Notation system for gamelismic temperaments

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Notation system for gamelismic temperaments

Post by Xen-Gedankenwelt »

Hi! :D

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  ...
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[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)
Note: If you don't want to learn the irregular forms, you can use ti and tu instead of chi and tsu, and si instead of shi.

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
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:

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
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:

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
Chromas / accidentals:
  • 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
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[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:

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   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
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 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[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(?)
P.S.: A good guitar tuning for gamelismic temperaments that work well with my notation is
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'
Last edited by Xen-Gedankenwelt on Sat Sep 26, 2015 9:57 am, edited 2 times in total.

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Dave Keenan
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Re: Notation system for gamelismic temperaments

Post by Dave Keenan »

Wow! That was quite an effort to read. It must have taken a lot more effort to research, develop and write.

I note that the :/ /|: and :\ \!: symbols are implemented. You can see them in the smilies palette in the post editing window, and click to insert them. But unfortunately double-slash and double-backslash already had other meanings for this forum software, and caused problems when I tried to use them in "smiley codes" for sagittals, and changing the forum software to allow them would have caused problems with other features of the forum such as automatically turning URLs into links. So we have to put a space between them. i.e. the smiley codes for :/ /|: and :\ \!: are ":/ /|:" and ":\ \!:" without the quotes.

I also note that although the precise sagittal notation of some commas may require diacritics (accent marks), it is quite permissible (and indeed suggested) to omit them from the staff when (a) your precise tuning is described elsewhere, and (b) this would not lead to a failure to distinguish different pitches.

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Re: Notation system for gamelismic temperaments

Post by cam.taylor »

Very interesting! Might take me a little while to process it all, and possibly try it out. I'm not really that familiar with slendric, and not at all with gamelan.

Is there any particular reason you used the consonants you did? For non-Japanese speakers, perhaps the "regular" lines where initial consonants don't change might be better than the irregulars, such as t- morphing to ch- with -i and ts- with -u, and s- morphing to sh- with -i.

These would be
[k] カ行:カキクケコ and [g] ガ行:ガギグゲゴ, perhaps as a comma alteration in the other direction?
[n] ナ行:ナニヌネノ
[m] マ行:マミムメモ
[r] ラ行:ラリルレロ, though this tends to be difficult for non-Japanese speakers

Possibly also バ行:バビブベボ and パ行:パピプペポ, but they would seem to be possible alterations of [h], which is irregular

Unsure if that's enough consonants for you but I think that's all the regular single symbol rows

Is there no way to notate gamelismic using Sagittal "the usual" way? Even if you use 5 nominals C G D A E and the same accidentals planned, in a similar manner to the proposed 10EDO notation? I guess things get messy when your chain gets long enough to include better 3-limit approximations, such as 9:8 notated as an alteration of 8:7, rather than how Sagittal usually notates it the other way round. I guess you also lose the "every note is a single syllable[symbol]" thing too, so I'm starting to see your point.

Good stuff!

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Re: Notation system for gamelismic temperaments

Post by Xen-Gedankenwelt »

Many thanks for taking the time to read through it, and for your feedback! :)

Yes, I think I read about double-slashes causing problems here, but forgot about it. Should be fixed now!

Omitting diacritics where it's unambiguous is probably a good idea. However, I think most linear temperaments will only need the 64:63 accidental pair, and maybe the 28:27 one. So unless the notation is used for the planar gamelan temperament, I assume this will rarely be relevant. I'll list them for now, though, to make sure there is no confusion.
Also, people should feel free to replace accidentals with standard accidentals for the EDO they're using, i.e. :/|\: and :\!/: instead of :|): and :!): in 31EDO, if appropriate. This will make it harder to use the score for other EDOs, since accidentals may have a different meaning there, but at the same time it will eliminate redundant information, which can be useful if a notation is needed for an EDO, but not for a more specific temperament.

I like your idea to use nominals like C G D A E as an alternative, Cam! :)

For example, イi ウu エe オo アa = 1/1 8/7 21/16 3/2 7/4 could be defined as D E:|): G:!): A C:!):, and simply written as D E G A C (with the accidentals omitted), if there is no risk of confusion. # and b (and possibly x, bb and a semisharp/-flat) could be used to derive other nominals, and pure Sagittal notation would be used for accidentals. For example, Valentine[15] could be notated as following:

Code: Select all

1/1 21/20 11/10 8/7 6/5 5/4 21/16 11/8 10/7 3/2 11/7 5/3 7/4 11/6 21/11
  D     D#     Eb     E    E#  Gb     G     G#    Ab    A    A#    Cb   C    C#     Db
...where D E G A C represent D E:|): G:!): A C:!): (i.e. 1/1 8/7 21/16 3/2 7/4), and # and b are a 21/20 above or below the nominal from which they're derived. A single chroma 64:63 is :|): or :!):, and a double chroma 28:27 (if ever needed) would be :.::(|\: or :'::(!/:.

I'll try to get back to the consonant choice topic tomorrow!

P.S.: I don't have a Facebook account, and I don't want to register one, so I can't see what people write in Xenharmonic Alliance and related groups. I assume the 10EDO notation you mentioned was proposed there?

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Re: Notation system for gamelismic temperaments

Post by Xen-Gedankenwelt »

Sorry, I still didn't manage to respond! I made considerable progress, and I think I'll be able to notate all important gamelismic linear temperaments for 10-, 11-, 15-, 16-, 20-, 21-, 25-, 26-, 31-, 36-, and 41-EDO, and some for 46-, 72-, 87- and other important EDOs.
I also have a slightly less vague concept how to use it with 5 of the standard note names, but still nothing is carved in stone.

Before replying, I want to think it a little more through, and it'll probably be a huge chunk of text, so I can't promise yet when I'll find the time...

P.S.: Valentine has become one of my favorite linear temperaments in 31-EDO, along with meantone and orwell. The combination of gamelan/portent, porwell/zeus and starling is awesome! :)

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