Tuning-Math Digests messages 9152 - 9176

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Message: 9152

Date: Thu, 15 Jan 2004 18:49:24

Subject: Re: The Atomischisma Scale

From: Carl Lumma

>This is the unique Fokker block you get by crossing a schisma with an 
>atom. I don't know if it is what Kirnberger got, because if it is, it 
>won't be in the Scala archive because the numers in the quotients are 
>too large to load (this provides and example of why monzos would 
>sometimes be useful.) I've presented it below as a Scala file, but it 
>isn't really. There is very little difference between this and 12-
>equal, proving that 12-et is really a form of 5-limit just intonation.
>
>! atomschis.scl
>Atom Schisma Scale
>12
>!
>156348578434374084375/147573952589676412928
>134217728/119574225
>1307544150375/1099511627776
>18014398509481984/14297995284350625
>10935/8192
>1709671705179880612640625/1208925819614629174706176
>16384/10935
>14297995284350625/9007199254740992
>2199023255552/1307544150375
>119574225/67108864
>295147905179352825856/156348578434374084375
>2

Rock!

-C.


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Message: 9153

Date: Thu, 15 Jan 2004 21:56:31

Subject: Re: summary -- are these right?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> >> By "unweighted" I probably mean a norm without coefficents for
> >> an interval's coordinates.
> >
> >?
> 
> The norm on Tenney space...
> 
> || |u2 u3 u5 ... up> || = log2(2)|u2|+log2(3)|u3|+ ... + log2(p)|up|
> 
> The 'coefficients on the intervals coordinates' here are log2(2),
> log2(3) etc.

So 'unweighted', 9 has a length of 2 but 11 has a length of 1 . . . :(

> >> This ruins the correspondence with
> >> taxicab distance on the odd-limit lattice given by 
Paul's/Tenney's
> >> norm,
> >
> >Huh? Which odd-limit lattice and which norm?
> 
> It's the same norm on a triangular lattice with a dimension for each
> odd number.

That's not a desirable norm.

> The taxicab distance on this lattice is log(odd-limit).

No it isn't -- try 9:5 for example.

> It's also the same distance as on the Tenney lattice, except perhaps
> for the action of 2s in the latter (I forget the reasoning there).

Try building up the reasoning from scratch.

> >> as on that
> >> lattice intervals do not have unique factorizations and thus a
> >> metric based on unit lengths is likely to fail the triangle
> >> inequality.
> >
> >Not following.
> 
> Hmm, maybe I was wrong.  I was thinking stuff like ||9|| = ||3|| = 1
> and thus ||3+3|| < ||3|| + ||3|| but that's ok.  It seems bad
> though, since the 3s are pointed in the same direction.

What lattice/metric was this about?


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Message: 9154

Date: Thu, 15 Jan 2004 22:02:43

Subject: Re: TOP on the web

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:
> On Mon, 12 Jan 2004 18:13:59 -0000, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> 
> >I've put up a TOP web page. It needs to have, at least, a 
discussion 
> >of equal and linear temperaments and Tenney complexity and badness 
> >added to it, but it should be valuable as a starter. Here it is:
> >
> >/root/tentop.htm *
> 
> I can plug the formulas into a program, and they seem to produce 
accurate
> results, but I have no idea how to generalize them to higher 
limits. There
> seems to be a pattern to them, but it'd take a long time and some 
guesswork
> to figure it out. Could you be more explicit about how this works 
(in
> language a non-mathematician programmer can understand)?
> 
> Also, how would this work for more than one comma?

Oh, so you were talking about one comma above? The formula is very 
easy then; my most recent posting of it is here:

Yahoo groups: /tuning/message/51762 *


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Message: 9155

Date: Thu, 15 Jan 2004 21:57:57

Subject: Re: Two questions for Gene

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > Hmm . . . that severely reduces the appeal of using epimericity 
to 
> > define our badness contour. In this case, we should probably use 
> > something that crosses zero error at some finite complexity.
> 
> A curious remark for a physics major to make. We know the list is 
> compelte, we just can't prove it, and if there were one more comma 
we 
> missed it would be of no concievable musical use anyway, as Dave 
> would be quick to point out.

Then why not define our badness contour to exclude anything of "no 
conceivable musical use"?


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Message: 9156

Date: Thu, 15 Jan 2004 22:08:41

Subject: Re: The Four Mistyschism Scales

From: Paul Erlich

Is Kirberger's approximation to 12-equal in Scala? It would be 
interesting if you actually had to go all the way out to atomic to 
maximize the number of Scala matches in this big 12-tone exercise of 
yours.

