137EDO

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cmloegcmluin
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137EDO

Post by cmloegcmluin »

I have been delving into Sagittal EDO notations lately. Very interesting stuff.

I don't see that 137 has been notated yet; prime EDOs have (with the exception of 73) been notated only up to 89. @Dave Keenan do you have any record of taking a shot at 137, and if so, do you have any notes to share about it?

Of course there are many more universally appealing EDOs to work on so I don't expect to divert attention to 137. I'm motivated to notate it because it does a good job at approximating the pitches in my Yer tuning, as you can see here in the results of Graham Breed's temperament finder. I also think that drafting a proposal for a Sagittal EDO notation will be a good way for me to get a sense for the process as described in the Xenharmonikon article:
These have been selected using several criteria, including a symbol’s prime factor limit, the division’s prime number errors and consistency, and validity of secondary comma roles. Consistency of symbol flag arithmetic for the single-shaft symbols has been strictly maintained, but occasional inconsistencies have been allowed for double-shaft apotome complements in cases where they are not likely to be noticed.
137 has 12 steps per apotome and 11 steps per limma, which I think puts it in the grey area of the periodic table. It has a pretty good fifth of 700.729927 cents. Notably 137*18 = 2466 which is remarkably close to the 2460edo which the extreme precision JI notation is based on. Perhaps I can lean on it when choosing my symbols.

Anyway, again, don't spend too much time on this please. I just thought I'd be remiss not to at least ask if anyone had tried their hand at 137 already.
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Re: 137EDO

Post by Dave Keenan »

I haven't heard of 137-edo being of interest to anyone, for any reason. We certainly haven't looked at a Sagittal notation for it.
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Re: 137EDO

Post by cmloegcmluin »

Not surprised :) Okay, I'll see what I can do!
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Re: 137EDO

Post by Dave Keenan »

EDOs that differ by 7 sometimes have the same notation, because they have the same number of steps per apotome, as you can see by following the lines of constant "#=" in the periodic table. We have a notation for 130-edo (137-7), in Figure 9 on page 17, and a notation for 144-edo (137+7) in figure 10 on page 19, of http://sagittal.org/sagittal.pdf.

I see you found lots of "limit" consistency resources via facebook:
https://www.facebook.com/groups/xenharm ... 147499800/

Here's my favourite. It does integer limits, allowing for tempered octaves. But when the EDO is a whole number, the odd-limit is just the largest odd that's not greater than the integer limit.
http://www.huygens-fokker.org/docs/consist_limits.html

But you may want subset limits, e.g. 19-odd-limit with no 3 or 5. The following spreadsheet looks at all odd subsets up to the 19-odd-limit. There's a column for each EDO, and a row for each subset. Where they intersect, if the EDO is consistent over the diamond of that subset, the cell contains the maximum error in cents, otherwise it is blank.
Consistency.zip
(1.05 MiB) Downloaded 268 times
[Edit: Uploaded new version of spreadsheet with correction of the error in the hastily-added 137-edo column. Thanks cmloegcmluin.]
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Re: 137EDO

Post by Dave Keenan »

Any symbol that represents the same degree in both 130 and 144 edos, will also be valid for that degree of 137-edo, unless it has different comma roles in 130 versus 144.
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Re: 137EDO

Post by Ash9903b4 »

I don't see why Spartan accidentals won't work for 137. Yes, 7 doesn't have a consistent mapping, but it's reasonable to redefine :|): here as the 65-comma (65/64).

1\137: :|(: (325k)
2\137: :/|: (5c)
3\137: :|): (65c)
4\137: :/ /|: (25S) or Athenian :|\: (55S)
5\137: :/|): (13M)
6\137: :/|\: (11M)
7\137: :(|\: (13L)
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Re: 137EDO

Post by cmloegcmluin »

I'll respond in detail later (gotta get back to my day job) but I'll just quickly say that in my work last night I got the same thing as @Ash9903b4. Well, I only had the :/ /|: (25S) for the 4th step, but then I corrected it to :|\: (55S) after reading @Dave Keenan 's suggestion to compare with 130edo.
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Re: 137EDO

Post by cmloegcmluin »

But you may want subset limits, e.g. 19-odd-limit with no 3 or 5. The following spreadsheet looks at all odd subsets up to the 19-odd-limit. There's a column for each EDO, and a row for each subset. Where they intersect, if the EDO is consistent over the diamond of that subset, the cell contains the maximum error in cents, otherwise it is blank.
Thanks for sharing this super cool tool, Dave. And also thank you for providing the column (well, columns) for 137! I was disappointed at first (and surprised, given how well 137edo does for the 2.11.13.17.19 prime limit) to see that 137edo didn't appear to be consistent in the {1, 11, 13, 17, 19} odd limit. However, I then noticed (although you say "The interesting stuff starts around cell P49") that in the upper section it was still calculating based on 64edo! So I'm delighted to say that 137edo has a maximum error of just 0.5068751535 cents in that odd limit. : D
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Re: 137EDO

Post by Dave Keenan »

cmloegcmluin wrote: Wed Apr 01, 2020 10:29 am Thanks for sharing this super cool tool, Dave. And also thank you for providing the column (well, columns) for 137! I was disappointed at first (and surprised, given how well 137edo does for the 2.11.13.17.19 prime limit) to see that 137edo didn't appear to be consistent in the {1, 11, 13, 17, 19} odd limit. However, I then noticed (although you say "The interesting stuff starts around cell P49") that in the upper section it was still calculating based on 64edo! So I'm delighted to say that 137edo has a maximum error of just 0.5068751535 cents in that odd limit. : D
Oh dear. So I typed the "137" into the wrong cell? Sorry about that. It's been 18 years since I made that spreadsheet, and I've forgotten how it works. I'm so pleased you were able to figure it out and recover from my error. I've uploaded a corrected version so it won't confuse anyone else in future.
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Re: 137EDO

Post by cmloegcmluin »

I did notice that 2002 on there and figured it might have a been a minute since you last worked with it! : D
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