Brian McLaren's Group Theory

>From Paul Turner

Brian McLaren's December 10 post on Group Theory needs some tidying up.

One doesn't like to be pedantic over idiosyncrasies of terminology, nor
risk giving offence by talking about minor inaccuracies. On the other
hand, to say nothing might be considered antisocial in another sense: both
xenharmonic theory and mathematics could fall into disrepute and good
people could get confused.

There are, I think, two main issues that need clarification. Much of
Brian's article depends on his interpretation of the mod function and on
the relevance of the concepts of relative primeness and compositeness.
______

To begin, a group is not a group until a set is defined together with a
binary operation on the elements of the set that satisfies the group
axioms. We must take it as read in Brian's article that the group under
consideration is the set of n-tet intervals measured in scale-steps
together with the binary operation of addition modulo n. It is important
to mention this because other ways of defining a group in this setting are
possible. (It is true, however, that other formulations may have
essentially the some group structure, even though the set elements and the
binary operation are defined differently.)

In speaking about the elements 5 and 7 in 12-tet, and also the elements 1
and 11, Brian claims that in modulo 12 arithmetic, 7 5 and 11 1. This
is not true. It is true that 12 5(mod7) and 12 7(mod5) but neither of
these facts is particularly helpful. Closer to the mark is the fact that
5 -7(mod 12).

What is needed here is the idea of the 'inverse'. In this group 5 and 7
are each other's inverse, as are 1 and 11.

The significance of this is that an element and its inverse each generate
the same cyclic subgroup. In fact they generate the elements of the
subgroup in opposite order to one another.

It should be noted that the idea of 'order' in the sense of 'permutation',
is not intrinsic to groups. Indeed, elements 1,5,7,11 generate exactly
the same subgroup, namely the group itself. But the sequence in which the
elements appear in the process of generating the subgroup is different in
each case, as Brian points out.

Now, every element of a group generates a cyclic subgroup of some size or
other (always dividing n). In the case of the particular type of group
that arises in the n-tet context, i.e. the cyclic groups of order n, the
size of the subgroup generated by a particular interval is n/d where d is
the greatest common divisor of n and the number of scale-steps in the
interval. E.g. in 12-tet, the element 6 generates a subgroup of order 2,
the element 10 generates a subgroup of order 6, and 5 generates a subgroup
of order 12 - the whole group.

Clearly, if the number of scale steps in the generating interval is 1 or a
number relatively prime to n, the whole group will be generated. Contrary
to Brian's assertion, the compositeness of the number of scale steps is
neither here nor there.

For example, in 16-tet, the 3-step interval generates the whole group but
so does the 9-step interval, notwithstanding the fact that 9 is composite.
Only the permutation of the elements is different:

0 3 6 9 12 15 2 5 8 11 14 1 4 7 10 13
0 9 2 11 4 13 6 15 8 1 10 3 12 5 14 7

Similarly, the number of different 'cycles of fifths' in an n-tet tuning
doesn't depend on the compositeness or otherwise of the numerical size of
the generating intervals. It is their relative primeness to n that
counts.
__________

The full significance of Brian's post probably becomes apparent when read
in conjunction with the paper by Gerald J. Balzano:

The Group-theoretic Description of 12-Fold and Microtonal Pitch Systems;
Computer Music Journal, Vol. 4, No. 4, Winter 1980; MIT

which, in turn, should be read in conjunction with an introductory text on
abstract algebra, if necessary. (The one by Frahleigh would be a good
choice.)

Brian's main thesis, if I have inferred it correctly, is that a
group-theoretic view can be helpful in making sense of the structural
possibilities in n-tet tunings. In this I thoroughly concur.

I have to admit defeat in trying to unravel Brian's statements about
'tiling'. I suspect there is a connection here with the Balzano article.
It may be that the relevant essence is the fact that some cyclic groups
are isomorphic to direct products of smaller groups and some are not: the
group corresponding to 12-tet _is_ while the groups corresponding to
16-tet and 19-tet are not. (This seems to have a bearing on the kinds of
chordal relationships that can be found.)

PT




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Jim Kukula's thoughts on similarities between rhythmic and frequency
proportions strikes me as creative thinking - applying ideas in one aspect
of music to another. That's valuable.

I suspect though that pitch and rhythm are fundamentally different when
it comes to temperament. Tempering pitches is meaningful because our ears
hear in a manner that could perhaps be described as simultaneously in
linear and logarithmic frequency scales.

For example, octaves have meaning both as 4/3s of a perfect fifth's span
of linear frequency, and also as (roughly) 12/7s of a perfect fifth's span
on a logarithmic frequency. That first formulation relates those two
intervals by their JI, whole-number-ratio basis, and the second formulation
relates the two by their perceived pitch distance as is useful for weighing
them in building a scale.

While we'll all agree that fundamental rhythmic structures show very
clear proportions on a linear time or note-rate scale, I doubt if there is
much intuitive meaning to a logarithmic time or rate scale.