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> A hoary eccelesiastical joke has it that mysticism should really be 
> spelled "misty schism", so this is how I am spelling the four 
Fokker 
> blocks which come from misty and the schisma. The two most notable 
> things about these scales is that Scala classifies all four of them 
> as well-temperments, and that this concludes the classification, 
> though I'll check the latter claim. Mistyschism2 is the same scale 
as 
> Scala's duoden12.scl, about which we read "Almost equal 12-tone 
> subset of Duodenarium".
> 
> ! mistyschism1.scl
> Mistyschism scale 2048/2025 67108864/66430125
> 12
> !
> 524288/492075
> 9/8
> 1215/1024
> 512/405
> 4/3
> 64/45
> 3/2
> 262144/164025
> 2048/1215
> 3645/2048
> 256/135
> 2
> 
> ! mistyschism2.scl
> Mistyschism scale 2048/2025 67108864/66430125 = duoden12.scl
> 12
> !
> 135/128
> 9/8
> 1215/1024
> 512/405
> 4/3
> 64/45
> 3/2
> 405/256
> 2048/1215
> 3645/2048
> 256/135
> 2
> 
> ! mistyschism3.scl
> Mistyschism scale 2048/2025 67108864/66430125
> 12
> !
> 135/128
> 9/8
> 1215/1024
> 512/405
> 4/3
> 1476225/1048576
> 3/2
> 405/256
> 2048/1215
> 3645/2048
> 256/135
> 2
> 
> ! mistyschism4.scl
> Mistyschism scale 2048/2025 67108864/66430125
> 12
> !
> 524288/492075
> 9/8
> 1215/1024
> 512/405
> 4/3
> 64/45
> 3/2
> 405/256
> 2048/1215
> 3645/2048
> 256/135
> 2


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Message: 9160

Date: Thu, 15 Jan 2004 22:22:57

Subject: Re: Question for Dave Keenan

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> > If a timbre has
> > 
> > 2nd partial off by < 10.4 cents
> > 3rd partial off by < 16.5 cents
> > 4th partial off by < 20.8 cents
> > 5th partial off by < 24.1 cents
> > 6th partial off by < 26.9 cents
> > 
> > does it 'hold together' as a single pitch, or does it fall apart 
into 
> > multiple pitches?
> 
> Or is it experienced as somewhere in between, having a single but
> poorly-defined pitch. I don't know. I trust your ears more than mine
> on such questions, but I'd like to hear it.

OK, I'll put something together when I have a chance . . .

> Note that my objection to your claimed 5-limit temperaments with 
very
> large errors was not a disagreement between yours and my perception,
> but more of an epistemological problem. I sense you are setting up
> something to try to catch me in a similar logical mistake. This is
> fun. :-)
> 
> > (I'll try to prepare some examples, playing random scales . . .)
> > 
> > If yes:
> 
> What does "yes" mean here?

the sound holds together as a single pitch.

> > If I take any inharmonic timbre with one loud partial and some 
quiet, 
> > unimportant ones (very many fall into this category), and use a 
> > tuning system where
> > 
> > 2:1 off by < 10.4 cents
> > 3:1 off by < 16.5 cents
> > 4:1 off by < 20.8 cents
> > 5:1 off by < 24.1 cents
> > 6:1 off by < 26.9 cents
> > 
> > and play a piece with full triadic harmony, doesn't it follow 
that 
> > the harmony should 'hold together' the way 5-limit triads should?
> 
> I don't know. What has the single loud partial got to do with it? Is
> this partial one of those mentioned above? 

No, it essentially determines the pitch of the timbre.

> We know that with quiet sine waves nothing special happens with any
> dyad except a unison, and that loud sine waves work like harmonic
> timbres presumably due to harmonics

and combinational tones . . .

> being generated in the
> nonlinearities of the ear-brain system.

quiet harmonic timbres don't generate combinational tones, so they 
won't "work like" loud sine waves.

> Don't we?

That also ignores virtual pitch. A set of quiet sine waves can evoke 
a single pitch which does not agree with any combinational tone . . . 
at certain intervals, the pitch evoked will be least ambiguous, which 
is certainly 'something special happening' . . .

The fact is that, when using inharmonic timbres of the sort I 
described, Western music seems to retain all it meaning: certain 
(dissonant) chords resolving to other (consonant) chords, etc., all 
sounds quite logical. My sense (and the opinion expressed in 
Parncutt's book, for example) is that *harmony* is in fact very 
closely related to the virtual pitch phenomenon. We already know, 
from our listening tests on the harmonic entropy list, that the 
sensory dissonance of a chord isn't a function of the sensory 
dissonances of its constituent dyads. Furthermore, you seem to be 
defining "something special" in a local sense as a function of 
interval size, but in real music you don't get to evaluate each 
sonority by detuning various intervals various amounts, which 
this "specialness" would seem to require for its detection.

The question I'm asking is, with what other tonal systems, besides 
the Western one, is this going to be possible in.


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Message: 9161

Date: Thu, 15 Jan 2004 11:36:29

Subject: Re: Question for Dave Keenan

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> If a timbre has
> 
> 2nd partial off by < 10.4 cents
> 3rd partial off by < 16.5 cents
> 4th partial off by < 20.8 cents
> 5th partial off by < 24.1 cents
> 6th partial off by < 26.9 cents
> 
> does it 'hold together' as a single pitch, or does it fall apart into 
> multiple pitches?

Or is it experienced as somewhere in between, having a single but
poorly-defined pitch. I don't know. I trust your ears more than mine
on such questions, but I'd like to hear it.

