Interdependent Science: September 2024

I work with musical tuning systems that divide octaves into some number of equal steps. Conventional tuning divides octaves into 12 equal steps. A principal feature of a tuning system is how well it approximates fundamental intervals. Since the systems I work with are all built from octaves, they can represent octaves perfectly. But other intervals, such as perfect fifths (3:2) and major thirds (5:4), are only approximated. Another interval that is fun to explore is the ratio 7:4. This interval is not commonly used in conventional music, so it doesn't have a conventional name.

This is a table that shows, for three tuning systems, those that divide octaves into 12, 31, and 43 equal parts, how well they approximate these fundamental intervals. The conventional 12 equal steps per octave has the chief downside that the major third is off by 13 cents, which is definitely noticeable. The 7:4 interval is so far off, 31 cents, that it can't practically be used in conventional tuning. Of course the conventional system has its advantages: 12 is a relatively small number which makes it feasible to build and play musical instruments that use this tuning; and of course these instruments are very widespread, together with music that uses the tuning, theory in terms of the tuning, etc.

31edo, the tuning that divides octaves into 31 equal steps, is very close to quarter comma meantone. The syntonic comma, the frequency ratio of 81:80, is about 21 cents in size. The perfect fifth in 31edo is a bit more than 5 cents flat, i.e. about a quarter of a comma. A perfect fifth in conventional tuning is only about a ninth of a comma flat, i.e. it is much more accurate. But by flattening the perfect fifth more, 31edo can represent the major third with great accuracy. By whatever fluke of mathematics, 31edo also approximates 7:4 very well.

43edo is similar to 31edo. It flattens the perfect fifth a bit less, at the cost of a greater error in the major third. The error in 7:4 is considerably greater, but still not too bad.

Another key feature of tunings is which commas are tempered out. All three of these tunings temper out the syntonic comma, and are therefore known as meantone tunings. Conventional music theory and musical notation is based on meantone tuning, whose history goes back maybe five hundred years. For each of these meantone tunings, moving four perfect fifths, e.g. from C through G, D, and A, to E, has the same endpoint as moving a major third and two octaves. The table above makes this easy to check. In conventional 12edo, 4*7 = 4 + 2*12 = 28. In 31edo, 4*18 = 10 + 2*31 = 72. In 43edo, 4*25 = 14 + 2*43 = 100. Conventional music does not distinguish between the E that is four perfect fifths from C and the E that is a major third and two octaves from C. So a tuning system that can be used with conventional music should map these two different combinations of intervals to the same final pitch.

The 7:4 interval is not used in conventional music. There are other commas, such as 225:224, that are formed by combining 7:4 with other fundamental intervals. Both 31edo and 43edo temper out this comma. Another, more esoteric, comma, is 12288:12005. In 43edo, if one moves by 7:4 four times, this is 4*35 = 140 steps. From there, move further, by a major third: 140+14 = 154. This is the same pitch one arrives at by moving a perfect fifth and three octaves: 25 + 3*43 = 154. This comma is not tempered out by 31edo: 4*25+10 = 110, while 18 + 3*31 = 111. I decided to build a piece of music around this esoteric comma 12288:12005.

An effective way to structure a piece of music is by using a scale. A tuning system provides a set of pitches; a scale defines a useful subset of these. For example, conventional tuning provides 12 pitches per octave; a diatonic scale picks out 7 of these. A scale will support traversing some commas that are tempered out by the tuning system, but not all such commas. So, for example, a diatonic scale supports traversal of the syntonic comma: there are diatonic scales that include all the notes C, G, D, A, E that traverse the syntonic comma. Another comma tempered out by conventional 12edo is 128:125 - moving three perfect fifths is the same as moving an octave. But there is no diatonic scale that includes all the notes in the traversal C, E, G#.

I have seen mention of systematic ways to construct scales, but I am very much a beginner in all this. I like to learn by exploring! I just try a variety of scale structures to see what might work. What I landed on here is a very regular structure: a scale that includes 21 notes per octave out ot the full set of 43. Pick a starting note, then take the note two steps higher, and again two steps higher, until one has 21 notes in the set. So the scale will be made almost entirely from steps of size 2, except for a single interval per octave of size 3.

I use tonnetz diagrams to work out how scales function. The standard tonnetz diagram is based on the fundamental intervals of the perfect fifth and major third:

Each number represents a pitch class. Moving right one cell corresponds to a perfect fifth, e.g. from 0 to 25. Moving up one cell corresponds to moving a major third, e.g. from 0 to 14. All the notes here are folded into a single octave. The numbers in the diagram repeat a lot: this is nature of tempered tunings. One can start at a 0 cell, move four steps to the right: 25, 7, 32, 14; and then down one cell to arrive at another cell with a 0. This is how a syntonic comma is traversed. The notes in this scale are marked in green: one can see that there is no way to traverse a syntonic comma using this scale. It is an unconventional scale!

On the other hand, starting at a cell marked 0, move two cells to the left and then down two cells. One arrives at a cell marked 8. Moving the interval 7:4 from there, i.e. 35 steps, brings one back to a cell marked 0. This is a traversal of the comma 225:224 and can be done completely within this scale. The scale supports the traversal of this comma.

The intervals 7:4 are not explicitly shown on this diagram. I want to construct music from three fundamental intervals, 3:2, 5:4, and 7:4. But a computer screen or a piece of paper only has two dimensions! So there is a bit more work involved to trace the pathways in the three dimensional space.

Here is an unconventional tonnetz diagram, a different perspective on the three dimensional network of notes and intervals involved here. This tonnetz diagram shows perfect fifths (25 steps, horizontal movement) and 7:4 (35 steps, vertical movement). Again the notes of the scale are highlighted in green. Start at a cell marked 0 and move down four cells and then to the right one cell. One arrives at a cell marked 14. This is a major third from a cell marked 0. Major thirds are not explicitly represented on this unconventional tonnetz diagram, but this tonnetz diagram shows that this scale supports the traversal of the esoteric comma 12288:12005.

Here is a third perspective on the space of notes and intervals, showing major thirds horizontally and 7:4 vertically.

Enough with the diagrams! Here is a new algorithmic composition in 43edo that uses this scale to traverse the comma 12288:12005.

Here is a new puzzle that was developed by a team I've been working with: www.gridlexic.com. I would like to outline here, by way of an example, what is required to solve this puzzle.

This puzzle is essentially a variation of sudoku. A sudoku solution is a 9x9 array of numbers; a gridlexic solution is a 5x5 array of letters. In both puzzles, one starts with some small subset of the array already filled in, e.g.

A gridlexic solution will have 5 distinct letters in the solution. The puzzle presents 9 possible letters. Part of solving the puzzle is to figure out which 5 are in the solution, and which 4 are to be left out. Of course, if a letter is in the initial set of clues, then it will certainly be in the solution!

Each of the 5 solution letters must occur exactly once in each row, column, and outlined sector. In sudoku, the sectors are 3x3 rectangles. In gridlexic, they are irregularly shaped regions containing 5 cells. In addition to these sudoku-like rules, the letters must form words in the horizontal or vertical highlighted regions. In this example puzzle there is one vertical word with 4 letters, and one horizontal word with 4 letters. These words overlap at one cell.

Each of the 5 distinct letters in the solution will occur in at least one of the words in the solution.

Here is the solution to this example puzzle: