Tuning-Math messages 702 - 726

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Message: 702

Date: Mon, 20 Aug 2001 04:26:27

Subject: Re: Hypothesis

From: carl@l...

> I looked at CLAMPITT.pdf, and it seems to me the argument that
> there is something interesting about WF scales is extremely
> unconvincing.  Can anyone actually *hear* this? I notice that
> when you talk about periodiciy blocks, you ignore this stuff
> yourself, as well you might so far as I can see.
> 
> What gives? Am I missing something?

Howdy, Gene!

I doubt the "synechdochic property" (the "self-similarity" at the
center of the Carey and Clampitt article) is significant, except
maybe in very special kinds of musical examples and with a lot of
training.  In my opinion the Carey and Clampitt article amounts to
some interesting ideas for algorithmic composition.

I don't think MOS itself means much for the perception of melody.
Rather, I think it works together, or is often confounded with
other properties:

() Symmetry at the 3:2.  The idea is that the 3:2 is a special
interval, a sort of 2nd-order octave.  When a scale's generator
is 3:2, MOS means that a given pattern can more often be repeated
a 3:2 away.  Chains of 5, 7, and 12 "fifths" are historically
favored, but where are all the MOS chains of 5:4, 7:4, etc.?  In
my experience, MOS chains of non-fifth generators can be special
too, but we should be careful not to give MOS credit for symmetry
at the 3:2.

() Myhill's property -- every scale interval comes in exactly two
acoustic sizes.  This may make it easier for listeners to track
scale intervals.  Consider a musical phrase that is transposed to
a different mode of the diatonic scale -- it is changed with
respect to acoustic intervals but unchanged with respect to scalar
intervals.  I think this is an important musical device that is
only possible with certain kinds of scales.  Myhill's property
may make it easier for the listener to access such a device, but
probably doesn't mean much if the scale can't support the device
in the first place.  Here, I believe a property called "stability"
comes into play.[1]  Fortunately, we can test this by listening
to un-stable MOS scales.  I've done some of this listening
informally.

-Carl

[1]
Rothenberg, David. "A Model for Pattern Perception with Musical
Applications. Part I: Pitch Structures as Order-Preserving Maps",
Mathematical Systems Theory vol. 11, 1978, pp. 199-234.
 
Rothenberg, David. "A Model for Pattern Perception with Musical
Applications Part II: The Information Content of Pitch structures",
Mathematical Systems Theory vol. 11, 1978, pp. 353-372.
 
Rothenberg, David. "A Model for Pattern Perception with Musical
Applications Part III: The Graph Embedding of Pitch Structures",
Mathematical Systems Theory vol. 12, 1978, pp. 73-101.


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Message: 703

Date: Mon, 20 Aug 2001 04:45:13

Subject: Re: Mea culpa

From: carl@l...

Forgive me for stepping in here guys, but I'm online and
figure that sooner is better...

> I must say I am surprised and pleased with the attitude around
> here.  The one time I tried to publish about music, the Computer
> Music Journal turned it down as "too mathematical", so I thought
> people  were a little allergic. I would like a copy of that paper
> now, and I  could put it up on a web page--I think I sent a copy
> to some just intonation library in San Francisco--does that ring
> any bells?

The Just Intonation Network is here in SF:

The Just Intonation Network *

>PB I presume means periodicity block, and MOS is some kind of
>jumped-up well-formed scale, I understand. Could you similarly
>define MOS (and WF while you are at it?)

MOS, WF, and Myhill's property are all equivalent.  They are
usually given as something like:

MOS or WF- any pythagorean-type scale in which the generating
interval always spans the same number of scale degrees.

While strict pythagorean scales are usually generated with
3:2's against 2:1's, MOS and WF allow any generator, and
sometimes the interval of equivalence is allowed to be non-2:1.

Myhill's property- all generic scale intervals have exactly
two specific sizes.

-Carl


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Message: 705

Date: Mon, 20 Aug 2001 07:03:57

Subject: Re: Mea culpa

From: carl@l...

>> The Just Intonation Network is here in SF:
>> 
>> The Just Intonation Network *
> 
> Thanks. Do you know if it has a library and if it would still
> have a paper I sent to it back in the mid-80's? People have been
> getting  copies somehow, I've heard, and I suspect it comes from
> there.

They do in fact have a tremendous library, mostly of stuff from
the 80's, when the Network was at its peak.  Unfortunately it
is very disorganized, to the point where the chance they'll know
if they have thing x is less than 50%, and it would take hours,
even days to say for sure.  Xeroxes, dot-matrix printouts abound,
in boxes in Henry Rosenthal's basement. 

