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.