Circulating and Traversing

Here is a new piece in 42edo. My main idea here was to use a relatively small scale. The scale here uses 9 notes per octave, out of the total of 42 per octave the tuning makes available. I like how four steps of the interval 35:24 comes very close to 32:7. This is rather like how four steps of the interval 3:2 comes very close to the interval 5:1. This second closeness is the syntonic comma, fundamental to conventional music. 35:24 is quite close to 3:2. Maybe the similarity of these two closenesses will let me make some music that doesn't sound too too unconventional?

The interval 48:25 (the inverse of 35:24) corresponds to 19 steps of 42edo. This table shows, in the second column, the sequence of pitch classes visited when moving by repeated intervals of 19 steps. The third and fourth columns show when this sequence comes close to the starting pitch (0). When the sequence comes closer than it has before, that marks a scale size that has the circulation property. Circulation means the scale can be shifted by just tweaking one of the included pitch classes. Scales of size 3, 5, 7, 9, 11, 20, and 31 circulate here. The scale used in this new piece is the scale of size 9, including the pitch classes 0, 7, 11, 15, 19, 26, 30, 34, and 38. The scale can be shifted in one direction by replacing 0 with 3, or in the other direction by replacing 26 with 23. These replacements correspond to the conventional sharps and flats of changing keys.

The other desired property of a scale is that it supports comma traversal. In this diagram, moving one cell to the right is moving a minor third, 6:5. Moving up a cell is moving by the interval 8:7. The nine pitch classes of this scale form a path that connects repeated appearances of pitch classes. This path corresponds to a comma traversal.

My algorithmic composition software is based on a consonance score for intervals. 3:2 gets a very good score; 35:24 does not get a very good score! 6:5 and 8:7 get reasonably good scores. This scale being based on a rather dissonant generator gave my software a bit of a challenge! I can tweak my consonance score calculator to some extent, but there are practical limits! So trying to build music using 35:24 as a parallel to 3:2 could only go so far! I tried to use a scale with only 7 notes per octave, the same as a conventional diatonic scale. But the resulting path really could not be traversed with reasonably consonant intervals! So I moved up to 9 notes per octave... it came out a lot better!

Generating Scales

Here is a new piece in 31edo. 31edo is very close to quarter-comma meantone, which was used in Europe int the 16th and 17th Centuries. So it is easy to make very conventional music in 31edo. The scale I have chosen for this piece, however, pushes the sound in a rather unconventional direction!

A simple way to construct a musical scale is to select an interval as a generator. Start at some first pitch in the scale, then move up by the generating interval and add to the scale the pitch one arrives at. Repeat this process some number of times, and one has a scale with some basic structure. For example, the conventional diatonic scale is generated with the perfect fifth. Starting at F, one visits C, then G, then D, A, E, and B.

How many notes should be in the scale? Key circulation provides an answer. When the next pitch in the sequence is so close to the starting pitch that none of the pitches in the scale so far are between the next pitch and the starting pitch, that is a good place to stop. Leave out the next pitch! Then for a key change, one can remove the starting pitch from the scale and add the next pitch. So, for example, after B the next pitch in the sequence would be F#. This is very close to the starting pitch of F. To shift the scale to a new key, remove the F and add the F#.

Another good stopping point would be at A. The next pitch is E, which is very close to F. This earlier stopping point yields a pentatonic scale. To shift the pentatonic scale, one removes the F and adds the E.

Another fertile property of a scale is its support for traversing one or more commas. When pairs of pitches in the scale are separated by intervals that are not simply stacks of the generating interval, so there are two different harmonic relationships between the two pitches, this means that the tunings system has tempered out a comma. For example, one can move from F to A by a stack of four perfect fifths. But one can move from F to A directly by a major third. The pentatonic and diatonic scales temper out the syntonic comma.

This scale for this new piece is not a diatonic scale! Instead of generating the scale with the perfect fifth, a 3:2 frequency ratio, this scale is generated with the interval 35:24. I see that this interval has been called a septimal sub-fifth. In 31edo this interval is represented by 17 microsteps. Moving by this interval 20 times brings one back very close to the starting point, so a circulating scale can be constructed with 20 pitch classes. This is the scale used in this piece.

