I've been working on a notation system for gamelismic temperaments lately, which is based on the Slendric[5] 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:

Code: Select all

```
...
... 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 ...
...
```

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[5]. 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[5] 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[11] notation)

Here are some scales in gamelan temperament that could serve as a basis for notation:

A 10-note scale:

Code: Select all

```
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
```

- c1 = L - m = 64/63: ,
- c2 = m - s = 25/24: ,
- c2 - c1 = 50/49: ,
- c1 + c2 = 200/189 (tempers to 35/33 in portent): ?

Code: Select all

```
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
```

- c1 = L - s = 64/63: ,
- c2 = m - s = 126/125: ,
- c1 - c2 = 875/864: , (tempers to 100/99 in portent)
- c1 + c2 = 128/125: ,

Code: Select all

```
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
```

- c1 = L - m = 64/63: ,
- c2 = m - s = 225/224: ,
- c1 - c2 = 2048/2025: ,
- c1 + c2 = 50/49: ,

Code: Select all

```
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
```

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[10] and [15], respectively. The 10-note scales are Lemba[10] MODMOSes, and the 15-note scale is a Superkleismic[15] MODMOS.

For gamelismic temperaments with an 8/7 generator, the Slendric[5] or Slendric[11] notation could be used. However, I'm not sure how to notate stacked chromas (consider rodan[46]), and whether it's a good idea to use Slendric[11] as a basis.

Alternatively, one of the first three scales could be used in conjunction with the Slendric[5] 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.

Staff notation

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:

Code: Select all

```
o トto
--o-- オo
o モmo
o テte
--o-- エe
o メme
o ツtsu
--o-- ウu
o ムmu
o チchi
--o-- イi
o ミmi
o タta
--o-- アa
o マma
```

Generalization

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 2.3.5.11-limit temperament with an 11/9 neutral third generator that tempers out 81:80 and 121:120)

Mohaha could use the same base note names as Meantone[7], since it is a Mohaha[7] "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[5], or 16-note scale derived from Starlingtet[4]
- Miracle (3\31): 10-note scale derived from Slendric[5], or one that uses altered tripod scale notes and adds a 10th note
- Nusecond (4\31): 8-note scale derived from Starlingtet[4]
- Hemithirds/hemiwur (5\31): "Hemimean"[6]
- Mothra (6\31): (as discussed before, i.e. derived from Slendric[5] or [11])
- Orwell (7\31): Tripod notation
- Myna (8\31): (some scale based on starlingtet)
- Mohajira/migration (9\31): Mohaha[7]
- 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[7]
- Semisept (12\31): 18-note scale based on "Hemimean"[6]
- Meantone (13\31): Mohaha[7] MODMOS (standard notation), or 12-note scale derived from Marveltri[3], Starlingtet[4] or "Hemimean"[6]
- Casablanca/cypress etc. (14\31): 20-note scale derived from Starlingtet[4], or from Slendric[5] if interpreted as necromanteion/oracle, or as an Orwell[9] MODMOS if interpreted as 2-orwell
- Tritonic (15\31): 15-note scale based on Marveltri[3], 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)

-------------------------------------------------------------------

*Edit:*

*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'*