Tuning-Math Digests messages 9178 - 9202

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Message: 9178

Date: Fri, 16 Jan 2004 00:41:56

Subject: Re: More on the naming convention

From: Carl Lumma

>Not to be confused with "miracle-21", which is a 21-note miracle tuning.
>You could have both a prefix and suffix number, such as "7-meantone-19"
>for a 19-note scale of 7-limit meantone.

The convention is square brackets for cards, "miracle[21]", etc.

-Carl


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Message: 9179

Date: Fri, 16 Jan 2004 23:19:03

Subject: Re: Question for Dave Keenan

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> > What does "yes" mean here?
> 
> the sound holds together as a single pitch.

My guess is that it will be experienced as a single pitch, but one
that cannot be accurately determined. The pitch will be fuzzy or vague
in a similar way to that of a harmonic note of very short duration.

> > > 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.

So the waveform is essentially sinusoidal? Why not use sinusoidal
waves for this thought experiment?

> > 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 . . .

Good point.

> > 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' . . .

How many sine waves in an approximate harmonic series do you need for
this to be experienced? And what arrangements work? I was only
speaking of dyads.

> 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.

If by "Western tonal systems", you mean any based on approximating
small whole number ratios of frequency, and by "something special" you
mean "consonance and dissonance between simultaneous tones when using
only sine waves", then I suppose the answer is "none".

What's your point?


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Message: 9180

Date: Fri, 16 Jan 2004 23:24:13

Subject: Re: Anyone care to name a temperament?

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> There have been complaints I name far too many temperaments around
> here, so I hope someone will help me out.

But I also complained that far too many temperaments get named far too
early (by anyone). And then later when more is learned about them
other names seem more appropriate and if we change we cut ourselves
off from existing material on it in the archives. Whereas if we just
stuck to _describing_ it until a name was really needed, we wouldn't
have this problem.


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Message: 9183

Date: Fri, 16 Jan 2004 10:38:38

Subject: Re: summary -- are these right?

From: Carl Lumma

>> 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 . . . :(

Unless you use odd-limit.

>> >> 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.

This is what you were claiming in 1999.

""
But the basic insight is that a triangular lattice, with
Tenney-like lengths, a city-block metric, and odd axes or
wormholes, agrees with the odd limit perfectly, and so is
the best octave-invariant lattice representation (with
associated metric) for anyone as Partchian as me.
""

>> 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.

Here's what I was trying to remember...

"""
The reason omitting the 2-axis forces one to make the lattice
triangular is that typically many more powers of two will be
needed to bring a product of prime factors into close position
than to bring a ratio of prime factors into close position. So
the latter should be represented by a shorter distance than the
former. Simply ignoring distances along the 2-axis and sticking
with a rectangular (or Monzo) lattice is throwing away
information.
//
... a weight of log(axis) should be applied to all axes, and
if a 2-axis is included, a rectangular lattice is OK. If a
2-axis is not included, a triangular lattice is better.
//
... in an octave-specific rectangular (or parallelogram)
lattice, 7:1 and 5:1 are each one rung and 7:5 is two rungs. In
an octave-specific sense, 7:1 and 5:1 really are simpler than
7:5; the former are more consonant.  7:4 and 5:4 are each three
rungs in the rectangular lattice, but they still come out a little
simpler than 7:5 since the rungs along the 2-axis are so short.
If you can buy that 35:1 is as simple as 7:5, then the octave-
specific lattice really should be rectangular, not triangular.
35:1 is really difficult to compare with 7:5 -- it's much less
rough but also much harder to tune . . .
"""

>> 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?

Unweighted odd-limit taxicab.  By "3+3" I meant adding two 3
vectors.  The equation is 1 < 2.  It violates...

""
The city block distance has the very important property that
if an interval arises most simply as the sum of two simpler
intervals, the metric of the first interval is the sum of the
metrics of the other two.
""

...where clearly you were impling log() weighting.

More golden oldies at:
[Paul Hahn] *

-Carl


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Message: 9185

Date: Fri, 16 Jan 2004 15:29:36

Subject: Re: 46 augmented scales

From: Carl Lumma

>> I'm getting more scales out of this than I expected; below are 46
>> distinct augmented scales obtained by reducing the 53 Fokker blocks to
>> augmented.
>
>There are so many of these I decided to do a little sorting out. By
>"diameter" I mean a sort of complexity measure, where the difference
>between the largest and smallest values

In cents?

>for each of the generators is
>found and the maximum taken. Here are the scales with diameter less
>than eight.

Can you think of another term?  Paul Hahn has used it in a graph-
theory sense...

Music (and Music Theory) *

-Carl


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Message: 9188

Date: Fri, 16 Jan 2004 23:52:38

Subject: Re: TOP history

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> Paul, could you tell us something about when and how you discovered 
> TOP? I'd like to add some history to my top page.

It's exactly what I've been pleading to you guys to help me figure 
out last year and probably even earlier, except without octave-
equivalence. The idea was to temper out commas uniformly over their 
length in the lattice, to see what error function this was optimal 
with respect to, and to then apply this same error function to 
optimize temperaments with more than one comma. The posts asking 
about this can be found in the archives here.

