Notating 5-limit pental with added 7th harmonics

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Dave Keenan
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Re: Notating 5-limit pental with added 7th harmonics

Post by Dave Keenan »

רועיסיני wrote: Thu Apr 13, 2023 9:16 am
Dave Keenan wrote: Thu Apr 13, 2023 8:43 am I note that 255/244 is strictly called the 25/7-kleisma in the systematic comma naming scheme used in the Sagittal documentation. The 25/7-comma would be larger by a Pythagorean comma.
Interesting. The interval I chose to notate for 25/7 is actually smaller than the pythagorean comma by a pythagorean kleisma, so it seems to be neither of those. I chose it because it equals (64/63)/(81/80)2.
I'm sorry. I gave a wrong description of the 25/7-comma (or 7/25-comma). It is in fact (64/63)/(81/80)2. It is indeed smaller than the pythagorean comma by a 25/7-kleisma, which is what I assume you meant to write above.

It is exactly notated as :,,::|~: and approximately notated in Athenian as :~|(: . You may find this spreadsheet useful, if you haven't found it already: viewtopic.php?p=4457#p4457
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Re: Notating 5-limit pental with added 7th harmonics

Post by רועיסיני »

Dave Keenan wrote: Thu Apr 13, 2023 12:54 pm
רועיסיני wrote: Thu Apr 13, 2023 9:16 am
Dave Keenan wrote: Thu Apr 13, 2023 8:43 am I note that 255/244 is strictly called the 25/7-kleisma in the systematic comma naming scheme used in the Sagittal documentation. The 25/7-comma would be larger by a Pythagorean comma.
Interesting. The interval I chose to notate for 25/7 is actually smaller than the pythagorean comma by a pythagorean kleisma, so it seems to be neither of those. I chose it because it equals (64/63)/(81/80)2.
I'm sorry. I gave a wrong description of the 25/7-comma (or 7/25-comma). It is in fact (64/63)/(81/80)2. It is indeed smaller than the pythagorean comma by a 25/7-kleisma, which is what I assume you meant to write above.

It is exactly notated as :,,::|~: and approximately notated in Athenian as :~|(: . You may find this spreadsheet useful, if you haven't found it already: viewtopic.php?p=4457#p4457
Yeah I meant the 25/7 kleisma. Thanks for the link to the spreadsheet, I indeed haven't found it yet. Unfortunately, the flags it shows for the 25/7 comma are quite large for its size in pental (because the syntonic comma is much smaller there than in JI), and for the 175 small diesis it shows only ://|: and alterations of it, so I think it won't be of help this time.
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Re: Notating 5-limit pental with added 7th harmonics

Post by רועיסיני »

@Dave Keenan I just noticed that in viewtopic.php?p=986#p986 you wrote
Dave Keenan wrote: Wed Apr 01, 2020 12:43 am When the fifths are not pure (Pythagorean), the untempered size of the default comma should be irrelevant. It's the tempered size that matters. The 700c fifth is 1.955c narrower than a pure fifth, so the tempered size in cents, of a comma with a 3-exponent of n differs from its untempered (JI) size by n × 1.955c. For example, the tempered apotome is 113.685 - 7×1.955 = 100 cents. So that's the size of :/||\: in 700c land.
Is this also relevant here? I used the original comma sizes here to justify my last choice for notation, but if the tempered fifth, which is about 0.7c smaller than just, is used, my original choice of :~|(: and :(|(: has closer sizes to the tempered sizes of the commas, with the benefit of being Athenian (and if I ditch the will for nice symmetry between the single and double shafts it's possible to choose :)~|: and :/|~:, which both have closer tempered sizes).
Which of the three (:~|(::(|(:, :~|::)//|: and :)~|::/|~: for the 25/7 comma and 175 small diesis respectively) do you think is best here?
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Re: Notating 5-limit pental with added 7th harmonics

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Yes. The only-3's-tempered size has some relevance here because you have generators that are fifths (or their octave equivalent or inversion). I overstated the case when I wrote that the untempered size was irrelevant. At least the relative untempered sizes of the symbols are important. We try to avoid having any pairs of symbols whose size order in the temperament is the reverse of their untempered size order. If we can't avoid it we try to minimise their number and the amount by which they cross over (Trojan has a few). This is to try to preserve the feature of Sagittal whereby the visual size of the symbol is approximately proportional to the size of the alteration.

I'm sorry I don't wish to take the time I would need to spend to offer a valid opinion on your 3 options.

But the primary way of designing a Sagittal regular temperament notation is to decide what generator-count vectors you need symbols for, then take the prime count vectors for all the extreme resolution (Olympian) symbols (including those with mina and schisma accents) and apply your temperament's mapping to them (matrix times vector) to obtain the generator count vector corresponding to each one, to see which symbols correspond to the GC-vectors you need symbols for. You ignore octaves. Then consider whether you can drop accents. And consider the other things we've already mentioned like consistent flag arithmetic, double-shaft flags sequence recapitulating part of single-shaft flags sequence, minimising size order reversals.

See this thread: viewtopic.php?f=6&t=530 It only gives rank-2 examples, but the same principles can be generalised to rank-3.

The commas for all the single-shaft extreme precision symbols (including those with accents) appear in quotient form in the above-linked JI Calculator spreadsheet. I'm sorry that it doesn't contain the vectors, but I expect you can compute the prime factorisations. Those are in a spreadsheet or text file somewhere. I just can't find it at the moment. Sorry. You can at least obtain the vectors for the unaccented symbols, from any of the "character map" spreadsheets linked from the Sagittal home page. https://sagittal.org/
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Re: Notating 5-limit pental with added 7th harmonics

Post by Dave Keenan »

You may find this interesting. viewtopic.php?p=807#p807
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Re: Notating 5-limit pental with added 7th harmonics

Post by רועיסיני »

Then I think I'll choose :~|: and :)//|:. They cause only one size reversal, where :|(: that comes right before :~|: has value of about 1 cent more than it, while :~|: leads to the double shaft :)//||:, which repeats the shape of :)//|:. The only symbol smaller than :|(: is :)|: which has tempered size of only 1c, much smaller than the needed ~8.8c.
Thanks!
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