kind of blind faith will not help us navigate any sort of coming collapse!
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com1tag:blogger.com,1999:blog-2688138290971915830.post-11521375134386681442020-01-08T11:58:00.000-08:002020-06-24T18:44:37.817-07:00Schismatic TuningI am fascinated by musical tuning, from the conventional 12 equal steps per octave to all sorts of wild possibilities. Sometimes when I am out playing on the fringes, I learn something that can bring me back into much more conventional possibilities. So here is a way to tune a conventional keyboard. I'll call it "schismatic tuning" but I don't doubt that it has been explored again and again in the past.
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Musical tuning is essentially a branch of mathematics, especially the way I approach it. Often in mathematics and science the focus is on novelty, on fresh discoveries, where fresh means not previously encountered by the human mind. This focus is unnecessary in math and science, is somewhat distracting or misleading, and will likely serve us much less well in the future. Since the time of Kepler and Galileo, math and science have expanded in a stunning fashion. Predicting the future is a fools game, but it seems unlikely that environmental constraints will permit continuous growth in extracting resources and dumping wastes. The modern trend of constant growth seems destined to end sooner rather than later. Math and science will be of great value in any post-growth society. To keep them alive, though, the focus will need to shift away from novelty.
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So here is a historical preface to the schismatic tuning I have (re)discovered: <a href="https://www.youtube.com/watch?v=-tyIvhv1hc0">Emilio de’ Cavalieri’s mysterious enharmonic passage</a> - a modern rendition of a renaissance recovery of an ancient Greek tuning! Paul Erlich has written a thorough discussion of tuning <a href="https://en.xen.wiki/w/Paul_Erlich">A Middle Path Between Just Intonation and the Equal Temperaments</a> - I have barely scratched the surface of this paper! I imagine that schismatic tuning is described in there somewhere! I would just like to share my (re)discovery here of this one small facet of the vast universe of tuning. I offer it as an invitation to explore further!
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A quick review of fundamentals. A musical interval is the relationship between two pitches, which can be analyzed as the ratio between their frequencies. If pitches are an octave apart, their frequencies are in a 2:1 ratio; a fifth apart, a 3:2 ratio; a major third apart, a 5:4 ratio. These ratios are ideal. Just Intonation is a tuning that uses these ideal ratios. But for a variety of practical reasons, it is often useful to adjust, or temper, these ratios. There is no perfect solution to the puzzle of temperament. Modern keyboard tuning adjusts the fifth to 2^(7/12) ~= 1.4983 and the major third to 2^(4/12) ~= 1.26. The human ear can detect reasonably well the difference between this tempered major third and the ideal of 1.25.
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Schismatic tuning is actually a family of tunings. I will present one version, based on dividing an octave into 53 equal steps, rather than the conventional 12. With 53 steps available, a fifth is tempered to ~1.499941 (31 microsteps) and a major third to ~1.248984 (17 microsteps). The fifth is improved, but the conventional tuning was already very good; the main improvement is in the major third.
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How can such an improved tuning be adapted to a conventional keyboard? Here is my proposed schismatic tuning:
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhjVqQhs6v2WSkVGDh-76ZljpZMhjtMgtuQo8WG8_LlgvIWL4d5CXJgUvlwtuZWn04MZ14hzWasVn-q-BCxOHUHDypYdHclczFu2EloTWdjW3X_q1opGSTZqGNJQQTA52RC4fAtLlIYE0/s1600/scale0.jpg" imageanchor="1" ><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhjVqQhs6v2WSkVGDh-76ZljpZMhjtMgtuQo8WG8_LlgvIWL4d5CXJgUvlwtuZWn04MZ14hzWasVn-q-BCxOHUHDypYdHclczFu2EloTWdjW3X_q1opGSTZqGNJQQTA52RC4fAtLlIYE0/s400/scale0.jpg" width="400" height="35" data-original-width="703" data-original-height="61" /></a>
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The top row names the keys on the keyboard. The second row gives the number of microsteps from the low C to the particular key of that column. The bottom row re-expresses that pitch in terms of cents. The conventional tuning would result in pitches of 0, 100, 200, 300, etc. cents. So this last row makes clear the difference in pitch between the schismatic tuning and conventional tuning, e.g. D is 3.774 cents sharper.
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Some points to observe:
<ul>
<li>Almost all of the fifths are 31 microsteps, i.e. very accurate. From D to A is only 30 microsteps, though.
<li>Four of the major thirds are the ideal 17 steps: C to E, F to A, G to B, and D to F#. The others are sharp by a microstep, i.e. closer to a pythagorean major third, 81:64.
<li>The sizes of the chromatic intervals in this tuning are not all the same: 4, 5, 4, 4, 5, 4, 5, 4, 5, 5, 4, 5.
<li>The syntonic comma is not tempered. E.g. moving by fifths up from C to E, one must cross the "wolf" fifth from D to A.
This is a distinctly unconventional tuning.
</ul>
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One can certainly play in any key signature with this tuning - none of the intervals is too far off. But certainly a piece of music will sound different when the key signature is changed. This tuning does allow though a simple dynamic shift as outlined in my post <a href="http://interdependentscience.blogspot.com/2010/08/dynamically-tuned-piano.html">Dynamically Tuned Piano</a>. With perhaps a push of a foot pedal, A can be sharpened by a syntonic comma:
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZXKwP8ILdN9ZEg0WfhXfODTOAfyMvdBcneU05RvIL3tILmTnFV1hTM6tzgTD9Nf36DclA3hq8AvSB3a2k0maS2zln4mQvkK3YwPUmP9WzHa0_uGJU2os8zORPZjvDYM4Uzov7JsDih1k/s1600/scale1.jpg" imageanchor="1" ><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZXKwP8ILdN9ZEg0WfhXfODTOAfyMvdBcneU05RvIL3tILmTnFV1hTM6tzgTD9Nf36DclA3hq8AvSB3a2k0maS2zln4mQvkK3YwPUmP9WzHa0_uGJU2os8zORPZjvDYM4Uzov7JsDih1k/s400/scale1.jpg" width="400" height="35" data-original-width="703" data-original-height="61" /></a>
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or a different pedal could instead flatten the D by a syntonic comma:
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJjTKbvw0sUr1NYMkkFkHX8vQb10uTqkxJcKJZFhU8-bxE3l4tbvTxIsoLOLJRfm0UUf8XOAtFf06x83K-xy9GCj3oiR276sP2n7gNiAgdvEE-9ZRug-eofLmyqW3ruPBUe4so-k6ncsQ/s1600/scale2.jpg" imageanchor="1" ><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJjTKbvw0sUr1NYMkkFkHX8vQb10uTqkxJcKJZFhU8-bxE3l4tbvTxIsoLOLJRfm0UUf8XOAtFf06x83K-xy9GCj3oiR276sP2n7gNiAgdvEE-9ZRug-eofLmyqW3ruPBUe4so-k6ncsQ/s400/scale2.jpg" width="400" height="35" data-original-width="703" data-original-height="61" /></a>
<p>
These shifts will move the wolf fifth up or down a fifth, and also rotate which major thirds are pythagorean, etc.
JimKhttp://www.blogger.com/profile/16167191806249119508noreply@blogger.com1