13-limit JI

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Dave Keenan
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Re: 13-limit JI

Postby Dave Keenan » Mon Nov 23, 2015 5:36 pm

This thesis by Jeff Snyder in 2010 is interesting on the subject of Ben Johnston's notation. See section 1.2.3, pages 19 through 25.
http://www.scattershot.org/full-dissertation.pdf
It confirms that the accidentals "7" and "19" are downward pitch alterations, while the other prime-numeral accidentals are upward.

This quote from a violinist was interesting.
The demands of recording Ben Johnston's string quartets are astounding. We've been doing this for six years, and we still find it to be overwhelming... In essence, Ben's stuff is unperformable. He wrote for instruments and humans that don't exist yet ... a hybrid of acoustic strings with real-time electronic/digital feedback of some sort and people with brains the size of HumVees... In rehearsal, we use microtuners and contact mics to let us know where on the dial we're playing -- X number of cents above or below tempered 'in tune'. (This is after we have laboriously translated Ben's notation into tempered-speak values -- his scores and parts as offered by his publisher are only half ready to be of any use because they're not written in a language anybody understands.) But in performance, darting the eye down to the tuner and back up to the page isn't going to work.

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Re: 13-limit JI

Postby Juhani » Sun Nov 29, 2015 9:13 am

The direction of 7 and 19 didn't need confirming; I've used Johnston's notation both in sketching on paper and in Finale, with playback, for 15 years. It's interesting that the Kepler violist says they calculated all the cent values and used meters in the recording but I'm sure that they do tune by ear the JI intervals they read and understand from the Johnston scores in cases when the ratios are relatively simple, such as in the 10th quartet that Cam transcribed. However, some of the quartets are among the most complex music written in the 20th Century; there are more than 1200 different pitch classes in the 7th Quartet, so surely a computer is needed for performing that, as is the case for the performance of, say, Carter's 3rd string quartet (for its polyrhytmic complexity) as well as the Johnston-notated choral masterpiece Chrysalid Requiem by Toby Twining. But these would look extremely complex in any notation system.

I do agree with the author of the dissertation that the 5-limit nominals are often confusing, especially to string players. In a lot of JI repertoire, on the other hand, it's quite natural to assume that thirds are pure so that an uninflected C major chord on the stave refers to a 4:5:6. It also seems a natural assumption for players of early music and baroque - string players included. The other advantage in 5-limit nominals is that the most common accidentals refer to ratios that are much simpler than Pythagorean ones and easier to use in calculations.
Still, with the confusion that I sometimes see the pluses and minuses of Johnston scores creating in musicians, I intend to start using a notation system with 3-limit nominals (Sagittal or Sabat) in my own music in JI.

Thanks for the interesting link; I must now hear some music by the author.

Here's a 1991 article on Ben Johnston which is still one of the best introductions to his notation and some of his composition techniques.

http://sacredrealism.org/catlamb/tuning ... reters.pdf

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Re: 13-limit JI

Postby Dave Keenan » Sun Nov 29, 2015 4:26 pm

Juhani wrote:The direction of 7 and 19 didn't need confirming; ...

I'm afraid they did for me. Particularly when Sabat and von Schweinitz use exactly the same symbol to lower by a 7-comma, that Johnston uses to raise by a 7-comma. And given that it looks like a half-arrow, I have to say I'm with S & vS on this one, except that since Johnston was there first, they really should have chosen something else, to avoid confusion. Our mnemonic for the Sagittal 7-comma symbol -- the right arc :|): -- is that it is a rainbow which, according to tradition, has 7 colours.

However, some of the quartets are among the most complex music written in the 20th Century; there are more than 1200 different pitch classes in the 7th Quartet, so surely a computer is needed for performing that, as is the case for the performance of, say, Carter's 3rd string quartet (for its polyrhytmic complexity) as well as the Johnston-notated choral masterpiece Chrysalid Requiem by Toby Twining. But these would look extremely complex in any notation system.

I strongly suspect that these pieces could be re-notated using no more than 217 different pitch classes, and if performed by humans, with instructions to tune by ear, no listener could tell which was the original, in a blind test.