Note that my objection to your claimed 5-limit temperaments with very
large errors was not a disagreement between yours and my perception,
but more of an epistemological problem. I sense you are setting up
something to try to catch me in a similar logical mistake. This is
fun. :-)

> (I'll try to prepare some examples, playing random scales . . .)
> 
> If yes:

What does "yes" mean here?

> If I take any inharmonic timbre with one loud partial and some quiet, 
> unimportant ones (very many fall into this category), and use a 
> tuning system where
> 
> 2:1 off by < 10.4 cents
> 3:1 off by < 16.5 cents
> 4:1 off by < 20.8 cents
> 5:1 off by < 24.1 cents
> 6:1 off by < 26.9 cents
> 
> and play a piece with full triadic harmony, doesn't it follow that 
> the harmony should 'hold together' the way 5-limit triads should?

I don't know. What has the single loud partial got to do with it? Is
this partial one of those mentioned above? 

We know that with quiet sine waves nothing special happens with any
dyad except a unison, and that loud sine waves work like harmonic
timbres presumably due to harmonics being generated in the
nonlinearities of the ear-brain system. Don't we?


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Message: 9164

Date: Thu, 15 Jan 2004 22:24:50

Subject: Re: A temperament naming convention?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> It seems to be that if a temperament in one prime limit has the same
> TOP tunings of the primes it includes as the tunings in the next
> higher prime limit, it ought to get the same name (I'm not 
suggesting
> the converse.) From that point of view, I've already screwed up 
once,
> with the meantone names. 
> 
> 5-limit meantone: [1201.698522, 1899.262910, 2790.257558]
> 
> septimal meantone: [1201.698521, 1899.262909, 2790.257556, 
3370.548328]
> 
> Septimal meantone and 5-limit meantone both are simply "meantone"
> under this naming convention.
> 
> The 11-limit 31&43, what I called "meantone", has a TOP tuning of
> [1201.611156, 1899.198965, 2790.351234, 3371.044615, 4145.302457].
> This would not necessarily get the name "meantone".
> 
> The 11-limit 31&50, which I called "meanpop", has a top tuning
> [1201.698521, 1899.262909, 2790.257556, 3370.548328, 4150.346670].
> Under the proposed convention, it should be called "meantone".
> 
> Should 31&50, with wedgie [1, 4, 10, -13, 4, 13, -24, 12, -44, -71],
> be given the name "meantone" and some other name be found for 31&43,
> with wedgie [1, 4, 10, 18, 4, 13, 25, 12, 28, 16]?

Sounds somewhat reasonable to me . . .


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Message: 9166

Date: Thu, 15 Jan 2004 22:27:35

Subject: Re: More on the naming convention

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> The 5-limit temperament with Ampersand's comma, 7-limit miracle, and
> 11-limit miracle all have identical TOP tunings of the primes they
> cover.

Wow. I'm assuming you gave this tuning in your message giving a few 7-
limit linear TOP tunings.

> Should the Ampersand temperament simply be called miracle?

If it's all the same pitches, I suppose that makes some sense . . .

> One might extend this convention to closest points among reasonable
> temperament choices. For instance, the 225/224-planar temperament 
has
> TOP tuning [1200.493660, 1901.172569, 2785.167472, 3370.211784]. The
> nearest reasonable TOP point in val space to this seems to be 11-
limit
> marvel, with TOP tuning [1200.508704, 1901.148724, 2785.132538,
> 3370.019002, 4149.558115]. The projection of this point into 7-limit
> val space is at a distance of 0.06867 cents from 225/224-planar TOP,
> which seems to me to justify my proposal to give the 7 and 11 limit
> planar temperaments the same name of "marvel".

But they have different pitches, right?


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Message: 9168

Date: Thu, 15 Jan 2004 22:29:03

Subject: Re: Two questions for Gene

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > Then why not define our badness contour to exclude anything 
of "no 
> > conceivable musical use"?
> 
> Fine, but then it would be worth mentioning that you really didn't 
> need to.

If there is some other comma out there satisfying the epimericity 
requirement, then you *did* really need to.


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Message: 9171

Date: Thu, 15 Jan 2004 22:31:30

Subject: Re: The Four Mistyschism Scales

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > Is Kirberger's approximation to 12-equal in Scala? It would be 
> > interesting if you actually had to go all the way out to atomic 
to 
> > maximize the number of Scala matches in this big 12-tone exercise 
> of 
> > yours.
> 
> Is atomic a comma of 12-et?

Yes. <12 19 28|161 -84 -12> = 0
(Did I notate this incorrectly?)


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Message: 9173

Date: Thu, 15 Jan 2004 22:58:12

Subject: Re: The Four Mistyschism Scales

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > Yes. <12 19 28|161 -84 -12> = 0
> > (Did I notate this incorrectly?)
> 
> Wow. Is the tuning list ready for this kind of stuff?

Well, this is probably the only way you're going to get Kirnberger's 
tuning (the one that's a JI approximation of 12-equal: a chain of 
schisma-flattened fifths), and that's the sort of tuning you might 
find in Scala.


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