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Notating 37EDO

Posted: Sun May 29, 2016 8:58 am
by cam.taylor
I am trying to make a 37edo template on Mus2 for Joe Monzo, but I didn't find any obvious solution to notating 37, except as a subset of 74edo's meantone (which I don't think he's interested in).

If notating with the native fifth, we have m2 = 1\37, A1 = 6\37, M2 = 7\37, etc.

The most obvious option next seemed to me to be to notate 2\37 as either :/|: or :/|\:, and it seems they're both equal here (so :|\: is tempered out?). I chose to use :/|: to keep primes small, and make major and minor thirds/sixths etc pretty intuitive.

As :|): is tempered out, and :/|: = :/|\:, the next highest prime to look at was 13, and I found that :(|\: corresponds to 3\37, along with :/|):. I picked :(|\:, but I guess it doesn't matter. Maybe the convention is to take the smaller alteration.

However, I was wondering what to do with notations for 1\37. Taking the difference between my accidentals for 2\ and 3\ I ended up with 65:64, but that's notated as :|)::.:, which I thought would be far too confusing as the septimal comma is tempered out. Using the other 13-diesis gave me 40:39, which is usually notated as ://|:
Another option was to use :)|: to notate prime 19, but then the symbol's size is inflated almost by a factor of 10.

What would you do?

Re: Notating 37EDO

Posted: Sun May 29, 2016 1:24 pm
by Dave Keenan
Yes, 37edo is pretty awful. It is not consistent past the 13-odd-limit and only gets past the 7-limit if you omit 9's. But it has an extraordinarily accurate 8:11 at 17°37. So it's a shame not to notate that as e.g. C:F:/|\:. But I feel the prohibition against using :/|\: when the half-apotome exists and :/|\: is not it, is too strong. The apotome :#: :b: is 6°37. AB, CD, DE, FG, GA are 7°37 and BC, EF are 1°37.

It might be wise to notate it with only 5 nominals, like the 5n-EDOs. In that case we could just adopt the symbols for 30edo described here: viewtopic.php?p=197#p197. As the 11x13-comma, :)~|: is valid for 1°37, and the other symbols are the same as those you have already chosen. We'd also need a symbol for 4°37. :/ /|: seems obvious, but it involves a size order reversal with :/|): relative to their JI sizes. So I'll go with :(|): here instead. e.g.
37edo: C  C:)~|: C:/|: C:/|): D:\!): D:\!: D:)~!: D  D:)~|: D:/|: D:/|): E:\!): E:\!: E:)~!: E  E:)~|: E:/|: E:/|): G:(!): G:\!): G:\!: G:)~!: G  G:)~|: G:/|: G:/|): A:\!): A:\!: A:)~!: A  A:)~|: A:/|: A:/|): C:(!): C:\!): C:\!: C:)~!:
By using B and F, you could avoid the 4°37 symbol.
37edo: C  C:)~|: C:/|: C:/|): D:\!): D:\!: D:)~!: D  D:)~|: D:/|: D:/|): E:\!): E:\!: E:)~!: E  F  F:)~|: F:/|: F:/|): G:\!): G:\!: G:)~!: G  G:)~|: G:/|: G:/|): A:\!): A:\!: A:)~!: A  B  B:)~|: B:/|: B:/|): C:\!): C:\!: C:)~!:
Alternatively we could use the symbol :|~: as 1°37, in a secondary role as the 13:35-comma. Its primary role being the 23-comma. We might then use :/|~: as 3°37 in a secondary role as 7:13-S-diesis, (primary role 5:23-S-diesis). Then there would be no size reversal with :/ /|: as 4°37.
37edo: C  C:|~: C:/|: C:/|~: D:\!~: D:\!: D:!~: D  D:|~: D:/|: D:/|~: E:\!~: E:\!: E:!~: E  E:|~: E:/|: E:/|~: G:\ \!: G:\!~: G:\!: G:!~: G  G:|~: G:/|: G:/|~: A:\!~: A:\!: A:!~: A  A:|~: A:/|: A:/|~: C:\ \!: C:\!~: C:\!: C:!~:
37edo: C  C:|~: C:/|: C:/|~: D:\!~: D:\!: D:!~: D  D:|~: D:/|: D:/|~: E:\!~: E:\!: E:!~: E  F  F:|~: F:/|: F:/|~: G:\!~: G:\!: G:!~: G  G:|~: G:/|: G:/|~: A:\!~: A:\!: A:!~: A  B  B:|~: B:/|: B:/|~: C:\!~: C:\!: C:!~:

