Tuning-Math Digests messages 1625 - 1649

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Message: 1625

Date: Mon, 24 Sep 2001 20:56:44

Subject: Re: semi-periodic scales

From: Paul Erlich

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:

> >Only an MOS scale generated by approximate 3:2's will have this 
> >property (only one note changes with transpositions by a 3:2). 
> 
> Yes but I'm interested in a general measure for any scale.  I'm
> thinking of a function which can take a scale and an interval as
> an input, and spit out the "extent" of the scale's symmetry at
> that interval.

Sounds good . . . for an MOS scale, it depends on how many generators 
make up the 3:2 . . . for example, any MIRACLE MOS will have 6 notes 
change when you transpose it by a 3:2.
> 
> >My omnitetrachordality property is closely related. If 
> >omnitetrachordality holds, moves of a 3:2 will only change the
> >notes within a single 9:8 span.
> 
> Is definition in your paper still the most recent?
> 
I don't use the word "omnitetrachordality" in my paper but the 
concept is there.


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Message: 1626

Date: Mon, 24 Sep 2001 03:23:18

Subject: Miracle postscript

From: genewardsmith@xxxx.xxx

Dave Keenan asked for primes to be expressed in the miracle fourth 
system; since this was an eminently reasonable querry I am 
reproducing the results here. If A is 5/3, flat by one cent, and B is 
4/3, sharp by 2 cents, then we have the following approximations:

2  = A^(-6)  B^13
3  = A^(-12) B^25
5  = A^(-11) B^25
7  = A^14    B^(-18)
11 = A^3     B^3


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Message: 1627

Date: Mon, 24 Sep 2001 22:06:25

Subject: Re: semi-periodic scales

From: Carl Lumma

>>Yes but I'm interested in a general measure for any scale.  I'm
>>thinking of a function which can take a scale and an interval as
>>an input, and spit out the "extent" of the scale's symmetry at
>>that interval.
>
>Sounds good . . . for an MOS scale, it depends on how many
>generators make up the 3:2 . . . for example, any MIRACLE MOS
>will have 6 notes change when you transpose it by a 3:2.

I wonder if Gene, or somebody else with any knowledge of group
theory could help.  Is there any notion like being 'partially
closed' under a certian transformations?

>>I don't use the word "omnitetrachordality" in my paper but the
>>concept is there.

Is there a place where you do use omnitetrachordality?

-Carl


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Message: 1628

Date: Mon, 24 Sep 2001 06:41:01

Subject: Re: Miracle theory

From: genewardsmith@xxxx.xxx

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

> 225:224 is also in the kernel for characteristic meantone, schismic 
and 
> "magic" scales.  There's nothing special about associating it with 
> miracles. 

I dunno--it's associated with miracles in something like the way 
81/80 is associated to meantone, and that is a completely defining 
way. But it seems you want more:

 225:224, 2401:2400 and 3025:3024, however, completely define 
> miracle temperament.

This means that h10 and h31 also completely define it; and we have 
h41 = h10 + h31 and h72 = h10 + 2 h31 in this miracle kernel.

  Add 81:80, and you have 31-equal (actually a 62 note 
> periodicity block).  So is 31-equal the grand unified jumping jack 
> temperament?

It would be better if it came out 31 and not 62, but you do get 2 h31 
out of the first four jumping jacks. I've been planning to write a 
program to find more of these, and it will be interesting to see what 
they lead to.

> I don't get this.  The temperament consistent with 441- and 612-
equal 
> divides the octave into 9 equal parts, with a generator mapping of 
[-2 -3 
> -2].  The generator is around 49 cents, 18/441 or 25/612 octaves.  
It 
> covers the 9-limit with 37 notes, with 0.2 cents accuracy.

Sounds interesting, I will check this out.

> I don't see why these are better views, other than for meantone.  
As I 
> make the last case, the interval of repetition is 3, and a 7:4 is 
two 
> generators.  5:3 and 7:5 are both a single generator.

