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Message: 1650 - Contents - Hide Contents

Date: Tue, 25 Sep 2001 13:46:52

Subject: Re: semi-periodic scales

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Carl, how about the autocorrelation of the scale with its
transposition to the degree representing 3/2?
In Scala, look up the interval class for the nearest interval
to 3/2, do SHOW TRANSPOSE and look at the value for this
interval class.
It wouldn't be defined if the scale is not CS though, in any
case not if there's more than one i.c. for the fifth.
Or we can think further for a way to define it in that case
too, which may not be useful.

Manuel

Message: 1651 - Contents - Hide Contents

Date: Tue, 25 Sep 2001 15:16 +0

Subject: More lattices

From: graham@xxxxxxxxxx.xx.xx

I've updated my lattice page to include the alternatives from a couple of 
months back.  Start with 
<Octave Equivalent Music Lattices * [with cont.]  (Wayb.)>.  I hope it all makes 
sense, please say if it doesn't.


                    Graham

Message: 1652 - Contents - Hide Contents

Date: Wed, 26 Sep 2001 17:00:23

Subject: Re: Miracle theory

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:


>  Since you 

>> don't have octave-equivalence...
> 

> But I do have octave equivalence--the proposal is to run the 4/3 
> around inside the 5/3 for however many times we want to do that, 
and 
> then run that pattern out to 2. We then use 2, not 5/3, as the 
> interval of equivalence.


Then you're breaking the pattern, and I think it's clearly 
demonstrable that you have fewer of the chords you want. I'm sure 
Manuel can count them easily using Scala . . .

Message: 1653 - Contents - Hide Contents

Date: Wed, 26 Sep 2001 18:26:46

Subject: Re: semi-periodic scales

From: Carl Lumma

>So if P(i) is scale pitch i and S(i) is the step P(i)-P(i-1) and
>n the number of notes and k the interval class representing the
>fifth, then the autocorrelation for the scale step pattern is
> 
>   n
>  sum  S(i)*S((i+k) mod n)
>  i=1
>  ------------------------
>   n
>  sum  S(i)*S(i)
>  i=1


Is S(i) a log-freq. size?


> But apparently you want a more general measure.


I'm not sure autocorr. is too special, but it breaks if there
isn't a well-behaved k for 3:2.  Unlike propriety, and in
particular transpositional coherence, we don't care about the
mapping between scale intervals and acoustics here.


>> () Tune up the scale.
>> () Find all the subset of it.
>> () Transpose each subset by a perfect 3:2.
>> () Measure how far outside of original complete scale
>> each transposed subset is.
>> () Take the mean of these.
> 

> The fourth step is unclear to me. What are you measuring if
> the scale doesn't contain just fifths? What's the criterion
> for deciding if the tones of the subsets are inside the
> original scale or not?


I left that intentionally vague, because there are many ways
to do it, none of which have any bearing on the point that
whatever procedure we use, it should be equivalent to looking
at all the subsets of the scale.

The two main families of #4 choices are:

4a; We allow appoximations of a 3:2 to be as good as a true
    3:2 early on, and then only measure the relative change
    in the pattern of scale steps after the transposition.
    That's what Paul and I are working with so far.

4b; We consider things to get smoothly worse as distance from
    the 3:2 increases, making comparisons in log-freq. space
    at the end.

-Carl

Message: 1654 - Contents - Hide Contents

Date: Wed, 26 Sep 2001 18:38:11

Subject: Re: Miracle theory

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> --- In tuning-math@y..., genewardsmith@j... wrote:

> Then you're breaking the pattern, and I think it's clearly 
> demonstrable that you have fewer of the chords you want. I'm sure 
> Manuel can count them easily using Scala . . .


Scala only counted chords from one note when I tried it, so maybe I 
need another tip from Manuel. However, even within the 16 notes of 
5/3 of the 21 note scale, we already find lots of chords, and adding 
some extra 5/3 relationships will add more.

Message: 1655 - Contents - Hide Contents

Date: Wed, 26 Sep 2001 18:44:42

Subject: Re: Miracle theory

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., graham@m... wrote:

> In-Reply-To: <9oo0rt+s4tp@e...>

which scale?

>>>> temper.Temperament(441,612,temper.primes[:-2])


> 43/117, 49.0 cent generator

> So which scale do you mean?


3^(494/970) = 3^(247/485), reduced mod 3.

