## 39-odd limit diamond in Olympian

cam.taylor
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### 39-odd limit diamond in Olympian

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.

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

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.

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

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.

Dave Keenan
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### Re: 39-odd limit diamond in Olympian

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 Zipf's law 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.

Dave Keenan
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### Re: 39-odd limit diamond in Olympian

Actually, it would be good to include single-Sagittal notations for all ratios in the 13-smooth 45 odd limit that can be notated exactly using a Spartan symbol and up to two right accents (schismina accents). And the note would say that these accents can be dropped from the performance notation if errors of 0.4 cents per accent are not significant.

This includes 5:13 and 11:13 as shown here: viewtopic.php?p=93#p93
It also includes 7:13 as C:A although people would probably prefer to notate it as a kind of major seventh rather than a kind of augmented sixth.
It excludes 5:11 and 7:11 as their exact single-Sagittal notation requires either left accents (5-schisma accents worth about 2 cents) or Athenian symbols. Likewise, we stop at 45, because the next 13-smooth odd number, 49, needs non-Spartan symbols, albeit ones that don't require any new flag shapes, and .
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.
George and I aren't so sure the Promethean level was a good idea (a bit like the Greek gods not thinking Prometheus' gift of fire to the mortals was a good idea ). You may find, somewhat paradoxically, that you actually need more core symbols in Promethean than you do in Olympian. That's because the most common ratios all have Olympian notations that use Athenian cores. I think that cryptic comment in George's brilliant JI notation spreadsheet, "The herculean & olympian levels are highly recommended", should actually read, "The promethean level is deprecated". After all, there's nothing wrong with Athenian. But the Promethean symbols have to exist, to be used for less common commas in Herculean and Olympian. See http://sagittal.org/SagittalJI.gif
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.
No commas are "tempered out" as such. As you descend to lower precision levels, a symbol still represents the same specific comma exactly. It just comes to represent more more-complex and less-common commas approximately. However there are large EDOs that are related to most levels. It is useful to first look at the related EDAs (equal divisions of the apotome).

The Spartan level is too uneven to relate to any EDA or EDO. It divides the apotome into 13 very-unequal parts, however there are subsets of Spartan that resemble 11eda and 12eda. These correspond to 125edo and 130edo. Herculean is also uneven. Other levels are more even and can be approximated as follows.

Spartan    (low precision) subsets approx  11/12 eda, 125/130 edo
Athenian   (medium precision)      approx  21 eda,    217/224 edo, step name: quarter-comma
Promethean (high precision)        approx  47 eda,    494 edo
Herculean  (v.high precision) subsets apx  58/59 eda, 612/624 edo, step name: schisma
Olympian   (extreme precision)     approx 233 eda,   2460 edo,     step-name: mina (derived from schismina)
The next logical level would be:
Magrathean (insane precision)      approx 809 eda,   8539 edo,     step-name: tina

In the design of Sagittal, computer searches for these highly-consistent EDAs were performed independently by George Secor, Gene Ward Smith and myself, to be sure we hadn't missed any. In effect, we used them as scaffolding, to build highly-consistent JI precision levels that would not contain any nasty surprises when it came to adding together the values of the flags making up a symbol. So far, the only use for 809 eda has been in determining the exact boundaries for the "symbol-buckets" at the Olympian level (and hence all other levels). The boundaries are on odd half-multiples of an 809 eda step.

The EDA consistency requirements were derived from the ratio occurrence statistics mentioned earlier. These cannot be described as a simple odd-limit, nor a prime-smooth odd-limit. But were instead described as a separate limit (or weighting) on the absolute value of the exponent of each prime greater than 3. For example, ratios involving, 125 = 5³, 49 = 7² and 13 = 13¹ are all about equally common (around 1.6% of occurrences in the Scala archive). Incidentally, 125 can be notated with only Spartan and right accents, e.g. C:B.
Last edited by Dave Keenan on Wed May 11, 2016 5:07 pm, edited 2 times in total.

Dave Keenan
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### Re: 39-odd limit diamond in Olympian

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.
Commas for ratios in popularity order
Total count		29403
Num	Denom	Count	Cum.cnt	%
1	1	7624	7624	25.93%	[no accidental required]
5	1	5371	12995	18.27%
7	1	3016	16011	10.26%
25	1	1610	17621	5.48%
7	5	1318	18939	4.48%
11	1	1002	19941	3.41%
35	1	875	20816	2.98%
125	1	492	21308	1.67%
49	1	463	21771	1.57%
13	1	447	22218	1.52%
11	5	339	22557	1.15%
11	7	324	22881	1.10%
17	1	318	23199	1.08%
25	7	312	23511	1.06%
49	5	246	23757	0.84%
13	5	205	23962	0.70%
175	1	168	24130	0.57%
19	1	166	24296	0.56%
245	1	165	24461	0.56%
13	7	145	24606	0.49%
625	1	143	24749	0.49%
23	1	136	24885	0.46%
49	25	134	25019	0.46%
55	1	119	25138	0.40%
77	1	111	25249	0.38%
17	5	108	25357	0.37%
19	5	97	25454	0.33%
35	11	92	25546	0.31%
13	11	89	25635	0.30%
31	1	80	25715	0.27%
343	1	70	25785	0.24%
29	1	67	25852	0.23%
125	7	62	25914	0.21%
55	7	61	25975	0.21%
17	11	55	26030	0.19%
77	5	55	26085	0.19%
19	7	52	26137	0.18%
385	1	52	26189	0.18%
55	49	51	26240	0.17%
17	7	50	26290	0.17%
1225	1	47	26337	0.16%
37	1	46	26383	0.16%

