Tuning-Math Digests messages 3025 - 3049

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Message: 3025

Date: Mon, 07 Jan 2002 05:00:16

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: clumma

>>>Right, it's the chinese pentatonic.  I threw it in for
>>>completeness.
>>> 
>>>Completeness of what?
>> 
>>Of the journey from North to South, and of the survey of
>>pentatonic scales in motivic ethnic music of southeast asia.
> 
>You listen to three selections and call that a survey?

Uh-huh.  I didn't make any statistical claims about it.  In
fact the only claims I make about it are:

() it's better than a 'survey' of the scala scl archive
() it's better than a 'survey' without the Thai recording.

>>>Give me a strict JI scale to try.
>> 
>>1/1 5/4 4/3 3/2 15/8
>> 
>>Sounds like a fine pelog to me.
> 
>Yuck -- reminds me of Charles Lucy.

Me too.

>Totally inauthentic.

But much closer than 5-of-7.

>>>>I think the tuning on these discs is closer to JI than 23-tET,
>>>>and I don't hear them avoiding a disjoint interval.  Do you?
>>> 
>>>Avoiding a disjoint interval? You mean you hear it as 5-of-7?
>>>It modulates that much??
>> 
>>Actually, it doesn't.  They seem to stick mostly to I, IV, and
>>III (diatonic) with the bass, if you consider those tonics.  But
>>the melodic stuff does center itself on every degree of the
>>scale -- it treats the "bad" 4ths the same as the perfect 4ths.
> 
>What "bad" 4th are you referring to??

The one between 15/8 and 4/3 in the scale above.

>>To rephrase the question one more time, in what sense are these
>>bass notes tonics?  Do they change anything about the melody?
>>That is, what used to be scale degree 4 is now 1?  I say they
>>don't.
> 
> Fine -- I don't see how this is related to anything I'm saying.

I thought somebody was claiming there was an impetus for 5-limit
temperament in Javanese music.

>>What I hear is a fixed 1.  The melody is a very slow
>>series of scale degrees above that 1.  On each note of the melody,
>>a bunch of ornamentation is hung, which is made of scale arpeggio
>>bits.  The bass starts and ends on 1, and goes to 3, 2, and
>>sometimes 4 (I-IV-III-V diatonic), to provide a sense of
>>tension/resolution.
> 
>How do you get I-IV-III-V out of a 5-tone scale? And these chords
>sound major or minor to you?

I-IV-III-V as if pelog was a 5-out-of diatonic scale.  Am I
forgetting the case of me numerals?

Would be 1-3-2-4 in pelog.  1 and 3 sound major, 2 minor.  I
can't tell 4.

-Carl


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Message: 3026

Date: Mon, 07 Jan 2002 09:21:35

Subject: Heuristic rules!

From: paulerlich

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > #1
> > 
> > 2^-90 3^-15 5^49
> > 
> > This is not only the the one with lowest badness on the list, it 
is 
> the smallest comma, which suggests we are not tapering off, and is 
> evidence for flatness.
> > 
> > Map:
> > 
> > [ 0  1]
> > [49 -6]
> > [15  0]
> > 
> > Generators: a = 275.99975/1783 = 113.00046/730; b = 1
> > 
> > I suggest the "Woolhouse" as a name for this temperament, because 
> of the 730. Other ets consistent with this are 84, 323, 407, 1053 
and 
> 1460.
> > 
> > badness: 34
> > rms: .000763
> > g: 35.5
> > errors: [-.000234, -.001029, -.000796]
> > 
> > #2 32805/32768 Schismic badness=55
> > 
> > #3 25/24 Neutral thirds badness=82
> > 
> > #4 15625/15552 Kleismic badness=97
> > 
> > #5 81/80 Meantone badness=108
> > 
> > It looks pretty flat so far as this method can show, I think.
> 
> How well do these results back up my now-famous (I hope) heuristic, 
> which involves only the size of the numbers in, and the difference 
> between numerator and denominator of, the unison vector? How might 
we 
> weight the gens and/or cents measures so that the heuristic will 
work 
> perfectly?