In exploring 42edo, which gives good approximations for the intervals 5:3 and 7:4, I noticed that the combination of these intervals, 35:24, supports traversing the comma 10616832:10504375. Moving by 35:24 four times, one arrives at a pitch that can also be reached directly from the starting pitch via the interval 7:4. 31edo supports the same scale, but approximates many intervals better than 42edo, so I decided to try this scale in 31edo instead.

Anyway, this is an unconventional scale in a conventional tuning system! The key properties of the scale are 1) it circulates, and 2) it supports traversal of commas.

Adventures in Tuning

Here's a new piece in 42edo, the tuning system that divides octaves into 42 equal intervals. This is my first attempt to make some music in this tuning system.

42edo first came to my attention a few weeks ago. I was thinking about tuning systems that provide good approximations to the just interval 7:4. Meantone temperament is a class of tuning systems that provide good approximations to the just interval 5:4. Perhaps an approach that mimics meantone could work!

Meantone evolved from Pythagorean tuning, which uses a chain of perfect fifths, of the just interval 3:2. The fundamental problem of temperament is shown in this graph. The marks with note name labels are a chain of just tuned perfect fifths. The green horizontal line is the just interval 5:4. The note E is the just interval 81:64. The gap between this and 5:4 is the syntonic comma. It is small enough that 81:64 tends to sound like a mis-tuned 5:4. Tempered scales tweak intervals to eliminate such annoying slight differences. Meantone tuning uses a slightly flattened perfect fifth, which pulls that E note closer to the 5:4 line.

The red horizontal line is the just interval 7:4. The Pythogorean Bb and A# are both somewhat close to 7:4. One could flatten the perfect fifths to pull that A# down closer to the 7:4 line. To pull the A# down so it lands exactly on the 7:4 line requires perfect fifths about 5.07 cents flat of just. Remarkably, to pull the E down so it lands exactly on the 5:4 line requires almost the same tweak, perfect fifths 5.38 cents flat of just. 31edo has perfect fifths 5.18 cents flat of just, so it accomplishes both goals splendidly. It is a popular alternative to conventional 12edo for very good reason!

So what about 42edo? Well, instead of A#, one could pull Bb down to the 7:4 line, by making perfect fifths about 13.6 cents sharp of just. This is clearly a bit of a wild idea, because it pushes E away from the 5:4 line. But still, it's fun to explore. 42edo has perfect fifths that are about 12.3 cents sharp of just. Hmmm.

Then, a total coincidence as far as I can tell, the XA - Monthly Tunings group on facebook voted to make 42edo the tuning of this month. Well, OK! I guess I do need to give it a whirl! This is the sort of thing my algorithmic approach to composition is for, to make it easy to try out novel tuning systems.

As a first step, I looked at how well 42edo approximates various just intervals. This chart gives the errors in terms of steps of 42edo, for just ratios with numerators along the top and denominators along the left side. For example, a just perfect fifth, 3:2, is 24.568 steps of 42edo, so it will be approximated by 25 steps. The error is 0.432 steps, as shown in the chart at the intersection of the column labeled 3 and the row labeled 1 (factors of 2 can be ignored, since 42edo provides exact octaves). This 0.432 corresponds to the 12.3 cent sharpness of the perfect fifth: each step of 42edo is 1200/42 = 28.57 cents.

In this chart, the boxes are grey when the numerator and denominator are not relatively prime, i.e. the entry is redundant. The boxes are green when the error is small. The green ratios would be good to use as building blocks in music in 42edo. The two simplest green boxes are 7:4 and 5:3. These could form a good foundation for musical structures.

My next step was to build a tonnetz diagram with these two building block intervals. 7:4 is represented by 34 steps of 42edo, and 5:3 by 31 steps. I like to use comma traversal as a large scale structure for composition. Commas tempered out by 42edo appear as repeated cell labels on this tonnetz diagram. If you start, say, at a cell labeled 0 and move 6 cells to the left, a chain of 5:3 intervals, you arrive at a cell labeled 18. You can also arrive at a cell labeled 18 by moving down 3 cells, a chain of 7:4 intervals. This coincidence corresponds to the fact that 42edo tempers out the comma 250047:250000.

