Re: Towards a hyper MOS and Dissertation on MOS, Steinhaus, etc,

John Chalmers wrote

>I found this ref to Norman Carey's dissertation and thought it might be
>of interest. Carey was a student of John Clough in Buffalo.

>Carey, Norman, A. "Distribution Modulo 1 and Musical Scales"

Hi John, looks very relevant.

I've just found this paper in the Huygens bibliography by
looking up your Carey ref there:

Carey, Norman and David Clampitt. "Self Similar Pitch Structures, Their Duals, and Rhythmic Analogs",
Perspectives of New Music vol. 34 no. 2, summer 1996, pp. 62-87.

http://depts.washington.edu/pnm/CLAMPITT.pdf

(ref in Huygens bib. http://www.xs4all.nl/~huygensf/doc/bib.html#C)

He proves that if a scale has Myhill's property, then it has an interval class d such that
one of the specific intervals of generic size d has only one instance in the scale.

From this result he can conclude that the scale will be well formed, and the other specific
interval of generic size d will be the generator of the scale.

Also, if the number of steps of the smaller of the two steps that make up the scale is g,
and N is the number of notes in the scale, he shows that (g,N)=1, (i.e. that they are co-prime)
and that d is the multiplicative inverse of g modulo N, i.e. d*g==1 (mod N).

He also shows that if you replace g by (N-g) = number of the larger of the two step sizes,
then you get d' = multiplicative inverse of (N-g) mod N, as another generic size which has
one specific interval with only one instance.

Then once more, the other specific interval for this class has N-1 instances, which gives the
other generator.

E.g. for diatonic scale, N=7, g=2, N-g=5, and so d=4 (4*2=8==1(mod 7))) and d' = 3 (3*5=15 == 1 (mod 7))

d=4 gives the fifth = generator of diatonic scale, or the diminished fifth = specific interval with only one
instance for this d.

d'=3 gives the fourth, and the augmented fourth.

Another relevant result he shows in the course of the proof:

In case of the Myhill scales, he shows that in each interval class the two specific sizes of interval d
must be obtained from each other by replacing one L by an S or vice versa.

This is easy to see, because as you go through all possible locations for the generic interval, there
must be a first scale degree where it changes from one of the specific interval sizes to the next.
The two intervals then overlap apart from two steps, one in the previous one only, and one in the
next one only. Since the intervals differ in specific size, the two steps in question can't be the same
size, so one is an L and one is an S.

Ex., :
L L - S L L - L S
L L S - L L L - S

The L L is the overlap between perfect fourth, and the next one, which is an augmented fourth
The two intervals differ by the preceding S of the perfect fourth, and the following L of the
augmented fourth.

Relevance to Tryhill results:

Consider the scale
wilson_17.scl
L M S L M S M L S M L M S M L S M
which we know is Tryhill.

Take for example, the interval class 5
First interval is
L M S L M
i.e.
LLMMS

How can we get three interval sizes?

As the generic interval is moved upwards in the scale one degree at a time, the number of
component steps can only change by replacing one step by another of another size
each time, then once you've gone round the circle of the scale, you have to get back to
where you started.

So it could change using the rule

L -> M -> S

or alternatively, using
L -> M, L -> M, M -> L, M ->L
i.e.
L L -> M L -> M M -> L M -> L L

(Or equivalent version with S instead of M)

However the only way the second one will work, if it can be done at all, is if you have the S
steps spaced out equally in the scale.

For instance, if the second rule worked for interval class 5, the Ss would have to be exactly
five steps apart.

S L L M M - S M M L L
to
S - L L M M S - M M L L
to
S L - L M M S M - M L L
to
S L L - M M S M M - L L
would work for instance, at least for intervals of class 5 (haven't tried to make the example scale Tryhill for the other generic interval sizes).

But then you have to have an S every four steps all the way round,
which only works if number of notes in the scale is a multiple of 5. (Otherwise, you end up making
all the steps of size S)

Similar argument applies for all the step sizes.

So if the scale happens to have a prime number of notes, we can rule out the second type
of change for all the interval classes, and the only possibility is
L -> M -> S

In particular, this will apply to the Wilson 17 tone scale.

So for instance, for class five,
L M S L M S M L S M L M S M L S M

the specific interval sizes are

LLMMS LMMSS and LMMMS
related by
LLMMS (L -> M) LMMSS (M ->S) LMMMS (S -> L) back to LLMMS

We have just shown that all the specific interval sizes have to be obtained
by this same pattern of substitutions for L, M and S.

Note that this is for _any_ Tryhill scale with a prime number of notes.

Now note that if you change M to L everywhere, all these examples of three specific interval
sizes will change to two specific interval sizes.

So, we can apply the Myhill property theorem. Change the relative sizes of the Ls and
Ms until both are the same size, and you have a scale of two step sizes, and two specific
intervals for each generic interval size

L L S L L S L L S L L L S L L S L

So it will have two generators. There are five Ss. So the generator will be
of size d where d*5==1 (mod 17), i.e. d = 7.
The other generator will be at n*12==1 (mod 17), i.e. d'=10

When you replace M by S
L S S L S S S L S S L S S S L S S
then there are 5 Ls, so the generators again are at d=7 and d'=10

When you replace S by L (the last possibility)
L M L L M L M L L M L L M L M L L M
then you have 7 Ms, so the generators are at d*7==1 (mod 17), i.e. d=5
or d'*10==1 (mod 17), i.e. d'==12

All of these have only one specific interval in one of the classes.