I'm not sure what you mean by this--7/4 and 3 are two generators, 
just as 3/2 and 2 are two generators or 5/3 and 4/3 are two 
generators. Wherein lies the difference?


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Message: 1629

Date: Mon, 24 Sep 2001 22:23:54

Subject: Re: semi-periodic scales

From: Paul Erlich

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:
> >>Yes but I'm interested in a general measure for any scale.  I'm
> >>thinking of a function which can take a scale and an interval as
> >>an input, and spit out the "extent" of the scale's symmetry at
> >>that interval.
> >
> >Sounds good . . . for an MOS scale, it depends on how many
> >generators make up the 3:2 . . . for example, any MIRACLE MOS
> >will have 6 notes change when you transpose it by a 3:2.
> 
> I wonder if Gene, or somebody else with any knowledge of group
> theory could help.

Help how? Aren't we done?

> >>I don't use the word "omnitetrachordality" in my paper but the
> >>concept is there.
> 
> Is there a place where you do use omnitetrachordality?

On the tuning list. It means the same thing as the property mentioned 
in my paper.


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Message: 1630

Date: Mon, 24 Sep 2001 08:26:00

Subject: Block generator theorem

From: genewardsmith@xxxx.xxx

Suppose we have two vals u and v which together with two intervals e 
and f define a tempered notation. We want to take e to be an interval 
of equivalence, and we want conditions on the vals; the frst is that 
the matrix

[u, v] = [u(e) v(e)]
         [u(f) v(f)]

be unimodular; that is that it have determinant +-1, so that 
[u,v]^(-1) is an integral matrix. The second is that [u, v] is to 
have positive values, the third that u(e) > v(e), the fourth that 
gcd(u(e), v(e)) = 1, and finally that gcd(u(e), u(f)) = 1. We want to 
construct a tempered block of u(e) notes in the temperament defined 
by this notation, hence we want a complete set of residues mod u(e) 
from the first coordinates of our block, which is to be sorted 
according to the second coordinate, and all of it to satisfy the 
conditions for a semiblock.

We first shift coordinates to the notation [u, w], where 
w = u(e) v - v(e) u. Since u(e) and v(e) are relatively prime, we can 
without loss of generality take a generator to be of the form [n 1], 
where n is relatively prime to u(e). However, not all such choices 
give us a note, since the determinant of [u, w] = 
u(e) (u(e)v(f) - v(e) u(f)) = +- u(e) because of the unimodularity 
condition. The sublattice of actual notes is generated by 
(e, f) = [u, v]^(-1) (in the u, v notation), transforming this to the 
[u, w] notation by multiplying by [u, w] gives us [u, v]^(-1) [u, w], 
which is 

s [1 -v(e)]
  [0  u(e)],

where s = det([u,v]) = +-1. This tells us that notes are of the form 
[n, m u(e) - n v(e)], so that modulo u(e) they are [n, - n v(e)]. 
Choosing a generator of the form [n 1] therefore means solving the 
congruence n = -1/v(e) (mod u(e)); since v(e) is prime to u(e), we 
may solve this congruence for a unique positive n less than u(e), 
giving us a generator. If we transform this generator back to the 
[u, v] notation by [a, b] = [n 1] [u, w]^(-1), we get

[ n 1]   [1/u(e)        0] = [p -s],
         [-s u(f)/u(e) -s]

where the value of p is an integer, and s of course is +-1. Hence 
[a, b] corresponds to an interval r = e^p f^(-s), so that modulo our 
interval of equivalence e it is f or 1/f. The canonical example would 
be where e is 2, f is 3, and the generator we get is, modulo octaves, 
a fifth or a fourth.

This argument is too convoluted and I should try to improve it, but 
the theorem is more widely applicable than it at first may seem.


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Message: 1631

Date: Mon, 24 Sep 2001 23:15:52

Subject: Re: semi-periodic scales

From: Carl Lumma

>>I wonder if Gene, or somebody else with any knowledge of group
>>theory could help.
> 
>Help how? Aren't we done?