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PAULE wrote:
>
> Gary wrote,
>
> >I suspect though that pitch and rhythm are fundamentally different when
> >it comes to temperament. Tempering pitches is meaningful because our ears
> >hear in a manner that could perhaps be described as simultaneously in
> >linear and logarithmic frequency scales.
>
> During my month away from the list, I too came up with the idea of rhythmic
> temperament, and so was pleasantly suprised to find that it had been
> discussed in my absence.
>
> Although Gary's point has some merit, perhaps he is forgetting that
> logarthmic relationships arise from application of a given multiplicative
> factor, in this case a ratio, several times. There is nothing in the nature
> of rhythm that prevents this from having significance.
>
> For example, one can, in the course of a piece, reinterpret triplets as
> quarter-notes, a rhythmic modulation by a factor of 3/2. Repeat this three
> more times, and you will have accelerated by a factor of 81/16. Now
> interpret every fifth note as the basic pulse, and you will be at 81/80 of
> the original tempo.

Check out a book by composer Henry Cowell titled, I think, _New Musical Resources_.
He goes into metric modulation, and stacked polyrhythms or rhythmic chords, based
on whole numbers.

> Of course, no one will notice that you have not returned to the original
> tempo.

I disagree. Play anything you know to a metronome which is marking 80 beats per
minute, then play it again at 81. Notice the difference in feel? I think the
audience would as well. I forget which of the mainstream music newsmagazines
lists the tempos of the most popular dance tracks in fractions of beats per
minute, like mm 125.2, so other musicians, and DJ's, etc. can tune in to the
narrow bands of most-fashionable tempos that people are responding to. A
tempo difference of 125/124 can apparently affect the popularity of a tune.

> But what if you had left the original tempo in place through the
> entire process, in one voice or a drumbeat? You would want to come back to
> it at the end. The polyrhythms in the middle of the process would be too
> complex to hear mathematically, anyway, so why not "temper" them, so that
> the 81/80 becomes a 1/1? Each time the triplets are introduced, they could
> be played a tiny bit too slow, the exact amount of tempering being identical
> to that of meantone tuning.

The problem with tempered rhythms is in their combination. Tempered triplets say
would drift if played against quarter notes; they wouldn't line up at the
measure length. If this is what you want, then cool. An irrationally related
duration set could give a nice effect of non-groundedness while retaining
self consistency.

If you want to do a series of metric modulations which end up at the same tempo,
without having to temper them, make sure the same numbers which occur in the
numerators also occur in the denominators.

2 3 2 1
t * - * - * - * - t
1 2 3 2

To make the progression not retrace the same steps, shuffle the order of numerators or
denominators or both (without exchanging numerator with denominator):

2 2 1 3
t * - * - * - * - t
1 3 2 2

I wrote a piece which did something like this for The Martin Dancers here in Los Angeles.
It repeated a "rhythmic progression" of meters related by common pulses.

Something I wish to do with tempos and with pitch is to write progressions which
repeat their link pattern but do not end up at the same place (using whole
numbers, not tempered). I think this would sound neat. The simplest example is
the "spiral" of continuous 3/2 fifth root motion, but nicer ones must be waiting
out there too.

Matt Nathan


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[I think I misfiled this and forgot to send it. If I already sent it, please ignore and forgive.]

PAULE wrote:
>
> Gary wrote,
>
> >I suspect though that pitch and rhythm are fundamentally different when
> >it comes to temperament. Tempering pitches is meaningful because our ears
> >hear in a manner that could perhaps be described as simultaneously in
> >linear and logarithmic frequency scales.
>
> During my month away from the list, I too came up with the idea of rhythmic
> temperament, and so was pleasantly suprised to find that it had been
> discussed in my absence.
>
> Although Gary's point has some merit, perhaps he is forgetting that
> logarthmic relationships arise from application of a given multiplicative
> factor, in this case a ratio, several times. There is nothing in the nature
> of rhythm that prevents this from having significance.
>
> For example, one can, in the course of a piece, reinterpret triplets as
> quarter-notes, a rhythmic modulation by a factor of 3/2. Repeat this three
> more times, and you will have accelerated by a factor of 81/16. Now
> interpret every fifth note as the basic pulse, and you will be at 81/80 of
> the original tempo.

Check out a book by composer Henry Cowell titled, I think, _New Musical Resources_.
He goes into metric modulation, and stacked polyrhythms or rhythmic chords, based
on whole numbers.

> Of course, no one will notice that you have not returned to the original
> tempo.

I disagree. Play anything you know to a metronome which is marking 80 beats per
minute, then play it again at 81. Notice the difference in feel? I think the
audience would as well. I forget which of the mainstream music newsmagazines
lists the tempos of the most popular dance tracks in fractions of beats per
minute, like mm 125.2, so other musicians, and DJ's, etc. can tune in to the
narrow bands of most-fashionable tempos that people are responding to. A
tempo difference of 125/124 can apparently affect the popularity of a tune.

> But what if you had left the original tempo in place through the
> entire process, in one voice or a drumbeat? You would want to come back to
> it at the end. The polyrhythms in the middle of the process would be too
> complex to hear mathematically, anyway, so why not "temper" them, so that
> the 81/80 becomes a 1/1? Each time the triplets are introduced, they could
> be played a tiny bit too slow, the exact amount of tempering being identical
> to that of meantone tuning.

