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Re: Microtonal Helmholtz-Ellis notation in Lilypond: fine-tuning

From: Hans Aberg
Subject: Re: Microtonal Helmholtz-Ellis notation in Lilypond: fine-tuning
Date: Thu, 10 Sep 2009 11:00:07 +0200

On 10 Sep 2009, at 10:26, Torsten Anders wrote:
[Your mail does not cc to the list - added: seems relevant.]

t is part of the font
distribution itself at

I also found

It seems to be that that staff indicates the Pythagorean tuning, with accidentals to indicate offsets relative that. Right?

Exactly: nominals (c, d, e...) and the "common accientals" (natural, #, b, x, bb) denote a spiral of Pythagorean fifths. Other accidentals detune this Pythagorean by commas etc. Multiple comma- accidentals can be freely combined for notating arbitrary just intonation pitches. The Sagittal notation ( ) follows exactly the same idea.

Yes, I thought so.

This is in contrast, for example, to the older just intonation notation by Ben Johnston (see David B. Doty (2002). The Just Intonation Primer. Just Intonation Network), where some intervals between nominals are Pythagorean (e.g., C G) and others are a just third etc (e.g., C E). Accidentals again denotes various comma shifts exactly. However, as the notation is less uniform music not notated in C is harder to read. I assume this experience led to the development of the Pythagorean-based approach of the Helmholtz- Ellis and Sagittal notation.

The Sagittal notation allows for an even more fine-grained tuning (e.g., even comma fractions for adaptive just intonation), and also provides a single sign for each comma combination. However, I find the Helmholtz-Ellis notation more easy to read (signs differ more, less signs).

The Western musical notation system is limited to what I call a diatonic pitch system (as "extended meantone" suggest certain closeness to the major third).

For a major second M and minor second m, this is the system of pitches generated by p m + q M, where p, q are integers. The case (p, q) = (0,0) could be taken to be the tuning frequency. Sharps and flats alter with the interval M - m.

I have implemented it into ChuCK, so that it can easily be played in various tunings. The Pythagorean and quarter-comma meantone are of course special cases. But also others, like the Bohlen-Pierce scale in which the diapason is not the octave.

Now, inspired by Hormoz Farhat's thesis on Persian music, I extended it by adding neutral seconds. For each neutral seconds n between M & m, one needs accidentals to go from m to n, and from M to n. This suffices in Farhat's description of Persian music (sori and koron). For Turkish music, one needs the "dual" neutral n' := M - n; the reason is that different division of the perfect fourth leads to negative n coefficients. So then one needs to more accidentals to go from m to n', and from M to n'.

In this kind of music notation, one just tries to extend the Pythagorean tuning with 5-limit intervals. So one neutral n is sufficient in this description. For higher limits, one needs more neutrals, and for notation, a way to sort out preferred choice and order.

Now, one advantage of this model is that, like the Western notation system, one does not need to have explicit values for these symbols, though one can do so.

Basically just a FYI.


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