## 13-limit JI

Dave Keenan
Posts: 239
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: 13-limit JI

Juhani wrote:I can't help thinking that there's an overwhelming number of difficult-to-decipher, difficult-to-write-in-shorthand symbols in the Promethean level and that there's an overwhelming number of symbols to memorize. I'm very uncomfortable with the multitude of commas that one is required to learn (in the single-symbol version). I doubt many Just Intonation composers and performers even consider those commas or think in terms of them; these commas have to do with the logic of the Sagittal system, and with JI and temperament theory.
Yes. Forget Promethean. If we use a single-Sagittal notation at all, it should be Olympian, which consists mostly of Spartan symbols with accents. I would not expect performers to think in terms of the comma that is exactly represented by any accented symbol. I'd expect them to ignore the accents and think in terms of the approximate pitch alteration implied by the core symbol, then adjust by ear if necessary.
I fear the multi-Sagittal accidentals would quickly become awkward. Johnston's notation does have the problem of long horizontal clusters of symbols but it uses also symbols as attachments ( would be used as the stem of instead of having them side by side etc.), so this only applies to multiples of the same symbol ( +++ for 81^3, 77 for 49 etc.).
I think we try it and see. I think we should begin with an almost symbol-for-symbol transliteration of Johnston's notation, with the exception of using the Pythagorean nominals instead of the JI major. There are many intermediate stages available in Sagittal between one-symbol-per-prime and one-symbol per note.

Of course Sagittal can combine comma symbols with sharps and flats to make a single symbol too, but you have to be willing to use the Sagittal form of the sharp and flat, and then you're into the pure Sagittal. We've never liked symbols like Gould's quartertones or Sabat & von Schweinitz' 5-commas because they make the accidentals so tall that you can't place them above one another in chords, and because they put the arrowheads a long way above or below the notehead they apply to, possibly beside another notehead entirely. I really don't think the horizontal stacks of Sagittals will end up much wider that those of the Johnstons, on average.
It's difficult to say if there are singers or string players who'd prefer the former [multi-Sagittal] over the latter [single-Sagittal], as there is no JI repertoire published in Sagittal, except in Sagittal study materials and demos (as far as I know).
All the more reason to try both forms of Sagittal (and possibly some in-between).
It is a fact, on the other hand, that the scores of Johnston, Sabat etc. are written for musicians who tune the intervals by ear or are at least expected to do so, note against note, which is why the notation tells the tuning path exactly. That's why in Johnston, for example, a different symbol is used for 33:32 and 36:35 even if they're both quarter-tones and differ only about 4 cents.
I assume you're aware these are different in Sagittal too. versus or .
They're to all practical purposes the same interval. Sometimes they're even combined to the same accidental so that they cancel each other out (33:32 up and 36:35 down). It would be just fine to sing the uninflected note, but the singer wouldn't know how to find it if her note is the 11/8 of the 7th harmonic that's been sounding in the bass, say.
Yes. That's a good point. So if that bass note was notated C or C it would make sense to notate the upper note as F or F. The Olympian single-symbol alternative would be F.
That's the philosophy behind these JI notation systems: it's quite different from that of Sagittal, as I understand.
I believe Sagittal can serve that JI notation philosophy just as well as Johnston and Sabat & von Schweinitz notations can. We always intended it could be used that way (as per the bold entries in Table 1). We just didn't spend any time on it in the paper, thinking it was obvious how to use it in that manner, and preferring to show the options that aren't available with other JI notations.
But a cellist friend of mine played a solo piece by Sims, in extended JI notated as 72-equal, and he found it quite annoying that the notation didn't tell which just interval each note represents. Originally, he tried finding the 72 equal steps and only later realized that the music is supposed to be played in pure intervals, but it took a lot of trial, error and analyzing. He'd prefer a Johnston-notated version of the Sims piece.
Good point. Sagittal can certainly be used, with multiple symbols per note, to show the JI structure of a chord or arpeggio.
The situation is somewhat similar here, the main difference being that in JI Sagittal (without the accents), the notation gives rational approximations for ratios, rather than the irrational ones of equal divisions.
Right. But that's only one option for how to use Sagittal accidentals for notating JI. It was actually me, not Cam, who suggested we should not use any approximations. If we're doing multi-Sagittal, we define to be exactly the 26:27 thirdtone and we use or (depending on context) for the almost-identical 7-limit thirdtone.

Dave Keenan
Posts: 239
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: 13-limit JI

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.]

