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.

A Tale of Two Scales

Here are three new pieces in 43edo:

1. quite conventional
2. less conventional
3. quite unconventional

These are all composed by my highly randomized and highly parameterized software. I had to tweak the program in a variety of ways to coax it to produce pieces that sound reasonably musical. So some of the difference in sound is due to the different tweaks involved. But the fundamental changes were the notions of consonance and the scales.

The first two pieces use the same scale. The tuning system 43edo provides 43 equally spaced notes in each octave. The scale in the first two pieces picks 19 of these notes for use in the composition. The third piece uses a very different subset of 21 notes out of the 43. The first piece differs from the others in the set of intervals considered consonant. The second and third pieces share an extended notion of consonance.

The consonance relationships between the notes of a scale can be visualized using a lattice diagram.

This is the scale lattice used in the first piece. The fundamental consonant intervals are very conventional: perfect fifths (green arrows) and major thirds (blue arrows). This diagram is similar to the circle of fifths, but here I have prioritized the major thirds. Moving four steps along a sequences of green arrows will bring one back around close to where one started, just a blue arrow ahead. This relationship between the green arrows and the blue arrows, between the perfect fifths and the major thirds, is due to the fact that 43edo tempers the syntonic comma. Tempered scales, such as 43edo, are at the foundation of classical European music. Musicians in Europe were experimenting with a variety of tempered scales, including 43edo, in the Baroque era, before the modern 12edo scale became dominant.

This is the scale lattice for the second piece. It uses the same 19 notes per octave, but adds new consonant relationships, based on the 7:4 frequency ratio. The best approximation for 7:4 in conventional 12edo is 10 steps, which gives a frequency ratio of 1.7818. This is quite a bit sharp, one of the reasons that 7:4 is not considered consonant. In 43edo the best approximation for 7:4 is 35 steps, which gives a frequency ratio of 1.7580. This is much close to the pure interval of 1.75, so it is a natural interval to include as consonant.

This is the scale lattice for the third piece. The geometric layout of these diagrams is not fixed: what matters is what notes are present and how they are connected by consonant intervals. But this third scale is indeed very different in those relationships. For example, the 19 notes of the first two scales are connected by a single continuous path of green arrows, of perfect fifths, while in this third scale there are no green paths longer than two steps!

My hope here is that, with three algorithmic compositions based on three different scale lattices, that the sounds of the different scales can start to become apparent.

Scale Balance

Ah, one more piece in 43edo. Earlier today I was working with a 43edo scale that had 11 pitch classes. I was considering removing one pitch class, to make it better balanced. It really looks like adding one pitch class works better! So this new piece has 12 pitch classes. You could map this to a piano keyboard! It would be mighty strange!

I rearranged the lattice and added the new pitch class. The balance is clear!

Scale Design

Here is a new piece in 43edo. This piece uses a scale with 11 of the 43 pitch classes of 43edo. I designed this scale to support traversal of the comma 12288:12005. A few months ago I posted a piece using a scale with 21 pitches classes out of 43. My idea with this larger scale was mostly to avoid supporting traversal of the syntonic comma. 43edo is a meantone tuning, so it will support traversal of the syntonic comma, i.e. it will support most conventional music. So my little project here is to take a tuning system that supports conventional music, but then to make a scale with it is unconventional. An example of this in conventional 12edo is a whole note scale. A whole note scale includes no perfect fifths or perfect fourths at all!

Instead of making as big a scale as possible that still blocks traversal of the syntonic comma, my idea here was to make a smaller scale that more narrowly focuses on traversing 12288:12005. Another feature of the bigger scale is that it can be constructed with a single generator: it was just a sequence of intervals of size 2. This new scale with 11 pitch classes does have a pretty regular structure: the scale intervals are mostly size 2 and 6, with a single size 3 interval. I have seen where people build scales with very exact mathematical order... the scale I am using here was just something I engineered in an ad hoc way.

Scale lattice diagrams helped me engineer this scale. I was looking at a lattice for the conventional 12edo diatonic scale:

Here the blue arrows show major third intervals and the green arrows show perfect fifths. Most chords are subgraphs connected by these consonant intervals. One could add further arrows for minor thirds, but these are implied by the arrows already in the lattice. A traversal of the syntonic comma appears as a loop on the lattice, e.g. 0, 7, 2, 9, 4, 0. Moving by four perfect fifths arrives at the same pitch class as moving by a major third.