The problem with tempered rhythms is in their combination. Tempered triplets say
would drift if played against quarter notes; they wouldn't line up at the
measure length. If this is what you want, then cool. An irrationally related
duration set could give a nice effect of non-groundedness while retaining
self consistency.

If you want to do a series of metric modulations which end up at the same tempo,
without having to temper them, make sure the same numbers which occur in the
numerators also occur in the denominators.

2 3 2 1
t * - * - * - * - t
1 2 3 2

To make the progression not retrace the same steps, shuffle the order of numerators or
denominators or both (without exchanging numerator with denominator):

2 2 1 3
t * - * - * - * - t
1 3 2 2

I wrote a piece which did something like this for The Martin Dancers here in Los Angeles.
It repeated a "rhythmic progression" of meters related by common pulses.

Something I wish to do with tempos and with pitch is to write progressions which
repeat their link pattern but do not end up at the same place (using whole
numbers, not tempered). I think this would sound neat. The simplest example is
the "spiral" of continuous 3/2 fifth root motion, but nicer ones must be waiting
out there too.

Matt Nathan


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>I forget which of the mainstream music newsmagazines
>lists the tempos of the most popular dance tracks in fractions of beats per
>minute, like mm 125.2, so other musicians, and DJ's, etc. can tune in to
the
>narrow bands of most-fashionable tempos that people are responding to. A
>tempo difference of 125/124 can apparently affect the popularity of a tune.

Er, I don't think so. Dance club DJ's like to "mix" different tunes together
and segue from one to another without disturbing the beat. They speed up or
slow down their turntables by considerably larger fractions than 125/124 to
do this.

>> But what if you had left the original tempo in place through the
>> entire process, in one voice or a drumbeat? You would want to come back
to
>> it at the end. The polyrhythms in the middle of the process would be too
>> complex to hear mathematically, anyway, so why not "temper" them, so that
>> the 81/80 becomes a 1/1? Each time the triplets are introduced, they
could
>> be played a tiny bit too slow, the exact amount of tempering being
identical
>> to that of meantone tuning.

>The problem with tempered rhythms is in their combination. Tempered
triplets
>say
>would drift if played against quarter notes; they wouldn't line up at the
>measure length. If this is what you want, then cool. An irrationally
related
>duration set could give a nice effect of non-groundedness while retaining
>self consistency.

One wouldn't necessarily have to keep the 3-against-2 going for long enough
to hear the phase drift at each modulation. In fact, one could just switch
to triplets suddenly, and later establish the basic pulse as an even number
of these notes, without any drift occuring.


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PAULE wrote:

> >[Matt Nathan]
> >I forget which of the mainstream music newsmagazines
> >lists the tempos of the most popular dance tracks in fractions of beats per
> >minute, like mm 125.2, so other musicians, and DJ's, etc. can tune in to
> the
> >narrow bands of most-fashionable tempos that people are responding to. A
> >tempo difference of 125/124 can apparently affect the popularity of a tune.
>
> Er, I don't think so. Dance club DJ's like to "mix" different tunes together
> and segue from one to another without disturbing the beat. They speed up or
> slow down their turntables by considerably larger fractions than 125/124 to
> do this.

Ah, but they do this to bring the tunes to within the critical tempo bands
that the dancers in their clubs prefer! :) Do a test yourself, like I did,
and listen to the local rap/urban/house/dance music station with a metronome
and write down the tempo of each tune that comes along. You'll see that they
fall into about 3 separate narrow bands, separated by wide bands in which
no tunes are played, something like the emission bands given off by elements
in a flame, if I might be so analogistic.

> >The problem with tempered rhythms is in their combination. Tempered
> triplets
> >say
> >would drift if played against quarter notes; they wouldn't line up at the
> >measure length. If this is what you want, then cool. An irrationally
> related
> >duration set could give a nice effect of non-groundedness while retaining
> >self consistency.
>
> One wouldn't necessarily have to keep the 3-against-2 going for long enough
> to hear the phase drift at each modulation. In fact, one could just switch
> to triplets suddenly, and later establish the basic pulse as an even number
> of these notes, without any drift occuring.

You might as well use real triplets or n-plets for each base tempo, and temper
only the relationships between base tempos when modulating.

On a related anecdote, I often hear drummers on recordings and those who
I play with live speed up when they play fills involving faster notes.
It's only the really good drummers who keep comfortably consistent
time no matter what rhythmic octave they are playing in.

Another anecdote, someone once wrote to me about the possibility of a software
module to process a quantized midi file to slow down the tempo at
parts where a real pianist would have to move one or both hands far across
the keyboard, the amount of slowing depending on the distance traversed,
to make the track more human-sounding.

There's also that process where you delay each note by a small amount relative
to the volume of the note, since some musicians who use their hands like percussion
players will bring their hands farther away from the instrument in preparation
for a harder striking, and take a little longer time to reach the instrument.
Good players will of course compensate for or avoid this unconcious habit

Matt Nathan



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