Heptatonic notation

Purpose and Principles

I^[who?] developed heptatonic notation in order to notate my microtonal music, which is written in a number of different edos. Some principles were followed in the development this notation system:

• It should be nearly consistent with standard modern music notation.
• It should be consistent between different edos.
• It should be as non-arbitrary as possible.

From these principles I decided that it should be based on nominal-accidental chains, with seven nominals to an octave. In addition, accidentals should both be based on those of standard notation and give an approximation of their displacement size. From these principles is derived the fundamental rule behind this notation system:

• For each edo, nominals are assigned to the first mode of the maximally even heptatonic MOS.

Nominals

By finding these particular MOS scales in each edo, we can find that they, and thus that edo's nominals, follow a number of patterns:

• (L=1, s=0)
  • 1\7: LLLLLLL
• (2, 1)
  • 1\8: ssssssL
  • 4\9: LssLsss
  • 3\10: LsLsLss
  • 3\11: sLsLsLL
  • 5\12: sLLsLLL
  • 2\13: LLLLLLs
  • 2\14: LLLLLLL
• (3, 2)
  • 2\15: ssssssL
  • 7\16: LssLsss
  • 5\17: LsLsLss
  • 5\18: sLsLsLL
  • 7\19: sLLsLLL
  • 3\20: LLLLLLs
  • 3\21: LLLLLLL

These periodic patterns occur because the sum of step sizes must equal the edo's order, and for any (L, s) MOS there is only one scale-step pattern.

Accidentals

For edos 7(n-1)+1 to 7n, the large step is n edo-intervals, so n-1 unique accidentals of each flat/sharp are required, plus a natural accidental. This ensures that the maximum or minimum accidentals always modify a nominal to a neighboring nominal, as in the double-flats and double-sharps of standard notation. I chose the following twenty-five accidentals to be usable with the Ekmelily extension to Lilypond, which I use to notate my music:

bb bb^ dbv db db^ bv b b^ dv d d^ nv n n^ ‡v ‡ ‡^ #v # #^ ‡#v ‡# ‡#^ xv x,

where bb is a double flat, db is a three-half flat, b is a flat, d is a half-flat, n is a natural, ‡ is a half-sharp, # is a sharp, ‡# is a three-half sharp, and x is a double sharp. The v and ^ are arrows which are attached to the bottom and top of the accidentals, respectively. Because there are twelve flats and twelve sharps, these accidentals allow notation up to 84edo with full enharmonicity, as described above. In order to notate edos which use fewer accidentals, the accidentals used are chosen to best approximate the required modification value, assuming that the above accidentals are in increments of perfect twelfth-accidentals. For example, suppose we wish to notate 45edo; 45 = 7*6+3, so seven sharp and seven flat accidentals are required. To determine, say, the three-sevenths sharp we multiply that value by twelve: 12*3/7 = 5.14 ≈ 5, so we use the five-twelfths sharp, or #v. Doing this for all sets of fewer than or equal to twelve accidentals per sharp/flat class, we find the following:

• For two accidentals: bb b n # x
• 3: bb db^ dv n ‡^ ‡#v x
• 4: bb db b d n ‡ # ‡# x
• 5: bb dbv bv b^ d^ n ‡v #v #^ ‡#^ x
• 6: bb dbv db^ b dv d^ n ‡v ‡^ # ‡#v ‡#^ x
• 7: bb dbv db bv b^ d d^ n ‡v ‡ #v #^ ‡# ‡#^ x
• 8: bb dbv db bv b b^ d d^ n ‡v ‡ #v # #^ ‡# ‡#^ x (Because this results in values exactly in between twelfths, some discretion in was used in rounding to determine these to be the most intuitive.)
• 9: bb bb^ db db^ bv b^ dv d nv n n^ ‡ ‡^ #v #^ ‡#v ‡# xv x
• 10: bb bb^ dbv db^ bv b b^ dv d^ nv n n^ ‡v ‡^ #v # #^ ‡#v ‡#^ xv x
• 11: bb bb^ dbv db db^ bv b b^ dv d d^ nv n^ ‡v ‡ ‡^ #v # #^ ‡#v ‡# ‡#^ xv x
• 12: bb bb^ dbv db db^ bv b b^ dv d d^ nv n n^ ‡v ‡ ‡^ #v # #^ ‡#v ‡# ‡#^ xv x

This would clearly require a significant amount of memorization to be at all efficient. However, two "shorthand" systems can be made, for three and five accidentals per class:

• 3: bb db d n ‡ ‡# x
• 5: bb db bv b^ d n ‡ #v #^ ‡# x

These simply remove extraneous arrows. Now, given that you know the four-accidental (quartertone) system, these two follow easily; three accidentals simply removes the sharp/flat and five splits them into up-arrow and down-arrow versions. These systems—two to five accidentals per class—are easy to memorize, intuitive, and able to notate up to 35edo, which is sufficient for a significant amount of music.

(To be continued. Feedback welcome, as I'm sure there are ways to make it better!)

Scores and score-videos

Palinkalin Viharo (Flowers in the Mist) by Jake Huryn (Score): 22edo