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Re: Lilypond's internal pitch representation and microtonal notation

 From: Hans Aberg Subject: Re: Lilypond's internal pitch representation and microtonal notation Date: Mon, 20 Sep 2010 19:11:13 +0200

```On 20 Sep 2010, at 18:08, Joseph Wakeling wrote:

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```One scan should be fine.  The first step is to convince people that
the representation needs to be extended, and Stone should be
sufficient for that.  The next step is for somebody actually code it.
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Sure. I'll try and follow up with Hans separately, just to explore his
```code and ideas and see if they might be useful here.
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The algebraic approach I described solves your problem - that is one reason for doing it. The staff system does not in itself impose enharmonic equivalences. One can add them afterwards, with a method that gives better control.
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In your example, you start off with a minor second m and a major second M, and adds a neutral second n representing the quarter-tone. The full set of abstract pitches is p m + r M + q n, where p, q, r are integers.
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In E24, m = 2, M = 4, n = 3. But the problem is that the staff system does not as such take any enharmonic equivalences into account. One can see this by computing degrees:
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```For example, if we are using the very common 'arrow' notation for
quarter-tones, there are two distinct accidentals that can be used to
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represent the alteration +1/4 (i.e. quarter-tone-sharp): the first is a
```natural sign with an up arrow, the second is a sharp sign with a down
arrow.  There is currently no effective, well-defined way to indicate
which of the two is desired at any given moment.
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For two pitches or intervals x, y, write the difference y - x more intuitively as x->y. Then a sharp is m->M and a flat M->m. In addition, there are two quarter-tone accidentals m->n, the up arrow, and M->n, the down arrow.
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There also another pair of accidentals that would be needed if you use up/down arrows meaning a small amount, not necessarily an exact quarter-tone: n->m and n->M. One then also needs arrows indicating a large amount but not fully a flat or sharp, in total four microtonal accidental symbols. Four microtonal accidentals are used in Turkish music, though not Arab and Persian music, which has to do with how intervals are divided.
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Now compute degrees, to see that applying an E24 enharmonic equivalence changes the degree, and therefore also the position of the note in the staff system:
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Accidentals are of degree zero, so adding them to a note does not change the degree. So a note x with m->n added has the same degree as x. If you want to write the E24 enharmonic equivalent, a semitone above x lowered a quarter-tone, then this is x + m + M->n, and its degree is one above that of x, as the degree of m is one, and deg(x + y) = deg x + deg y.
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So these notes are different. If one wants enharmonic equivalencies, then those should be applied afterwards.
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