Dave Keenan
Posts: 239
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: 13-limit JI

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.
```.
E    5 (80:81)
B   7 (63:64)
F  11 (32:33)
A  13 (26:27)
D  17 (4096:4131)
E  19 (512:513)
F  23 (729:736)
B  29 (256:261)
C  31 (31:32)```

Juhani
Posts: 31
Joined: Sun Sep 06, 2015 12:14 am

### Re: 13-limit JI

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).

cam.taylor
Posts: 51
Joined: Thu Sep 03, 2015 11:55 am

### Re: 13-limit JI

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.
Attachments
p2
12255544_1144442618918124_128034945_o.jpg (144.6 KiB) Viewed 1318 times
p1
12250538_1144442522251467_1393488332_o.jpg (137.81 KiB) Viewed 1319 times
Last edited by cam.taylor on Thu May 26, 2016 10:07 am, edited 3 times in total.

Dave Keenan
Posts: 239
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: 13-limit JI

cam.taylor wrote: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.
Awesome! I'm dying to be able to compare the two. I have increased the attachment size limit for the forum from 256 KiB to 1 MiB. Let me know if you need more, or alternatively you might try scaling the image down to 75% or 50% assuming it remains legible.

You used throughout, as what? 13-thirdtone, 35-thirdtone or 125-thirdtone?
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.
We encourage the dropping of accents when it does not lead to ambiguity. You then simply define as 26:27 (or whatever) for that particular score, so there is no approximation, and you do not use it for any other comma.

Dave Keenan
Posts: 239
Joined: Tue Sep 01, 2015 2:59 pm
Location: Brisbane, Queensland, Australia
Contact:

### Re: 13-limit JI

There is an explanation of Johnston's system, up to prime 13, starting on page 6 of this article by Marc Sabat.
http://www.marcsabat.com/pdfs/EJItext.pdf
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
http://www.smufl.org/files/smufl-1.18.pdf
(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).
http://smufl-discuss.50501.x6.nabble.co ... td772.html

We can represent Ben Johnston's accidentals in plain text as:
```80:81   +  -
24:25   #  b
35:36   L  7 (note that "7" is the downward alteration)
32:33   ^  v
64:65   13 e1
50:51   17 L1
95:96   61 19 (I'm guessing "19" is the downward alteration)
45:46   23 eZ
144:145 29 6Z
30:31   31 1e```
Determining the most direct Sagittal equivalents is complicated by the fact that the chain of fifths that includes C natural is
```Fb   Cb   Gb   Db   Ab  Eb  Bb  F  C  G  D  A  E  B  F#   C#   G#   D#   A#    E#    B#    in Sagittal, and
Fb-- Cb-- Gb-- Db-- Ab- Eb- Bb- F  C  G  D  A+ E+ B+ F#++ C#++ G#++ D#++ A#+++ E#+++ B#+++ in Johnston.```
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.
```.
E     5
7bB   7
^F    11
13bA  13
17#C  17
19bE  19
23+#F 23
29bB  29
31B   31```
The direct Sagittal equivalents are:
```.
E    5 (80:81)
B   7 (63:64)
F   11 (32:33)
A  13 (1024:1053)
C  17 (2176:2187)
E  19 (512:513)
F  23 (729:736)
B  29 (256:261)
B   31 (243:248)```
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.

Juhani
Posts: 31
Joined: Sun Sep 06, 2015 12:14 am

### Re: 13-limit JI

"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!).

cam.taylor
Posts: 51
Joined: Thu Sep 03, 2015 11:55 am

### Re: 13-limit JI

Well, Cam already notated 13/8 as an alternation of 27/16 (A+ in Johnston when C=1/1) rather than 8/5 (Ab)
I actually thought to notate the 13:8s as a minor sixth raised by , as "mutually intelligible" with Johnston, but the Sagittal convention as I understand it is to have the least number of accidentals, so if you can write an interval using only one simple accidental instead of two simple accidentals, it should be done. They are equivalent though right? is equal to , right? That was how I understood it, using these symbols in their 13-prime meaning. 13:8 from E would be C while 13:8 from C would be A . Let me know if my understanding was wrong

**Edit: here are the last two pages
Attachments
p3
12236582_1144442718918114_1553768152_o.jpg (138.59 KiB) Viewed 1317 times
p4
12255390_1144442828918103_513910689_o.jpg (138.99 KiB) Viewed 1317 times
Last edited by cam.taylor on Thu May 26, 2016 10:08 am, edited 1 time in total.

Dave Keenan