Let's start with #5 and work our way up to #1:

                 U            V           W      X       Y      Z
unisonvector  rms(oct)  |n-d|/(d*log(d)) V/U     g     log(d)  X/Y 
------------  --------   --------------- ----  ----    ------  ---
80/81         0.003517     0.002809      .799  2.944   4.394   .67
15552/15625  0.0008583    0.0004838      .564  4.546   9.657   .471
24/25         0.024083     0.012427      .516  1.414   3.219   .439
32768/32805  0.00013475   0.00010847     .805  6.976  10.398   .671
[90 15 -49]  6.358e-007   3.44e-007      .541  35.5   78.862   .450

Our current "g" measure is clearly too large when the generator is a 
fifth, as I've been trying to complain for quite a while now and only 
Dave Keenan has changes his ways accordingly, and the comparison with 
the heuristic, though agreeable, suggests that the heuristic is in 
fact better than our current measure. What if we weighted the 
intervals unequally in both the g and in the "rms" calculations? 
Could we get the heuristic to work perfectly? I think that would be 
very interesting for our paper.

Moving on to some relatively "bad" examples . . . Gene, 
your "Enneadecal" comma should have a power of 2 equal to 14, not 15 
as you said, right?

                 U            V           W      X       Y      Z
unisonvector  rms(oct)  |n-d|/(d*log(d)) V/U     g     log(d)  X/Y 
------------  --------   --------------- ----  ----    ------  ---
[52 17 -34]  2.864e-005    1.5102e-005   .527  24.042  54.721  .439
128/135       0.01508        0.010571    .701  2.94    4.905   .599
[14 19 -19]  8.733e-005     5.314e-005   .608  15.513  30.579  .507
648/625      0.009217        0.005716    .620  3.27    6.438   .508
[8 14 -13]   0.0002305      0.0001463    .635  11.045  20.923  .528

Here the heuristic seems to work even better (and the V/U and X/Y are 
well within the range of their values for the top 5).


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Message: 3027

Date: Mon, 07 Jan 2002 02:18:22

Subject: Re: tetrachordality

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >>The interval pattern stuff (the L-L-s of conventional theory) is
> >>a relative pitch thing... ?
> > 
> >Not sure what you mean.
> 
> I'm trying to understand the psychoacoustical basis for the
> version in your paper, and recent posts about ethnic scales
> on the main list (x+x+y, etc.).

You mean when I talk about tetrachordal (now homotetrachordal) 
octave species of ethnic scales?

> And I'm trying to understand
> the lack, if any, of a psychoacoustical basis for my stuff
> (absolute pitches being transposed by a 3:2).

Psychoacoustic basis? Well, when you hear the step pattern repeated 
at a 4:3 or a 3:2, it spells a more coherent tonal organization than 
when you hear it repeated at some other interval. The patterns sound 
more similar to one another in the first case, so the information 
can be "compressed" more efficiently, if that makes any sense . . .


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Message: 3028

Date: Mon, 07 Jan 2002 05:51:58

Subject: A 72-et decatonic

From: genewardsmith

There were 66 scales of this type, but one was something of a standout, so I'll give just it:

[0, 5, 14, 19, 28, 33, 42, 49, 58, 63]
[5, 9, 5, 9, 5, 9, 7, 9, 5, 9]
edges 11 24 34   connectivity 0 4 6


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Message: 3029

Date: Mon, 07 Jan 2002 09:23:16

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >> I've never heard a voice
> > 
> > Voice?
> 
> As in, part or parts in the music sharing the same rhythm.

So what does the sentence,

"I've never heard a voice in the music that was triads, Paul."

mean? You haven't heard parallel triads? Me either!


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Message: 3030

Date: Mon, 07 Jan 2002 02:31:26

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >>
> >>s represents the blur of the spectral components coming in.
> >>How could an inharmonic timbre change that?
> >
> >When we're dealing with a dyad consisting of complex tones, and 
> >trying to apply harmonic entropy to that dyad, s is decreased
> >below the value that sine waves in place of the complex tones
> >would imply.  The more inharmonic the timbre, the less s is
> >decreased below the sine-wave case.
> 
> Still doesn't explain how.  You need a way for data from the
> combination-sensitive stuff to improve the spectral stuff coming
> off the cochlea.

There's less ambiguity as to the possible ratio-intepretations.

> I don't think it works that way.

It does. Devise a baby mathematical model for it and you'll see.

> The "accuracy"
> of the "fundamental" is improved as the spectral components get
> closer to just, as the harmonic entropy calc. itself correctly
> models.  But to change s in this way is a fudge, in my opinion.