Dave Keenan
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### Re: 39-odd limit diamond in Olympian

Popularity of commas chart.gif

Dave Keenan
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### Re: 39-odd limit diamond in Olympian

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

cam.taylor
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Joined: Thu Sep 03, 2015 11:55 am

### Re: 39-odd limit diamond in Olympian

It's going to take me a little while to digest all your comments. Thank you for such detailed replies.

What I meant by Sagittal notation "tempering out" certain commas is in the extended diamond, I was seeing a lot of "approximations" (using schismas and schisminas) rather than exact representations, e.g. notating a higher prime ratio as being a much lower prime one modified by accent marks, e.g. 37:25 as a fifth, that is, 40:27 lowered by a schisma, or, indistinguishable from a diminished sixth, or 37:33 notated as a major second, the same as 28:25 without the lowered schisma, and the same as 21:20 plus an apotome. 21:20, a minor second and 20:19, a minor second, are also separated notationally by a schisma, which is a 5-prime interval, though the difference here involves prime factors 2, 3, 5, 7 and 19 [400:399=(2^4.5^2)/(3.7.19)]. Pretty much my point about (virtual) "tempering" is, the notation generally uses combinations of simple lower prime symbols to represent higher prime identities, meaning some unison vector is being assumed (generally within the realm of 0-0.2c or so in JI). As seen in 2460edo, where the mina represents one of any variety of prime factors between very close ratios.

Also, any reason why 494 was chosen as the EDO to represent Promethean? The schisma is not tempered out, and exists as 1\494. The schismina seems to be tempered out though. 612 looks very handy for Herculean however, with the schisma very closely approximated by and mapped to 1\, incredibly near-just fifths, 12edo available, as well as meantones a tiny bit lower than 1/3-comma~19edo, higher than 2/7-comma~50edo, lower than 1/6-comma~55edo, and 12edo, etc, and gentle fifths rather close to where Margo Schulter likes them, with a virtually just 22:21 minor second, and close 13:11, 13:7, 21:13, etc. I quite like having the ability to add or subtract schismas to help with well temperaments, timbre/colour manipulation and scale structure, but I'm not that concerned with schisminas, so this seems like a good balance. I started obsessing over 581edo earlier this year, but this may be another handy cousin, and one that ties in better with the notation.

Are all primes above 23 only "approximatable"? That is, there exist symbols to exactly represent intervals involving primes up to and including 23, but as far as I can see, symbols involving primes 29, 31, etc are approximations, using other primes' symbols inflected by accent marks, again this is part of the microtempering I was trying to explain.

Interesting point about Promethean, I was thinking the Promethean notation for the diamond would be available simply by removing all accent marks, as Herculean would be from removing only schisminas. Is this not the case?

I'll think about making a more practical chart for 13-prime JI, and 13-prime 45-odd limit seems like a good idea, though my own interests have led me to generally include primes 17, 19, 23 and occasionally 29 and 31 in how I think about JI. It surely will simplify the total number of unique symbols in the chart, which might be one reason it took so long. I agree that intervals with primes higher than 13 have currently a rather limited fanbase, but I didn't want to exclude them here in this first jumbo attempt.
Actually, it would be good to include single-Sagittal notations for all ratios in the 13-smooth 45 odd limit that can be notated exactly using a Spartan symbol and up to two right accents (schismina accents). And the note would say that these accents can be dropped from the performance notation if errors of 0.4 cents per accent are not significant.
You don't mean Athenian here do you? I see no reason to limit ourselves to Spartan symbols when notating extended 13-prime JI, I find 14:13, 13:11, 33:28, 33:26, 14:11, etc, pretty core ratios that should be able to be notated how they're often used as seconds, thirds, etc. But I do see your point in catering for the 70-90%, rather than getting too technical trying to cover 99.9%.

Multi-spartan is definitely an option, I can work on that. But I'm sticking to D as 1/1 for now, because it makes the most sense notationally, and it's in the middle of the gamut of keys people usually use, in the middle of the most popular 12-tone meantone gamuts Eb-G# and Ab-C#, and in the middle of a very popular 17-tone chain of fifths from Gb-A# that has occupied the vast majority of notated music. Notating things from C will get you a few too many flats in general, plus the diamond won't be notationally symmetric making calculations harder. I wish Scala allowed you to change 1/1 to keys other than C, D would be a great one.
I fear that many will find the number of different symbols used, including diacritics, to be overwhelming.
Agreed. But I decided to start here at the complex level, and from there I can simplify. Makes more sense for me than going the other way, where I'd need to redefine quite a few ratios as I went.