Message: 1656 - Contents - Hide Contents

Date: Wed, 26 Sep 2001 18:48:01

Subject: Re: Miracle theory

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

>> --- In tuning-math@y..., genewardsmith@j... wrote:
> 
>> Then you're breaking the pattern, and I think it's clearly 
>> demonstrable that you have fewer of the chords you want. I'm sure 
>> Manuel can count them easily using Scala . . .
> 

> Scala only counted chords from one note when I tried it, so maybe I 
> need another tip from Manuel.


I think, at least with a recent enhancement by Dave Keenan, Scala 
should be able to count the chords of a given type no matter where 
they appear in the scale. Manuel?


> However, even within the 16 notes of 
> 5/3 of the 21 note scale, we already find lots of chords, and 
adding 
> some extra 5/3 relationships will add more.


Agreed. But if you count the number of 1:3:5, 1:3:7, 1:5:7, and 3:5:7 
triads, and 1:3:5:7 tetrads, and their inverses, I'm betting that 
Blackjack will look better than your 21-tone scale. If I'm wrong, I'm 
going to have to spend a lot of time understanding why I was wrong, 
which will include making color lattices for your scale, getting into 
the periodicity block intepretations of it, and making Joseph compose 
with it :)

Message: 1657 - Contents - Hide Contents

Date: Wed, 26 Sep 2001 22:16:40

Subject: Re: Miracle theory

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:


> I think, at least with a recent enhancement by Dave Keenan, Scala 
> should be able to count the chords of a given type no matter where 
> they appear in the scale. Manuel?


I got it to count chords by means of the "location" command, and 
you're right--while my scale has a lot of chords, Blackjack does even 
better--for instance 8 major and 8 minor triads vs 7 major and 7 
minor, 7 supermajor and subminor triads vs 6 each, and so forth. 
About all my scale has going for it in the face-off is the presence 
of familiar subscales such as major and minor diatonic and Paul's 10-
note PBs.

Oh well, back to the drawing board--it works, but it isn't better 
than Blackjack, which I thought it would be. The ship didn't sink, 
but it didn't beat the record for fastest time to New York, either.

Message: 1658 - Contents - Hide Contents

Date: Wed, 26 Sep 2001 00:40:25

Subject: Re: semi-periodic scales

From: Carl Lumma

>Suppose we look at 12-note scales of the form abababaababa, formed
>by iterating fifths within octaves. We have 7 a and 5 b, and if we
>take scale steps such that a=2, b=1, we get the 19 et, a=3, b=2
>gives the 31 et, a=4 b=5 the 53 et, and a=5 b=4 the 55 et. One way
>of measuring self-similarity would make all of these the same,


All of these are the same under the measure Paul and I are pushing.


>however we could also look at it in proportion to the average number
>of et intervals making up a scale step. In all cases a shift of a
>fifth interchanges an a and a b; for the 19-et that means an
>exchange of 2 and 1, in a sitation where the average step size is
>19/12, whereas for the 55-et it is and exchange of 5 and 4 compared
>to an average size of 55/12; both are an exchange of one et interval
>so we might measure the goodness of the first by 12/19 and the
>second by 12/55.


Hmm... what would that do... this may be useful for comparing one
scale to another when they otherwise share the same value on the 
measure Paul and I suggest (as in the example).  In general, though,
I was after a way to compare a scale only with itself.  So the
answer is, I don't care about the proportions.
 

>This is related to the approximation of the fifth of the et by
>7/12; a standard measure here would measure the relative goodness
>of the approximation to x by the reduced fraction p/q by
>q^2 |x - p/q|. If we do this, we find 12^2 |7/12 - 11/19| = 12/19
>and 12^2 |7/12 - 32/55| = 12/55, etc. If a scale of n steps is
>generated from a single cycle of a generator in an m-et, this
>suggests n^2 |a/n - b/m|, where b/m is the generator and a/n is
>what we might call the scale-step generator, as a measure of self-
>similarity.


Self-similarity?  I'm not sure I'm making that connection.  I
should warn you that I'm pretty slow when it comes to the
language of math.  I can do simple algebra, but it takes me a
long time.  I'm not afraid of it, but maybe I should be.  That
sort of thing.


>We can also adapt it to more than one cycle of the generator; for 
>instance a 12-cycle in the 22 et gives us 12^2 |7/12 - 13/22| =
>12/11 as a measure of self-similarity, whereas if we have two
>cycles of six tones a half-ocatave apart, we have two cyles each
>of which has a self-similarity measure of 6^2 | 1/6 - 2/11| = 6/11,
>suggesting the total self-similarity should be the same at 12/11.