It would be nice to derive the change in s mathematically, and any 
baby model will do so. However, until the model is more fully 
developed, I'm hesitant to put forward any exact formulas about how 
s changes.

> With harmonic timbres, h.e. on the fundamentals is a good
> approximation of things, but with inharmonic timbres, all spectral
> components need to be put in to the h.e. calculation.  Jacking up
> s may approximate this, but it would be a fudge.

Again, if the inharmonicities are only about 50 cents, I thing it's 
a pretty darn good one.

> Anyway, there is now psychoacoustic evidence for harmonic entropy.
> In fact, it looks like it perfectly models what happens in
> populations of "combination-sensitive" neurons in the inferior
> colliculus.  At least in bats.  I plan on posting to
> harmonic_entropy on this as soon as I can get the citations
> together.

You might want to mention this to J on the main tuning list, who 
seems to think that Terhardt and I are the only two people in the 
world who believe there is such a thing as "virtual pitch".

> >>>>Yes, to me, pelog sounds like a I and a III with a 4th in
> >>>>the middle.  But the music seems to use a fixed tonic, with
> >>>>not much in the way of triadic structure.
> >>>
> >>>How about 5-limit intervals?
> >>
> >>Not sure what you're asking.
> >
> >Not much in the way of 5-limit intervals?
> 
> I think the large 2nd approximates a 5:4, and the perfect
> 4th a 3:2, with some tempering to reduce the roughness of
> these intervals on the instrumentation used (as opposed to
> tempering to improve the consonance of these interval in
> different modes, to distribute any commas, etc.).

Why don't you experiment with this with a number of timbres. I know 
you can't, but I can, and believe me, reducing roughness does 
absolutely nothing to capture an authentic Gamelan sound.

> >>Right, it's the chinese pentatonic.  I threw it in for
> >>completeness.
> > 
> >Completeness of what?
> 
> Of the journey from North to South, and of the survey of
> pentatonic scales in motivic ethnic music of southeast asia.

You listen to three selections and call that a survey?

> And it was informative; nothing about the music changed as
> we went from Pelog, to the hybrid, to the chinese pentatonic
> _except_ the scale.  You could transcribe the notes and wind
> up with the same stuff, more or less.

Probably to people in that part of the world Beethoven and the blues 
sound the same. Go live in Southeast Asia for twelve years and then 
get back to me.

> >>Western music uses progressions of four fifths and expects
> >>to wind up on a major third.
> > 
> >These don't have to be triadic, harmonic progression.
> 
> I guess not.  But there's a big difference in how this stuff
> is used.  The Indonesia music is motivic, not modal.  At least,
> I follow the pitches and their positions in the scale, not the
> intervals of the scale and there relation to one another.

Right, those, you might say, are "incidental". Still, they give the 
sound a certain texture.

> The
> harmonic motion is used to render some consonance, and some
> tension/release action, but that's it.  It's a backdrop to the
> motivic material.

The same is true in Western music, in a certain sense.

> >>I didn't notice anything like this
> >>for the [1 -3] map (right?) on the cited discs.
> > 
> >[3 1]. It's not something you should expect to hear as a triadic
> >harmonic progression. It's simply the way the 5-limit intervals
> >fit together in the scale. If they didn't, the scale, and the
> >music that depends on it, wouldn't work.
> 
> [3 1]?  I thought these maps expressed each odd identity, from
> three to the limit, increasing from left to right, in numbers
> of generators.  Thus up one 3:2 for the 3:2, and down three 3:2s
> for the 5:4.

I though you meant the unison vector. OK ,[1 -3] map looks right.

> >>But I think a lot of the other stuff that goes along with
> >>harmonic music is missing from this music.  Western music
> >>requires meantone.  The pelog 5-limit map is far more extreme,
> >>but what suffers in this music as we change the tuning from
> >>5-of- 7, to 23, to 16, all the way to strict JI?
> > 
> >23 and 16 give you the Pelog sound. 7 doesn't.
> 
> By gods, you're right!  7-of-5 doesn't sound like pelog at all.

5-of-7?
> 
> >Give me a strict JI scale to try.
> 
> 1/1 5/4 4/3 3/2 15/8
> 
> Sounds like a fine pelog to me.

Yuck -- reminds me of Charles Lucy. Totally inauthentic.

