45 odd-limit 13-prime limit diamonds in Athenian and beyond

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cam.taylor
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45 odd-limit 13-prime limit diamonds in Athenian and beyond

Postby cam.taylor » Wed May 18, 2016 11:53 am

As per Dave and others' suggestions, I've made a few revised diamonds, only involving the primes up to 13, but extending the odd limit to also include ratios of 45, which can be pretty important.

This one's Athenian-level precision, so small intervals like the 13-schismina ("mina") and 5-schisma are effectively "virtually tempered out", that is, notationally ignored, so that 5/4 from D can be notated as either F:\!::#: or as G:b: (with schismas, usually G:'::|::b:)

The Athenian set also uses quite a few less symbols, so no :|~: :)/|: :/|~: :)|: :~|: :)~|: :~~|: :)|~: :)|): :~|): :~|\: :)/ /|: :(|~: :(/|: :)/|\: :|\): :|\ \: :)|\ \: for anyone interested, those are all simplified to Athenian precision symbols.

Hope there are no errors in this one, just been through my other two diamonds and found quite a few little mistakes, mostly involving accidentally reversing up- and down-shafts.
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cam.taylor
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Re: 45 odd-limit 13-prime limit diamonds in Athenian and beyond

Postby cam.taylor » Wed May 18, 2016 3:00 pm

Here is another version.

The same 13-prime, 45-odd limit diamond is represented in Herculean, adding some non-athenian flags, as well as the schisma accidentals to distinguish some interval pairs ~2c apart.

I have also made a 37-prime 45-odd limit diamond in both Olympian and Multi-Sagittal, but will start another thread for that, as there are quite a few more symbols for the higher primes involved - 17, 19, 23, 29, 31, 37.
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Dave Keenan
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Re: 45 odd-limit 13-prime limit diamonds in Athenian and beyond

Postby Dave Keenan » Wed May 18, 2016 3:09 pm

That's great Cam. Thanks again!

I suggest completely omitting any alternative spellings from the Athenian chart that involve a schisma (approx 2c) approximation.

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cam.taylor
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Re: 45 odd-limit 13-prime limit diamonds in Athenian and beyond

Postby cam.taylor » Wed May 18, 2016 3:14 pm

I thought about that, but it dawned on me that Athenian level notation already uses some other "approximations", possibly of a similar order of comma size. I had the schismatic spellings in brackets initially, but decided I'd include them as other approximations are surely already filling the diamond in other spots.

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Dave Keenan
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Re: 45 odd-limit 13-prime limit diamonds in Athenian and beyond

Postby Dave Keenan » Wed May 18, 2016 4:09 pm

cam.taylor wrote:I thought about that, but it dawned on me that Athenian level notation already uses some other "approximations", possibly of a similar order of comma size. I had the schismatic spellings in brackets initially, but decided I'd include them as other approximations are surely already filling the diamond in other spots.

That's one of the beauties of Athenian. In the the 45-odd-13-prime-limit diamond, I think the only one that actually needs a schisma approximation is 25/7, and of course its inverse. And I don't think there are any double-mina approximations needed. See the 13-odd-limit diamond in Athenian in Figure 7 on page 14 of http://sagittal.org/sagittal.pdf. The filled notes are the approximations (all single mina, 0.4 c).

It would be good if you could indicate which spellings involve approximations, and approximately how large the error is. e.g. colouring the nominal with saturated colours:
minas
0 black
1 blue
2 cyan
3 green
4 yellow
5 red (1 schisma)


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