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Message: 1275 - Contents - Hide Contents

Date: Mon, 20 Aug 2001 02:38:38

Subject: Re: Mea culpa

From: Paul Erlich

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


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


>> I think you're jumping the gun with that intepretation. 
> 


> You were right--my method doesn't work for finding the homomorphism 
> in these cases. 



Whew! I thought I was going crazy :)



> 
> I'll try to sort this out tomorrow if I have time, but the image 
> under the homomorphism has to have a 2-torsion part. I think the deal 
> is h([a, b, c]) gets sent to [12*a+19*b+28*c, a+b+c (mod 2)].



Well, I look forward to the explanation. Let me 
know if there's a reference I should study to make 
this all more comprehensible.
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Message: 1276 - Contents - Hide Contents

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: 1277 - Contents - Hide Contents

Date: Mon, 20 Aug 2001 02:42:15

Subject: Re: Microtemperament and scale structure

From: Paul Erlich

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


> 
> [-12  0  0]
> [-19 -3  1]
> [-28  0  4]
> 
> The left hand column is the "homomorphism column vector" above (sign isn't 
> important as long as it's consistent).  It's identical to Gene's formula 
> by the definition of matrix inversion.  The 12 is Fokker's determinant.
> 
> 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?
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Message: 1278 - Contents - Hide Contents

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: 1279 - Contents - Hide Contents

Date: Mon, 20 Aug 2001 02:46:56

Subject: Re: Hypothesis

From: Paul Erlich

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


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


>> MOS is almost synonymous with WF (well-
>> formed) and that concept is explained in many 
>> papers, such as
>> 
>> 404 Not Found * [with cont.]  Search for http://depts.washington.edu/~pnm/CLAMPITT.pdf in Wayback Machine
>> 
>> except that in an MOS, the interval of repetition 
>> (which Clampitt calls interval of periodicity) can be 
>> a half, third, quarter, etc. of the interval of 
>> equivalence, and not necessarily equal to it.
> 


> 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?



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. The whole point of my 
Hypothesis is to show that these scales, and 
perhaps ultimately the entire interest of WF/MOS 
scales, in fact has a deeper basis in just intonation 
and periodicity blocks.
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Message: 1280 - Contents - Hide Contents

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: 1281 - Contents - Hide Contents

Date: Mon, 20 Aug 2001 02:59:42

Subject: Re: Microtemperament and scale structure

From: Paul Erlich

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



>> So--what's the claim?
> 


> The concept of generator is defined in the Carey/Clampitt paper that 
> Paul's already pointed you towards.  Or maybe it's only referred to, but 
> you'll get the idea.  I'm claiming I can uniquely define a generated scale 
> from a set of unison vectors.  The full process is defined by a Python 
> script.  It's something like:
> 
> Put the octave at the top of the matrix and the chromatic unison vector 
> next.



You still haven't told Gene what the claim is

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".

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).

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.
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Message: 1282 - Contents - Hide Contents

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: 1283 - Contents - Hide Contents

Date: Mon, 20 Aug 2001 03:35:48

Subject: Re: Microtemperament and scale structure

From: genewardsmith@xxxx.xxx

--- 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?



> 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. 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.

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.



> 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?)
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Message: 1284 - Contents - Hide Contents

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 * [with cont.]  (Wayb.)> is all about.  
> <Unison vectors * [with cont.]  (Wayb.)> is a list of examples.
> 
>                    Graham



I meant for the particular case which you erased above.
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Message: 1286 - Contents - Hide Contents

Date: Mon, 20 Aug 2001 03:39:21

Subject: Re: Hypothesis

From: genewardsmith@xxxx.xxx

--- 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?
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Message: 1287 - Contents - Hide Contents

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

Subject: Re: Mea culpa

From: carl@xxxxx.xxx


>> >OS, 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: 1288 - Contents - Hide Contents

Date: Mon, 20 Aug 2001 04:00:33

Subject: Re: Microtemperament and scale structure

From: genewardsmith@xxxx.xxx

--- 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!



> Tell me what JT means. 



To me, something defined in terms of rational numbers. What does it 
mean to you?
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Message: 1289 - Contents - Hide Contents

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

Subject: Re: Hypothesis

From: carl@xxxxx.xxx


>> >) 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: 1290 - Contents - Hide Contents

Date: Mon, 20 Aug 2001 04:07:50

Subject: Re: Mea culpa

From: genewardsmith@xxxx.xxx

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



> Whew! I thought I was going crazy :)



I knew my method didn't always work, but I had concluded it worked in 
the "interesting" cases. You came up with an "uninteresting" case of 
a kind I hadn't thought about, which turned out to be interesting.



> Well, I look forward to the explanation. Let me 
> know if there's a reference I should study to make 
> this all more comprehensible.



I hope the red-green show made some kind of sense. An introductory 
textbook on abstract algebra would be the place to start if you want 
to learn this stuff.

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?
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Message: 1291 - Contents - Hide Contents

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

Subject: Re: Hypothesis

From: carl@xxxxx.xxx

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: 1292 - Contents - Hide Contents

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

Subject: Re: Hypothesis

From: carl@xxxxx.xxx


> 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: 1294 - Contents - Hide Contents

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

Subject: Re: Mea culpa

From: carl@xxxxx.xxx

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 * [with cont.]  (Wayb.)



>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: 1296 - Contents - Hide Contents

Date: Mon, 20 Aug 2001 05:33:32

Subject: Re: Mea culpa

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., carl@l... wrote:



> The Just Intonation Network is here in SF:
> 
> The Just Intonation Network * [with cont.]  (Wayb.)



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.
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Message: 1297 - Contents - Hide Contents

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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Message: 1298 - Contents - Hide Contents

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

Subject: Re: Mea culpa

From: carl@xxxxx.xxx


>> >he Just Intonation Network is here in SF:
>> 
>> The Just Intonation Network * [with cont.]  (Wayb.)
> 


> 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: 1299 - Contents - Hide Contents

Date: Mon, 20 Aug 2001 20:31:55

Subject: Re: Hypothesis

From: Paul Erlich

--- In tuning-math@y..., carl@l... 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.
> 


Roughly, the reasoning is that slicing the lattice with parallel, 
hyperplanar slices is likely to minimize the number of "wolves" or 
broken consonances relative to using "bumpy" slices.
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