> >>I think the tuning on these discs is closer to JI than 23-tET,
> >>and I don't hear them avoiding a disjoint interval.  Do you?
> > 
> >Avoiding a disjoint interval? You mean you hear it as 5-of-7?
> >It modulates that much??
> 
> Actually, it doesn't.  They seem to stick mostly to I, IV, and
> III (diatonic) with the bass, if you consider those tonics.  But
> the melodic stuff does center itself on every degree of the
> scale -- it treats the "bad" 4ths the same as the perfect 4ths.

What "bad" 4th are you referring to??

> To rephrase the question one more time, in what sense are these
> bass notes tonics?  Do they change anything about the melody?
> That is, what used to be scale degree 4 is now 1?  I say they
> don't.

Fine -- I don't see how this is related to anything I'm saying.

> What I hear is a fixed 1.  The melody is a very slow
> series of scale degrees above that 1.  On each note of the melody,
> a bunch of ornamentation is hung, which is made of scale arpeggio
> bits.  The bass starts and ends on 1, and goes to 3, 2, and
> sometimes 4 (I-IV-III-V diatonic), to provide a sense of
> tension/resolution.

How do you get I-IV-III-V out of a 5-tone scale? And these chords 
sound major or minor to you?


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Message: 3031

Date: Mon, 07 Jan 2002 06:20:48

Subject: Re: Optimal 5-Limit Generators For Dave

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > > Wouldn't it be nice to say whether or not Fokker's methods 
would 
> > have worked if he 
> > > had run into torsion?
> 
> > I'm pretty sure the answer is no. Gene?
> 
> I don't know they are. What would he have done in the case of the > 
24-note business which was our first example?

I'd guess he would just leave it as a 24-tone JI scale, but we'll 
never know, as he only ever published 12-, 19-, 22-, 31-, 41-, and 53-
tone PBs (and left them all as JI scales).


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Message: 3032

Date: Mon, 07 Jan 2002 09:33:54

Subject: Distinct p-limit intervals and ets

From: genewardsmith

Here's an interesting computation--the first ets which distinguish all
the p-limit intervals for p = 5,7,11,13 and 17. Anyone who cares to
may join in the fun by filling in the gaps, or extending the list of
p-limits.

5: 9,11,12,14,15,16,18,19 ...

7: 27,31,35,36,37,40,41,42 ...

11: 58,65,72,73,80,84,87,89 ...

13: 87,94,95,103,111,113,118,120 ...

17: 149,151,159,161,163,169,170,171 ...


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Message: 3033

Date: Mon, 07 Jan 2002 06:24:47

Subject: Re: Some 12-tone, 2-step 46-et scales

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > Sure. Now are there some 46-tET commas you did not take into 
account? 
> > You didn't answer my "hyper-torus" point on the tuning list 
yet . . .
> 
> I wasn't clear what you meant, but there are topological 
>considerations which come into graph theory. A graph can be a planar 
>graph, for instance, or a graph on a quotient (cylinder or torus), 
>so it can have a genus--it might be a graph on something with 
>negative curvature. I plan on reading some graph theory and seeing 
>if anything I run across suggests some application.

It shouldn't be so complicated. You can draw a big giant graph for 
all of 72-tET, can't you?


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Message: 3034

Date: Mon, 07 Jan 2002 09:38:40

Subject: Re: Distinct p-limit intervals and ets

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Here's an interesting computation--the first ets which distinguish 
all the p-limit intervals for p = 5,7,11,13 and 17.

There are an infinite number of 5-prime limit intervals. You must 
mean odd limit, in which case you should include 9 and 15.

>Anyone who cares to may join in the fun by filling in the gaps, or 
>extending the list of p-limits.
> 
> 5: 9,11,12,14,15,16,18,19 ...
> 
> 7: 27,31,35,36,37,40,41,42 ...
> 
> 11: 58,65,72,73,80,84,87,89 ...
> 
> 13: 87,94,95,103,111,113,118,120 ...
> 
> 17: 149,151,159,161,163,169,170,171 ...

This is what we call "uniqueness", and Manuel has compiled vast 
tables of both this and "consistency" for both odd and integer 
limits, and fractional ETs. Please see

Definitions of tuning terms: unique, (c) 1998 by Joe Monzo *


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Message: 3035

Date: Mon, 07 Jan 2002 06:28:36

Subject: Re: Some 12-tone, 2-step 46-et scales

From: genewardsmith

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

> It shouldn't be so complicated. You can draw a big giant graph for 
> all of 72-tET, can't you?

