Scales for Traversing Commas

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.

Gridlexic

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:

Each row has exactly one occurrence of each of the 5 letters in the solution, e.g.:

Each column has exactly one occurrence of each of the 5 letters in the solution, e.g.:

Each sector has exactly one occurrence of each of the 5 letters in the solution, e.g.:

The vertical highlighted cells form a word:

The horizontal highlighted cells form a word:

We certainly hope it is a fun puzzle to play!

Freedom and Constraint

Interesting things happen in the space where freedom and constraint play with and against each other. In my musical explorations, with algorithmic composition and CSound synthesis as my vehicles, I have several mechanisms for defining this space of play.

Tuning and consonance are fundamental. I can constrain pitch selection to a scale, to a subset of the full set of pitches provided by the tuning system. Vertical relationships can be regulated, requiring chords to conform to some set of shapes. A variety of horizontal relationships, adjacency in a voice but also across longer scale repetition structures, can be guided more or less rigidly to some set of consonant intervals.

The large scale repetition structure of the piece is another aspect of constraint. Low dimensionality means few horizontal relationships, allowing greater freedom. High dimensionality introduces many horizontal relationships, clusters of clusters, which constrain the pitch selections.

The thermodynamic approach of my algorithm provides a temperature parameter. High temperature allows more freedom, low temperature imposes more constraint. There is generally a transition where long range order emerges, with fractal fluctuations at the transition.

In this new piece I don't target the phase transition. I gave the piece a high dimensionality, so it was tending to jump into a very orderly state. To forstall this, I initialized it randomly and then cooled it just enough to let a moderate amount of order emerge... that's another dimension of the freedom-order interplay: how the pitches are initialized, and how long the consonance optimizer is run.

This piece is in 171edo and uses the same chord shape constraint as the piece I posted a few days ago. But this piece has three voices instead of five. This gives the piece more freedom to move harmonically. My idea was that this would reduce the tendency to fall into a highly ordered state... but it didn't seem to work that way! I thought I could get away with increasing the dimension; I did keep the higher dimension, but just reduced the amount of pitch optimization jostling to preserve some of the initial freedom.

Chord Progression

Here is a new piece in 171edo. 171edo, the tuning system that divides octaves into 171 equal steps, provides very precise approximation to the just intervals 3:2 (a perfect fifth), 5:4 (a major third), and 7:4 (an unconventional interval). 3:2 is approximated by 100 steps of 171edo, 5:4 by 55 steps, and 7:4 by 138 steps. If one starts at any pitch, and moves up a perfect fifth, then up five major thirds, and then up again by a 7:4 interval, the total movement will be 100 + 5*55 + 138 = 513 steps, which is exactly three octaves, equivalent to the starting point. This piece moves around this loop 36 times, once per 63 seconds. All 36 cycles are superimposed in this score:

This piece has five voices, which form relatively complex chords. In constructing this piece, the chord shapes have been constrained:

This is a fragment of the Tonnetz diagram for 171edo. It shows the three dimensional network of relationships among the pitch classes. Horizontal neighbors are connected by perfect fifths, vertical neighbors by major thirds, and the third dimension, in and out of the page, shows pitch classes related by 7:4. The green and purple boxes here have that same shape: the purple box is simply shifted to the right. Each box encloses 8 pitch classes. These boxes represent the constraint on chord shape. At any instant in time, the pitch classes assigned to the five voices must be contained in a box of this size and shape. Picking 5 points out of a total set of 8 allows for 56 different chord shapes.

What fascinates me at the moment is the relationship between the chord constraint and the harmonic movement driven by the 63 second cycle. With the five voices often starting and stopping at different times, much of the time the pitch class of just one voice will change at a time. The cube shaped chord constraint used here will allow unbounded harmonic movement even with this kind of overlap. The green box and the purple box in the diagram include four pitch classes in their intersection: 7, 40, 123, and 156. A five note chord might add pitch class 78, which would be allowed because all five pitch classes are in the green box. But then the voice sounding the 78 could switch to pitch class 52, which would be valid because all the pitch classes are in the purple box. The other voices could all move within the purple box to set up another move to the right. The same tactic works for movment in the other directions.

Consonance and Dissonance

Here is a new algorithmic piece in 50edo. 50edo, the tuning system that divides octaves into 50 equal parts instead of the conventional 12 equal parts, is still a quite conservative tuning system. It is very close to 2/7-comma meantone, whose history goes back to the 16th Century. I was inspired to create this piece from some discussion about diminished chords, chords built by stacking minor thirds. In conventional 12edo, four minor thirds add up to an octave: each minor third is 3 steps of 12edo, and 4*3=12, the number of steps in an octave in 12edo. In 50edo, a minor third is 13 steps, so four minor thirds adds up to 52 steps, 2 steps sharper than an octave. In just intonation a minor third is a 6:5 frequency ratio, so four minor third combine to make 1296:625, sharper than an octave by 648:625. In this piece I wanted to explore what kind of rich chord structure is made available by the greater precision of 50edo.

This piece has five voices, enabling quite complex chords. I didn't want the chords to get too wild, so I constrained the structure of the chords. The diagram above has a green template imposed on a Tonnetz diagram for 50edo. Since 50edo is a meantone tuning, the conventional names for notes can be used; but note that with 50edo, e.g. C# and Db are distinct pitches.