Sorry, I've written this all in bits and pieces in the wrong order, it might not make too much sense. Still very keen to hear more from you and anyone else interested and/or knowledgeable, and try a few more things out.

Dave Keenan
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### Re: 39-odd limit diamond in Olympian

cam.taylor wrote:It's going to take me a little while to digest all your comments. Thank you for such detailed replies.

What I meant by Sagittal notation "tempering out" certain commas is ...
... the notation generally uses combinations of simple lower prime symbols to represent higher prime identities, meaning some unison vector is being assumed (generally within the realm of 0-0.2c or so in JI). As seen in [Olympian], where the mina represents one of any variety of prime factors between very close ratios.
I guess it's just a slightly different way of looking at it, which perhaps has more to do with politics than music or mathematics. If you mention temperament in association with a notation for JI, people may jump to the conclusion that you are notating some large EDO and using it to approximate rational intervals. Sagittal doesn't do that.

First let's define some terms that George and I have found useful. A Sagittal symbol consists of a core and 0 to 3 diacritics (also called accents). A core consists of 1 to 4 shafts and 0 to 3 flags. The cores are also symbols in their own right. But a core symbol with diacritics added is also considered a single symbol. This has nothing to do with how, for practical reasons, some components might be represented as separate glyphs/characters/code-points in a font. Not every possible combination of core and diacritics is a valid symbol. At this point in time there are "only" 148 valid single-shaft symbols (and their inverses) as listed in the sa_ji4.par file that comes with Scala, and in George's JI notation spreadsheet, and on the Olympian level of the JI notation diagram. Each of these symbols is defined to represent exactly one rational ratio.

But yes, if you try to determine an exact value for each component, i.e. each shaft, each flag and each diacritic—in effect solving all the simultaneous equations represented by the definitions of all the symbols—then you will indeed find that a few of these components represent slightly different ratios depending on what other components they appear with. And, as you say, the variation in size of these components is of the order of 0 to 0.2 cents.
Also, any reason why 494 was chosen as the EDO to represent Promethean?
It's hard to remember the reasons for some of these decisions. It may in fact be that Promethean chose 494 edo rather than the other way 'round—in the sense that we had certain flags used in Athenian and we tried all possible combinations of them to see which of them give us commas that get us further down the ratio popularity list, then we see that there are some large gaps that look like they need filling, eventually we end up with something relatively even and find it has 47 symbols to the apotome and is related to 494 edo.
Are all primes above 23 only "approximatable"? That is, there exist symbols to exactly represent intervals involving primes up to and including 23, but as far as I can see, symbols involving primes 29, 31, etc are approximations, using other primes' symbols inflected by accent marks, again this is part of the microtempering I was trying to explain.
Again, just because a symbol has accent marks doesn't make it in any sense approximate. Olympian happens to contain symbols to exactly represent primes 31 and 37.

31 31M (31:32)
37 37M (36:37)
And in multi-Sagittal (one-symbol-per-prime) we don't need any symbols for combinations of primes so we can redefine some symbols as commas for other primes, e.g.
29 29S (256:261)
George and I have recently defined preferred symbols for one comma per prime up to 61. I need to post about it soon.
[Edit: Done, here]
But yes, only 23 and below do not have accent marks.
Interesting point about Promethean, I was thinking the Promethean notation for the diamond would be available simply by removing all accent marks, as Herculean would be from removing only schisminas. Is this not the case?
It is not the case.
Removing mina accents from Olympian gives you Herculean, and so removing schisma accents from Herculean is equivalent to removing all accents from Olympian. And removing schisma accents from Herculeans with non-Athenian Promethean cores gives you the correct Prometheans. But removing schisma accents from Herculeans with Athenian cores doesn't always give you the correct Promethean. You can see the following exceptions on the Herculean level of the JI notation diagram. etc. where the background colour does not extend down into the Promethean level.
Actually, it would be good to include single-Sagittal notations for all ratios in the 13-smooth 45 odd limit that can be notated exactly using a Spartan symbol and up to two right accents (schismina accents). And the note would say that these accents can be dropped from the performance notation if errors of 0.4 cents per accent are not significant.
You don't mean Athenian here do you? I see no reason to limit ourselves to Spartan symbols when notating extended 13-prime JI, I find 14:13, 13:11, 33:28, 33:26, 14:11, etc, pretty core ratios that should be able to be notated how they're often used as seconds, thirds, etc. But I do see your point in catering for the 70-90%, rather than getting too technical trying to cover 99.9%.
I did mean Spartan (plus right accents). Just to show what can be done with that small readily-learnable starter set, and avoid freaking out newcomers with symbols for ratios they may never need. But I guess the real starting point for learning Sagittal for JI should be the 13-limit multi-Sagittal that uses only 4 of those 8 Spartan symbols.
Last edited by Dave Keenan on Wed May 11, 2016 10:22 pm, edited 2 times in total.