Certainly. What are you getting at, though?


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Message: 3036

Date: Mon, 07 Jan 2002 09:43:02

Subject: Re: Distinct p-limit intervals and ets

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > Here's an interesting computation--the first ets which distinguish 
> all the p-limit intervals for p = 5,7,11,13 and 17.
> 
> There are an infinite number of 5-prime limit intervals. You must 
> mean odd limit, in which case you should include 9 and 15.

Thus my mention of gaps.


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Message: 3037

Date: Mon, 07 Jan 2002 06:40:56

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >>>Right, it's the chinese pentatonic.  I threw it in for
> >>>completeness.
> >>> 
> >>>Completeness of what?
> >> 
> >>Of the journey from North to South, and of the survey of
> >>pentatonic scales in motivic ethnic music of southeast asia.
> > 
> >You listen to three selections and call that a survey?
> 
> Uh-huh.  I didn't make any statistical claims about it.  In
> fact the only claims I make about it are:
> 
> () it's better than a 'survey' of the scala scl archive

Maybe. We still haven't heard which scales Dave used, and where they 
came from. There's lots of other research -- on the internet alone.

> >>>Give me a strict JI scale to try.
> >> 
> >>1/1 5/4 4/3 3/2 15/8
> >> 
> >>Sounds like a fine pelog to me.
> > 
> >Yuck -- reminds me of Charles Lucy.
> 
> Me too.
> 
> >Totally inauthentic.
> 
> But much closer than 5-of-7.

5-of-7 gives you neutral thirds.

> >>>>I think the tuning on these discs is closer to JI than 23-tET,
> >>>>and I don't hear them avoiding a disjoint interval.  Do you?
> >>> 
> >>>Avoiding a disjoint interval? You mean you hear it as 5-of-7?
> >>>It modulates that much??
> >> 
> >>Actually, it doesn't.  They seem to stick mostly to I, IV, and
> >>III (diatonic) with the bass, if you consider those tonics.  But
> >>the melodic stuff does center itself on every degree of the
> >>scale -- it treats the "bad" 4ths the same as the perfect 4ths.
> > 
> >What "bad" 4th are you referring to??
> 
> The one between 15/8 and 4/3 in the scale above.

So the fact that they're not avoiding it helps prove I'm right! It's 
a GOOD fourth in Pelog, since 135:128 vanishes! It's a bad fourth in 
JI, so _if_ the tuning were JI, _then_ there might be a tendency to 
avoid that interval.

> >>To rephrase the question one more time, in what sense are these
> >>bass notes tonics?  Do they change anything about the melody?
> >>That is, what used to be scale degree 4 is now 1?  I say they
> >>don't.
> > 
> > Fine -- I don't see how this is related to anything I'm saying.
> 
> I thought somebody was claiming there was an impetus for 5-limit
> temperament in Javanese music.

Still don't see any relationship.

> >>What I hear is a fixed 1.  The melody is a very slow
> >>series of scale degrees above that 1.  On each note of the melody,
> >>a bunch of ornamentation is hung, which is made of scale arpeggio
> >>bits.  The bass starts and ends on 1, and goes to 3, 2, and
> >>sometimes 4 (I-IV-III-V diatonic), to provide a sense of
> >>tension/resolution.
> > 
> >How do you get I-IV-III-V out of a 5-tone scale? And these chords
> >sound major or minor to you?
> 
> I-IV-III-V as if pelog was a 5-out-of diatonic scale.  Am I
> forgetting the case of me numerals?

I think so.

> Would be 1-3-2-4 in pelog.  1 and 3 sound major,

So C major and E major ???

> 2 minor.  I
> can't tell 4.


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Message: 3038

Date: Mon, 07 Jan 2002 09:43:03

Subject: Re: Distinct p-limit intervals and ets

From: paulerlich

> Definitions of tuning terms: unique, (c) 1998 by Joe Monzo *

Monz, the links to the tables are outdated. Manuel, could you provide 
the updated links?


