ET tuning and rhythm

Here is a question that arose in the context of a mathematical classification
theory of musical motifs by G. Mazzola. It states a dependency between
(equal temparent) tuning and the time measure a piece of music is written in.

Background is that the process of octave identification of pitches (in n-TET:
calculating mod n) has an equivalent in the time parameter when we consider
only what happens inside one bar of music: measuring time coordinates in
integer multiples of a smallest time unit and a bar containing m such units,
the time coordinates with respect to the beginning of the bar are calculated
mod m.
Musical motifs in n-TET tuning and in m time meter can thus be represented
as sets in Zm x Zn. For the properties and classification of sets in this
space, the numbers n and m (their prime factorization in particular) are of
high importance. Roughly said: less the numbers are relatively prime (the
larger their gcd), the better it goes. (I skip the details here, but I can give
them, of course, if wanted.)

Now, among jazz musicians it is said that bar-exceeding improvisation 5/8
time signature (on "Take five", e.g.) is more difficult than in 3/4, 4/4 or 6/8.
This appears to be a striking coincidence with the mathematical result stated
above (5 and 12 are relatively prime, but multiples of 2, 3 or 4 are not) - but I
hesitate to assert this since another explanation could be that we just prefer
"simplicity" (not only concerning interval ratios but also time ratios!).

Anyway, there is a possibility to verify or falsify this question: if Mazzola's
supposition holds, there would be an obvious consequence: in music where
pitches are not in 12-TET but in 10-TET or 5-TET, the case should be exactly
the other way round: improvisation and memorization of motifs on 5/8 should
be easier than 3/4, 4/4 or 6/8!

So I would like to ask the experts for music in other tunings here: did you
ever notice a thing like that - that, depending on tuning, certain time
signatures appeared to be "easier" than others? I would really like to know!

Hans Straub

--- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:

> Background is that the process of octave identification of pitches
(in n-TET:
> calculating mod n)

i don't think many people hear this way. for example, in western
diatonic music, intervals and motifs are predominantly classed into
SEVEN categories, not twelve, and a fixed interval can sound
completely different depending on the context which dictates which of
the seven categories it "sounds like" -- for example minor thirds vs.
augmented seconds.

but ignoring that . . .

> Anyway, there is a possibility to verify or falsify this question:
if Mazzola's
> supposition holds, there would be an obvious consequence: in music
where
> pitches are not in 12-TET but in 10-TET or 5-TET, the case should
be exactly
> the other way round: improvisation and memorization of motifs on
5/8 should
> be easier than 3/4, 4/4 or 6/8!
>
> So I would like to ask the experts for music in other tunings here:
did you
> ever notice a thing like that - that, depending on tuning, certain
time
> signatures appeared to be "easier" than others? I would really like
to know!
>
> Hans Straub

absolutely not. tunings very close to 5-equal and 7-equal are very
common all over the world, and those cultures show no predilection
for quintuple or septuple meters -- let along a predilection for
correlation those meters with the respective tunings!

i feel that something's deeply wrong with mazzola's supposition or
your interpretation of it . . . let me think . . .

--- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:

> Musical motifs in n-TET tuning and in m time meter can thus be
represented
> as sets in Zm x Zn. For the properties and classification of sets
in this
> space, the numbers n and m (their prime factorization in
particular) are of
> high importance.

is mazzola influenced by balzano in this respect? i have a feeling a
number of "set-theorists" are falling into a major trap, seduced by
certain coincidences involving the number 12, while in reality the
number 12 is of peripheral importance, if any, to our music (play any
pre-beethoven common-practice western, thus inherently HEPTATONIC,
piece of music in 19-equal or 31-equal or golden meantone instead of
12, and no one complains. where's 12? nowhere).

> I
> hesitate to assert this since another explanation could be that we
just prefer
> "simplicity" (not only concerning interval ratios but also time
ratios!).

that's more like it! (btw, the two processes are similar, but not
exactly analogous.)

> --- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:
>
> > Musical motifs in n-TET tuning and in m time meter can thus be
> represented
> > as sets in Zm x Zn. For the properties and classification of sets
> in this
> > space, the numbers n and m (their prime factorization in
> particular) are of
> > high importance.
>
> is mazzola influenced by balzano in this respect? i have a feeling a
> number of "set-theorists" are falling into a major trap, seduced by
> certain coincidences involving the number 12, while in reality the
> number 12 is of peripheral importance, if any, to our music (play any
> pre-beethoven common-practice western, thus inherently HEPTATONIC,
> piece of music in 19-equal or 31-equal or golden meantone instead of
> 12, and no one complains. where's 12? nowhere).
>

Mazzola knows Balzano, AFAIK; whether this has relevance in this special
case, I don't know.

The formula, of course, applies only for music in 12-TET, hence indeed not
for pre-Beethoven. (Mazzola explicitly mentions jazz.)

But what you write about the number 7 is a good point - for that would foster
7-rhythms - and those are even rarer than 5-rhythms...

>> So I would like to ask the experts for music in other tunings here: did you
>> ever notice a thing like that - that, depending on tuning, certain time
>> signatures appeared to be "easier" than others? I would really like to
>>know!
>absolutely not. tunings very close to 5-equal and 7-equal are very
>common all over the world, and those cultures show no predilection
>for quintuple or septuple meters -- let along a predilection for
>correlation those meters with the respective tunings!

Ah well, this was what I wanted to know... Thanks for the informations.

--- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:
> > --- In tuning-math@y..., "Hans Straub" <straub@d...> wrote:

> The formula, of course, applies only for music in 12-TET, hence
indeed not
> for pre-Beethoven.

viols and lutes and guitars were tuned in 12-equal centuries before
beethoven.

meanwhile, though 12-equal was fully worked out mathematically in
1585 by Simon Stevin and in 1636 by Marin Mersenne, it would have to
wait until 1802-1817 (1850s in england and spain) to become a de
facto standard. post-beethoven, essentially.