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

From: Torsten Anders
Subject: Re: Microtonal Helmholtz-Ellis notation in Lilypond: fine-tuning
Date: Thu, 10 Sep 2009 09:44:22 +0100

On 10.09.2009, at 09:03, Hans Aberg wrote:
On 10 Sep 2009, at 01:26, Torsten Anders wrote:

This notation is explain in detail in a paper that is part of the
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.

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

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


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