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Message: 3039

Date: Mon, 07 Jan 2002 06:42:11

Subject: Re: A 72-et decatonic

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> There were 66 scales of this type, but one was something of a 
standout, so I'll give just it:
> 
> [0, 5, 14, 19, 28, 33, 42, 49, 58, 63]
> [5, 9, 5, 9, 5, 9, 7, 9, 5, 9]
> edges 11 24 34   connectivity 0 4 6

Are you now taking into account _all_ the consonances of 72-tET?


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Message: 3040

Date: Mon, 07 Jan 2002 09:54:07

Subject: Re: Heuristic rules!

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>                  U            V           W      X       Y      Z
> unisonvector  rms(oct)  |n-d|/(d*log(d)) V/U     g     log(d)  X/Y 
> ------------  --------   --------------- ----  ----    ------  ---
> 80/81         0.003517     0.002809      .799  2.944   4.394   .67
> 15552/15625  0.0008583    0.0004838      .564  4.546   9.657   .471
> 24/25         0.024083     0.012427      .516  1.414   3.219   .439
> 32768/32805  0.00013475   0.00010847     .805  6.976  10.398   .671
> [90 15 -49]  6.358e-007   3.44e-007      .541  35.5   78.862   .450

I was going to ask what this was, but I see when I follow-up the table makes more sense.

> Our current "g" measure is clearly too large when the generator is a 
> fifth,

Why? 

 as I've been trying to complain for quite a while now and only 
> Dave Keenan has changes his ways accordingly, and the comparison with 
> the heuristic, though agreeable, suggests that the heuristic is in 
> fact better than our current measure.

Better for what? If you mean as a badness measure, g isn't one.

 What if we weighted the 
> intervals unequally in both the g and in the "rms" calculations? 

We could, but I was never clear how exactly you wanted this done.


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Message: 3041

Date: Mon, 07 Jan 2002 06:43:49

Subject: Re: Some 12-tone, 2-step 46-et scales

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> 
> > It shouldn't be so complicated. You can draw a big giant graph 
for 
> > all of 72-tET, can't you?
> 
> Certainly. What are you getting at, though?

If you do that, you take _all_ the commas of 72-tET into account.


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Message: 3042

Date: Mon, 07 Jan 2002 10:08:01

Subject: Re: Heuristic rules!

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> >                  U            V           W      X       Y      Z
> > unisonvector  rms(oct)  |n-d|/(d*log(d)) V/U     g     log(d)  
X/Y 
> > ------------  --------   --------------- ----  ----    ------  ---
> > 80/81         0.003517     0.002809      .799  2.944   4.394   .67
> > 15552/15625  0.0008583    0.0004838      .564  4.546   
9.657   .471
> > 24/25         0.024083     0.012427      .516  1.414   
3.219   .439
> > 32768/32805  0.00013475   0.00010847     .805  6.976  
10.398   .671
> > [90 15 -49]  6.358e-007   3.44e-007      .541  35.5   
78.862   .450
> 
> I was going to ask what this was, but I see when I follow-up the 
table makes more sense.
> 
> > Our current "g" measure is clearly too large when the generator 
is a 
> > fifth,
> 
> Why? 

Because of two systems with the same "g" measure, the one which has a 
generator of a 3:2 is musically simpler. Movements by 3:2 are easier 
to hear than movements by 5:3 or 5:4.

>  as I've been trying to complain for quite a while now and only 
> > Dave Keenan has changes his ways accordingly, and the comparison 
with 
> > the heuristic, though agreeable, suggests that the heuristic is 
in 
> > fact better than our current measure.
> 
> Better for what? If you mean as a badness measure, g isn't one.

I mean as a measure of the musical complexity of the system.

> >  What if we weighted the 
> > intervals unequally in both the g and in the "rms" calculations? 
> 
> We could, but I was never clear how exactly you wanted this done.

The suggestion I gave you in the past was, in the g measure, to 
weight the 3:2 by 1/log(3), 5:4 by 1/log(5), and 5:3 by 1/log(5). 
This is what Dave Keenan is now using. It's unclear to me that an 
root-mean-square calculation for g is exactly what we want, though 
it's probably a good approximation.

But my nagging suspicion is that some perfectly musically reasonable 
formulations of g, and of 'rms' or the cents-error measure, should 
conform to my heuristics _exactly_, or at least way better than the 
current measures are. You're the mathematician, can't you figure this 
out? Read my original justification for the heuristics again if you 
need to (message #1437), and consider the van Prooijen lattice: 
Searching Small Intervals * and 
Searching Small Intervals * except for the last graph. This 
should be doable!