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Re: 13-limit JI

Postby Dave Keenan » Thu Dec 03, 2015 12:59 am

Here's something I just created to help folks remember the basic 13-limit Sagittal symbols. It shows a 4:5:6:7:9:11:13 chord. It's root is G because that's the only root that conveniently avoids all sharps and flats for a complete 13-limit otonal heptad.

You can see that the 13-diesis symbol is made from a straight flag and a curved flag, just as the numeral "13" is made from a straight digit and a curved digit, while the 11-diesis symbol has two straight flags just as the numeral "11" has two straight digits. The 7-comma symbol is the 7-colour rainbow. The 5-comma symbol is by far the most common symbol, being for the lowest prime that needs a symbol, and so it is visually the simplest.

G45679113.png

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Re: 13-limit JI

Postby Xen-Gedankenwelt » Thu Dec 03, 2015 7:36 am

Thanks, much appreciated! :)

Viewing :(!/: as a 13 rotated by 180° helps me.

I used ^ and v for quartertones 33:32 by my own before seeing this notation elsewhere, because it's an obvious character pair, and the German word for quartertone is 'Viertelton', so for me it was an obvious choice.

:/|: and :\!: make perfect sense for me: The first important books about microtonality that I read were by Martin Vogel, who used / and \ for syntonic comma 81:80 alterations. Like Hermann von Helmholtz (who was in turn inspired by others), he also used overlines and underlines, and / and \ are convenient alternatives that retain the line aspect, while pointing in the direction of the alteration if you read them from left to right - which implies that the stem should be to the right, s.th. the arrow direction makes sense.

:!): is intuitive for me, because it reminds me of the 63/64 accidental used by Martin Vogel, which looks like :!/:, and similar to a 7 rotated by 180° (according to him, it was used "since Tartini's time"). However, it is logical to use :!/: for 54/55 instead, s.th. :\!: + :!/: = :\!/:, so using a slightly different accidental for 63/64 is perfectly understandable, and the rainbow comparison helps me even more.

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Re: 13-limit JI

Postby Juhani » Thu Dec 03, 2015 7:40 am

"I'm afraid they did for me." [need confirming] Right, I meant that I'd already confirmed them (in this thread, Mon Nov 16, 2015 6:25 pm)
"Sabat and von Schweinitz use exactly the same symbol to lower by a 7-comma, that Johnston uses to raise by a 7-comma"
Not exactly, neither of the versions (raise / flatten) is a 7 or 7 upside down, they're mirror images of those.
I agree that Johnston's exceptions to the direction may be confusing or annoying but on the other hand, his notation for the harmonic seventh is a seven, and musicians generally think of that interval as a narrow (flattened) version of a minor seventh. You say to a musician singing a minor seventh, please make that a natural seventh, so he flattens it; a seven marked in the score would seem a, well, natural, way of notating that.

"I strongly suspect that these pieces could be re-notated using no more than 217 different pitch classes, and if performed by humans, with instructions to tune by ear, no listener could tell which was the original, in a blind test." Of course. Again, no-one expects anyone to play or hear such distinctions. Much less than 217 pitch classes may very well do the job. But Johnston wants to show the modulations on the lattice, and the tuning chains. The violist was saying that they don't play like that, that they rather learn individual pitches and finger positions using meters and other electronic gadgets. Even so, the original notation is needed to show what the actual ratios between the sounding pitches are. These are mostly simple in Johnston's music, including these hyper-complex works, and the ideal, in some extreme cases perhaps unattainable, is that they tune to each other's pitches.

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Re: 13-limit JI

Postby Dave Keenan » Thu Dec 03, 2015 9:49 am

Juhani wrote:Right, I meant that I'd already confirmed them (in this thread, Mon Nov 16, 2015 6:25 pm)

Sorry, Juhani. So you did.

"Sabat and von Schweinitz use exactly the same symbol to lower by a 7-comma, that Johnston uses to raise by a 7-comma"
Not exactly, neither of the versions (raise / flatten) is a 7 or 7 upside down, they're mirror images of those.

