Euler (in english!)

Consonance of intervals
***********************

Euler rated the sweetness of an interval (or chord) according to its
"exponent". We would say that he evaluated consonance according to the
least common multiple (LCM) of a chord.

A chord is "complete" if we cannot had a note to it without changing its
LCM. For example,
the chord 1 3 5 (C1 G2 E3) has a LCM of 15, it can be completed by adding
the note with frequency ratio 15, yielding the complete chord 1 3 5 15 (C1
G2 E3 B4). Of course, the chord will be "more complete", but, according to
Euler's model, it will have the same consonance.
Euler's genus musicums (or musical modes) are the octave repetition of the
complete chords. They are always rectangular on a square lattice and are
caracterised by their LCM.

The music of A. D. Fokker and Jan Van Dijk makes use of the euler genera
(modes).
They built a 31TET organ to better express the septimal comma. (Recherches
musicales, theoriques et pratiques, Archives du musee Teyler, 1951)

See this page for musical examples:
http://www.interlinx.qc.ca/~kami/fokker.html

The notation Eu(x,y,z) refers to the LCM of the scale. For example,
Canzone is Eu(2,0,1), meaning 3^2 * 5^0 * 7^1, or (3.3.7) in Fokker's
notation.
This would yield a complete chord of 1 3 7 9 21 63. After octave
reduction, this is
G 7Bb 7C C D 7F (where 7 is used to lower the pitch by 64/63).

Interpretation of a dominant seventh chord
******************************************

Classical interpretation: 1/1 5/4 3/2 9/5 (36 45 54 64)
Euler's interpretation: 1/1 5/4 3/2 7/4 (36 45 54 63, or 4 5 6 7)

D'Alembert says that the seventh chord is "too arbitrary and far from the
true principles of harmony", but Euler says that the ear tolerates small
deviations in the proportions of consonances: "everytime this happens, he
says, the perceived proportion is simpler that the one that is played, and
the difference is so small (64/63) that it is not perceived." Therefore we
can suppose that the ear substitutes 7/4 for 9/5, "so that all the factors
are divisible by 9, yielding the 4 5 6 7 chord, whose perception is less
embarassing.

"Maybe _this_ is the foundation of the rules about the preparation and
resolution of dissonance, to warn the auditors the the same sound will
express two different musical entities." (1)

"The musical instruments tuned according to the principles of harmony use
only proportions composed of the prime numbers 2, 3 and 5. But if my
conjecture is right, says Euler, we can say that composers already imply
the prime number 7 in their music and that the ear is already accustomed
to it. If 7 is a perfection in composition (a consonance), maybe we should
build instruments that support it."

***********************

(1) As is the case in any ET where 64/63 vanishes (like 12 and 22).

I accept constructive criticism, please correct any typo or error in this
post.

BTW, my regular web page is currently offline, because I am working on a
webpage for a customer.

-Kami

If someone has a translated version of the the euler. please post. How
are you translating the latin. I am especially interested in the
paragraph where he mentions the fibonacci series.
-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

KAMI ROUSSEAU wrote:

> Interpretation of a dominant seventh chord
> ******************************************
>
> Classical interpretation: 1/1 5/4 3/2 9/5 (36 45 54 64)
> Euler's interpretation: 1/1 5/4 3/2 7/4 (36 45 54 63, or 4 5 6 7)
>
> D'Alembert says that the seventh chord is "too arbitrary and far from the
> true principles of harmony", but Euler says that the ear tolerates small
> deviations in the proportions of consonances: "everytime this happens, he
> says, the perceived proportion is simpler that the one that is played, and
> the difference is so small (64/63) that it is not perceived."

> I accept constructive criticism, please correct any typo or error in this
> post.

The difference between the two interpretations above is 36/35, not 64/63.

There's a lot more on that French Euler page. Can we gain any insight into his GS
function?