How can I notate my own temperament?

[Dannyu NDos]

I made my own temperament of the 12-tone system. Its objective was to, while maintaining versatility of 12edo, express different characteristics for different keys.

My sketch was to, for each notes, give the geometric mean of some septimal just ratios. As such, I'd say it's a "pseudo-septimal" tuning.

• C = 1/1 = 0¢
  C♯/D♭ = (25/24 * 21/20 * 16/15 * 15/14)^¼ = (5/4)^¼ ≈ 97¢
  D = (35/32 * 10/9 * 28/25 * 9/8 * 8/7)^⅕ = (7/4)^⅕ ≈ 194¢
  D♯/E♭ = (7/6 * 25/21 * 6/5)^⅓ = (5/3)^⅓ ≈ 295¢
  E = (60/49 * 49/40 * 5/4 * 32/25 * 9/7)^⅕ = (108/35)^⅕ ≈ 390¢
  F = (64/49 * 21/16 * 4/3 * 49/36 * 48/35)^⅕ = (64/15)^⅕ ≈ 502¢
  F♯/G♭ = (2/1)^½ = 600¢
  (And their octave complement will give the rest)

As far I've tested, the most xen-sounding major key is E, having noticeably sharp major third and perfect fifth.

Yet I'm unsure what commas this tuning tempers out. If this temperament were to be notated in sagittal, how?

[yahya.abdal-aziz]

Well, the simplest way to notate this temperament - simplicity is good! - would be to only use sharps () or flats (), each according to your preference for - for example, calling a key "F" or "G". An accompanying textual note to explain the actual tunings of each note you use would assist a performer who wanted to recreate your piece of music.

Of course, there are more complex ways you could notate these notes - but what practical advantage would that give?

[cmloegcmluin]

Hi @Dannyu NDos. Good to see you again here on the forum.

Re: notation, I agree with Yahya that the best way for you to notate these pitches would be to use the same notation Sagittal uses for standard tuning (AKA 12 Equal Divisions of the Octave, or 12-EDO), which is equivalent to standard notation if you use the Evo flavor of Sagittal, and if you use the Revo flavor of Sagittal instead you would just replace with and with  .

Sagittal symbols are based on JI, and the ways EDOs distort JI. But your pitches' ratios are irrational (roots of numbers like ^⅓, ^¼ and ^⅕ are all irrational). And JI is just the world of only rational pitches. And so, your pitches do not call out directly for the information encoded into Sagittal symbols.

But Sagittal symbols can and should also be used to simply indicate sizes of deviations (such as measured in cents), so you can use them that way too. One popular way is with the 12-Relative notation, meaning relative to 12-EDO. But your pitches are so close to 12-EDO, in fact, that even if you used the 12-Relative notation at its highest precision level (essentially notating as if 192-EDO!) it would only affect half of your pitches, and never by more than its 2nd smallest symbol:

Evo	Revo	¢
C	C	0
C, D	C, D	100 - 96.578 = 3.422
D	D	200 - 193.765 = 6.235
D, E	D, E	300 - 294.786 = 5.214
E	E	400 - 390.145 = 9.855
F	F	500 - 502.346 = -2.346
F, G	F, G	600

I also want to echo Yahya's point that unless this gives some practical advantage, I'd stick with the simpler notation. Specifically, why have a D when there's no plain D that it needs to contrast with? Personally, I tend to drop unneeded symbols from my notations like this whenever I can. The better solution in this case seems not to be found within notation, but a simple key at the beginning of your piece explaining that all D's are 6¢ flat from standard tuning, etc.

As for which commas this tuning tempers out: that's not a straightforward question to answer here. "Tempering out" is a term associated with regular temperament theory (RTT). A regular temperament is described by a mapping, which is a matrix of numbers that takes JI pitches in and spits tempered pitches out, and if the tempered pitch is 0, then you say that JI pitch is tempered out. But I don't think we could describe your pitch system here using such a mapping matrix. Your thought process behind your pitches is certainly interesting, and I think it's appropriate to call it a "temperament", but I think you took a quite different approach here from regular temperaments. I wonder, have you given my RTT How-To a read yet? That might help. I myself just started studying this stuff 6 months or so ago, so as a fellow beginner, I might share your perspective on things and be able to help in that way — if you have more questions on any of this stuff, please just let me know. Dave Keenan on the other hand not only co-created Sagittal but also co-founded RTT, so he's definitely our resident expert.