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Quadritikleismic (68&72 13-limit)

Posted: Mon Mar 08, 2021 7:35 am
by Henrik Ljungstrand
Hello everyone, i am pretty new to Sagittal, less new to xenharmonic alliance or the regular tuning paradigm. Some musical experience, mostly from church choir and amateur playing, nothing professional. Would say i have a good musical ear and a keen mathematical mind though.

After having noticed that standard Sagittal notation for 13-limit JI uses the symbols :\!: , :!): , :/|\: , and :(!/: for 81/80, 64/63, 33/32 and 27/26 (down, down, up, down) as standards, and noticing that 33/32 is almost exactly a fourth root of 9/8 and 27/26 is almost exactly a cube root of 9/8, thus being likely to be perceived as a quarter tone and a third tone respectively, i wondered about what temperament we would get by setting 27/26 = 26/25 = 25/24 (thus (27/26)^3 = 9/8) and setting 36/35 = 33/32 and 36/35 * 33/32 = 35/33 (thus (33/32)^4 = 9/8).
The answer is Quadritikleismic temperament, a rank 2 (linear) 13-limit temperament, supported by 72edo, 68edo and 140edo.
Having commas 676/675, 625/624, 385/384 and 9801/9800 tempered to unison, among many others.

I now wonder which accidentals would be best for using in this temperament, besides :/|\: and :(!/: ? It should preferrably be a good compromise between 72edo and 68edo, and possibly 140edo.
Are :\!: and :!): still good alternatives for 5 and 7 commas, or are there better ones? Should the 11 and 13 commas also be changed perhaps?

The idea is for this temperament to possibly be a good microtemperament for 13-limit music performance, especially real time performance, if there is any interest for such in 13-limit almost JI based on diatonicism plus pretty simple extensions.
Just an idea, i haven't really had the opportunity to train myself very much in any instrument except voice yet, although planning to learn recorder, piano/synth and guitar better, and hopefully be able to retune such instruments to e.g. Quadritikleismic, and other tunings/temperaments.

Re: Quadritikleismic (68&72 13-limit)

Posted: Mon Mar 08, 2021 11:54 am
by Dave Keenan
Hi Henrik. Welcome to the Sagittal forum.

I include this link to your related facebook thread in case some readers here haven't seen it.

My understanding of your facebook post is that you weren't particularly asking about a notation for it, but just wanted to know if anyone had any information about a temperament that treated:
the 5-comma 81/80 as 1/9-tone,
the 7-comma 64/63 as 1/8-tone,
the 11-M-diesis 33/32 as 1/4-tone,
the 13-L-diesis 27/26 as 1/3-tone,
where 1 tone is 9/8.

422ef-ET certainly covers it (having 72 steps to the 9/8) but of course it begs the question: Why not use the best approximations of 33/32 and 27/26 at 19 and 23 steps (rather than the exact 1/4-tone and 1/3-tone at 18 and 24 steps)?

So understandably you'd like a higher-rank temperament that would effectively let you choose a subset of 422ef-ET that still gives say a 13-limit diamond, or some useful number of 13-limit consonances, without too many notes.

I'm not convinced that Quadritikleismic is what you want here. Can you explain in what sense it satisfies your above specification? I see that it satifies the last two criteria (re 11 and 13). But I don't see that it satisfies the first two (re 5 and 7).

Quadritikleismic
POTE generator: 316.989
Map: [ 4 0 4 7 17 0], 0 6 5 4 -3 14] ]
EDOs: 68, 72, 140, 212

But I decided to have a look at a notation for 140-edo anyway.

I assume you've seen the standard Sagittal notations for 68 and 72, e.g. on the Periodic Table.

140edo has the same best fifth as 70edo and has CD=24, EF=10, #=14. The following proposal has valid flag arithmetic. I was forced to use primes 19 and 17 to get symbols for 1 and 2 steps.