The readers of this blog post can be divided roughly into two categories: those who know more about tuning systems than I do, and those who know less. The distribution of tuning expertise is surely a good example of fat tails, which is to say that most people who know more than I do almost all know a lot more, and those who know less know a lot less. As I explore and learn about tuning, I am constantly amazed that I never, for all practical purposes, discover anything new. Those who know a lot less than I might be skeptical. Surely I am out into uncharted territory, at best bordering on nonsense, if not deeply plunged! But notice... this comma 250047:250000 was named the Landscape Comma almost twenty years ago!

Once I spot an interesting comma to traverse, a typical next step is to define a scale that supports the traversal. One fun feature of scales is when they have a period that is a fraction of the octave. With tunings like 31edo that divide octaves into a prime number of equal steps, this kind of scale is impossible. But 42 = 2x3x7 is not prime! I noticed that the path from a 0 cell to a 0 cell has nice stepping stones of 14 and 28. So a scale with a period of 14 looks very nice. I put 5 notes of the the 14 in the scale, as highlighted in green on the tonnetz diagram. These form a nice path for traversal of the Landscape Comma. The music linked at the top of this post uses this scale.

Health Science

The exact reason for the assassination of UnitedHealthcare CEO Brian Thompson may never be known, but it has certainly brought into focus the widespread frustration with medical insurance. The reluctance of insurers to pay for medical care care often enough means that people fail to get proper medical care, leading to pain, disability, and early death. Medical expenses drive many people to bankruptcy, despite their having medical insurance that on the surface should have prevented financial disaster. The medical insurance industry is generating tidy profits. Thompson's compensation was $10 million per year, and no doubt other top executives are similarly well off. There is a clear imbalance between the benefits to executives and stockholders, versus their customers.

Of course, many corporations beyond the medical insurance industry are also biased toward shareholder profits and executive compensation, at the cost of the quality of the products delivered to their customers. What makes medical insurance special is the degree of need. Medical care is often a matter of life and death. But medical insurance is just one component of medical care. U.S. health care expenditures total $4.5 trillion. Private insurers paid about 30% of this, or about $1.4 trillion. About $0.25 trillion went to the insurance companies themselves, for expenses and profits. So medical insurance is about a 6% overhead on medical spending. The small size of this number doesn't reflect all the costs of the insurance industry: for example, they have a lot of control over who gets what treatment, and may make inefficient decisions. And of course the real bottom line is not financial, but people's health.

I have seen charts that compare U.S. healthcare costs and outcomes with those of other countries. It looks quite clear that the U.S. spends an enormous amount on health care, but does not get commensurate health care results. People often post medical bills on social media, with astounding charges, often for relatively routine procedures. Pharmaceutical prices are often prohibitively expensive. To some extent, the medical insurance companies are caught in the middle. People cannot afford to pay huge insurance premiums, but they often need very expensive medical care. The insurers have somehow to allocate available funds to the most necessary care. No doubt they do an imperfect job of it, but the task is really impossible.

Health care corporations are like any other corporation: they are controlled by shareholders whose goal is profit, return on their investment. What incentives can be put in place to motivate corporations to provide high quality products at fair, affordable prices... this is a general problem. Part of the solution is the competitive nature of markets. Part of the solution is regulation by governments or other supervisory bodies. Part of the solution is cultural, a matter of changing understandings and expectations among all the various players.

Health care is a vast network of businesses and practices. Ultimately, it incorporates everything we do. For example, workplace conditions create stresses that often result in strains and injuries requiring medical attention. If workplaces were less physically stressful, there would be less need for medical care. Health care starts at home. If people help each other catch problems when they are small, there would be less need for medical care. How much of our increased medical expense is due simply to the increase in people living in greater isolation, away from extended families, away from people who catch problems before they need professional attention?

My particular interest here is in the scientific facet of healthcare. Medicine is the supreme example of a science where pure science and applied science are inseparable. Modern science is founded on physics, the supreme example of a science where pure science and applied science are separable. This separation of science from its use bears much responsibility for our global environmental crisis. Modern science is essentially driving itself out of business. The environmental crisis and the healthcare crisis are both pointing to the need for a new kind of science, a new vision of science, where pure science and applied science are understood to be inseparable. Medicine is thus the proper foundation for this new science. The healthcare crisis represents the birth pangs of a new science.