So we predict that the Wilson scale will have only one specific interval size for the
following classes:

for M-> L and M -> S
class 7 : two cases
class 10 : two cases

For S -> L
class 5: one case
class 12: one case

Referring to the SCALA results for this scale, (see post:)
we find:

Interval class, Number of incidences, Size:
5: 8 6/5 315.641 cents minor third
5: 1 8192/6561 384.360 cents Pythagorean diminished fourth
5: 8 5/4 386.314 cents major

7: 1 6561/5120 429.326 cents
7: 15 4/3 498.045 cents perfect fourth
7: 1 10935/8192 499.999 cents fourth + schisma, 5-limit approximation to ET fourth

10: 1 16384/10935 700.001 cents fifth - schisma, 5-limit approximation to ET fifth
10: 15 3/2 701.955 cents perfect fifth
10: 1 10240/6561 770.674 cents

12: 8 8/5 813.686 cents minor sixth
12: 1 6561/4096 815.640 cents Pythagorean augmented fifth
12: 8 5/3 884.359 cents major sixth, BP sixth

as predicted.

You can also see the two generators that were used to make the scale in these results.

The eight instances of 5/4 and 6/5 are there because the scale was constructed by alternating
the generators 5/4 and 6/5.

We see that it also has 15 3/2s. That isn't surprising since each 5/4 + 6/5 contributes a 3/2,
and so does each 6/5 + 5/4.

So any scale of 2n+1 notes constructed using n applications of two generators that turns out
to be Tryhill will follow a similar pattern.

It will have two interval classes that have (n, n, 1) (in some order) as the numbers of instances
of the three specific intervals.

It will also have two interval classes with specific intervals (1, 1, 2n-1)

However what about an arbitrary Tryhill scale with a prime number of notes.

Many scales with a prime number of notes do follow this pattern. But are there any exceptions?

Here is an interesting seven note scale from the SCALA archive:
arist_penh2.scl | Permuted Aristoxenos's Enharmonion, 3 + 24 + 3 parts
0 cents 50 cents 450 cents 500 cents 700 cents 750 cents 1150 cents 1200 cents

It is Tryhill, and here is the analysis:
1: 4 50.000 cents
1: 1 200.000 cents
1: 2 400.000 cents
2: 1 100.000 cents
2: 2 250.000 cents
2: 4 450.000 cents
3: 1 300.000 cents
3: 4 500.000 cents
3: 2 650.000 cents
4: 2 550.000 cents
4: 4 700.000 cents
4: 1 900.000 cents
5: 4 750.000 cents
5: 2 950.000 cents
5: 1 1100.000 cents
6: 2 800.000 cents
6: 1 1000.000 cents
6: 4 1150.000 cents

One can see that for each interval class, there is one specific interval which has only one
instance in the scale.

So it doesn't fit the model of scales constructed using the alternating generator method, since for
that to be the case, one of the interval classes would have to have two specific intervals
sizes each with only one instance. So there is no way to make this scale using alternating generators.

So there are at least some Tryhill scales with prime numbers of notes that can't be constructed using the
alternating generator method. They will still be Myhill after the three types of substitution, and so will
still have six '1's in the second column of the SCALA results.

The method of analysis here could also be used to work back and find the two generators for
a scale.

Take one of my favourites:

Japanese Koto scale
1/1 9/8 6/5 3/2 8/5 2/1

It turns out to by tryhill:

Class, instances, size
1: 2 16/15
1: 1 9/8
1: 2 5/4
2: 1 6/5
2: 3 4/3
2: 1 45/32
3: 1 64/45
3: 3 3/2
3: 1 5/3
4: 2 8/5
4: 1 16/9
4: 2 15/8

First look for a class with two '1's in the second column. Candidate classes are class 2 and class 3.

We see the sum of the two generators has to be 4/3 or 3/2.

Now look to see what the two generators are from the other ones with 1 in the second column.

We have a choice of 16/15 + 5/4 (class 1) or 8/5 + 15/8 (class 4) for the alternating generators

steps 5/4 16/15 (= 1/1 5/4 4/3) gives the inversion of this scale:

1/1 5/4 4/3 5/3 16/9 2/1

while steps 8/5 15/8 (= 1/1 8/5 3) gives the Japanese Koto scale

This of course leaves many questions open.

How is the arist_penh2.scl scale to be understood, and is there a method of generating it that
can be used to make other Tryhill scales?

We saw that a Tryhill scale with a prime number of steps becomes Myhill on setting any
two of the interval sizes equal to each other.

Proof of this depended on observing that the three interval sizes for all the classes are related
by changing one L -> M ->S
as in LLMMS (L -> M) LMMSS (M ->S) LMMMS (S -> L) back to LLMMS

Can a scale with a non prime number of steps which is Tryhill have three interval sizes
for some of its interval classes related to each other instead by the rule
L L -> M L -> M M -> L M -> L L?
(or equivalent rules, any permutation of L, M, S).

Such a scale wouldn't necessarily become Myhill on setting any two of the interval sizes equal to each
other, and so if there are any such scales, they would form yet another class of Tryhill scales.

Answer is yes - the S L - M L - L - M L - M L Tryhill scale I constructed in 22 -tet using
L -> M L, M -> S L, S -> L, and which Paul has just turned into a 31-tet scale can't be Myhill
three ways over, because when you look at the Scala analysis, it has unique specific intervals
for class 1, 2, 7 and 8, one for each, and it needs six such.

So let's look for the substitutions:

2 4 3 4 4 3 4 3 4
S L M L L M L M

We find the exceptional case straight away using two step intervals:
SL LM LL

It works because there is only one S, so ipso facto, equally spaced.

Is there any way to predict which alternating generators will make Tryhill scales, and to derive
the numbers of notes in some simple way from the sizes of the two generators?

Thanks,

Robert

That's great Robert.

Can you tell me a simple way to calculate the numbers of notes in
Myhill's scales with a given generator and interval of equivalence?

Regards,
-- Dave Keenan