Re: accidentals for just intonation

Right then. Let me include a few files to show what I mean.

The basic principle is this: the fraction that makes up a ratio in just intonation can be broken down to its prime factors, as I said in the previous message. In just intonation, this is what matters for determining harmonic relationships. So, the accidentals are graphic representations of these primes. The basic concept of an accidental is a thin vertical line, with different parts attached to it: say, a one-quarter-sharp sign is usually a single vertical line, with two thick angled lines; the others follow similar principles (see here: http://www.rpmseattle.com/of_note/wp-content/uploads/2012/06/03_QT_Symbols.png). So I want to preserve that basic "thin vertical line with stuff attached" design principle in my own set. For brevity, let's call that thin vertical line the scaffold, and everything attached to it a "module."

Second, I want the simplest ratios to have the simplest visual design. This makes them visually uncluttered. Third, they should (roughly) resemble corresponding relationships with which we are already familiar.

So, the relationship "3" attaches a single thick horizontal line to the "scaffold", since ascending by pure fifth (remember, a perfect fifth in just intonation is a 3/2 relationship) usually results in sharps (thus a rough visual similarity to the basic idea of what a sharp sign looks like). Each additional upward fifth adds another horizontal line. It is also possible to go downward by fifth, but here's the difference relative to normal accidentals: a downward leap uses the same symbol, flipped vertically and placed at the bottom of the "scaffold." See the attached files for 3/2 and 9/8. The next prime is 5, and gets the next simplest symbol: also a thick line, but now angled. See the attached file for 5/4. You can combine both in various ways: the ratio 6/5 (a minor third) is the symbol for 3 over the (flipped) symbol for 5; the ratio 15/8 is really (5*3)/8, so both 5 and 3 appear at the top.

I'm not going to include any more files, but you get the idea. These are really low-grade scans of stuff I did by hand; obviously a path I draw in Inkscape or whatever would be much cleaner.

So, the just-intonation accidental engraver would be able to break down the ratio given for a pitch into its prime factors (or expect that they are already thus formatted), and then place the paths on the accidental in the correct positions, and then typeset the resultant accidental into music notation in place of regular accidentals.

I hope that clarifies a bit what I'm going after. Thanks again for the help.

Cheers,

On Tue, Dec 1, 2015 at 10:20 AM, Urs Liska <address@hidden> wrote:

Am 01.12.2015 um 10:10 schrieb N. Andrew Walsh:

Hi List,

this is a somewhat specialist request, and more of a long-term project, but I'm hoping you nice people can help me with something I'd like to do with Lily someday.

If you've been watching the OpenLilyLib repository, you'll see that Urs has been working on a set of tools for rendering music in just intonation. He (quite modestly) says that it isn't ready for production,

Well, it isn't ...

but there are already some impressive things it can do: for one, the interface allows to input a fraction and get back a nearest-semitone pitch with a deviation in cents *automatically*, which is something the commercial programs don't offer in any way (every composer I know who works with JI just inputs text entries manually for every note, with no change in, for example, MIDI output for ability to handle transpositions).

Yes, I also find this exciting prospects, and I don't expect the limitations there are so far to be really problematic. But basically you can now use the functionality to produce individual notes, in a monophonic context.
Probably it will be more clear once the second and third part of my latest blog post are out.

There's something I'd very much like to do with this, largely out of my own (admittedly rather opinionated) view on the best means of producing accidentals for just intonation. I'm going to assume some familiarity with just intonation concepts, but (in short) it works like this: the relationship between two pitches is defined in terms of the frequency relationship, given usually as a fraction. For example, the interval of a perfect fifth may be rendered as 3/2: that is, if I play notes with base frequencies of 200 and 300Hz, we hear them as a (very purely tuned) fifth. The equal-tempered one you have on a piano (ie, 7 semitones) is about two 1/100th of a semitone (called "cents" logically enough) too narrow to be pure (ie, a 3/2 fifth is about 702 cents).

Here's my thing: I believe that the most appropriate type of accidental for such a system is one that reflects the harmonic ratio, not the number of steps on a scale. Flats and sharps tell us whether a pitch is lowered or raised from its "natural" position in the scale, and just intonation doesn't have those positions. So, I designed accidentals that graphically reflect the harmonic ratio between a note and the tonic.

I'd like to be able to put these into Lily, and Urs tells me it can be done by calling a draw function to draw a path. I can relatively easily make up some paths with Inkscape and save them as SVGs, but is there a better way to do this? The NR describes (here: http://lilypond.org/doc/v2.19/Documentation/notation/formatting-text#graphic-notation-inside-markup) the means to include eps files into a markup, which presumably could be used to replace the accidental.

That's not exactly what I meant. I was rather thinking of the \path command on http://lilypond.org/doc/v2.19/Documentation/notation/graphic and have LilyPond *generate* the paths on-the-fly. This will make the modularity of your accidentals pretty straightforward to achieve.

But I don't have significant experience with paths (although I'd be very interested in learning), others have done more on the topic, and I think there are quite some helper tools and functionality already availabe that I can't point you to.

There are some potential complicating factors here. First, the accidentals I use change depending on the prime factorization of the ratio involved: for example, the ratio 9/8 (a type of whole tone) would comprise two of the symbol for 3 (because "9/8" is really "(3*3)/8" ), which means that Urs' interface for JI ratios would need an add-on to do prime factorization of the ratios (which is also computationally intensive, even for relatively simple numbers) or a means to encode ratios as lists of primes that are then calculated to return the value in cents (that is, do the process in reverse, starting from "(3*3)/8" and getting 9/8, which might be easier to do).

The advantage here, though, would be this: one of the interesting things about just intonation is that there is no theoretical limit to what kinds of ratios you use. You could theoretically have unique signs for all the primes you want, and then the draw function could build them on the fly. The accidentals become modular, scaling to whatever level of complexity the composer wants. Harry Partch writes music that tops out at the 13th overtone, but La Monte Young has pieces with primes in the upper 300s.

So, List: this is, as I said, a somewhat long-term project, but would any of you be willing to help me learn/do the programming necessary to develop a system like this? I also have in mind a more general add-on to the OLL just-intonation library: I'd like to see a set of different .ily files, each with different sets of accidentals, which a composer could \include into the score as needed. For example, I could write the ratios using my system, or use a system that shows accidentals approximated to the nearest 12th-tone, with cents deviation for more exact tuning (which might be of more relevance to keyed instruments).

I think this will also become clearer after the third part of the post-series.

I can send a few hand-drawn mock-ups of the accidentals to show what I mean; I've been doing them by hand, but I'd really like to see them engraved.

This would be extremely useful because I'm pretty sure noone has a clear idea what you are talking about yet.

Urs
Thanks for the help.

Cheers,

A
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From:	N. Andrew Walsh
Subject:	Re: accidentals for just intonation
Date:	Tue, 1 Dec 2015 10:47:31 +0100