Big-Small

A comma in musical tuning is a ratio made of small primes which is very close to 1. Two classic commas are the syntonic comma, 81:80, and the Pythagorean comma, 531441:524288. These are quite similar in size. The ratio between them is even closer to 1, the comma known as the schisma, 42467328:42515280.

Commas are important in music because consonant intervals have frequency ratios built from simple primes. Combining consonant intervals then generates more complex ratios that are still built from simple primes. Commas thus correspond to combinations of consonant intervals that are very close to unison. This closeness has potential to cause trouble and potential also to cause delight; in any case, managing this closeness is an important musical task. The main tool for this is temperament, adjusting intervals slightly so that when they are combined the result is never awkwardly just slightly different than unison: it is either exactly unison, or distinctly different.

Conventional tuning, twelve tone equal temperament, tempers intervals so that the syntonic comma and the Pythagorean comma both vanish, i.e. the corresponding combinations of tempered consonant intervals results in exact unison. But there are other musical possibilities!

Here is a new piece: 612edo scale 82.

This is in the tuning system that divides octaves into 612 equal steps. Any tuning with such small steps will be extremely precise. 612edo is one of the most precise, for intervals like perfect fifths and major thirds, among other tuning systems with similarly small steps. I am using it here, though, to explore commas. The Pythagorean comma is 12 steps of 612edo; the syntonic comma is 11 steps. Thus the schisma, the difference between these, is 1 step. 612edo is so precise that it does not temper out the usual commas.

I don't know a name for this comma:

450359962373049600:450283905890997363

but 612edo tempers it out! Factored into primes, this is 2^54 * 5^2 : 3^37. It is about 0.3 cents off of unison.

This piece uses a scale with 82 notes per octave. A perfect fifth is 358 steps of 612edo; the octave and the perfect fifth have a greatest common divisor of 2, which means that there are two cycles of fifths. The scale I used here is made of sequences of 41 perfect fifths, one for each cycle of fifths.

This piece traverses this big-small comma 25 times.

Magic

Here's a new piece of algorithmic music: 19edo 2026.

This piece is in 19edo, the tuning system that divides octaves into 19 equal steps. 19edo is a meantone tuning, so conventional note names work well. The tonnetz diagram above shows how 19edo pitch classes are related by the fundamental intervals of perfect fifths, perfect fourths, major thirds, minor thirds, major sixths, and minor sixths. Start at a C note and move four perfect fifths, to G, D, A, and E. From E move a minor sixth to C, the pitch class at the start of this sequence. This combination of intervals is a syntonic comma (81:80). The fact that the combination returns to the starting pitch class is due to 19edo tempering out the syntonic comma, which is why it is a meantone tuning.

Another comma tempered out by 19edo is the magic comma (3125:3072). Start from the C pitch class and move five major thirds, to E, G#, Cb. Eb. and G. From there, move a perfect fourth to return to the starting pitch class of C.

The conventional diatonic scale is built by stacking perfect fifths. The piece posted above is built from an unconventional scale, built from stacking major thirds. The scale diagram above is like a piece of the tonnetz diagram, but then folded into a loop to show how the magic comma is tempered out.

Extended Consonance

Here's a new piece: 494edo scale 17.

This piece is in a 17 note scale, in the 494edo tuning system. The scale is diagrammed above. Green arrows, e.g. from pitch class 459 to pitch class 254, represent perfect fifths, a frequency ratio of 3:2, or at least the best approximation available in the 494edo tuning system. Blue arrows, e.g. from 300 to 459, represent major thirds, 5:4. Red arrows, e.g. from 459 to 364, represent the 7:4 interval which is not so conventional. Orange arrows, e.g. from 0 to 227, represent the even less conventional 11:8 interval. Dark purple arrows, e.g. from 148 to 0, represent the yet less conventional interval 13:8. I didn't use a strict division between consonant vs. dissonant intervals in constructing this piece, but a more flexible scoring system. For this piece, 13:8 is treated as more consonant than e.g. 81:50, which is a more complex interval but built up from simpler primes.