Re: Notating 37EDO

Posted: Wed Jun 01, 2016 2:00 am
by herman.miller
I've considered notating 37EDO as <37, 59, 86, 104], which gives you a number of options. For one step, it looks like the best option might be :~|): 49/48. The options for two steps are :/|: 81/80, :/|): 36/35, and :)||(: 25/24, but :/|: 81/80 is smaller than :~|): 49/48. Probably the best option is :)||(: 25/24. Three steps could be :)/|\: 405/392 or :.::||): 21/20, which could also be written :)/||: in a 7-limit temperament like this.

Re: Notating 37EDO

Posted: Wed Jun 01, 2016 9:45 am
by Dave Keenan
Good to hear from you Herman. Sorry about the delay in approving your first post. Standard anti-spam measure, with me as the only moderator and in the East Australian time zone.

Those are excellent suggestions. I was stuck on :/|: as 1 degree and trying to find others that fit with that. However I don't like the idea of using double-shaft symbols for alterations that are less than or equal to the half-apotome. It becomes problematic if the user prefers to use mixed-Sagittal. And :)/|\: as the half-apotome is very desirable, given that we can't use :/|\:, and so your set :~|): :/|): :)/|\: is brilliant, and it allows us to add :(|\: for 4 degrees as needed if we choose to use only 5 nominals. I note for Cam's benefit that this uses :/|): in its 7-limit primary role where it is 2°37, as opposed to its 13-limit secondary role where it would be 3°37. This set of symbols from Herman are all primary role, or putting it another way, they don't require accent marks to be dropped.
:~|): :/|): :)/|\: :(|\: :~~||: :/||\:
1 2 3 4  5 6
:~|): :/|): :)/|\: :(|\: :~!)::#: :#:

Re: Notating 37EDO

Posted: Wed Jun 01, 2016 11:49 am
by Dave Keenan
Some other primary-role-only options are:
:~~|: :/|): :)/|\: :(|\: :~||): :/||\:
1 2 3 4 5 6
:~~|: :/|): :)/|\: :(|\: :~~!::#: :#:
:)~|: :/|): :)/|\: :(|\: :~||\: :/||\:
1 2 3 4 5 6
:)~|: :/|): :)/|\: :(|\: :)~!::#: :#:
We try to use the lowest precision-level set that does the job, but all 37edo proposals so far require Promethean.

And we try to use the lowest prime limit that does the job. However we might consider the 11-limit notation above because, in the 7-limit notation, :~|): and :/|): are not very distinct visually, and it's not immediately clear (by looking at the area inside the head) which one represents the smaller alteration.

Another consideration is whether the sequence of flag-combinations on the double-shaft symbols recapitulates part of the sequence on the single-shaft symbols. But there's no chance of that with 37edo.

Another is whether there is valid symbol arithmetic. i.e. if there is a symbol that can be obtained by graphically combining two or more other symbols (and removing a shaft from all but one) then do their degrees add correctly? There are no such combinations in any of the proposals, so we don't have to worry about this one.

A more strict requirement is whether there is valid flag arithmetic. In other words, can we assign a non-negative (preferably non-zero, preferably integer) number of degrees to each flag type (and to the second shaft), such that they add up to the correct number of degrees for every symbol. The fact that all proposed 1-degree symbols have two flags means that either they will both be fractional or one will be zero. In the case of :~~|: we have no choice but to assign :~|: = 0.5. And in all cases we are forced to assign 0 to either :|): or :|\: or both. So there's little point in worrying about flag arithmetic below the level of whole symbols.

