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13-limit JI - Page 3 - The Sagittal forum

13-limit JI

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Dave Keenan
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Re: 13-limit JI

Postby Dave Keenan » Sun Nov 15, 2015 2:04 pm


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Re: 13-limit JI

Postby Dave Keenan » Sun Nov 15, 2015 2:36 pm

Here's the translation of all those 13-limit opposed pairs from Spartan multi-Sagittal to Olympian single-Sagittal (no approximations).

:\!::|): = :|(: 5:7-kleisma
:\!::/|\: = :|\: 55-comma
:\!::(|\: = :/ /|::'::': 5:13-diesis (fifthtone)
:!)::/|\: = :.::|)::': 77-comma
:!)::(|\: = :(|(::.::.: 7:13-diesis (fifthtone)
:\!/::(|\: = :|(::.::.: 11:13-kleisma

Only the 7:13 diesis symbol has a non-Spartan core. [Edit: Actually, the 55-comma symbol is non-Spartan too.]

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Re: 13-limit JI

Postby Dave Keenan » Sun Nov 15, 2015 3:15 pm

Cam, Do you have notation software that can place multiple accidentals against a note as multiple characters? Or am I going to have to make a special font, with any combination we need being a single character in the font?

Juhani, If you tell me which commas Johnston used to notate primes 17, 19, 23, 29, 31, I can give the corresponding Sagittals. Here's one Sagittal spelling for the prime harmonics of C from 5 to 31.

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Re: 13-limit JI

Postby Juhani » Mon Nov 16, 2015 4:42 am

Johnston accidentals exist for
80:81
24:25
35:36
32:33
64:65
50:51
95:96
45:46
144:145
30:31
and that's it. So there are 20 different Johnston accidentals (the above 10 up and down). Theoretically the system could exceed up to even higher primes but Johnston has never used them in his music so the symbols have not been not defined by him (although we know what they'd look like and could easily define them).

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Re: 13-limit JI

Postby cam.taylor » Mon Nov 16, 2015 7:59 am

OK so I went ahead and translated Johnston quartet 10, II, by hand, and it was incredibly simple, and very intuitive. Perhaps because there aren't too many unique pitches. I have used :(|\: throughout, without accent marks, because I thought the score would be a lot more intuitive without them. Very neat, the only accidentals used being :/|:, :|):, :/|\:, :(|\: along with their inverses and sharps and flats. Oh, and one case of :\ \!::#: for 25/16.

I kind of understand your point about the :(|\: symbols, but I quite like using them, without alterations.If we stick accents on them, unaccustomed players might think we need an alteration of some basic pitch, and may try to compensate. Simple symbols for simple ratios, complex symbols for perhaps more complex ratios. If you want to chuck Olympian accents on it then fine, but I quite like the way this looks without them.

Anyway here's the draft of that Johnston fugue. I'll write it out on a notation program at some point too. I'll just say that sagittal looks way easier so far.

Ah, apparently photo files are too large. Here are the first two pages.
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Re: 13-limit JI

Postby Dave Keenan » Mon Nov 16, 2015 11:46 am


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Re: 13-limit JI

Postby Dave Keenan » Mon Nov 16, 2015 1:41 pm

Thanks Juhani. For other readers:
There is an explanation of Johnston's system, up to prime 13, starting on page 6 of this article by Marc Sabat.

However, he appears to be in error when he writes, "In Johnston’s notation, septimal intervals above a note are indicated by adding a small 7 accidental. An inverted 7 simply means that the septimal interval was generated downward." He later contradicts this by writing, "The 7 sign alters a 9/5 interval (the 5-Limit minor seventh) downward by approximately a quarter-tone to produce the septimal minor seventh 7/4." This agrees with the SMuFL document linked below.

Johnston's symbols, to 13, can be seen, starting on page 113 of the SMuFL document, here

(Sagittal starts on page 119)

In other words, if you interpret Johnston's 7-comma symbol as a kind of half-arrow you will get entirely the wrong idea about the direction of its pitch alteration.

The following post explains why SMuFL does not include Johnston's symbols beyond prime 13, and what they would look like, namely small versions of the numerals themselves (presumably for the harmonic), and their 180-degree rotation (presumably for the sub-harmonic).


We can represent Ben Johnston's accidentals in plain text as:

Determining the most direct Sagittal equivalents is complicated by the fact that the chain of fifths that includes C natural is

So we do not simply want the Sagittal symbols for those commas listed above. All but the 11-comma include factors of 5 whose job is, in effect, to cancel out + and - signs. In the following, I assume that we want to choose Sagittals that preserve Johnston's choice of nominal (and hence staff position) plus sharp or flat, for each prime harmonic of C, although an alternative is just to use whatever is most natural in Sagittal.

So how would those prime harmonics be notated in Johnston's notation? Using the above commas I get the following. Can you confirm this Juhani? (or anyone else) I'm not confident of those above 13. In particular, the requirement for a + sign in notating the 23rd harmonic seems strange.

The direct Sagittal equivalents are:

That seems like a perfectly reasonable prime set to me. Compared to the set I gave earlier, it favours smaller commas at the expense of nominals-plus-sharps-or-flats that are slightly more distant from C on the chain of fifths. It differs from it only for primes 13, 17 and 31. A minor consideration is that it uses :/|): for 13 instead of :(!/: and this is more prone to confusion with the 7-limit combination :/|::|):. But then such a misreading wouldn't matter very much since the two differ by less than half a cent.

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Re: 13-limit JI

Postby Juhani » Tue Nov 17, 2015 2:25 am

"Can you confirm this Juhani? (or anyone else) I'm not confident of those above 13. In particular, the requirement for a + sign in notating the 23rd harmonic seems strange."
Yes, those are correct. I found 23 being an inflection of F#+ strange, too, when I learned it, as I'd assumed the idea was to notate the overtones of C without any + and - signs. I intended to ask about this from Bob Gilmore, who edited the collection of Johnston's writings that includes Johnston's article on his notation, but, sadly, I never got to that. But I believe it's because the "comma" from F# (25/18) would be a large one (diesis), 60 cents, and he found the 45/32 a less complex and more common pitch for the starting point for tuning, when attempting to find the 23/16. Johnston always has tuning by ear in his mind.

"I assume that we want to choose Sagittals that preserve Johnston's choice of nominal (and hence staff position) plus sharp or flat, for each prime harmonic of C, although an alternative is just to use whatever is most natural in Sagittal."
Well, Cam already notated 13/8 as an alternation of 27/16 (A+ in Johnston when C=1/1) rather than 8/5 (Ab); Johnston writes that "the resulting interval is much closer to a minor sixth than a major sixth---" and apparently finds the 27 cent comma up more intuitive than 27:26 or 40:39 down, but that large 13-chroma will surely be learned easily enough, it's still on the same staff position. 13 is already a high prime-limit, and I personally don't worry at all if the symbols for anything above 23, or even 13, are not the simplest and most intuitive. Those intervals are rarer in JI because they're unlikely to be tuned by ear. So in their case the notation is more for the composer than the performer, I'd say. Johnston only used them in his piano tunings, and when he has them for strings in his 9th Quartet, he's notating, rather than high-limit harmonies, a chromatic scale in the fifth octave of the harmonic series. But it's important that the symbols for 5, 7, 11, and 13 are clear and easy and unequivocal ("You then simply define :(|\: as 26:27 (or whatever) for that particular score, so there is no approximation" - yes!).

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Re: 13-limit JI

Postby cam.taylor » Mon Nov 23, 2015 8:01 am

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Re: 13-limit JI

Postby Dave Keenan » Mon Nov 23, 2015 10:58 am



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