There is an awful lot of good music that can be built from this diatonic scale. The lattice doesn't look so complicated, so my idea here was to make a scale in 43edo whose lattice doesn't look too terribly much more complicated:

This lattice includes red arrows for the interval 7:4. Traversing the comma 12288:12005 appears on the lattice as a loop, e.g. 0. 8, 16, 24, 32, 18, 0.

This scale looks related to 5edo. It has 5 clusters of pitch classes: {40, 0, 2}, {8, 10}, {16, 18}, {24, 26}, and {32, 34}. Hmmm, maybe it would be more symmetric if pitch class 40 was ommitted. But that would leave a pretty big hole in the scale, from 34 to 43, 9 steps. More to explore!

Tempering 4096:4095

Here is a new piece in 140edo.

This piece traverses the comma 4096:4095 eighteen times. The repeating staircase pattern in the above score is indicative of this looping structure.

I wanted to construct a piece that traversed the comma 4096:4095, and went searching for a good tuning system, a good equal division of octaves, to do this.

This table guided me to 140edo. I looked through thousands of possible equal divisions of the octave. First I filtered out just those that temper out 4096:4095. 4096 is a power of 2. Most of my software treats octaves as equivalent notes, so powers of 2 tend to disappear. The graphical score above works this way: the vertical axis is pitch class, i.e. the pitch with the octaves erased. 4095 = 3*3*5*7*13, so the search for a good tuning system involves those primes. In 140edo, the prime number 3 is approximated by 222 steps, 5 by 325 steps, 7 by 393 steps, and 13 by 518 steps. So the composite number 4095 is represented by 222 + 222 + 325 + 393 + 518 = 1680 steps. But 1680 = 12 * 140, i.e. 1680 is exactly 12 octaves. 4095 and 4096 are both exactly 1680 steps in 140edo, so we say that 140edo tempers out 4096:4095.

This table also shows the error involved in the various tunings. E.g., the prime number 3 is actually 221.8948... steps of 140edo, which the tuning will approximate by 222 steps. The error for the prime 3 is thus 0.1052 of a step. The table shows the errors for all the primes involved, and then a combined score. I sorted the tunings by this combined score. 441edo is a bit better than 140edo, but then it is also nice to have a smaller division of the octave, so I decided to use 140edo instead.

Harmonic Layers

Here's a new piece in 270edo. 270edo is the tuning system that divides octaves into 270 equal parts.

The main thing I am exploring here is how the meaning of unusual intervals can be clarified by a rich harmonic context. This piece pushes my usual pattern in two ways. This piece has six voices; usually I limit myself to four. Also, this piece uses intervals that approximate frequency ratios built with the prime numbers 2, 3, 5, 7, 11, and 13; I rarely go beyond 7. Intervals such as 14:13 and 15:14 have been permitted in the construction process of this piece. 14:13 is approximated by 29 steps of 270edo; 15:14 by 27 steps. So the tuning system has the precision to distinguish between these intervals. I suspect that if these intervals were heard in isolation, they would be practically impossible to distinguish. But if a richer context is provided, perhaps in the form of tetrads 9:11:13:14 and 9:11:14:15, the ear would have more information and would be able to hear the difference. Anyway, that's what I am trying to do here. With six voices, the more esoteric intervals will occur in combination with less esoteric intervals, and the ear will be able to make some sense of what is happening.

This is a graph of the piece, with time in seconds on the horizontal axis, and pitch class on the vertical axis. The pitch class is the pitch folded into a single octave. The vertical axis labels give the fraction of that single octave.

This graph shows that there is further structure to this piece, that should help clarify the meaning of the intervals. This piece traverses the comma 2080:2079 twelve times.

270edo is such a precise tuning system, one might think it to be effectively equivalent to just intonation, where the intervals are exact rational frequency ratios, rather than approximations. But 270edo does temper out many commas, as indeed any edo, any equal division of the octave, must. Another way to think about tempering out 2080:2079 is that 270edo maps 77:65 and 32:27 to the same interval.