Hellegouarch

Can anyone post a summary of the material covered in this
paper?

http://smf4.emath.fr/Publications/Gazette/1999/81/smf_gazette_81_25-39.pdf

-Carl

Carl Lumma <carl@lumma.org> wrote:
> Can anyone post a summary of the material covered in this
> paper?
>
> http://smf4.emath.fr/Publications/Gazette/1999/81/smf_gazette_81_25-39.pdf

This is "Gammes naturelles I" so Gene and I have been
through it before. I'll see what sense I can make of
it tonight.

Preamble:

Something about G# being a bit higher than Ab.

This text is a revised version of the one we haven't found
yet.

Something about harmonics I suppose.

Something about the Farey sequence and he thinks he can get
up to the 13-limit on his instrument?

Picture with circles showing something or other.

1. Musical introduction:

Stuff we know about ratios and temperament.

Examples:
1) 12-equal is a discrete(?) subgroup of real numbers.
2) The scale of Serge Cordier, 7 equal divisions of 3/2,
is also a tempered scale and is also a discrete subgroup
of the real numbers.
3) The Pythagorean a subgroup of the real numbers
generated by 2 and 3 but isn't a discrete subgroup.
4) Zarlino's scale (5-limit JI) is a subgroup of the
real numbers generated by 2, 3 and 5 and isn't a
discrete subgroup.
5) The meantone scale is a subgroup of the reals
generated by 2 and 5**(1/4) and isn't discrete.
6) In a famous work, Euler pronounced that 7-limit JI
wasn't interesting for musicians. [Is "travail"
correct here? I thought it should be "oeuvre".]
Examples: The Pythagorean and Zarlino scales are natural,
tempered scales aren't.

Stuff about ratios. The Pythagorean comma.

In mathematical terms, the musician imposes the relation
Pythagorean comma identical with 1 in a free abelain group
<2,3> a strict subset of the rationals; the quotient group
is nothing more than the integers. So, by applying Euler's
principle to find a system of representatives(?) of the
classes of <2, 3> modulo <Pythagorean comma> we find the
Pythagorean scale written in books of music history.

2. Commas

Not sure about this, but it's commas.

3. Commas of rank 2 groups

Translating "rang" as "rank" makes some sense but I may be
completely wrong.

Mentions the formula for a Pythagorean interval in terms of
logs of 2 and 3.

Continued fraction convergents for 3:1 give ETs including
5, 12, 41, 53.

Theorem 2: relates commas to the convergents to the log of
a ratio.

Lots of stuff, don't know if it's important, skipped over
it.

4. Quotient groups of scales for rank 2

This is about hobbit-like scales that are constant
structures with each scale step within the octave being the
simplest possible ratio within the prime limit.

Pythagorean scale as an example and tables for 41 and 53
degrees.

Theorem stating something about this.

Note about the second part of the article being in the next
edition of the Gazette. It's out there but I don't have an
official URL for it. Search for "Gammes naturelles
suite". Covers:

5. Limits of the Pythagorean scale as the number of notes
tends to infinity.

6. Scales of rank 3 groups. 5-limit JI, example from
Mozart. Matrices.

7. Harmonic attraction, expressive JI?

Conclusion, appendix, bibliography.

Graham

Hi Graham,

>This is "Gammes naturelles I" so Gene and I have been
>through it before. I'll see what sense I can make of
>it tonight.

Is there a Gammes naturelles II?

>Preamble:
>Something about G# being a bit higher than Ab.
>
>This text is a revised version of the one we haven't found
>yet.

Gammes naturelles 0?

>Something about harmonics I suppose.
>
>Something about the Farey sequence and he thinks he can get
>up to the 13-limit on his instrument?
>
>Picture with circles showing something or other.

I hosted the html translation by google for a while, but
nobody seems to have checked it out. Here it is again:
http://lumma.org/temp/GammesNaturellesI.html

>In mathematical terms, the musician imposes the relation
>Pythagorean comma identical with 1 in a free abelain group
><2,3> a strict subset of the rationals; the quotient group
>is nothing more than the integers. So, by applying Euler's
>principle to find a system of representatives(?) of the
>classes of <2, 3> modulo <Pythagorean comma> we find the
>Pythagorean scale written in books of music history.

That sounds good.

>3. Commas of rank 2 groups
>Translating "rang" as "rank" makes some sense but I may be
>completely wrong.
>Mentions the formula for a Pythagorean interval in terms of
>logs of 2 and 3.
>Continued fraction convergents for 3:1 give ETs including
>5, 12, 41, 53.
>Theorem 2: relates commas to the convergents to the log of
>a ratio.
>Lots of stuff, don't know if it's important, skipped over
>it.

There's something about classifying intervals as commas
or not. I couldn't quite follow it, spurring my query.

>4. Quotient groups of scales for rank 2
>This is about hobbit-like scales that are constant
>structures with each scale step within the octave being the
>simplest possible ratio within the prime limit.

That sounds good.

>Note about the second part of the article being in the next
>edition of the Gazette. It's out there but I don't have an
>official URL for it. Search for "Gammes naturelles
>suite". Covers:
>5. Limits of the Pythagorean scale as the number of notes
>tends to infinity.
>6. Scales of rank 3 groups. 5-limit JI, example from
>Mozart. Matrices.

Sounds good...

Thanks for taking the time,

-Carl