I won't speak for everybody else, but I'm not done before I have
a metric.  Something that can return more than a boolean... as in,
"such-and-such scale in x % symmetrical at the 3:2", or something.

What makes sense here?  I've got:

"The percentage of a scale's modes containing a 3/2, in which the
pattern of intervals generating scale degrees between 1/1 and 3/2
generates no pitches outside the scale when started at 3/2."

Which adds a modal aspect to the metric.  But I wonder if there's
a raw measure of symmetry?

>>>I don't use the word "omnitetrachordality" in my paper but the
>>>concept is there.
>> 
>>Is there a place where you do use omnitetrachordality?
> 
>On the tuning list. It means the same thing as the property
>mentioned in my paper.

Cool.  I've seen it on the lists.  Just wondering if a Monzo
dictionary entry, or something, had been made.

-Carl


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Message: 1632

Date: Mon, 24 Sep 2001 11:45 +0

Subject: Re: Miracle theory

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9omkht+amc8@xxxxxxx.xxx>
Gene wrote:

> > 225:224 is also in the kernel for characteristic meantone, schismic 
> and 
> > "magic" scales.  There's nothing special about associating it with 
> > miracles. 
> 
> I dunno--it's associated with miracles in something like the way 
> 81/80 is associated to meantone, and that is a completely defining 
> way. But it seems you want more:

81:80 completely defines meantone because it converts the three 
dimensional 5-limit into a two-dimensional linear temperament.  (Or 2-D 
into 1-D with octave equivalence.)  One dimension is being lost, so you 
need one commatic unison vector.  225:224 is a 7-prime limit interval, and 
can't of itself define a linear temperament.


>  225:224, 2401:2400 and 3025:3024, however, completely define 
> > miracle temperament.
> 
> This means that h10 and h31 also completely define it; and we have 
> h41 = h10 + h31 and h72 = h10 + 2 h31 in this miracle kernel.

Yes, that's right.


> > I don't see why these are better views, other than for meantone.  
> As I 
> > make the last case, the interval of repetition is 3, and a 7:4 is 
> two 
> > generators.  5:3 and 7:5 are both a single generator.
> 
> I'm not sure what you mean by this--7/4 and 3 are two generators, 
> just as 3/2 and 2 are two generators or 5/3 and 4/3 are two 
> generators. Wherein lies the difference?

No, 7:4 and 3 are two intervals, but they aren't generators of this scale. 
 The 3 is wrong anyway.  I meant to say that the interval of equivalence 
is 1/9 of an octave, whereas you seemed to say 1/3.  5:3 and 7:5 are both 
equivalent to each other and to the generator in this system.  7:4 is not 
a generator -- it's represented by two generator steps, the same as 9:8 in 
meantone.  The system you describe is different to the one I get by 
following your instructions, so I don't understand where it comes from.


              Graham


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Message: 1633

Date: Mon, 24 Sep 2001 23:24:14

Subject: Re: semi-periodic scales

From: Carl Lumma

I wrote...

> "The percentage of a scale's modes containing a 3/2, in which the
> pattern of intervals generating scale degrees between 1/1 and 3/2
> generates no pitches outside the scale when started at 3/2."
> 
> Which adds a modal aspect to the metric...  But I wonder if there's
> a raw measure of symmetry?

One try is to root each mode of the scale on a random pitch, then
mulitply all those pitches by 3:2, then count the number of
new pitches and divide by the number of notes in the scale.

-Carl


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Message: 1634

Date: Mon, 24 Sep 2001 11:45 +0

Subject: Re: Miracle theory

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9olut9+7mhg@xxxxxxx.xxx>
Gene wrote:

> The miracle generator certainly does better than the meantone, which 
> may be why it is a miracle, but I am not convinced. The JI diatonic 
> scale requires 25 miracle steps from 5/3 at -13 to 9/8 at 12, whereas 
> the compass is 14 for the fourth-in-5/3 version. Paul was actually 
> worried that Jacky's Blackjacky sounded too diatonic, an issue which 
> would not even arise in the other system.