This scale contains loops such as 459, 300, 205, 432, 227, 22, 170, 459. This loop traverses the comma 2080:2079. The loop travels along three green arrows, one red arrow, and one orange arrow in the forward direction, and along one blue and one purple arrow in the reverse direction. If these intervals were all tuned to precise rational intervals, the loop would not return to the start, but would have shifted by that comma 2080:2079. The tempered scale 494edo approximates these intervals, adjusting them slightly so the loop returns to the starting pitch class.

494edo might seem like a rather arbitrary choice for a tuning system, but the above diagram shows how it is not. The diagram shows the tuning errors for a variety of intervals. For example, in the column labeled 5 and the row labeled 3 appears the number 0.061. This is the error in the approximation of 494edo for the interval 5:3. This error number is given in terms of a single step of 494edo. 494edo divides octaves into 494 equal steps, so these steps are very small. But the error for 5:3 is only 0.061 of one of these small steps. This table has many such small errors. The way I chose 494edo was quite simple: I just computed these errors for a wide range of edo possibilities, and then searched through the results for the tuning system that had small errors for all the intervals I wanted to use. The table shows that 494edo does not approximate e.g. 17:16 very well, but that is not an interval I wanted to use here.

Porwell 15

Here's a new piece: 99edo porwell 15.

This diagram shows the scale used by the piece. Green arrows are perfect fifths (3:2 frequency ratio), blue arrows are major thirds (5:4), and red arrows are the less conventional 7:4 interval. Tracing paths in the diagram, there is a direct connection, a perfect fifth, by which one can move from pitch class 32 to pitch class 90. One can also get from 32 to 90 by way of two red arrows and three blue arrows, e.g. moving through pitch classes 64, 96, 77, and 58. This convergence of paths is how 99edo tempers out the Porwell comma, 6144:6125.

This scale has 15 notes per octave, with spaces between notes of sizes 7, 6, 6, 7, 6, 7, 6, 6, 7, 6, 7, 6, 6, 7, and 9 steps of 99edo, i.e. quite evenly spaced.

Toward Beauty

One more swing at 87edo: 87edo scale 10b.

More staring at scale diagrams. Move that 28 to 26. The scale steps are still nicely enough spaced: 11, 6, 11, 6, 11, 6, 11, 8, 9, 8. But now there aren't pitch classes hanging out in the remote country:

I changed the rhythmic structure of the piece, too. The piece from this morning was a 7x7x7x array of short measures. This new piece is a 12x12 array of longer measures.

The More I Look, the More I See!

Yet another piece in 87edo: 87edo scale 10.

Some of my fascination here just comes from the comma 1029:1024. It's a very small comma but it is also quite simple.

As I was staring at the scale diagrams from my last post, I thought it ought to be easy enough to make a scale with a nicer structure:

The step sizes in this scale are also nice: 11, 6, 11, 6, 11, 6, 11, 6, 11, 8.

A Loop of Pentatonics

Here's a new piece in 87edo: 87edo pentaloop.

I was mulling over the 26 note scale of the pieces I posted recently. This scale has five bands of closely spaced notes. What if I made pentatonic scales by picking one note from each band? There could be a sequence through these scales that traverses a comma. So that's what I made here, a traversal of the comma 1029:1024 using sixteen pentatonic scales, with a total of sixteen notes among those scales, chosen out of the full 26 note scale. Moving from one scale to the next shifts just one note of the scale. So the piece is a bit like a chord progression, but the full five notes of the pentatonic scale played all together would probably not make a very pleasant chord!

This new piece has 256 measures. The sixteen scale traversal is repeated sixteen times.

The green horizontal arrows represent perfect fifths; the blue vertical arrows represent major thirds.

The red diagonal arrows represent the not very conventional interval 7:4.

The shape of this scale is the same as the starting scale. All the notes have been shifted by 8:7. The next scales shift in the same pattern by another 8:7.

We are again back at the starting shape. A third move of 8:7 will bring us back to the actual starting scale, but the sequence will also have to move by 4:3 to form the traversal of 1029:1024.

This next step is where the 4:3 move begins:

From here, pitch class 66 can be shifted to pitch class 64, which completes the loop back to the starting scale.

This is a diagram of the full set of sixteen notes from all the pentatonic scales here.