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Message: 3043

Date: Mon, 07 Jan 2002 06:55:56

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: clumma

>>I thought somebody was claiming there was an impetus for 5-limit
>>temperament in Javanese music.
>
>Still don't see any relationship.

I'm trying to show that the things in Western music that led
to temperament are absent in Indonesian music.

>>>How do you get I-IV-III-V out of a 5-tone scale? And these chords
>>>sound major or minor to you?
>> 
>>I-IV-III-V as if pelog was a 5-out-of diatonic scale.  Am I
>>forgetting the case of me numerals?
> 
>I think so.
> 
>>Would be 1-3-2-4 in pelog.  1 and 3 sound major,
> 
> So C major and E major ???

C and F major.

>> 2 minor.

E minor.

>>I can't tell 4.

G something.

-Carl


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Message: 3044

Date: Mon, 07 Jan 2002 10:09:58

Subject: Re: Heuristic rules!

From: paulerlich

I wrote,
 
> Searching Small Intervals * and 
> Searching Small Intervals *

Oops -- the second link should be

lattice orientation *


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Message: 3045

Date: Mon, 07 Jan 2002 07:01:14

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >>I thought somebody was claiming there was an impetus for 5-limit
> >>temperament in Javanese music.
> >
> >Still don't see any relationship.
> 
> I'm trying to show that the things in Western music that led
> to temperament are absent in Indonesian music.

So what? They may have had their own reasons, and inharmonicity makes 
the situations rather different. Gamelan intervals are "pastelized", 
as Margo Schulter says.

> 
> >>>How do you get I-IV-III-V out of a 5-tone scale? And these chords
> >>>sound major or minor to you?
> >> 
> >>I-IV-III-V as if pelog was a 5-out-of diatonic scale.  Am I
> >>forgetting the case of me numerals?
> > 
> >I think so.
> > 
> >>Would be 1-3-2-4 in pelog.  1 and 3 sound major,
> > 
> > So C major and E major ???
> 
> C and F major.

How did the note A get in there?

> >> 2 minor.
> 
> E minor.
> 
> >>I can't tell 4.
> 
> G something.

You mean you can't hear which notes make up the chord?


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Message: 3046

Date: Mon, 07 Jan 2002 10:15:45

Subject: Enneadecal?

From: paulerlich

Gene, you didn't reply:

'Gene, your "Enneadecal" comma should have a power of 2 equal to 14, 
not 15 as you said, right?'

I'm referring to your results in message #2121.


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Message: 3047

Date: Mon, 07 Jan 2002 07:02:26

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: paulerlich

Carl, it seemed you yourself just gave evidence for the 135:128 
vanishing, didn't you?


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Message: 3048

Date: Mon, 07 Jan 2002 10:52:25

Subject: Re: Enneadecal?

From: genewardsmith

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> Gene, you didn't reply:
> 
> 'Gene, your "Enneadecal" comma should have a power of 2 equal to 14, 
> not 15 as you said, right?'

If that's what makes it small, of course. If not, obviously not.


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Message: 3049

Date: Mon, 07 Jan 2002 07:03:41

Subject: Re: Some 12-tone, 2-step 46-et scales

From: genewardsmith

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> I was facinated to discover that the 7,5 system did a little better than the completely symmetrical 6,6 system.

Here are the graphs. Looking at these, 12 might be a good place to center.

> [0, 4, 8, 12, 16, 20, 23, 27, 31, 35, 39, 43]
> [4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 3]
> edges   24   24   40   connectivity   3   3   6

Yahoo groups: /tuning-math/files/Gene/graph/g5_1.GIF *
Yahoo groups: /tuning-math/files/Gene/graph/g7_1.GIF *
Yahoo groups: /tuning-math/files/Gene/graph/g11_1.GIF *

> [0, 4, 8, 12, 16, 20, 24, 27, 31, 35, 39, 43]
> [4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 3]
> edges   24   25   41   connectivity   3   3   6

Yahoo groups: /tuning-math/files/Gene/graph/g5_2.GIF *
Yahoo groups: /tuning-math/files/Gene/graph/g7_2.GIF *
Yahoo groups: /tuning-math/files/Gene/graph/g11_2.GIF *


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