Why is the JI diatonic being taken as the standard?  Even so, your 
calculation only seems to cover one octave.  Merely to represent two 
octaves with the 5:3 equivalence, you need 26 fourths.  With octave 
equivalence, any number of octaves can be represented with the same number 
of miracle steps.  With the 5:3 equivalence, an 11:9 is 43 fourths, as 
compared to 3 Secors.  So a neutral third scale will come out a lot worse 
in your 5:3 system.

I thought Jacky's piece sounded too diatonic because the retuning wasn't 
working.

> > But the value for quarter comma meantone is the minimax for the 5-
> limit.  
> > There doesn't seem to be an analogy here.
> 
> I don't know what you mean by "the" minimax, nor why you could not do 
> a similar calculation here.

The 5-limit minimax is the tuning where the largest 5-limit error 
(relative to JI) is as small as possible.  The relevant limit for miracle 
is 11, and indeed Dave Keenan did this calculation a long time ago.  The 
result is a 116.7 cent generator giving a worst error of 3.3 cents.


                           Graham


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Message: 1635

Date: Mon, 24 Sep 2001 23:35:46

Subject: Re: semi-periodic scales

From: Paul Erlich

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:
> >>I wonder if Gene, or somebody else with any knowledge of group
> >>theory could help.
> > 
> >Help how? Aren't we done?
> 
> I won't speak for everybody else, but I'm not done before I have
> a metric.  Something that can return more than a boolean... as in,
> "such-and-such scale in x % symmetrical at the 3:2", or something.
> 
> What makes sense here?

How about the number of notes that stay the same, divided by the 
total number? So for the diatonic scale, it's 6/7 . . . for 
blackjack, 15/21 . . . for canasta, 25/31 . . . my decatonics would 
be 4/5 . . . one can easily calculate this as

1-(number of generators in 3:2)/(number of notes).


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Message: 1636

Date: Mon, 24 Sep 2001 23:37:02

Subject: Re: semi-periodic scales

From: Paul Erlich

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:

> One try is to root each mode of the scale on a random pitch, then
> mulitply all those pitches by 3:2, then count the number of
> new pitches and divide by the number of notes in the scale.

I don't see what the mode has to do with it. What changes when you 
label one note as the root, as opposed to not doing so?


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Message: 1637

Date: Tue, 25 Sep 2001 20:04:31

Subject: Re: semi-periodic scales

From: Carl Lumma

>> Which adds a modal aspect to the metric.  But I wonder if there's
>> a raw measure of symmetry?
> 
> Do you care about the relative proportions of the scale steps?

Not sure what you mean... sounds like the answer would be yes...
maybe an example would help make it clear.

-Carl


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Message: 1638

Date: Tue, 25 Sep 2001 20:08:20

Subject: Re: semi-periodic scales

From: Carl Lumma

> 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.

Don't really know what how autocorrelation is calculated.  How
many lines of code does it take?  You can take your pick between
posting sample code and attempting a verbal explanation.

> 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.

What I'm after here should definitely be independent of CS.

-Carl


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Message: 1639

Date: Tue, 25 Sep 2001 20:34:15

Subject: Re: semi-periodic scales

From: Paul Erlich

Carl, you and I independently came up with the same measure here, so 
aren't we done? That is, once you realize that mode is irrelevant.


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Message: 1640

Date: Tue, 25 Sep 2001 22:00:53

Subject: Re: Miracle theory

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9oo2ic+tpk6@e...>
> Gene wrote:

> > What would you suggest 
> > as a standard?

> The 11-limit tonality diamond.  