-Carl


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Message: 706

Date: Mon, 20 Aug 2001 08:25:16

Subject: The hypothesis

From: genewardsmith@j...

I found a posting by Paul over on the tuning group, and it seems I 
may be closing on a statement of the Paul Hypothesis.

"In fact, a few months ago I posted my Hypothesis, which states that 
if you temper out all but one of the unison vectors of a Fokker 
periodicity block, you end up with an MOS scale. We're discussing 
this Hypothesis on tuning-math@y..."

Sounds like we may be getting there, but there seems to be some 
confusion as to whether 2 counts as a prime, and so whether for 
instance the 5-limit is 2D or 3D. Most of the time it makes sense to 
treat 2 like any other prime.

"A temperament can be 
constructed by tempering out anywhere from 1 to n unison vectors. If 
you temper out n (and do it uniformly), you have an ET. If you temper 
out n-1, you have a linear temperament. If you temper out n-2, you 
have a planar temperament (Dave Keenan has created some examples of 
those)."

From my point of view, the 5-limit is rank (dimension) 3, and the 7-
limit 4, and so forth. If you temper out n-1 unison vectors which 
generate a well-behaved kernel, then you map onto a rank-1 group, and 
get an equal temperment. So "codimension" 1 (one less than the full 
number of dimensions) leads to a rank-1 group. In the same way, 
codimension 2 for the kernel leads to a rank 2 group, etc. If for 
instance you temper out 81/80 in the 5 limit, the kernel has 
dimension 1 and codimension 2, and leads to a rank 2 image group.

We can tune the rank 1 group any way we like so long as the steps are 
of the same size, which means that our ET can have stretched or 
squashed octaves if we so choose. In the same way, we can tune the 
rank 2 group any way we like, except that we need to retain 
incommensurability of two generators (or at least to ignore the fact 
if they are not.) If we make the octaves pure in our example where 
the kernel is generated by a comma, we could for instance make the 
fifths pure also, leading to Pythagorean tuning. Alternatively, we 
could make the major thirds pure, leading to 1/4 comma mean tone 
temperment. (Pythagorean tuning is not considered a temperment, since 
the fifth isn't tempered, but it is the same sort of thing 
mathematically as 1/4 comma mean-tone temperment.) Other choices lead 
to other results, and all we need to do is to ensure the circle of 
fifths does not close--or at least to pretend otherwise it if it does.

A rank 3 image group, coming from a kernel of codimension 3, is what 
people have been calling a 2D temperment. I hope that clarifies 
things (as it does for me) rather than further confuses them!


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Message: 708

Date: Mon, 20 Aug 2001 11:45 +0

Subject: Re: Microtemperament and scale structure

From: graham@m...

In-Reply-To: <9lq0ik+d44m@e...>
In article <9lq0ik+d44m@e...>, genewardsmith@j... () wrote:

> At last we are making progress! I don't see much role for 
> the "chromatic" element here, though. If the n-1 unison vectors are 
> linearly independent, we've already seen recently how to tell if they 
> generate a kernel of something mapping to Z: compute the gcd of the 
> determinant minors, and see if it is 1 or not. If they have no common 
> factor, then they define such a mapping, and the "chromatic vector" 
> will go to a certain number of steps in this mapping--hopefully 1, 
> but perhaps 2, 3, 4 ... etc.

Yes, the chromatic UV should be redundant.  That reminds me of two 
hypotheses I didn't get round to implementing in Python code:

1) If you use an octave-invariant matrix, with the chromatic UV at the 
top, the left hand column of the adjoint matrix is the mapping by 
generator, like the second column was before.  (May need to be divided 
through by the GCD.)  That makes the adjoint matrix a list of generator 
mappings (the octave being a special case where it's specific).

2) Any simple interval will do for the chromatic UV, so long as all the 
UVs are linearly independent.  I don't have strict criteria for 
"simple" here, any more than criteria for what work as unison vectors in 
the first place.  But trying [1 0 0 ...], [0 1 0 ...], etc until something 
works (non-zero determinant) should do the trick.  I don't think such 
things can be described as unison vectors, but I believe they do work in 
this context.

So are you saying you have an equivalent to (2)?  I don't think the GCD 
has to be 1.

> As for temperment, that has to do with tuning and you cannot draw any 
> conclusions about tuning unless you introduce it into your statement
> somewhere--nothing in, nothing out.

I suppose it depends on how you define "temperament".  Is "meantone" a 
temperament or a class of temperaments?  The chromatic UV is used to 
define the tuning.  If you want to push the definition and make a third a 
unison vector, you can define quarter comma meantone by setting it just.  
So the commatic UVs define the temperament class and the chromatic UV is 
used to define the specific tuning.  I make the octave explicit for the 
same reason.