Hrm.  This sounds interesting, but I'm not sure how it relates
to symmetry at the 3:2.  If I had to brute-force the problem, here's
how I'd do it:

() Tune up the scale.
() Find all the subset of it.
() Transpose each subset by a perfect 3:2.
() Measure how far outside of original complete scale
   each trasposed subset is.
() Take the mean of these.

Keep in mind that I need to be able to look at scales which are
not MOS, not even a periodicity block at all.  Is there a standard
way to measure partial symmetry?  Come to think of it, how do we
even write normal octave equivalence for a non-PB?  Write a seed,
and then a transformation?

-Carl

Message: 1659 - Contents - Hide Contents

Date: Wed, 26 Sep 2001 11:20:51

Subject: Re: semi-periodic scales

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

This autocorrelation value measures the scale step similarity.
From the help file:

[...] the normalised autocorrelation values for the logarithmic
intervals between consecutive pitches. It is a measure how similar the
interval sequence is when the scale is transposed to the given key. A
value of 1.0 means identical, a value of 0.0 means no similarity.

So if P(i) is scale pitch i and S(i) is the step P(i)-P(i-1) and n the
number of notes and k the interval class representing the fifth,
then the autocorrelation for the scale step pattern is

  n
 sum  S(i)*S((i+k) mod n)
 i=1
 ------------------------
  n
 sum  S(i)*S(i)
 i=1

But apparently you want a more general measure.


>() Tune up the scale.
>() Find all the subset of it.
>() Transpose each subset by a perfect 3:2.
>() Measure how far outside of original complete scale
>   each transposed subset is.
>() Take the mean of these.


The fourth step is unclear to me. What are you measuring if
the scale doesn't contain just fifths? What's the criterion
for deciding if the tones of the subsets are inside the
original scale or not?

Manuel

Message: 1660 - Contents - Hide Contents

Date: Thu, 27 Sep 2001 11:03:28

Subject: Re: Miracle theory

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

>I think, at least with a recent enhancement by Dave Keenan, Scala
>should be able to count the chords of a given type no matter where
>they appear in the scale. Manuel?


Not a recent enhancement, use SHOW LOCATIONS. You can use integer
notation for chords, like 1:3:5:7.
I don't know Gene's scale, so can he do the comparison with Miracle
himself.

Manuel

Message: 1661 - Contents - Hide Contents

Date: Thu, 27 Sep 2001 11:29:12

Subject: Re: semi-periodic scales

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Carl wrote:

>Is S(i) a log-freq. size?

Yes.

And earlier:
>Namely, every possible interval
>pattern in the scale is generated, transposed by a fifth, the number
>of changing notes is divided by the length of the pattern in each
>case, and then the whole lot of results is averaged.


Is the fact that you want to give each subset equal weight the reason
for evaluating each subset?
And if you'd give each tone equal weight instead, wouldn't it be
necessary anymore to evaluate subsets?

Manuel

Message: 1662 - Contents - Hide Contents

Date: Thu, 27 Sep 2001 19:07:57

Subject: Re: semi-periodic scales

From: Carl Lumma

> Is the fact that you want to give each subset equal weight the
> reason for evaluating each subset?

Yes.

> And if you'd give each tone equal weight instead, wouldn't it be
> necessary anymore to evaluate subsets?


As far as I can tell, giving each tone equal weight is equivalent
to giving each subset equal weight, sinc the set of all subsets
should have the same proportion of intervals as a single instance
of the complete scale.

-Carl

Message: 1663 - Contents - Hide Contents

Date: Thu, 27 Sep 2001 22:06:59

Subject: Tetrachordality and Scala (was Re: semi-periodic scales)

From: Carl Lumma

Paul,

Looks like the measure we both came up with can be done in
Scala, like this:

() LOAD a scale, or make one
() use KEY to rotate the scale until a 3/2 shows up
() SHOW TRANSPOSE
() observe the row corresponding to the key with the 3/2
() observe the first number in the third column of that row
() divide that number by the number of tones in the scale

There's a problem, though.  If there are different approx.
3:2's in the scale, you may get diff. results depending on
which row you observe from SHOW TRANSPOSE.  It seems this
is a reason to go over to something like autocorrelation...
unfortunately, it enforces degree order (but it wouldn't
have to, right Manuel?).

Maybe SHOW DIFFERENCE can help:


>If the scales do not have the same number of notes, the amount
>compared is the maximum of the two scale sizes.

?

