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

Dave,

You can find the relevant result on page 80 (of pages 62-87) of the Carey and Clampitt article
I'm reading

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

Basically, the numbers are the denominators of the best approximating fractions for the
real number that generates the scales.

First you find the continued fraction for the number, using the usual algorithm

The basic idea of the algorithm is:
suppose the generator is g.
Step 1
Find a = integer part of g, and zeta = remainder on dividing g by a.

Step n+1 from n:
a(n+1)=integer part of 1/zeta(n)
zeta(n+1) = remainder on dividing 1/zeta(n) by a(n+1)

but you prob. know that one.

So for example the continued fraction for golden ratio is
[1,1,1,1,1....]
i.e. 1+1/(1+1/(1+....))
The best approximations are 1, 2, 3/2, 5/3, 8/5,..

So the numbers of notes are 1, 2, 3, 5, 8, 13,...

This will also apply to the two generator method, but this time, not all the numbers
produce Tryhill scales. In fact, usually only the first few, and I wonder if perhaps all
the two generator sequences may turn out to be finite, even the ones that keep
going for quite a way, as they all seem to work by gradually filling in gaps, and
slowly changing some of the ratios.

----------------------------------------------------------------------

Here is another idea for alternating generators:

Use two generators, (g-d)*1200 cents and (g+d)*1200 cents
where g = golden ratio, and d is varied according to number of
notes at MOS, getting smaller with more notes, always smaller than
the S for the scale.

Result is, the usual golden ratio scales, but some of the intervals are made
a bit larger, and others made a bit smaller

So for instance,
L S
S L - L
L - L S - L S
S L - S L - L - S L - L
L - L S - L - L S - L S - L - L S - L S
S L - S L - L - S L - S L - L - S L - L - S L - S L - L - S L - L

(here using L -> LS alternately with L -> S L
as happens if you make the sequence using the
generator 1 g*1200 cents)

becomes

S M L
L - L S - L M
(this row doesn't work)
M - L S - M - L S - M S - L - M S - L S
S L - M L - L - S L - M L - L - S L - L - M L - S L - L - M L - L

Thinking of the points round the circle:

This makes a three step size scale provided each new point at new level which is even numbered
precedes an even numbered point of prev. level,and each odd numbered point
precedes an odd numbered point of prev. level.

That is why it doesn't work for the third row. It works for all the rest up to 233.

I don't yet know why they should be Tryhill (or indeed if they always are for this construction), and don't
know why the two generator method is so good at making Tryhill scales - the last post just showed that
if the two generator method scales are Tryhill, and have a prime number of steps, they have a particularly nice
structure.

Robert

Dave,

Sorry, forgot to send it as html, and it made nonsense of the equations etc:

You can find the relevant result on page 80 (of pages 62-87) of the Carey and Clampitt article
I'm reading

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

Basically, the numbers are the denominators of the best approximating fractions for the
real number that generates the scales.

First you find the continued fraction for the number, using the usual algorithm

The basic idea of the algorithm is:
suppose the generator is g.
Step 1
Find a = integer part of g, and zeta = remainder on dividing g by a.

Step n+1 from n:
a(n+1)=integer part of 1/zeta(n)
zeta(n+1) = remainder on dividing 1/zeta(n) by a(n+1)

but you prob. know that one.

So for example the continued fraction for golden ratio is
[1,1,1,1,1....]
i.e. 1+1/(1+1/(1+....))
The best approximations are 1, 2, 3/2, 5/3, 8/5,..

So the numbers of notes are 1, 2, 3, 5, 8, 13,...

This will also apply to the two generator method, but this time, not all the numbers
produce Tryhill scales. In fact, usually only the first few, and I wonder if perhaps all
the two generator sequences may turn out to be finite, even the ones that keep
going for quite a way, as they all seem to work by gradually filling in gaps, and
slowly changing some of the ratios.

----------------------------------------------------------------------

Here is another idea for alternating generators:

Use two generators, (g-d)*1200 cents and (g+d)*1200 cents
where g = golden ratio, and d is varied according to number of
notes at MOS, getting smaller with more notes, always smaller than
the S for the scale.