I think you may be misremembering the S & vS 7-comma symbols. See them here http://www.marcsabat.com/pdfs/notation.pdf. In both extended Helmholtz-Ellis notations (Sagittal and S & vS), the direction of alteration is changed by flipping or mirroring the 7-comma symbol vertically, and so they have their flag always on the right of their shaft. This is in contrast to Johnston's notation where the direction of alteration is changed by rotating the symbol 180 degrees. So both Johnston's upward symbol and S & vS's downward symbol resemble an "L".

I agree that Johnston's exceptions to the direction may be confusing or annoying but on the other hand, his notation for the harmonic seventh is a seven, and musicians generally think of that interval as a narrow (flattened) version of a minor seventh. You say to a musician singing a minor seventh, please make that a natural seventh, so he flattens it; a seven marked in the score would seem a, well, natural, way of notating that.

Yes, quite natural. But unfortunately what is natural to some is unnatural to others. It depends on what they have already learned. http://www.asktog.com/papers/raskinintuit.html. Johnston apparently decided to rotate the symbol slightly so the top line of the "7" does not get lost against the staff lines (this a problem for the "+" and "-" symbols too) and it then becomes natural for some to read it as a (highly misleading) half-arrow.

But Johnston wants to show the modulations on the lattice, and the tuning chains. The violist was saying that they don't play like that, that they rather learn individual pitches and finger positions using meters and other electronic gadgets. Even so, the original notation is needed to show what the actual ratios between the sounding pitches are. These are mostly simple in Johnston's music, including these hyper-complex works, and the ideal, in some extreme cases perhaps unattainable, is that they tune to each other's pitches.

Agreed. But to me, this argues for multiple notations of the same piece, for different purposes. A one-symbol-per-prime notation for composition and analysis and for some mathematically inclined performers and instruments. A one-or-two-symbol-per-note notation with pythagorean nominals for other performers and instruments, and a one-or-two-symbol-per-note notation with 12-edo nominals (and possibly cent offsets) for yet other performers and instruments. Sagittal has all three of these options covered within a single consistent system.

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Re: 13-limit JI

Postby Dave Keenan » Thu Dec 03, 2015 10:34 am

Thanks Xen. You have well described the evolution of the Sagittal notation from the pre-existing comma slashes and quartertone arrows.

Here's something else about Sagittal. At least when notating JI, and temperaments whose nominals are in a chain of near-just fifths, you can determine the relative sizes of the alterations merely by considering the area enclosed by the flags and the shaft and a horizontal line joining them, as filled in yellow below. i.e. the area of yellow plus the black parts of the symbol that enclose it.

G45679113yellow.png

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Re: 13-limit JI

Postby Juhani » Fri Dec 04, 2015 3:11 am

"both Johnston's upward symbol and S & vS's downward symbol resemble an "L"." Right, so they do, sorry. Johnston's inverted numbers are 'upside down' mirrors that you get when you turn your music paper upside down.

It might be that some sort of computerized, electronic score (something like Joe Monzo's Tonescape) might be the best format for such complex works as Johnston's 7th Qt or Twining's Requiem.

"Johnston apparently decided to rotate the symbol slightly so the top line of the "7" does not get lost against the staff lines (this a problem for the "+" and "-" symbols too) and it then becomes natural for some to read it as a (highly misleading) half-arrow."
Looking at his scores, the 7-accidental is (hand-)written in numerous ways, sometimes resembling a half-arrow, sometimes a seven, sometimes it's a straight line. The + and - getting lost against the staff lines is often mentioned as a serious problem - but it's only a problem in computer notation, and should be avoidable there, too; it's never a problem in hand-written music, and most of Johnston's published works are indeed hand-written. The minus symbol should never be written on the stave. A combination of three minuses does get awkward, though (so does a combination of three half-arrows in multi-Sagittal). Here my preference is Sabat-v.S.
As to the misleading half-arrow and the intuitiveness: yes, you're right, but who's used to half-arrows? Can't think of any published scores of any well-known (in contemporary music circles) music that uses them.