Re: Notating 37EDO

Posted: Fri Jun 03, 2016 3:27 pm
by George Secor
I haven't read most of this thread, but I do have a native notation for 37-EDO in my notes, which Dave Keenan & I must have agreed to sometime in the past. The symbol sequence is:
:)~|: :/|: :)/|\: :||\: :~||\: :/||\:

Re: Notating 37EDO

Posted: Sat Jun 04, 2016 12:47 am
by Dave Keenan
Thanks George. With those symbols, I was able to find it in my email exchanges with you, from 2012-Apr-26. I had earlier failed to find it because it did not contain "et" or "edo" in association with "37", and "37" is too prevalent otherwise. It turns out that in 2012 I said I'd get back to you on that (and on a notation you gave for 135-edo, because it would give 45-edo as a subset) as soon as possible, but I never did. Sorry.

I can now say that's a fine notation for 37-edo. Cam and Herman, now that we've considered the options, can we come to a consensus?

BTW, the other notations George proposed in that email, that I have never checked because they are very high-numbered and no one has ever asked for them, are:
128-edo: :)|: :~|(: :/|: :(|(: :~|\: :/|\: :(|): :)~||: :~||(: :||\: :(||(: :(||~: :/||\:
135-edo: :)|(: :~~|: :|): :~|): :/ /|: :/|\: :(|): :)||(: :~~||: :||): :~||): :/ /||: :/||\:
140-edo: :)|(: :~~|: :/|: :~|): :(|(: :/|): :)/|\: :(|\: :~||(: :~~||: :||\: :~||): :/ /||: :/||\:
141-edo: :/|: :/ /|: :(|(: :(|): :(|\: :/|\: :~||(: :)||(: :||\: :/||\:
147-edo: :)|(: :~|(: :/|: :|\: :~|): :/|): :/|\: :(|\: :~~||: :/||: :||\: :(||(: :/ /||: :/||\:
205-edo: :)|: :)|(: :~|(: :/|: :|\: :/|~: :(|(: :/ /|: :/|\: :)/|\: :(|): :)||(: :~||(: :||~: :/||: :||\: :(||(: :/ /||: :(||~: :/||\: (GS :)/|\: = 13:17M)
270-edo: :|(: :)|(: :~|(: :~~|: :/|: :|): :|\: :~|): :(|(: :/ /|: :/|): :/|\: :)/|\: :(|): :(|\: :)||(: :~||(: :~~||: :/||: :||): :||\: :~||): :(||(: :/ /||: :/||): :/||\:

Re: Notating 37EDO

Posted: Sat Jun 18, 2016 11:27 am
by cam.taylor
These 205 and 270 edo notations are very useful!

I've been looking into 205 (in relation to H-Pi, and to 41edo), 270, 311 and 270+311=581 as master tunings for JI and regular/irregular temperaments alike. Now I know your recommended 270 notation I can probably work out how most of 581 works, but maybe that's for another thread.

Thanks George also for the 37edo notation. I haven't actually notated anything in 37 yet, but having played in it using all 37 tones on my axis keyboard, and a smaller 17-tone set on another keyboard, I've found it strange not quite knowing how to refer to pitches except by degree numbers and approximate cents, which I don't tend to use very often.

Re: Notating 37EDO

Posted: Sat Jun 18, 2016 12:23 pm
by George Secor
Finding a symbol set for 581-EDO is problematic, to say the least, beginning with how to notate a single degree. The 5-schisma (32768:32805) :'::|: and the 65:77-schismina (2079:2080) :|::'::': [Edit: Now :``::|: ] are each 2 degrees, while the 455-schismina (4095:4096) :|::': [Edit: Now :`::|: ] vanishes.

Re: Notating 37EDO

Posted: Wed Jun 03, 2020 3:41 pm
by Dave Keenan
For the latest 37-EDO notation, see viewtopic.php?p=841#p841 and the following post (on the next page).