>   /NEAREST
>
>Show the differences not between corresponding scale degrees but
>between the pitch of the current scale and its nearest counterpart
>in the given scale.


That sounds like what we want.


>At the end the number of pitches which are different is given


Ehhhxcellent.  Manuel, there are a couple of precision variables
in Scala (TOLERANCE with SHOW LOCATIONS, for example).  Does this
final number go through any of them?

-Carl

Message: 1664 - Contents - Hide Contents

Date: Fri, 28 Sep 2001 13:55:45

Subject: Re: semi-periodic scales

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

>As far as I can tell, giving each tone equal weight is equivalent
>to giving each subset equal weight, sinc the set of all subsets
>should have the same proportion of intervals as a single instance
>of the complete scale.


Ah yes, that should be right.

Manuel

Message: 1665 - Contents - Hide Contents

Date: Fri, 28 Sep 2001 14:11:25

Subject: Re: Tetrachordality and Scala (was Re: semi-periodic scales)

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

>() use KEY to rotate the scale until a 3/2 shows up


Or use SHOW LOC 3/2 to see where one is right away.


>There's a problem, though.  If there are different approx.
>3:2's in the scale, you may get diff. results depending on
>which row you observe from SHOW TRANSPOSE.


Yes it only looks for exactly corresponding intervals.


>It seems this
>is a reason to go over to something like autocorrelation...
>unfortunately, it enforces degree order (but it wouldn't
>have to, right Manuel?).


Then it probably needs a more complex definition. Perhaps an
average value for transpositions by a fifth starting on all
keys of the scale.


>> If the scales do not have the same number of notes, the amount
>> compared is the maximum of the two scale sizes.


Ah, that's unclear. What's meant is that the smaller scale is
implicitly octave-extended to the size of the larger scale.
So that's not what you want.


>Ehhhxcellent.  Manuel, there are a couple of precision variables
>in Scala (TOLERANCE with SHOW LOCATIONS, for example).  Does this
>final number go through any of them?

No.

Manuel

Message: 1667 - Contents - Hide Contents

Date: Fri, 28 Sep 2001 19:37:56

Subject: Re: 34-tone ET scale

From: Paul Erlich

--- In tuning-math@y..., "Steve Cullinane" <m759@p...> wrote:

> The 53-tone equal-temperament (ET) scale is well-known as giving a 
> very good approximation to "just" intervals.  The following 
websites 
> make the case that the next-best ET scale is the 34-tone ET scale.
> 
> The Harmony Problem -- 
> The Harmony Problem * [with cont.]  (Wayb.)
> 
> Natural Temperament --
> Natural Temperament * [with cont.]  (Wayb.)
> 
> Comments are welcome.
> 
> -- Steve Cullinane (m759)


34-tET was discussed by Larry Hanson and others before 1980. But it 
unfortunately suffers from the famous "comma problem" in diatonic 
common-practice music. Therefore it has not been very important 
historically. Tunings that eliminate the comma, such as 19-tET, 31-
tET, 43-tET, 55-tET, have been much more important historically, with 
19- and 31-tone keyboards dating from as early as the 16th century.

Message: 1668 - Contents - Hide Contents

Date: Fri, 28 Sep 2001 20:33:25

Subject: Tetrachordality and Scala (was Re: semi-periodic scales)

From: Carl Lumma

>> >t seems this is a reason to go over to something like
>> autocorrelation... unfortunately, it enforces degree
>> order (but it wouldn't have to, right Manuel?).
//

>Perhaps an average value for transpositions by a fifth
>starting on all keys of the scale.


Can you illustrate with an example?


>Then it probably needs a more complex definition.
//
>  n
> sum  S(i)*S((i+k) mod n)
> i=1
> ------------------------
>  n
> sum  S(i)*S(i)
> i=1


This is exactly what we want.  Except, instead of comparing
the scale and its rotation to a degree near a 3:2, we compare
it with its transposition by an exact 3:2, simply taking the
lowest value for all rotations of the transposed version,
using a simple ordered pairing of intervals in each case.

  n
 sum   S(i) * S(i')
 i=1
 -------------------
  n
 sum   S(i)^2
 i=1

...where i' is an interval from the transposed scale.

-Carl

Message: 1669 - Contents - Hide Contents

Date: Fri, 28 Sep 2001 20:46:31

Subject: Re: 34-tone ET scale

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:


> 34-tET was discussed by Larry Hanson and others before 1980.