> > The strong form says that if you construct the 
> > Fokker (hyperparallelepiped) periodicity block 
> > from the n unison vectors, and again 1 is 
> > chromatic and n-1 are commatic, then the notes in 
> > the PB form an MOS scale.
> 
> PB I presume means periodicity block, and MOS is some kind of jumped-
> up well-formed scale, I understand. Could you similarly define MOS 
> (and WF while you are at it?)

Whatever they mean, MOS and WF are the same thing: a generated scale with 
only two step sizes.  It may be a useful property for alternative systems 
of tonality, but we don't have enough examples to pronounce on that yet.  
It's certainly useful to think about when designing generalized keyboards 
or alternative notation systems.  (See 
<http://www.anaphoria.com/xen3b.PDF - Ok *> if you haven't already.)

                 Graham


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Message: 709

Date: Mon, 20 Aug 2001 11:45 +0

Subject: Re: Microtemperament and scale structure

From: graham@m...

In-Reply-To: <9lpte7+27ap@e...>
Paul wrote:

> > The other columns happen to be the generator mappings for the 
> > equivalent column being a chromatic unison vector.
> > I don't think there's a proof for 
> > this always working yet, but it does.
> 
> Can you show with examples?

It's what <Unison vector to MOS script *> is all about.  
<Unison vectors *> is a list of examples.

                   Graham


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Message: 711

Date: Mon, 20 Aug 2001 18:28:26

Subject: Re: Microtemperament and scale structure

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > Gene -- first of all, start with a set of n unison 
> > vectors. The unison vectors that are tempered out 
> > or completely ignored are called "commatic unison 
> > vectors". The unison vectors that amount to a 
> > musically significant difference, but not (often) 
> > large enough to move you from one scale step to 
> > the next, are called "chromatic unison vectors".
> 
> Thanks! I'd guessed that was what it meant. I think you are adding 
to 
> the confusion by calling both of them "unison vectors", though--why 
> not unison and step vectors instead?

Three reasons:

1) The number of notes in the scale should be (normally) the 
determinant of the matrix of unison vectors. One has to include both 
the chromatic and the commatic unison vectors in this calculation.

2) In the "prototypical" case, the commatic unison vector is "the 
comma", 81:80; and the chromatic unison vector is "the chromatic 
unison" or "augmented unison", 25:24. These define a 7-tone 
periodicity block: the diatonic scale. You see how the terminology is 
just a generalization of this case.

3) "Step vectors" would refer to something else. In the prototypical 
example, the step vectors would be 16:15, 10:9, and 9:8.
> 
> > The weak form of the hypothesis simply says that 
> > if there is 1 chromatic unison vector, and n-1 
> > commatic unison vectors, then what you have is a 
> > linear temperament, with some generator and 
> > interval of repetition (which is usually equal to the 
> > interval of equivalence, but sometimes turns out to 
> > be half, a third, a quarter . . . of it).
> 
> At last we are making progress! I don't see much role for 
> the "chromatic" element here, though.

You're right . . . it plays no role here.

> If the n-1 unison vectors are 
> linearly independent, we've already seen recently how to tell if 
they 
> generate a kernel of something mapping to Z:

No -- you did that with n unison vectors -- I'm not counting the 2 
axis as a "dimension" here.
> 
> > The strong form says that if you construct the 
> > Fokker (hyperparallelepiped) periodicity block 
> > from the n unison vectors, and again 1 is 
> > chromatic and n-1 are commatic, then the notes in 
> > the PB form an MOS scale.
> 
> PB I presume means periodicity block, and MOS is some kind of 
jumped-
> up well-formed scale, I understand. Could you similarly define MOS 
> (and WF while you are at it?)

MOS means that there is an interval of repetition (normally equal to 
the interval of equivalence [usually octave], but sometimes it comes 
out as half, third, quarter . . . of the IE). The scale repeats 
itself exactly within each interval of repetition. Within each, there 
is a generating interval, which is iterated some number of times such 
that the scale has two step sizes.

Examples:

The diatonic scale (LsssLss) is MOS: the IoR is an octave, and the 
generator is L+s+s.

The melodic minor scale (LssssLs) is not MOS: there is no generator 
that produces all the notes and no others.


