I've been working on a notation system for gamelismic temperaments lately, which is based on the Slendric scale. Then I realized its similarity to Graham Breed's tripod notation, and that it could be generalized as a class of notation systems that can cover a lot more temperaments.
Some of the ideas are still unfinished, and a lot of concepts are probably hard to understand if you aren't familiar with regular temperament theory (or if I explained something poorly / didn't provide an explanation), so ... proceed at own risk.
First of all:
What is slendric / gamelan?
Slendric is a linear temperament with a septimal whole tone generator 8/7. It tempers out the gamelisma 1029:1024, which is the difference between the septimal subfourth 21/16, and 64/49. Hence, stacking three 8/7 generators gives a perfect fifth 3/2.
Slendric is a 2.3.7-limit, or a "no 5s" 7-limit temperament, meaning that all ratios are composed of prime numbers 2, 3 and 7. As a consequence, there are no major thirds 5/4 in this subgroup temperament.
The generator chain for slendric looks like following:
... 14/9 16/9 49/48 7/6 4/3 32/21 7/4 1/1 8/7 21/16 3/2 12/7 96/49 9/8 9/7 ...
If we add a vertical axis of major thirds 5/4, we get the planar 7-limit temperament gamelan:
... 35/24 5/3 40/21 35/32 5/4 10/7 80/49 15/8 15/14 ...
... 7/6 4/3 32/21 7/4 1/1 8/7 21/16 3/2 12/7 ...
... 28/15 16/15 49/40 7/5 8/5 64/35 21/20 6/5 48/35 ...
Note: Instead of 5/4, we could have used any ratio from the first or third row for the vertical axis, like 21/20 or 16/15.
Some EDOs that support slendric, gamelan, and gamelismic temperaments:
5, 10, 11, 15, 16, 20, 21, 25, 26, 31, 36, 41, 46, 72, 77, 87, 103, 113, 118, 128
About the actual notation
The core idea is to use the vowels a i u e o (or ア イ ウ エ オ with Japanese katakana) to notate Slendric. Specifically, a i u e o / ア イ ウ エ オ refers to the scale 1/1 8/7 21/16 3/2 12/7. Typically, I'll use the mode i u e o a / イ ウ エ オ ア as a basis, though, which denotes 1/1 8/7 21/16 3/2 7/4. I will write single notes as "アa イi ウu エe オo", i.e. the katakana symbol followed by it's romanization.
The Slendric chroma, represents 49/48 or 64/63, so we can use and , or and for it. Since there is no difference if 1029:1024 is tempered out, and the latter pair of accidentals is simpler, I will use that. Two chromas represent 28/27, the corresponding accidentals are and .
So, why did I introduce Japanese katakana? Well, most of the characters denote either the vowels a i u e o, or syllables like 'ro' or 'ha' that consist of a consonant which is followed by one of those vowels. It's actually a little more complicated, and there are exceptions like 'shi' instead of 'si' (and some other concepts like Youon which probably aren't relevant here), but since I will list the kana together with their romanization, you don't have to learn them if you don't want.
Anyway, adding a consonant before a vowel allows us to denote 2-dimensional scales in an intuitive way. With the help of katakana, we can denote complete syllables with a single character, which can be convenient. However, it's not mandatory to use them, and everyone is free to write the syllables however they want.
I didn't fully decide yet which consonants to use, and how, but here is an example how it could work:
- m (マma ミmi ムmu メme モmo) denotes a 21/20 below, and t (タta チchi ツtsu テte トto) denotes a 21/20 above the vowel
- s (サsa シshi スsu セse ソso) denotes a 16/15 below the vowel
- k (カka キki クku ケke コko) denotes a 64/63 below the vowel, and ヨyo denotes a 64/63 above o (used for Slendric notation)
Here are some scales in gamelan temperament that could serve as a basis for notation:
A 10-note scale:
1/1 35/32 8/7 5/4 21/16 10/7 3/2 5/3 7/4 40/21
イi ムmu ウu メme エe モmo オo マma アa ミmi
Chromas / accidentals:
- c1 = L - m = 64/63: ,
- c2 = m - s = 25/24: ,
- c2 - c1 = 50/49: ,
- c1 + c2 = 200/189 (tempers to 35/33 in portent): ?
A 15-note scale:
1/1 21/20 35/32 8/7 6/5 5/4 21/16 48/35 10/7 3/2 63/40 5/3 7/4 64/35 40/21
イi チchi ムmu ウu ツtsu メme エe テte モmo オo トto マma アa タta ミmi
Chromas / accidentals:
- c1 = L - s = 64/63: ,
- c2 = m - s = 126/125: ,
- c1 - c2 = 875/864: , (tempers to 100/99 in portent)
- c1 + c2 = 128/125: ,
Another 10-note scale:
1/1 15/14 8/7 60/49 21/16 45/32 3/2 105/64 7/4 15/8
イi スsu ウu セse エe ソso オo サsa アa シshi
Chromas / accidentals:
- c1 = L - m = 64/63: ,
- c2 = m - s = 225/224: ,
- c1 - c2 = 2048/2025: ,
- c1 + c2 = 50/49: ,
1/1 9/8 8/7 9/7 21/16 72/49 3/2 32/21 12/7 7/4 63/32
イi クku ウu ケke エe コko オo ヨyo カka アa キki
Chroma: 54/49 (the large neutral second between 7/6 and 9/7)
Note: Most of the accidentals listed here are only relevant for the planar temperament gamelan, and will vanish if a linear gamelismic temperament is used.
