103EDO and 113EDO

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cam.taylor
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103EDO and 113EDO

Post by cam.taylor » Sun Dec 06, 2015 11:55 am

103EDO and 113EDO have been topics of Scott Dakota's posts over on the XA Math facebook page, and I'm quite interested in exploring them as 103=31+72 and 113=41+72, so they're both Miracle friendly and very compatible with material from 31, 41 and 72EDOs as well as being excellent competitors at high prime limits (I believe Scott's been looking into the 47-limit)

I was thinking of comparing these two with 94, how interval relations work, and how things might "translate" across these moderately fine-grain equal divisions of the octave. It would be wonderful to know what the recommended Sagittal set would be for 103 and 113, as I'd imagine they'd look similar to each other and also 72... any thoughts?

(I have, of course, checked the Sagittal pdf for EDO notations, and neither 103 or 113 is listed, which is why I'm asking)

Thanks in advance!

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Xen-Gedankenwelt
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Re: 103EDO and 113EDO

Post by Xen-Gedankenwelt » Mon Dec 07, 2015 9:53 am

cam.taylor wrote:I was thinking of comparing these two with 94, how interval relations work, and how things might "translate" across these moderately fine-grain equal divisions of the octave.
A good basis for 94EDO and tunings that support miracle is 11-limit marvel temperament, or unimarv.

Marvel tempers out the marvel comma 225:224, which is the difference between the following intervals:
- 16/15 and 15/14 (-> 8/7 can be split into 2 of those semitones which are called 'secors')
- 25/16 and 14/9 (-> allows to divide the octave into two 5/4s, and one 9/7)
- 9/8 and 28/25 (the latter is the whole tone between 5/4 and 7/5)
- 25/24 and 28/27
- 81/80 and 126/125
- 8/7 and 256/225 (the latter is a "diminished third" generated by stacking two 16/15 semitones)
- 7/6 and 75/64 (the latter is the "augmented second" between 16/15 and 5/4)
- 35/32 and 49/45

In the 11-limit, marvel also tempers out the keenanisma 385:384 (difference between 35/32 and 12/11, and between 48/35 and 11/8), the swetisma 540:539 (difference between 49/45 and 12/11, and between 49/36 and 15/11), and 4125:4096 (difference between 33/32 and 128/125).

A good basis is the marvel-tempered scale Gypsy; some example modes:
- E:\!: F G:)||(: A:\!: B:\!: C D:)||(: E:\!: = 1/1 16/15 5/4 4/3 3/2 8/5 15/8 2/1
- A:\!: B:\!: C D:)||(: E:\!: F G:)||(: A:\!: = 1/1 9/8 6/5 7/5 3/2 8/5 15/8 2/1
- F G:)||(: A:\!: B:\!: C D:)||(: E:\!: F = 1/1 7/6 5/4 7/5 3/2 7/4 15/8 2/1
...with :\!: = 80/81, and :)||(: = 25/24.

There's also Graham Breed's Tripod scale, which nicely fills the 7/6 gaps with a 16/15 and a 35/32 (or here: 12/11) step, and features some 11-limit intervals.

In the 13-limit, 94, 103 and 113 share a marvel extension for which 3/2 and 14/13 can be used as generators, where three 14/13 generators give a 5/4. In case you're interested, here's the temperament finder link:
http://x31eq.com/cgi-bin/rt.cgi?ets=94+ ... 3&limit=13

I didn't explore higher prime limits.

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Dave Keenan
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Re: 103EDO and 113EDO

Post by Dave Keenan » Mon Dec 07, 2015 12:12 pm

I wonder if Scott is has considered the consistency of their JI approximations. When you look at that, neither of those divisions is particularly remarkable, compared to others in the vicinity, such as 94, 111 and 121.
Strictly-speaking 103edo isn't even 1,3,9-consistent, meaning that, as a chain of fifths, C : G : D is not it's best approximation of 1:3:9, although that can be repaired by stretching its octave by 0.2 to 1.9 cents, when its integer consistency limit becomes 15. 113 edo has an integer consistency limit of 14 (which doesn't increase with any stretch or compression of the octave).
See http://www.huygens-fokker.org/docs/consist_limits.html

A notation for 113edo is straightforward: :)|(:=1, :/|:=2, :|):=3, :/ /|:=4, :/|\:=5, :#:=10. An alternative would be :|(:=1.
In the case of 103edo we have: :/|:=1, :|):=3, :/|\:=4, :#:=8. But 2 degrees is difficult.
2 degrees of 103edo could be :|(: or :)|(: but their untempered size is very small, and less than :/|:. 2 degrees of 103edo could also be :/ /|:, which has the opposite problem, but I prefer it for the obviousness of :/|: + :/|: = :/ /|:.

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Xen-Gedankenwelt
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Re: 103EDO and 113EDO

Post by Xen-Gedankenwelt » Mon Dec 07, 2015 6:21 pm

Dave Keenan wrote:In the case of 103edo we have: :/|:=1, :|):=3, :/|\:=4, :#:=8. But 2 degrees is difficult.
2 degrees of 103edo could be :|(: or :)|(: but their untempered size is very small, and less than :/|:. 2 degrees of 103edo could also be :/ /|:, which has the opposite problem, but I prefer it for the obviousness of :/|: + :/|: = :/ /|:.
If untempered size is more important than avoiding accents, 2\103 could be interpreted as :|)::.: = 65/64. To avoid confusion with :|): , something else could be used for 3\103, like :|\: = 55/54 (with :/|: + :|\: = :/|\:), or :~|): = 49/48.

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