J. Murray Barbour mentions it in his classic "Tuning and 
Temperament", published in 1951, and before that in "Music and 
Ternary Continued Fractions", American Mathematical Monthly vol. 55, 
1948, p. 545. Barbour's analysis has the same two features--it 
considers only the 5-limit, and it ignores the diatonic comma. One 
can however consider not being meantone a feature, not a bug, and the 
34 division has its own individual merits; it is also worth looking 
at it from the point of view of the 17-et contained within it, or 
putting it inside of a 68-et (inside of a 612-et, a division which 
Barbour also mentions, is going a little too far.)

Message: 1670 - Contents - Hide Contents

Date: Fri, 28 Sep 2001 21:06:34

Subject: Re: 34-tone ET scale

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:


> One 
> can however consider not being meantone a feature, not a bug,


Absolutely -- if one is not simply composing according to common-
practice schemata (which the use of conventional notation ABCDEFG#b 
tends to imply).


> it is also worth looking 
> at it from the point of view of the 17-et contained within it, or 
> putting it inside of a 68-et


Exactly -- things I mentioned in the "bicycle chain" discussion.


> (inside of a 612-et, a division which 
> Barbour also mentions, is going a little too far.)


Ha -- I never noticed that 612 was 36*17!

Message: 1671 - Contents - Hide Contents

Date: Fri, 28 Sep 2001 08:14:09

Subject: Blocks as the lifting of an et

From: genewardsmith@xxxx.xxx

I mentioned that one can lift a musical piece to JI if one picks a 
version in each of a number of suitably chosen ets equal to the 
number of primes in the p-limit in question. So, for example, one can 
lift to the 5-limit if one can find a version in the 12, 22, and 31 
ets, and to the 7-limit if we add 55.

One way to think of the "scale along a line" of a block is that it is 
the lifting of an n-et to JI or to some tempering, where we simply 
pick the closest availble note in each of the other ets to the et we 
are trying to lift, allowing ourselves any starting point, not 
necessarily the unison, for the et we are lifting.

Message: 1672 - Contents - Hide Contents

Date: Fri, 28 Sep 2001 08:37:28

Subject: Complexity

From: genewardsmith@xxxx.xxx

I think Paul had the right idea with a logarithmic weighting, if we 
want 9 to be exactly half as important as 3, so that 2n steps to 9 
and n to 3 amount to the same thing. However, what about taking the 
weight over a tonality diamond, and using Paul's L1 metric?

We can make Np into a lattice (in fact, we can make all positive 
rational numbers into a lattice in an infinite-dimensional Banach 
space if we want to) by setting ||q|| where q = 2^e2 3^e3 ... p^ep to

||q|| = |log(2) e2| + |log(3) e3| + ... + |log(p) ep|

The base of the logarithm doesn't matter--we are free to use octaves 
or cents if we stick to that choice. Once we have a norm (for this 
defines a norm on the real vector space with basis {e2, e3, ..., ep}) 
we have a metric, defined by d(p, q) = ||p/q||. (This is usually 
written additively, as d(p, q) = ||p - q||, but our "addition" is 
multiplication here.)

We can define a measure associated to a generator g and interval of 
equivlance E by

max_{q in diamond} |ord(q)|/||q||,

where ord(q) is the exponent such that g^ord(q) ~ q mod E. This is 
pretty close to what Paul suggested.

Message: 1674 - Contents - Hide Contents

Date: Sat, 29 Sep 2001 19:57:39

Subject: Re: Digest Number 124

From: Paul Erlich

--- In tuning-math@y..., jon wild <
wild@f...> wrote:

> 
> I've had to skip quite a bit on these lists recently. Is there anything
> already written that someone could point me to, that would help me start
> to understand the couple of things I mention below?
> 
> Graham wrote, responding to Gene:
> 
>>> 5-limit
>>> 

>>> 16/15, 25/24, 81/80
>> 

>> And these are the usual intervals for defining a 5-limit diatonic
>> scale!
> 

> I know where these intervals crop up in comparing combinations of
> intervals in the usual J.I.  scale to one another, but how, precisely, do
> the three intervals "define" the scale? Are they like a "basis" for
> 5-limit diatonic space, and I could just as well take three other linearly
> independent intervals like 9:8 10:9 4:3 to define the scale, in a similar
> way to however Gene's three do?


The two intervals 25:24 and 81:80 
"define" the diatonic scale in a 
very precise sense. Have you 
studied the "Gentle Introduction 
to Periodicity Blocks"? If that 
doesn't help, perhaps we should 
meet for coffee and I'll give you a 
copy of my paper, _The Forms of 
Tonality_.

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