Toroidal Scales

Here's a new piece of algorithmic music: 99edo scale 25.

This is in the 25 note per octave scale diagrammed above, in the 99edo tuning system. This scale can be generated by the 16\99 interval, which corresponds to a 28:25 frequency ratio. But that's not how I came up with it!

I am working to develop a systematic approach to scale construction, using a toolbox of path-oriented methods. The main idea is to start with a path that traverses some comma, and then to augment this path in various ways until a scale emerges that invites musical composition. Augmentation might be adding another path between two notes in the scale. The new path could consist of the same intervals as an existing path, but simply reordered. Or, the new path could be topologically distinct, so that the old path and the new path form a traversal of some comma. Or, a complete comma traversal could be added to any note in the scale.

My idea with this 25 note scale was start with a traversal of a comma, and then add to each point in the scale a traversal of a second comma. For this piece, I chose the commas 3136:3125 and 6144:6125, both of which are tempered out by 99edo. The first comma can be traversed by a path of seven notes; the second by a path of six notes. For each note in the first path, I added five more notes to make a traversal of the second comma. The second notes of each of these again form a traversal of the first comma, as do the third notes, the fourth notes, etc. This is a very simple property of intervals: if two notes are separated by an interval X, and we raise both notes by the same interval Y, then the raised notes are again separated by the interval X.

Seven traversals of length six could produces as many as 42 notes, but there were many duplicates. The seven traversal produced 23 distinct notes. These were quite nicely spaced, mostly with gaps of 3\99 and 7\99. There were a couple gaps of size 10\99. By adding a couple more notes, bringing the total to 25, a regular pattern of gaps was produced. This same pattern can be generated using a chain of 16\99 intervals. Why the toroidal construction results in the same notes as this simpler method of generation... probably somebody in the microtonal world knows, but I sure don't!

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Here is a new piece of music: 270edo scale 23.

This music is in the 270edo tuning system which divides octaves into 270 equal parts. This provides very precise tuning for a wide range of intervals. The music uses a scale with just 23 of the possible 270 pitches in each octave. The diagram above shows the pitches of the scale and their harmonic relationships. Pitches are labeled by integers, from 0 to 269 in each octave, and then starting up with 0 again for the next octave. In ascending order, the pitches in the scale are: 0, 10, 24, 36, 45, 62, 70, 77, 87, 97, 114, 122, 136, 148, 158, 168, 182, 199, 209, 228, 235, 245, and 260. The intervals between the successive pitches vary between 7 and 19 microsteps of the tuning system, so the spacing is reasonably even.

The arrows between pitches show the harmonic relationships. A green arrow represents a 3:2 frequency ratio (the best approximation in the 270edo tuning system), a perfect fifth in conventional tuning. Blue arrows are 5:4 ratio, major thirds in conventional tuning. Red arrows are 7:4, purple are 11:8, and yellow are 13:8; none of these are conventional intervals.

This scale supports traversal of a family of commas. The diagram above shows a traversal of the comma 1001:1000.

This is a traversal of 4096:4095.

This is a traversal of 41503:41472.

It looks to me like these three commas form a basis for all the commas that can be traversed in this scale. With six prime numbers involved in the tuning, the total set of commas tempered out by 270edo should be a five dimensional space. This scale supports traversal of a three dimensional subspace of that full set.

This traversal of 123201:123200 is a combination of the basis set of traversals.

I don't have a systematic way to construct scales like this, where the scales steps are reasonably even, where the graph of harmonic relationships is reasonable dense, and where comma traversals are supported. The set of commas tempered out by a tuning system already tend to have an irregular structure, so these sorts of scales probably won't be very regular either!

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Here is a new piece: 22edo scale 7

This piece uses a seven note scale in 22edo, as diagramed above. Green arrows represent perfect fifths, 3:2 frequency ratios. Blue arrow represent major thirds, 5:4. Purple arrows represent the unconventional interval 11:8. There are paths in this scale that traverse the commas 100:99 and 55:54.