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Message: 712

Date: Mon, 20 Aug 2001 18:39:04

Subject: Re: Hypothesis

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > There are a tremendous number of arguments as 
> > to why there is something interesting about WF or 
> > MOS scales in the literature. Personally, I buy very 
> > few of them, if any. But there are some very 
> > powerful WF/MOS scales around, especially, of 
> > course, the usual diatonic scale, and the usual 
> > pentatonic scale. 
> 
> Unless I am missing something (highly likely at this point!) the 
> pentatonic and diatonic scales are WF in mean tone intonation but 
not 
> in just intonation. Is that right? If it is right, doesn't that 
serve 
> to make the whole idea seem fishy?

Strict, fixed-pitch just intonation has almost never been used in 
actual music with these scales. This is because of the so-
called "comma problem". Don't let the JI advocates fool you: 
Pythagorean tuning and various meantone-like temperaments have been 
far more important than fixed-pitch 5-limit just intonation for the 
actual performance of these scales -- even in China!

One thing I forgot to mention about the hypothesis: if you don't use 
the parallelepiped, you might end up with a scale that is not MOS, 
but has the same number of notes as the MOS that comes from the 
parallelepiped. I conjecture that in some precise sense, the MOS has 
more consonant structures (intervals and/or chords) than the 
corresponding non-MOS. This is seen, for example, in the decatonic 
case, where the chromatic unison vector is one member of the set 
{25:24, 28:27, 49:48}, and the commatic unison vectors are two 
members of the set {50:49, 64:63, 225:224}. The MOS is LssssLssss, 
which has 8 consonant 7-limit tetrads (4 4:5:6:7s and 4 
1/7:1/6:1/5:1/4s); while a melodically superior non-MOS scale, 
LsssssLsss, has only 6.


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Message: 713

Date: Mon, 20 Aug 2001 18:41:50

Subject: Re: Microtemperament and scale structure

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > But, in this case, if you temper out the schisma and 
> > the diesis, you're tempering out their sum, which 
> > means you're tempering out _two_ syntonic 
> > commas . . . which means that you're either 
> > tempering out the syntonic comma, or setting it to 
> > half an octave.
> 
> I'm afraid that is where the "torsion" I was talking about comes 
in. 
> Suppose you color all 5-limit notes either green or red, by making 
> [a,b,c] green if a+b+c is even, and red if it is odd. Then two reds 
> add up to a green, a green and a red to a red, and two greens a 
green.
> 
> Your two generators are green, but the comma is red. The generators 
> generate only greens, but you need two reds to get a green. Hence 
the 
> image under the homomorphism goes to a 12 et note, but there is a 
red 
> keyboard and a green keyboard!

Are you saying that both keyboards are tuned identically, or that 
there may be an offset?
> 
> > Tell me what JT means. 
> 
> To me, something defined in terms of rational numbers. What does it 
> mean to you?

I was just asking what it stood for. "Just Tuning"?


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Message: 714

Date: Mon, 20 Aug 2001 18:45:05

Subject: Re: Hypothesis

From: Paul Erlich

--- In tuning-math@y..., carl@l... wrote:
> 
> I don't think MOS itself means much for the perception of melody.
> Rather, I think it works together, or is often confounded with
> other properties:
> 
> () Symmetry at the 3:2.  The idea is that the 3:2 is a special
> interval, a sort of 2nd-order octave.  When a scale's generator
> is 3:2, MOS means that a given pattern can more often be repeated
> a 3:2 away.  Chains of 5, 7, and 12 "fifths" are historically
> favored, but where are all the MOS chains of 5:4, 7:4, etc.?  In
> my experience, MOS chains of non-fifth generators can be special
> too, but we should be careful not to give MOS credit for symmetry
> at the 3:2.

Did you get this from me? 'Cause you know I agree. But see the 
message I just posted about why MOSs appear to be _harmonically_ 
special for the class of scales with given step sizes and number of 
notes.


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Message: 715

Date: Mon, 20 Aug 2001 18:46:18

Subject: Re: Mea culpa

From: Paul Erlich

--- In tuning-math@y..., carl@l... wrote:
> 
> MOS, WF, and Myhill's property are all equivalent.

This is not quite true -- for example, LssssLssss is MOS but not WF 
and doesn't have Myhill's property.


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Message: 716

Date: Mon, 20 Aug 2001 18:49:32

Subject: Re: The hypothesis

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> I found a posting by Paul over on the tuning group, and it seems I 
> may be closing on a statement of the Paul Hypothesis.
> 
> "In fact, a few months ago I posted my Hypothesis, which states 
that 
> if you temper out all but one of the unison vectors of a Fokker 
> periodicity block, you end up with an MOS scale. We're discussing 
> this Hypothesis on tuning-math@y..."
> 
> Sounds like we may be getting there, but there seems to be some 
> confusion as to whether 2 counts as a prime, and so whether for 
> instance the 5-limit is 2D or 3D. Most of the time it makes sense 
to 
> treat 2 like any other prime.