Result is, the usual golden ratio scales, but some of the intervals are made
a bit larger, and others made a bit smaller

So for instance,
L S
S L - L
L - L S - L S
S L - S L - L - S L - L
L - L S - L - L S - L S - L - L S - L S
S L - S L - L - S L - S L - L - S L - L - S L - S L - L - S L - L

(here using L -> LS alternately with L -> S L
as happens if you make the sequence using the
generator 1 g*1200 cents)

becomes

S M L
L - L S - L M
(this row doesn't work)
M - L S - M - L S - M S - L - M S - L S
S L - M L - L - S L - M L - L - S L - L - M L - S L - L - M L - L

Thinking of the points round the circle:

This makes a three step size scale provided each new point at new level which is even numbered
precedes an even numbered point of prev. level,and each odd numbered point
precedes an odd numbered point of prev. level.

That is why it doesn't work for the third row. It works for all the rest up to 233.

I don't yet know why they should be Tryhill (or indeed if they always are for this construction), and don't
know why the two generator method is so good at making Tryhill scales - the last post just showed that
if the two generator method scales are Tryhill, and have a prime number of steps, they have a particularly nice
structure.

Robert

Thanks Robert.

Wherever you had "remainder on dividing <thing> by <other thing>" you
should have had simply "fractional part of <thing>".

I've implemented it in an Excel 97 spreadsheet so you can type in the
generator and interval of equivalence and see all the cardinalities of
the Myhill's scales for that generator.

http://dkeenan.com/Music/MyhillCalculator.xls

It seems that the denominators of the convergents give the
cardinalities of the strictly proper scales, while the
semi-convergents give improper (or merely proper) scales. Do you know
whether that is always the case?

Thank you so much. This has greatly simplified a project I've had on
hold for some time, enumerating all the Myhill's MOS scales of (and
all the ET approximations of) the seven good 5-limit generators
(single-chain). The results are included in this spreadsheet.

If anyone without Excel wants the results, let me know.

Regards,
-- Dave Keenan

Hi David,

>It seems that the denominators of the convergents give the
>cardinalities of the strictly proper scales, while the
>semi-convergents give improper (or merely proper) scales. Do you know
>whether that is always the case?

Norman Carey has replied to this via John Chalmers.

Answer is yes, and he cites pages 150ff and exs III-9 and III-10
of his thesis. Says he uses term "generically ordered" there for strictly
proper.

1a Any well-formed scale determined by a full convergent
is strictly proper
2a Any well-formed scale determined by an intermediate
convergent is usually improper, possibly proper.

Using R = ratio of large to small step sizes, his result is

1b Whenever R < 2, the well-formed scale is strictly proper
2b Whenever R = 2, the scale is proper
3b Whenever R > 2, the scale is improper.

Also, if the generator is irrational, there can be no ambiguities -
i.e. if two intervals have the same size, they have the same
step composition. For ambiguities to arise, R has to be
an integer, and the generator has to be rational.

>Thank you so much. This has greatly simplified a project I've had on
>hold for some time, enumerating all the Myhill's MOS scales of (and
>all the ET approximations of) the seven good 5-limit generators
>(single-chain). The results are included in this spreadsheet.

Glad to be of help!

I've yet to see and hear your Excell animation, but will take it with me
over
Xmas holidays in case any of my relatives have Excell or know someone
who has it on their machine at home.

Robert

My so-called Myhill calculator (an Excel spreadsheet)

http://dkeenan.com/Music/MyhillCalculator.xls

is really a general purpose tool for obtaining rational approximations
of real numbers. So you can use it in the linear/harmonic domain as
well as the logarithmic/melodic.

e.g. If you want to know what frequency ratios come near to 4 steps of
13-tET, then for the "Interval of equivalence" put 1, and for the
"Generator" put =2^(4/13) and see what happens.

In this application one could add a column after every
numerator/denominator pair, that tells you by how many cents (and in
which direction) the ratio differs from the real number.

Something to be aware of is the fact that I somewhat arbitrarily cut
off the semi-convergents at number 11. It might be worthwhile taking
them out to 19 in the harmonic application.

Regards,
-- Dave Keenan