When I made that 'waterfall' plot, I thought about replacing the 
former 'octave-equivalent' quantities, that Gene called 'heuristics', 
with their octave-specific versions in the Tenney lattice. Sometime 
shortly after that, and after I posted an example of the kind of 
tempering that makes the new 'heuristics' exact, I was at home and 
grasped that there was no 'limit' to the set of intervals satisfying 
the particular error function, minimax tenney-weighted error, that 
was being optimized here.


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Message: 9189

Date: Fri, 16 Jan 2004 23:54:07

Subject: Re: The Atomischisma Scale

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> 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,

If it's a chain of 11 schisma-flattened fifths, it is. Monzos would 
help.


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Message: 9190

Date: Fri, 16 Jan 2004 23:54:59

Subject: Re: TOP on the web

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
> <paul.hjelmstad@u...> wrote:
> 
> > What does BP stand for? Thanks
> 
> I'm using it to mean music which excludes 2s, but originally it 
> referred to dividing the 12th into 13 parts.

Better stick to the original definition, or anything that pertains to 
this labyrinthine set of webpages:

The Bohlen-Pierce Site *


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Message: 9192

Date: Fri, 16 Jan 2004 17:56:56

Subject: Re: Question for Dave Keenan

From: Carl Lumma

>> 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' . . .
>
>How many sine waves in an approximate harmonic series do you need for
>this to be experienced? And what arrangements work?

Only 2 in many cases.

-Carl


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Message: 9193

Date: Fri, 16 Jan 2004 17:54:23

Subject: Re: summary -- are these right?

From: Carl Lumma

>> >> 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 . . . :(
>> 
>> Unless you use odd-limit.
>
>Please elaborate on how that's 'unweighted' in your view.

On a unit-length odd-limit lattice both 9 and 11 have length 1.
I'm not claiming anything necessarily good about this, note.
I am asking for comments about it however.

>> >> >> This ruins the correspondence with taxicab distance on
>> >> >> the odd-limit lattice given by Paul's/Tenney's norm,
//
>> >> It's the same norm on a triangular lattice with a dimension
>> >> for each odd number.
>> > 
>> > That's not a desirable norm.

Why not?

>> >> The taxicab distance on this lattice is log(odd-limit).
>> > 
>> > No it isn't -- try 9:5 for example.
>> 
>> This is what you were claiming in 1999.
>> 
>> ""
>> But the basic insight is that a triangular lattice, with
>> Tenney-like lengths, a city-block metric, and odd axes or
>> wormholes, agrees with the odd limit perfectly, and so is
>> the best octave-invariant lattice representation (with
>> associated metric) for anyone as Partchian as me.
>> ""
>
>Right -- you need those odd axes, which screws up uniqueness,
>and thus most of how we've been approaching temperament.

But does the metric agree with log(odd-limit) or not?
For 9:5, log(oddlimit) is log(9).  If you run it through
the "norm" you get... 2log(3) + log(5).  Not the same,
it seems.  However if you followed the
lumma.org/stuff/latice1999.txt link, apparently Paul Hahn
did present a metric that agrees with log(odd-limit).

>> >> 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.
>> 
>> Here's what I was trying to remember...
>
>citation?

You wrote it in 1999.  I'm afraid I can't tell you anything
more specific than that.

>> """
>> The reason omitting the 2-axis forces one to make the lattice
>> triangular is that typically many more powers of two will be
>> needed to bring a product of prime factors into close position
>> than to bring a ratio of prime factors into close position. So
>> the latter should be represented by a shorter distance than the
>> former. Simply ignoring distances along the 2-axis and sticking
>> with a rectangular (or Monzo) lattice is throwing away
>> information.
>> //
>> ... a weight of log(axis) should be applied to all axes, and
>> if a 2-axis is included, a rectangular lattice is OK. If a
>> 2-axis is not included, a triangular lattice is better.
>> //
>> ... in an octave-specific rectangular (or parallelogram)
>> lattice, 7:1 and 5:1 are each one rung and 7:5 is two rungs. In
>> an octave-specific sense, 7:1 and 5:1 really are simpler than
>> 7:5; the former are more consonant.  7:4 and 5:4 are each three
>> rungs in the rectangular lattice, but they still come out a little
>> simpler than 7:5 since the rungs along the 2-axis are so short.
>> If you can buy that 35:1 is as simple as 7:5, then the octave-
>> specific lattice really should be rectangular, not triangular.
>> 35:1 is really difficult to compare with 7:5 -- it's much less
>> rough but also much harder to tune . . .
>> """

To my mind the good thing about the Tenney/rectangular approach
is that it gives log(n*d).

>> >> 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?
>> 
>> Unweighted odd-limit taxicab.
>
>In which 9 has its own axis . . . so the following:
>
>> By "3+3" I meant adding two 3
>> vectors.  The equation is 1 < 2.
>
>does not apply.

Sure it does.  As you say, the 9 appears in two places.  If
the metric comes out the same either way, I don't see how this
fact would "screws up uniqueness, and thus most of how we've
been approaching temperament."

When I said "ok" above, I meant it does not violate the
triangle inequality.  But it does 'screw up uniqueness'.

-Carl


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Message: 9197

Date: Fri, 16 Jan 2004 02:16:57

Subject: Re: TOP history

From: Carl Lumma

> Paul, could you tell us something about when and how you
> discovered TOP? I'd like to add some history to my top page.

Well you can see from my recent post that Dave was pretty
close for fixed scales.

Also of excerpts from the list in 1999 may be of interest,
forthcoming in a forthcoming post of mine.

-Carl


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