Response to McLaren's kind remarks

Several kilobytes ago - TUNING 783 - Brian McLaren posted some agreable
remarks on an idea I had communicated to him about applying group theory
to understanding some properties of et_s. Brian mentioned some references
to work that had already been done along these lines, for which I'm grateful.
(I'd be even more grateful if I knew exactly _which_ number of the Computer
Music Journal contained Gerald G. Bolzano's paper "The Group Theoretic
Structure of 12-fold and Microtonal Tunings".)

It might be worth saying here what the idea was that attracted Brian's
comments. Quite likely I'm covering more or less familiar material with this
and, in any case, the mathematics is not deep. Maybe others will have useful
comments.

*****

We say that two frequencies u and 2u differ by an octave and that a scale of
n steps from u to 2u is equal-tempered if the frequencies
2^(1/n)u, 2^(2/n)u, ... , 2^(n-1/n)u
are interpolated between u and 2u.

The set of numbers {1 = 2^(0/n), 2^(1/n), ... , 2^((n-1)/n)} representing
the multipliers of the frequency u, forms a group under the operation of
adding the numerators of the respective indices, modulo n.
I.e. 2^(p/n) * 2^(q/n) = 2^((p+q)/n), where p+q is reduced mod n.

Identifying the numerators 0,1,...,n-1 with pitch-class numbers, we have
the familiar group of pitch-classes under addition modulo n.

The group formed in this way is the cyclic group of order n. It has (cyclic)
subgroups of every order dividing n. This means, for example, that the groups
corresponding to, say, 14-et, 21-et and 35-et all have subgroups of order 7
and hence these et_s possess _all_ the structure of 7-et. In fact they include
7-et both as a subgroup and as cosets of the subgroup of order 7.

[E.g. in 21-et, the set of pitch classes {0, 3, 6, 9, 12, 15, 18} constitutes
the subgroup of order 7, while the sets {1, 4, 7,...,19}, {2, 5, 8,...,20},
etc. are the cosets.]

Similarly, 8-et contains transpositions of the 'diminished seventh' chord
found in 12-et, and 9-et contains transpositions of the 'augmented' triad
from 12-et, due to shared subgroups of orders 4 and 3 respectively.

Practical consequences of this way of looking at et_s might include the
recognition of:
(1) the availability of smooth transitions between tunings by using a common
subgroup as a kind of 'pivot', and
(2) the need to avoid certain combinations or progressions nominally
available in a given tuning, in order to express the _uniqueness_ of that
tuning.

In n-et, where n is not prime, consider the factorisation pqr...-et, where
p,q,r,... are prime powers and n = pqr... . One might want to avoid the
'trap' of expressing p-et or q-et or r-et etc. rather than whatever is special
about n-et itself.

Of course, one might _not_ want to avoid these commonalities: indeed, one
might want to exploit them. But to be aware of their existence, at least,
seems important.

The question arises as to where the specialness of any n-et, for a composite
n, resides. Consider a sequence, of sufficient length, of pitches a,b,c,... .
If the sequence does not lie entirely within any particular coset of any
proper subgroup, then n is the smallest order of et_s that contains the
sequence. The sequence is unique to n-et if we discount the fact that there
are higher order et_s that also contain it. But what might 'of sufficient
length' mean? And how difficult does it become to meet this criterion for
uniqueness when n is highly composite (so that there are subgroups of many
orders)?

For example, in 12-et, a passage in the key of C-major consisting of the tones
F-G-A-B, expresses 6-et until the tones -C-D-E are appended. Note that if
C,D,E belong to one coset of the subgroup of order 6, then F,G,A,B belong to
the other. Likewise, if F,B,D belong to one coset of the the subgroup of order
4 then C,A and E,G belong to the other two cosets respectively; and the
sets {C,E}, {G,B}, {F,A}, {D} contain representatives of the cosets of the
subgroup of order 3. It looks as if structures like the major scale express
the unique features of 12-et rather well, while unadorned permutations of the
sets {C,D,E,F#,G#,A#} or
{C,D#,F#,A} or
{C,E,G#} or
{C,F#}
or their transpositions, might not.

*****

All of this is knowable through other approaches. It might even be trivial.
But it seems to me that group theory provides a relatively tidy way of
dealing with at least some tuning issues.

Paul Turner





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