The chords in this piece, the combinations of notes that are sounded at the same time, are constrained by the rule that there should be some positioning of the template that covers all the notes in the chord. In the position shown for the template, the notes of the C major chord C-E-G are all covered, so the C major chord is allowed. An Fb major chord Fb-Ab-Cb is not covered by the template in the position shown, but the template can be shifted down two rows to cover the Fb major chord, so a Fb major chord is allowed. The template defines the shapes of the allowed chords. If a shape is allowed, it is allowed however it might be transposed.

An example of a forbidden chord is a two semitone chord such as D#-E-F. There is no way to slide the template to cover these notes together. Some common tetrads are allowed, such as a major seventh and a dominant seventh. A diminished triad is allow, but a diminished tetrad is not. The question that inspired this piece was about diminished tetrads, so their exclusion here is a bit of a disappointment, but I wanted to keep the template reasonably bounded in hopes of creating some coherent music!

Exotic Intervals

Here's an algorithmic composition in 270edo, the tuning system that divides octaves into 270 equal steps. This precision allows exotic intervals to be introduced with great clarity. This piece uses intervals such as 7:4 and 11:8, and more complex combinations such as 11:7 or 11:10 or 9:7.

Any tuning system with discrete steps will temper out commas, i.e. will map multiple just-tuned intervals to the same pitch. 270edo maps both 96:55 and 110:63 to 217 steps. This new composition is based on a pattern where pitch class 0 moves to pitch class 217 by a sequence of intervals corresponding to one of these just ratios, and then returns by a sequence corresponding to the other.

This composition is built mostly from just 15 pitch classes, out of the full set of 270. This diagram shows the pitch classes used, and the fundamental intervals that relate them to each other. More complex ratios such as 6:5 or 7:6 etc. are not shown.

This pieces progresses clockwise around this diagram 16 times, each cycle taking about 177 seconds.

This score superimposes all 16 cycles to show the general pattern.

Compound Traversal

Here's a new algorithmic composition, in the tuning system 34edo, which divides octaves into 34 equal steps rather than the conventional 12 equal steps.

Changing the tuning system like this does two things. A tuning system makes available some collection of intervals, the building blocks of music. Sometimes this collection of intervals is unconventional. But here I am using very much the same basic intervals in 34edo that conventional musics uses in 12edo, most fundamentally the major third and the perfect fifth, along with the octave. This new composition is not something that would work in 12edo, but the difficulty is not with the basic intervals being used.

The other characteristic of a tuning system that matters musically is how the intervals combine to form more complex intervals. With a tempered tuning system, some complex combinations end up being equivalent to a unison. Such a complex combination is known as a comma; when the tuning system makes it equivalent to unison, then the tuning system is said to temper out the comma.

The conventional 12edo system tempers out the syntonic comma: combine four perfect fifths and a minor sixth and, in 12edo, you will end up three octaves above your starting point. 34edo does not temper out the syntonic comma: the same combination of intervals ends up slightly sharper than three octaves.

The most basic commas tempered out by 34edo are the diaschisma, four perfect fifths and two major thirds, and the Würschmidt comma, eight major thirds and a perfect fourth. This new algorithmic composition is built from traversals of these two commas. A comma traversal, sometimes called a comma pump, is simply a sequence of the intervals that compose a comma. When the comma is tempered out by the tuning system, the comma traversal will return to the starting pitch. Thus one can repeat the comma traversal and the pitch of the music will not drift up or down.

Conventional 12edo does not temper out the Würschmidt comma. That's why this piece would not translate to 12edo. The same sequence of intervals could be used, but in 12edo the piece would drift in pitch.

The piece here is predominantly 27 traversals of the diaschisma. But every third traversal, the sequence is shifted by a major third. With 9 such triple traversals, separated by 8 major thirds, the first and last triples are separated by a perfect fourth. The end of the piece ties back to the beginning.

The way the piece is constructed algorithmically, it starts with just this simple sequence of intervals. The piece is almost 27 minutes long, so each traversal of the diaschisma is about a minute in duration. There are six intervals in the diaschisma. So the initial structure of the piece has pitches being repeated over about ten seconds, then shifting by a perfect fifth or a major third, then another ten second repetition, etc.

The algorithm I use is based on statistical mechanics, with temperature as a key parameter. Consonance is treated as low energy, and dissonance as high energy. At low temperature, pitches will be chosen to be maximally consonant. At higher temperature, more dissonance will be allowed. In a system like this, there will typically be a phase transition, a temperature below which the system will have long range consonance, and above which the overall consonance will break down. Right around the phase transition, there are typically fractal variations. The hope is that such fractal variations in consonance and dissonance will provide musical interest.

This is a plot of the temperature and energy computed in the course of the construction of this piece. The thermodynamic simulation is initiated with the simple comma traversal structure and with low temperature. The temperature is gradually raised; this involves reassigning pitch values to moderately less consonant alternatives. A phase transition is characterized by a rapid rise in energy with a small change in temperature. The composition process ends when this rapid rise is detected.