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Re: Helmholtz-Ellis notation

From: Heikki Johannes Junes
Subject: Re: Helmholtz-Ellis notation
Date: Thu, 14 Jul 2005 03:21:02 +0300 (EEST)
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Dear John,

John Wiedenhoeft wrote:

> Dear all,
> I know Lilypond is mainly a program for writing classical scores. But for the
modernists among us, I really think support for the so-called "Extended
Helmholtz-Ellis JI Pitch notation" should be included.
> It's nothing special, just some accidentals used for microtonal composition.
To my opinion this is the most convincing way of writing microtonal pitches,
however it's much more convincing than the system currently used in Lilypond.

This notation is the first one which clearly has a mathematically sound origin!
I give an example (for others reading the mails).

All notes have upper harmonics. Consider frequency 442 Hz, which is c'. Its
upper harmonics are multiples of the basic frequency: 2*442 Hz = 884 Hz, 3*442
Hz = 1322 Hz, etc.

Let us say that you want to play a Perfect Fifth, with c' and g' played
together. In that case, the 3rd harmonics of c', 3*442 Hz, and the 2nd harmonics
of g', 2*662 Hz, exactly match at 3*442 Hz = 2*662 Hz = 1322 Hz. And, there is
no wobbling in the sound!

Let us also calculate what frequencies that interval would correspond in Equal
Tempered Semitones (logarithmic scale). The (approximated) frequencies of the
scale are (calculated using GNU Octave)

octave:1> 442 * 2.^([0:24]/12)
ans =

 Columns 1 through 8:
   442.00   468.28   496.13   525.63   556.89   590.00   625.08   662.25
 Columns 9 through 16:
   701.63   743.35   787.55   834.38   884.00   936.57   992.26  1051.26
 Columns 17 through 24:
  1113.77  1180.00  1250.16  1324.50  1403.26  1486.70  1575.11  1668.77
 Column 25:

of which the frequencies of our interest are

octave:2> 442 * 2.^([0 7 12 19]/12)
ans =

   442.00   662.25   884.00  1324.50

Now we got 1324.50 Hz. This is by 2.50 Hz larger than in a perfect fifth. What
does this mean?

If you have had two tuning forks, one of them at 442 Hz, and the other at 440
Hz, you may guess what happens. The sound appears and disappers two times in a
second, which means that the sound waves interfere either constructively
(summing up to maximum) or destructively (summing up to zero). A similar
phenomenon will happen when you play the organ.

> Please refer to for a chart, or for further information (fonts
are provided there).

I really like the idea behind this system. The quarter tones themselves are
rarely interesting, but microtonal music, when it tries to make interval more
sound and pure, is valuable in everyday life. If you prefer to sing pure major
thirds, it is not a bad idea to show explicitly that in this place and in that
place you should sing the pure major third in stead of an equally tempered and
wobbling major third.

Referring to the PDF-file, seems like the notation for the 3-limit (Pythagorean)
intervals and the 11-limit (undecimal) intervals is already included in 

We would need to add the notation for the 5-limit (Ptolemaic) intervals, which
provides for example a pure major third (with matching 5th and 4th harmonics)
and a pure minor third (with matching 6th and 5th harmonics). In addition, the
notations for the 7-limit (Septimal) intervals, the 3-limit (Tridecimal)
intervals, and the irrational-and-tempered intervals need to be included in
LilyPond. The 'extended' part of the notation is for tones which match at at
primes higher than 13 (which are 17, 19, 23, 29, 31, 37, etc.).

> It can also be used for writing just intonation. For example, research is
going on on intonation of Bach pieces... it really sounds impressive!

Bach used to play the organ. The organ produces very pure single sounds. When
you add two pure sound together, you may hear wobbling of upper harmonics which
is characteristic to the tempering in hand. Bach must have been aware of
different temperings; he wrote his Preludes and Fugues für ``Das wohltemperierte
Klavier'', for the well-tempered piano.

For me, it is an interesting question who did tune the organ Bach played? And,
how much Bach was involved in the tuning of the organs he played?

> I'd appreciate your opinion on this request, 

> Best regards,
> John Wiedenhoeft
In my opinion, there are four requests, of which at least three are easily

  (1) the microtonal notation of pitches with a number representation showing
cents to be added or removed, which is Alexander Ellis' part of the pitch 

  (2) the microtonal notation of pitches for irrational-and-tempered intervals,
one of the extensions,

  (3a) the microtonal notation of pitches with accidentals which are based on
primes, which is Hermann von Helmholz's part of the pitch notation,
  (3b) the extended Hermann von Helmholz's notation which is based on primes
higher than 13.

Implementation could be the following:

  (1) make a command called \cents, which produces a text markup.

  Example: <c' g'\cents{-2.5}>.

  (2) make a command called \tempered, which only ensures that there is an
accidental of the irrational-and-tempered type.

  Example: <c' g'\tempered>

  (3a) make commands called \syntonic, \septimal, \undecimal, and \tridecimal,
which bring the accidental symbols.

  Example 5: <c' g'\syntonic{-1}> = a pure fifth, (allowed range for the
argument is -3 ... +3)
  Example 7: c'\septimal{+1}, (allowed range for the argument is  -2 ... -2)
  Example 11: c'\undecimal{+1} = cih', (allowed range for the argument is -4 
... +4)
  Example 13: c'\tridecimal{+1}, (allowed range for the argument is -1... +1)

  (3b) the extended part is tricky, I could not figure out the system at first

In conclusion, commands \cents, \tempered, \syntonic, \septimal, \undecimal, and
\tridecimal, can be implemented rather straightforwardly. No big changes will
appear in the code, as long as only markup commands are implemented.

Best wishes,
Heikki Junes

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