Well I've been treating 5-limit as 2D, following Fokker. In many 
contexts, it's important to keep 2 as an additional dimension -- but 
not in this context.

> I hope that clarifies 
> things (as it does for me) rather than further confuses them!

Well it certainly seems that you understand what we're talking about!


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Message: 717

Date: Mon, 20 Aug 2001 18:55:06

Subject: Re: Microtemperament and scale structure

From: Paul Erlich

--- In tuning-math@y..., graham@m... wrote:
> 
> I suppose it depends on how you define "temperament".  
Is "meantone" a 
> temperament or a class of temperaments?  The chromatic UV is used 
to 
> define the tuning.

You mean the commatic UVs (81:80 in the case of meantone)?

> If you want to push the definition and make a third a 
> unison vector, you can define quarter comma meantone by setting it 
just.  

Now I think you're pushing definitions too far. Let's not forget the 
strong form of the hypothesis!

> So the commatic UVs define the temperament class and the chromatic 
UV is 
> used to define the specific tuning.

Hmm . . . perhaps one _can_ define things this way, but it's by no 
means universal. How would one define LucyTuning in this way??
> 
> Whatever they mean, MOS and WF are the same thing: a generated 
scale with 
> only two step sizes.

Not the same thing. Clampitt lists all the WFs in 12-tET, and there 
is no sign of the diminished (octatonic) scale, or any other scale 
with an interval of repetition that is a fraction of an octave. These 
are all MOS scales, though.


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Message: 718

Date: Mon, 20 Aug 2001 18:55:44

Subject: Re: Microtemperament and scale structure

From: Paul Erlich

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9lpte7+27ap@e...>
> Paul wrote:
> 
> > > The other columns happen to be the generator mappings for the 
> > > equivalent column being a chromatic unison vector.
> > > I don't think there's a proof for 
> > > this always working yet, but it does.
> > 
> > Can you show with examples?
> 
> It's what <Unison vector to MOS script *> is all about.  
> <Unison vectors *> is a list of examples.
> 
>                    Graham

I meant for the particular case which you erased above.


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Message: 719

Date: Mon, 20 Aug 2001 19:20:53

Subject: Re: Mea culpa

From: carl@l...

>> MOS, WF, and Myhill's property are all equivalent.
> 
> This is not quite true -- for example, LssssLssss is MOS but not WF 
> and doesn't have Myhill's property.

What single generator produces the scale?

-Carl


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Message: 720

Date: Mon, 20 Aug 2001 19:23:18

Subject: Re: Hypothesis

From: carl@l...

>> () Symmetry at the 3:2.  The idea is that the 3:2 is a special
>> interval, a sort of 2nd-order octave.  When a scale's generator
>> is 3:2, MOS means that a given pattern can more often be repeated
>> a 3:2 away.  Chains of 5, 7, and 12 "fifths" are historically
>> favored, but where are all the MOS chains of 5:4, 7:4, etc.?  In
>> my experience, MOS chains of non-fifth generators can be special
>> too, but we should be careful not to give MOS credit for symmetry
>> at the 3:2.
> 
> Did you get this from me? 'Cause you know I agree.

Absolutely -- I've long credited you with it, even in a pre-send
version of that post.

> But see the message I just posted about why MOSs appear to be
> _harmonically_ special for the class of scales with given step
> sizes and number of notes.

I didn't catch the why, but I am of course familiar with the
example you gave.

-Carl


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Message: 721

Date: Mon, 20 Aug 2001 19:26:19

Subject: Re: Hypothesis

From: carl@l...

I wrote...

>> But see the message I just posted about why MOSs appear to be
>> _harmonically_ special for the class of scales with given step
>> sizes and number of notes.
> 
> I didn't catch the why, but I am of course familiar with the
> example you gave.

I mean, I caught that they are non-parallelpiped PBs, but not
why this should translate into fewer harmonic structures (do
you mean only complete chords? total consonant dyads?).

-Carl


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Message: 724

Date: Mon, 20 Aug 2001 20:30:09

Subject: Re: Mea culpa

From: Paul Erlich

--- In tuning-math@y..., carl@l... wrote:
> >> MOS, WF, and Myhill's property are all equivalent.
> > 
> > This is not quite true -- for example, LssssLssss is MOS but not 
WF 
> > and doesn't have Myhill's property.
> 
> What single generator produces the scale?
> 
> -Carl

One possibility is s -- here the interval of repetition is the half-
octave.


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