140edo
Steps   Symbol  Comma  Other valid symbols and their commas
-----------------------------------------------------------
1	:)|:	19-schisma
2	:~|:	1/17-kleisma
3	:/|:	1/5-comma		:|): 1/7C	:|\: 55C
4	:)/|:	19/5-comma		:)|): 19/7C
5	:(|:	7/11-comma
6	:/|\:	11-medium-diesis	:/|): 1/35M, 13M
7	:)/|\:	5/49-medium-diesis
8	:(|):	1/11-large-dieisis	:(|\: 35L, 1/13L

But that's not a notation for Quadritikleismic and I don't think it really helps you get to one.

Re: Quadritikleismic (68&72 13-limit)

Posted: Mon Mar 08, 2021 1:07 pm
by herman.miller
One thing to note if you're wanting to notate quadritikleismic is that :|): (64/63) and :|\: (55/54) are equivalent in this temperament: both work out to be [+1, -16>. This is also the case with the combinations :/|): (36/35) and :/|\: (33/32) = [+0, +3>, as well as :(|): (729/704) vs. :(|\: (8505/8192) = [-2, +39>. So you might want to pick one or the other flag in each of these three pairs to make the addition easier to remember. But note that :||): and :||\: map to different intervals.

I haven't looked into quadritikleismic, but with a generator of 17 cents, you're going to need a lot of accidentals for the small intervals. Here's some options:

[+0, +1> (17.09): 99/98 :~~|:
[+0, +2> (34.17): 49/48 :~|):
[+0, +3> (51.26): 36/35 :/|): or 33/32 :/|\:
[+0, +4> (68.35): 25/24 :)||(:
[+0, +5> (85.43): 21/20 :.::||):
[+0, +6> (102.52): 35/33 :.::/ /||: (approx.)

Note that at least with the TOP tuning of 17.086 cent generators with a 300.00 cent period, the schisma accent is not negligible, but it's still a relatively small interval at 5.584 cents. You might be able to leave out the schisma accents if you don't stray too far from your reference pitch.

Re: Quadritikleismic (68&72 13-limit)

Posted: Mon Mar 08, 2021 1:40 pm
by Dave Keenan
Henrik Ljungstrand wrote: Mon Mar 08, 2021 7:35 am I now wonder which accidentals would be best for using in [Quadritikleismic], besides :/|\: and :(!/: ? It should preferrably be a good compromise between 72edo and 68edo, and possibly 140edo.
With 4 periods to the octave and 6 generators to the fifth, notation of Quadritikleismic, based on diatonic (chain-of-fifth) nominals, will require up to two sagittals per note (in addition to any sharps or flats). The first will be chosen from a set of 2 up/down pairs of symbols to specify the number of periods modulo 4. And the second will be chosen from a set of 3 up/down pairs to specify which interleaved chain of fifths it's on. So, quite complex.
Are :\!: and :!): still good alternatives for 5 and 7 commas, or are there better ones?
I'm not really sure what you're asking here. Those are the symbols for 81/80 and 64/63 which we call the 5-comma and 7-comma (or if we care about the direction, the 1/5-comma and the 1/7-comma). But you may be using "comma" in the generic sense. Sagittal also has accented symbols :'::|: for the 5-schisma 32805/32768, :'::/|): for the 7-M-diesis 59049/57344 and :.::(|\: for the 7-L-diesis 28/27. But we normally wouldn't use accented symbols for tunings with less than 200 notes to the octave.
Should the 11 and 13 commas also be changed perhaps?
You will find other 11 and 13 commas and their symbols here:
http://sagittal.org/SagittalJI.gif
The idea is for this temperament to possibly be a good microtemperament for 13-limit music performance, especially real time performance, if there is any interest for such in 13-limit almost JI based on diatonicism plus pretty simple extensions.
There is definitely interest in that. But my feeling is that when people are willing to depart from strict JI at all, they would typically be willing to put up with larger errors (maybe around 3 cents) in return for lower complexity. Such as with Myna temperament whose notation is being discussed here:
viewtopic.php?f=6&t=532