If you look at the 11-limit diamond, it takes an average of 10.4 
generator steps to get its elements using (7, 72) and 9.1 using 
(30, 53) within 5/3; however we need to adjust that value by 72/53, 
which gives 12.36; by this measure (7, 72) is better. However, where 
(30, 53) really shines is in generating harmonies; we have triads in 
a compass of 1, which adjusts to 1.358, compared to 13 for the 
miracle system; we have 7-chords in a compass of 4, which adjusts to 
5.434, compared to 13 again, and 11-chords in a compass of 6, 
adjusting to 8.151, as compared to 22. The (30, 53) system is jam-
packed with harmonies, which to my mind makes it more "normal"; it 
certainly helps make it practical.

> What???  So if it's practical, presumably you have music made with 
it to 
> show us ...

Save this sort of thing for Page Wizard or someone of that ilk, don't 
try it on me. :)

You are well-equipped by knowledge and ability to evaluate this scale 
without needing to write a piece of music in it, and so am I. Having 
said that, I did spend yesterday working on getting myself to the 
point of actually being able to produce music, and indeed plan on 
writing a piece to demonstrate this scale. The relevance of that to 
this discussion is more or less like the relevance of reading on 
April 10, 1912 on that Thomas Andrews was going to be on the maiden 
voyage of the Titanic--essentially nil, unless there is a design flaw.

You seem to have responded to a posting I canceled shortly after 
posting it--I didn't like my discussion of your minimax stuff, and 
was going to find the original and re-do it; however I looked through 
the archives and didn't locate it.


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Message: 1641

Date: Tue, 25 Sep 2001 22:15:04

Subject: Re: Miracle theory

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., graham@m... wrote:
> > In-Reply-To: <9oo2ic+tpk6@e...>
> > Gene wrote:
> 
> > > What would you suggest 
> > > as a standard?
> 
> > The 11-limit tonality diamond.  
> 
> If you look at the 11-limit diamond, it takes an average of 10.4 
> generator steps to get its elements using (7, 72) and 9.1 using 
> (30, 53) within 5/3; however we need to adjust that value by 72/53, 
> which gives 12.36; by this measure (7, 72) is better. However, 
where 
> (30, 53) really shines is in generating harmonies; we have triads 
in 
> a compass of 1, which adjusts to 1.358,

I don't understand the compass or the adjustment. What kind of triads 
are you talking about? How can you get three notes with only 1 
iteration of a generator??


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Message: 1642

Date: Tue, 25 Sep 2001 22:19:44

Subject: Re: Miracle theory

From: Paul Erlich

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

> However, where 
> (30, 53) really shines is in generating harmonies; we have triads 
in 
> a compass of 1,

Oh . . . I understand this now . . . but that's cheating! Since you 
don't have octave-equivalence, you're quite constrained in terms of 
the inversions of the triads you can use . . . this fact can't go 
unpenalized. In particular, to make nice music you'll often want to 
be able to use a bass note which is the implied fundamental of the 
chord.


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Message: 1643

Date: Tue, 25 Sep 2001 23:32:25

Subject: Re: semi-periodic scales

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:

> > Do you care about the relative proportions of the scale steps?

> Not sure what you mean... sounds like the answer would be yes...
> maybe an example would help make it clear.

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, 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.

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. 

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.


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Message: 1644

Date: Tue, 25 Sep 2001 23:38:02

Subject: Re: Miracle theory

From: genewardsmith@xxxx.xxx

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

> Oh . . . I understand this now . . . but that's cheating!

It would be cheating except for the fact that I adjusted by a factor 
of 72/53, which makes them directly comperable.

 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.


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Message: 1645

Date: Tue, 25 Sep 2001 23:38:52

Subject: Re: semi-periodic scales

From: Carl Lumma

>Carl, you and I independently came up with the same measure here,
>so aren't we done? That is, once you realize that mode is
>irrelevant.

Yeah, we may be done.  Just so long as nobody comes along and says
that the effect of a transposition on the entire scale doesn't
nec. represent the effects of transpositions on all of its subsets.

Also, I'm interested to find out how the autocorr. thing works in
Scala.  And maybe gene has a group-theoretical way of looking at
it.

