## 13-limit JI

Dave Keenan
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### Re: 13-limit JI

Juhani wrote:A JI cello piece of mine, originally written in Johnston's notation is going to be published and I was thinking of including additional versions in both Sabat's system (popular in Germany) and Mixed Sagittal but for the latter, I have to see if I can notate even the fairly simple ratios that are used in it, and how it looks like.

If you don't mind things like (which you can't avoid in Johnston or Sabat and von Schweinitz), this is all you need for 13-limit JI.
```.
5-comma  (80:81)
7-comma  (63:64)
11-diesis (32:33)
13-diesis (26:27)```
Same number of symbols as Johnston or Sabat and von Schweinitz, using the same commas (except for the 5-limit complication of Johnston).

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

### Re: 13-limit JI

So two steps on the 7-axis would be notated as ? That is, would the accidentals in this system be combined into potentially long horizontal strings of symbols, as in Johnston?
What's the recommended accidental for 49 in mixed Sagittal?

Thanks for your offer to help with the cello piece. I'll look into it; it's not complex.

Juhani

Dave Keenan
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### Re: 13-limit JI

Juhani wrote:So two steps on the 7-axis would be notated as ? That is, would the accidentals in this system be combined into potentially long horizontal strings of symbols, as in Johnston?
Yes, although I would tend not to put a space between them, so in one-symbol-per-prime Sagittal (perhaps we should call it "multi-Sagittal") 49 would be .

Wouldn't they be combined into potentially long horizontal strings of symbols in Sabat and von Schweinitz notation too?
What's the recommended accidental for 49 in mixed Sagittal?
It is shown in this diagram http://sagittal.org/SagittalJI.gif.
=
It's unfortunate that it's not as straightforward as
=
but we tried various designs for a more literal symbol and they just didn't work, for various reasons. And the diesis for the combination differs only by 0.13 of a cent, and has an easily-remembered apotome-complement .

We also have a symbol for 48:49. It is .

Dave Keenan
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### Re: 13-limit JI

Given

5-comma (80:81)
7-comma (63:64)
11-medium-diesis (32:33)
13-large-diesis (26:27)

Here are single symbols for all the opposed pairs of those symbols, in the Promethean-level JI notation. That's the highest precision-level that does not use diacritics (accent marks). Approximations are shown with a tilde "~". The worst approximation is within 0.8 of a cent.

= 5:7-kleisma
= 55-comma
= ~5:13-small-diesis
= ~77-comma
= ~7:13-small-diesis
= ~11:13-kleisma

Again, the best reference is http://sagittal.org/SagittalJI.gif.

A composer, or a reader who wants to analyse the score in terms of the prime lattice, may prefer the multi-Sagittal form. A performer may prefer the single-Sagittal form, depending on the instrument.

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

### Re: 13-limit JI

Juhani, I'd be keen to try and notate some of the "JI classics", like some extracts of Johnston's quartets, in Sagittal. I learned the Johnston system after I first stepped into JI, and it would indeed prove an interesting comparison. I'd like to think Sagittal scores would come off looking cleaner and more logical for the most part. One even has some choices in Sgittal where there aren't any in Johnston - one can notate borderline intervals fairly easily in multiple nominals depending on melodic and harmonic context, just like Dave has mentioned 49 here - either as an alteration of a fifth, using the 49:48 accidental, or as a minor sixth lowered by two septimal commas (or one ), depending on whether one wants some kind of "fifth" or "sixth", etc.

Have you got a score available of anything you'd like notated? I'd agree that any of the Johnston quartets could be potentially very useful.
I absolutely love the fugue ("Solemn") in his 10th quartet if you happen to have a copy of any of that?

Dave Keenan
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### Re: 13-limit JI

That would be great, Cam. But I'm thinking it might be considered sacrilegious to use any kind of approximations. Perhaps we should use Olympian notation. And it would also be good to see a multi-Sagittal version for comparison.

Incidentally, I notice Sabat & von Schweinitz notation has symbols for a triple syntonic comma. In Olympian Sagittal this is:
= 125-large-diesis.
I remind you that Olympian for the 13-large-diesis (26:27) is .
The grave and acute right accents indicate that these are respectively 0.4 cents below and 0.4 cents above the 35-large-diesis = .

And there are symbols for two other 125 dieses, for other spellings. The apotome complement of the one given above is
= 125-medium-diesis.
The third one uses a left accent that indicates a 5-schisma (about 2 cents).
125-small-diesis
So the note three just major thirds up from C can be spelled in these 3 ways:
B
B
C

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

### Re: 13-limit JI

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.

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.).

"A composer, or a reader who wants to analyse the score in terms of the prime lattice, may prefer the multi-Sagittal form. A performer may prefer the single-Sagittal form, depending on the instrument." It's difficult to say if there are singers or string players who'd prefer the former over the latter, as there is no JI repertoire published in Sagittal, except in Sagittal study materials and demos (as far as I know). 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. 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. That's the philosophy behind these JI notation systems: it's quite different from that of Sagittal, as I understand. Even if the accidental combinations get complicated, they're still preferable to numbers over or under the notes; all those off-stave markings are slow and difficult to read, and if the numbers are ratios, they require calculating, and if they're factorizations, they take up a lot of space. It's true that some performers (and composers) prefer 72-equal (Sims et al., the Austrian 72-toners) or 1200-equal (Johnny Reinhard, who notates in 24-equal with cent deviations above the notes) even for JI music. 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. 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.
Last edited by Juhani on Sun Nov 15, 2015 9:08 am, edited 3 times in total.

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

### Re: 13-limit JI

Cam, "I'd agree that any of the Johnston quartets could be potentially very useful.
I absolutely love the fugue ("Solemn") in his 10th quartet if you happen to have a copy of any of that?"

Great! Good choice. I'll scan it for you. We might also try something very complex (a passage from the 5th quartet perhaps).

J

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

### Re: 13-limit JI

Great! Good choice. I'll scan it for you. We might also try something very complex (a passage from the 5th quartet perhaps).
That would be amazing. I've been looking forward even just to see a score for those two pieces, they're probably my favourite by Johnston. Really beautiful, intricate harmonies, well thought-out, but I haven't given them a serious listen for a couple of years now, and since then I'm sure I've learned a whole lot about JI and more that would be helpful in all this.

If you could somehow get me the score for some of that stuff I'd be more than happy to undertake a Sagittalising of it all!

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

### Re: 13-limit JI

Cam, as you said above, you should use the precise notation with the accents, to be faithful to the composer who does not accept any edo-approximations in notating his music, so he wouldn't approve rational approximations either, no matter how accurate. Again - it's not that he or anyone wants that precision in the tuning of the notes, but rather that the notation shows how the intervals are to be arrived at, by ear. So which ratio is which has to be unambiguous.

Those quartets are my favorites, too, along with #4 and #9. # 4 is extremely complex rhythmically but not so much in the pitch domain. Maybe a passage from it would be nice to see. I think the first pages of it would be absolutely devoid of accidentals in Sagittal, just white notes, because it starts in Pythagorean, then moves on to 5- and 7-limit. #9 would be interesting, on the other hand, in its unique use of higher primes, mostly in hyperchromatic scales, up to 31.

Will send you some scans.