The second scale contains the first scale twice, and becomes the 15-note MOS in valentine temperament, which divides 8/7 in three equal 21/20 steps. The third scale becomes the 10-note MOS in miracle temperament, which divides 8/7 in two secors 16/15 or 15/14. All of the first three scales temper to Blacksmith and , respectively. The 10-note scales are Lemba MODMOSes, and the 15-note scale is a Superkleismic MODMOS.
For gamelismic temperaments with an 8/7 generator, the Slendric or Slendric notation could be used. However, I'm not sure how to notate stacked chromas (consider rodan), and whether it's a good idea to use Slendric as a basis.
Alternatively, one of the first three scales could be used in conjunction with the Slendric chroma: 21/20 equals 1 chroma in gorgo, 2 chromas in mothra, and 3 chromas in rodan temperament. For any of those, adding a chroma to 21/20 gives 16/15.
A typical 5-line staff can be used for アa イi ウu エe オo, each note corresponding to a line. If 10 or 11 notes are used, the additional notes are between the lines. For 15 notes, the notation would look similar to Graham Breed's tripod notation:
If a 5-note notation is used, it could be useful to use a line and the space below for the same base note. For example, if you have a mothra scale with root イi that contains both 5/4 and 21/16, both are derived from エe, and the former is flattened by a (28/27). In that case, the note head for エe could be placed on the fourth line, and the note head preceded by for エe in the space below that line.
First of all, the same note names could be used for other temperaments that divide the octave into 5 almost equal steps. For example, the linear 2.3.7-limit temperaments that temper out 28/27, 49/48 or 64/63 could serve as a similar basis as slendric in my notation.
In addition, similar notations that divide the octave into n almost equal steps could be developed. For example, in 31-EDO, the following subgroup temperaments could serve as a basis:
- 3 notes: Marveltri (the 2.5.9/7-limit temperament with a 5/4 major third generator that tempers out 225:224)
- 4 notes: Starlingtet (the 2.5/3.7/3-limit temperament with a 6/5 minor third generator that tempers out 126:125)
- 6 notes: "Hemimean" (the 2.5.7-limit temperament with a 28/25 whole tone generator that tempers out the hemimean comma 3136:3125
- 7 notes: Mohaha (the 184.108.40.206-limit temperament with an 11/9 neutral third generator that tempers out 81:80 and 121:120)
Using marveltri as a basis would lead to something equivalent to tripod notation, but could also be used for 6-note or 12-note scales.
Mohaha could use the same base note names as Meantone, since it is a Mohaha "MODMOS" (if we allow an MOS to be called a MODMOS).
Specifically, the following temperaments could be notated as following:
(31-EDO generator in brackets)
- Valentine (2\31): 15-note scale derived from Slendric, or 16-note scale derived from Starlingtet
- Miracle (3\31): 10-note scale derived from Slendric, or one that uses altered tripod scale notes and adds a 10th note
- Nusecond (4\31): 8-note scale derived from Starlingtet
- Hemithirds/hemiwur (5\31): "Hemimean"
- Mothra (6\31): (as discussed before, i.e. derived from Slendric or )
- Orwell (7\31): Tripod notation
- Myna (8\31): (some scale based on starlingtet)
- Mohajira/migration (9\31): Mohaha
- Würschmidt (10\31): (some scale based on marveltri; notation could be easily modified for the non-marvel temperament worschmidt)
- Squares (11\31): 14-note scale based on Mohaha
- Semisept (12\31): 18-note scale based on "Hemimean"
- Meantone (13\31): Mohaha MODMOS (standard notation), or 12-note scale derived from Marveltri, Starlingtet or "Hemimean"
- Casablanca/cypress etc. (14\31): 20-note scale derived from Starlingtet, or from Slendric if interpreted as necromanteion/oracle, or as an Orwell MODMOS if interpreted as 2-orwell
- Tritonic (15\31): 15-note scale based on Marveltri, maybe(?)
イi - エe - アa - ウu - オo - イi = 1/1 21/16 7/4 16/7 3/1 4/1
(I think Cryptic Ruse used the 5-EDO-tempered version where strings are a 2\5 interval apart)
- added hyperlinks for EDOs
- listed the katakana for each consonant, and a note that irregular forms like 'chi' are optional, and can be replaced with 'ti' etc.
- fixed the 128/125 accidentals and
- added generator information under 'Generalization'