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39-odd limit diamond in Olympian - The Sagittal forum

39-odd limit diamond in Olympian

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cam.taylor
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39-odd limit diamond in Olympian

Postby cam.taylor » Sun May 08, 2016 2:16 pm

Hi all. Thought I would write up a chart for most common use intervals in JI, showing the minute differences in intonation amongst the 39-odd limit ratios. The 20x20 diamond seems to have 317 unique pitches, so it's taking a bit of a while, but I'm getting there, and will definitely post here and on facebook when I'm finished, to show people how well Sagittal can handle almost all intervals. I haven't really heard of people using primes higher than 37 in their work (except perhaps Johnny Reinhard) but this should be a good test. One can of course, continue a chain of fifths from any note in the diamond to get >99% of the pitches ever used or conceived of in JI, uniquely identified using Olympian level symbols. One can of course leave off some diacritics to make things quicker, but then a whole bunch of pitches are conflated. Using this chart, pitches in any of Ben Johnston's string quartets or other works of similar JI complexity can be read/written/spoken/understood at a reasonable fluency, and pitch height is always close at hand, unlike many other JI notations.

I'd like to see something like this done with Sabat or Johnston to see how they cope with a similar complexity.

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cam.taylor
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Re: 39-odd limit diamond in Olympian

Postby cam.taylor » Sun May 08, 2016 2:37 pm

I've got a whole bunch of enharmonics too, so people can, for example, choose to spell 13:7 as an alteration of either a major seventh or diminished octave.

I'm doing the diamond on D, because it's notationally symmetrical. Should make things a bit easier.
Last edited by cam.taylor on Mon May 09, 2016 5:27 pm, edited 1 time in total.

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cam.taylor
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Re: 39-odd limit diamond in Olympian

Postby cam.taylor » Mon May 09, 2016 5:11 pm

Here it is!!! Harmonics 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 running along the top (horizontal), and subharmonics running down the side (vertical).

Let me know if there are any mistakes, or revisions that should be made. It took me longer than I expected, but then again, there are 400 pitch notations here (~320 or so unique).

I'd like to make a Promethean one too soon, which would simplify things a whole lot, reduce the total number of symbols, and merge several ratios together, creating a kind of interesting high level meta-temperament of JI. Speaking of that, does anyone have on-hand, some of the commas that are tempered out by the various levels of Sagittal notation? Could point us to interesting (large) EDO tunings as frameworks for each level of accuracy.
Attachments
39-ODD LIMIT DIAMOND IN SAGITTAL.docx
For the symbols I used the images from the forum so it should be legible for all regardless of font libraries, etc.
(151.5 KiB) Downloaded 64 times

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Dave Keenan
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Re: 39-odd limit diamond in Olympian

Postby Dave Keenan » Tue May 10, 2016 1:39 pm

Wow! Well done Cam!

However I fear that many will find the number of different symbols used, including diacritics, to be overwhelming.

The fact is that ratios in JI compositions obey something like where the second most common ratio is used only about half as often as the most common, and the next used only a third as often, and so on. We used ratio statistics from Manuel Op de Coul's Scala scale archive in designing Sagittal—kindly provided by Manuel.

So a chart like this, while great to show what Sagittal can cover, gives a new-comer a false impression of what you need to learn in order to use Sagittal.

Would it be too much to ask you to also make a cut-down version that is a diamond of only the 13-smooth odd numbers up to 39 (possibly adding 45), and omits all spellings requiring diacritics, apart from the single right accent required to distinguish 13 from 35. This would show just how much can be done using only the Spartan set of 8 symbols. We would note that the right accents can be omitted if an error of 0.4 cents is of no concern. The 13-smooth odds to 45 are 1 3 5 7 9 11 13 15 21 25 27 33 35 39 45. Omitting 17 19 23 29 31 37 41 43.

Limiting to Spartan will mean that some squares will be empty, but that's OK because they will represent rarely used ratios such as 13/11 and 33/13. The ratio of two such high primes is almost as rare as their product 11 × 13 = 143. I calculate that this will cover about 70% of the ratio occurrences in the Scala archive. But this doesn't take into account how often a given scale is used for a composition. Scales containing more common ratios will be used more often, so the ratios notatable with Spartan may constitute 90% of ratio occurrences in compositions.

Another request is to do the same in multi-Sagittal, i.e. one-symbol-per-prime, 5 :\!:, 7 :!):, 11 :/|\:, 13 :(!/: —using the symbols always in that left to right order whether on the staff or in text, since it corresponds to increasing size of alteration. These might be shown as alternative spellings on the same chart.

Yet another request is to make a version that uses C = 1/1.

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Dave Keenan
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Re: 39-odd limit diamond in Olympian

Postby Dave Keenan » Tue May 10, 2016 5:39 pm

Last edited by Dave Keenan on Wed May 11, 2016 5:07 pm, edited 2 times in total.
Reason: Corrected misremembered info re Herculean being specifically related to 58 eda

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Re: 39-odd limit diamond in Olympian

Postby Dave Keenan » Tue May 10, 2016 6:34 pm

Here's the first part of those Scala archive statistics I mentioned—up to the first occurrence of prime 37. Powers of 2 and 3 have been removed from the ratios because these are taken care of by octaves and nominals-plus-sharps-and-flats. So each line (other than the first) shows something we need a symbol for. You can see that Spartan covers the first 7 entries (totaling 70.8% of ratio occurrences), without any approximations, and with some redundancy of apotome-complements to reduce the requirement for two symbols altering in opposite directions (a sharp or flat and a Sagittal). The fact that the same set of 8 symbols is also capable of notating around 50 EDOs seems almost too good to be true. Or maybe it is telling us something about human psychology.


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Dave Keenan
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Re: 39-odd limit diamond in Olympian

Postby Dave Keenan » Tue May 10, 2016 7:22 pm

Popularity of commas chart.gif

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Re: 39-odd limit diamond in Olympian

Postby Dave Keenan » Tue May 10, 2016 9:42 pm

We can see here that, once we get beyond the ratios notated exactly by the Spartan set, the usage falls off faster than Zipf's law. The Zip's Law predictions are the red bars, the actual frequencies are the blue.

Popularity of commas Zipfs law.gif

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cam.taylor
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Re: 39-odd limit diamond in Olympian

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


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Dave Keenan
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Re: 39-odd limit diamond in Olympian

Postby Dave Keenan » Wed May 11, 2016 8:25 pm

Last edited by Dave Keenan on Wed May 11, 2016 10:22 pm, edited 2 times in total.


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