Anyway, back to Sagittal. High-quality computer notation in multi-Sagittal is difficult without a font that has a character for each compound symbol. Perhaps the best solution would be a font that has the most common symbol combinations; extra symbols would be inserted manually when needed in the case of more complex ratios. This is possible in Finale, if somewhat awkward. I suggest we go through some important and representative Johnston scores with Cam so we get an idea what compound symbols are in active use in this repertoire.

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Re: 13-limit JI

Postby Dave Keenan » Fri Dec 04, 2015 11:43 am

Juhani wrote:It might be that some sort of computerized, electronic score (something like Joe Monzo's Tonescape) might be the best format for such complex works as Johnston's 7th Qt or Twining's Requiem.

That still seems to assume that there is a single best format for all readers, all purposes and all instruments. There is no reason why modern music notation software could not allow us to enter the music once, then allow us to automatically generate versions of it in various notations. But sadly, the designers of most notation software appear to have little understanding of microtonal notation requirements. Is there any notation software available now that allows multiple accidental characters against a single note? I have great hopes for Steinberg's new product when it arrives -- at least they had the foresight to consult George and I regarding microtonal accidentals in general, not merely Sagittal. We convinced them to include many historical accidentals that have nothing to do with Sagittal, and we ensured many others were correctly described and attributed, including Johnston's and Sabat and von Schweinitz'.

Looking at his scores, the 7-accidental is (hand-)written in numerous ways, sometimes resembling a half-arrow, sometimes a seven, sometimes it's a straight line.

Thanks for explaining that. It appears I have been misled by the glyphs for the Johnston symbols shown in the SMuFL documentation and Bravura font. See page 113 of http://www.smufl.org/files/smufl-1.18.pdf.

The + and - getting lost against the staff lines is often mentioned as a serious problem - but it's only a problem in computer notation, and should be avoidable there, too;

Yes. If the Sagittal 5-comma symbols are unacceptable for some reason, we recommend Erv Wilson's plus and minus 5-comma symbols which can be seen as U+E47B and U+E47C on page 143 of the above SMuFL document. Their otherwise horizontal strokes are thickened and slanted in the manner of Bosanquet's 5-comma slashes, which is the same manner as the flags on the Sagittal 5-comma symbols. In fact one can visualise a simple continuous geometrical transformation that would morph Johnston's 5-comma symbols via Wilson's to the Sagittals.

The minus symbol should never be written on the stave.

Where should it be written?

A combination of three minuses does get awkward, though (so does a combination of three half-arrows in multi-Sagittal). Here my preference is Sabat-v.S.

We would never recommend combining symbols vertically in the manner of Gould's quartertones or S & vS's 5-commas. It moves the visual center of the symbol, multiple staff positions away from its intended target, which could be quite confusing when notating chords. And using a full (left and right barbed) arrowhead for something as small as a 5-comma just seems wrong IMO. At least Sagittal and Johnston agree on the appropriate uses of the full arrow.

As to the misleading half-arrow and the intuitiveness: yes, you're right, but who's used to half-arrows? Can't think of any published scores of any well-known (in contemporary music circles) music that uses them.

That's a very small musical category. :) Even if it's not perceived as a half-arrow, :!/: might naturally be perceived as a modified flat symbol and therefore assumed to indicate a flattening by some fraction of a semitone. The naturalness of such an interpretation led to such a symbol being used with this meaning in the 18th century by van Blankenburg and Tartini, and in the 20th century by Fokker and the International Musicological Society.

But independent of this, one reason we don't recommend using the prime numbers themselves as accidentals is because we want a notation system that is not restricted to JI. For example, although the fundamental meaning of the :!): symbol in the Sagittal system is the 7-comma down, it needs to be possible for users to think of it as being associated with numbers other than 7. For example, it is 1/6th-tone in the 12-edo-relative notation, 1 degree of 21, 36, 43 and 51 edo, 2 degrees of 72 edo, and the chroma for the decimal notation of Miracle temperament.

I suggest we go through some important and representative Johnston scores with Cam so we get an idea what compound symbols are in active use in this repertoire.

Yes. That's an excellent idea. I'd be happy to create the necessary font, provided the number of combinations doesn't run into the hundreds. :)


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