-Carl


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Message: 1646

Date: Tue, 25 Sep 2001 00:11:34

Subject: Re: semi-periodic scales

From: Carl Lumma

>>One try is to root each mode of the scale on a random pitch, then
>>mulitply all those pitches by 3:2, then count the number of
>>new pitches and divide by the number of notes in the scale.
> 
>I don't see what the mode has to do with it. What changes when you 
>label one note as the root, as opposed to not doing so?

Quite possibly nothing... I'm cautious because I'm already thinking
through a layer of abstraction -- specifically, we don't actually
care about pitches moving (as we do for transpositional coherence),
we care about how much of an interval pattern is destroyed when we
_aren't_ allowed any new pitches.  I'm trying to think if this is
really equivalent to counting the new pitches we'd need.  I think
it is, but a lot of what I've thought has historically failed when
tested against "unusual" (improper, non-just) scales...

Assuming counting the changing pitches works, then we need to ask
if looking at a single mode, or even all the modes, is equivalent
to what we really care about.  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.

-Carl


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Message: 1647

Date: Tue, 25 Sep 2001 05:26:45

Subject: Re: semi-periodic scales

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:

> Which adds a modal aspect to the metric.  But I wonder if there's
> a raw measure of symmetry?

Do you care about the relative proportions of the scale steps?


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Message: 1648

Date: Tue, 25 Sep 2001 11:11 +0

Subject: Re: Miracle theory

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9oo0rt+s4tp@xxxxxxx.xxx>
Gene wrote:

> > No, 7:4 and 3 are two intervals, but they aren't generators of this 
> scale. 
> 
> They are the generators of a system of scales I mentioned, so I need 
> a context for "this scale"--which scale?

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

43/117, 49.0 cent generator

basis:
(0.1111111111111111, 0.040838783185694151)

mapping by period and generator:
[(9, 0), (15, -2), (22, -3), (26, -2)]

mapping by steps:
[(612, 441), (970, 699), (1421, 1024), (1718, 1238)]

unison vectors:
[[-5, -1, -2, 4], [-1, -7, 4, 1]]

highest interval width: 3
complexity measure: 27  (45 for smallest MOS)
highest error: 0.000170  (0.204 cents)
unique


So which scale do you mean?



                    Graham


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Message: 1649

Date: Tue, 25 Sep 2001 11:11 +0

Subject: Re: Miracle theory

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9oo2ic+tpk6@xxxxxxx.xxx>
Gene wrote:

> > Why is the JI diatonic being taken as the standard?
> 
> Mostly because of the Blackjacky discussion. What would you suggest 
> as a standard?

The 11-limit tonality diamond.  Or the 7-limit diamond, because miracle 
temperament works well with that too.

>   Even so, your 
> > calculation only seems to cover one octave.  Merely to represent 
> two 
> > octaves with the 5:3 equivalence, you need 26 fourths. 
> 
> That is why I suggested beginning the second 5/3 repetion, taking it 
> out to an octave, and then repeating that scale within an octave. I 
> think the result is entirely practical.

What???  So if it's practical, presumably you have music made with it to 
show us ...

> I take it you are using Paul's definition of 5-limit? The way I use 
> the word, this doesn't mean anything, but you can for instance look at
> 3^a 5^b where a^2+ab+b^2 < 2, which I recall you doing. I don't see 
> why the result cannot be used for either scale.

It can be, provided yours repeats at the octave.  But that isn't a 
calculation I've seen you doing.

> However, if you add 2 to the mix, you could consider 2^a 3^b 5^c with 
> a^2 + b^2 + c^2 + ab + ac + bc < 2, and get a closely related linear 
> programming exercise; the previous one being what one gets if we fix 
> 2 to its exact value. One could certainly fix 5/3 instead. Similar 
> comments apply to a least squares opitimization.

Don't know.  This is looking complicated.  The usual rule is that the 
numerator shouldn't exceed a particular integer limit.  In this case, 
limits from 7 to 12 would be appropriate.  You can apply that to any 
generator and period you like, but you haven't applied it here, so you 
don't have anything to show that your method is of value.


                            Graham


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