Tuning-Math Digests messages 3125 - 3149

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Message: 3125

Date: Fri, 11 Jan 2002 12:33:48

Subject: Re: updated "positive" and "negative" definitions

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Well I wouldn't object to still calling a schismic
temperament "meantone", but you should make the distinction
between negative systems and negative temperaments
and positive systems and positive temperaments clear to
avoid this confusion.

Manuel


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Message: 3126

Date: Fri, 11 Jan 2002 19:34:50

Subject: Re: More proposed definitions

From: monz

> From: genewardsmith <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Friday, January 11, 2002 2:24 PM
> Subject: [tuning-math] Re: More proposed definitions
>
>

> More to the point would be definitions of vector,
> vector space, lattice, bilinear form, group,
> abelian group, homomophism, kernel, equivalence
> relation, equivalence class, quotient group, graph,
> wedge product, and determinant, but this would definately
> start to look like mathematics.


Yes, well ...  I think we might be agreeing that there
should be a separate area in the Dictionary for the
heavy math definitions.

But I already have definitions for vector, lattice,
matrix, and determinant -- so it *is* already beginning
to look a lot like a math dictionary.  I just want to
make sure that the focus stays on musical concepts.

I'd like to hear from some others besides Gene and Paul
... should these math terms go directly into the
Tuning Dictionary, or should they live "off-campus"?


-monz


 



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Message: 3127

Date: Fri, 11 Jan 2002 00:09:49

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: clumma

>>>>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!
>> 
>>I've never heard a voice that played triads, one after the
>>other.  -C.
>
>Fine. But I never mentioned triads in these discussions, only 
>intervals.

You asked if it sounded triadic.

-Carl


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Message: 3128

Date: Fri, 11 Jan 2002 13:49:25

Subject: Re: Distinct p-limit intervals and ets

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

>OK, that's done, and it's been uploaded.  I just wasn't sure
>if the other link should have been in the definition.

You still need to change the second link, it's the same as the
first one.

>Anyway, my request still stands: can you please explain these
>tables in more detail?  I don't quite understand what's in them.

Let's take 31-tET as example. If you do EQUAL/DATA 31 in Scala,
then you see
Highest harmonic represented consistently: 12
This is the first and fourth column in the table. If you find
31.0 in the third column, you see it's in the range between
30.85557 and 31.07329 with consistency 12.
Next in Scala you get
Highest harmonic represented uniquely: 9
This is the fifth column. In 31-tET it's 9, because it
rounds to 5 steps. One ratio with the 10th harmonic, 10/9,
rounds to the same 5 steps so it doesn't have a unique representation.
Next you get
Highest harm. represented uniquely inv. equiv.: 8
which means with inversional equivalence. In 31-tET it's 8,
because a ratio with the 9th harmonic, 9/8, rounds to 5 steps and
therefore its inverse to 31-5=26 steps. However 9/5 also rounds
to 26 steps and so 9 is not unique.

The other file contains the same information in a different
arrangement.

Manuel


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Message: 3129

Date: Fri, 11 Jan 2002 00:13:09

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> You asked if it sounded triadic.

Well, we're going in circles.

Tune up a scale generated by 677-cent fifths, using only power-of-two 
partials and a marimba-like decay, and play some Gamelan-like music 
with it. Tell me if it sounds right to your ears.


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Message: 3132

Date: Fri, 11 Jan 2002 20:13:53

Subject: Re: More proposed definitions

From: monz

> From: unidala <JGill99@xxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Friday, January 11, 2002 8:05 PM
> Subject: [tuning-math] Re: More proposed definitions
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > 
> > I'd like to hear from some others besides Gene and Paul
> > ... should these math terms go directly into the
> > Tuning Dictionary, or should they live "off-campus"?
> > 
> > 
> > -monz
> 
> 
> J Gill: While it might not be the easiest task to implement,
> how about presenting it as non-esoterically as possible in
> the main definition, with links (much as Monz allready does)
> within that text which (heirarchically) enter the realms of
> complexity (deeper and deeper) if the reader is so inclined?
> 
> That way the information is pre-compiled in levels of detail.


Hey J, thanks for your input ... and I think, thanks to your
suggestions, that I've already hit on the right way to do this:
simply include Gene's defintions "as is" as individual entries
in the Dictionary, but then have a "see also" link at the bottom
of each, which leads to an elementary tutorial webpage which
explains this brand of tuning math.

Feedback?


-monz


 



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Message: 3133

Date: Fri, 11 Jan 2002 00:35:32

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: clumma

>>You asked if it sounded triadic.
>
>Well, we're going in circles.
>
>Tune up a scale generated by 677-cent fifths, using only power-
>of-two partials and a marimba-like decay, and play some Gamelan-
>like music with it. Tell me if it sounds right to your ears.

(Midway between 16 and 23?)  Okay, I'll do it!  

-Carl


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Message: 3134

Date: Fri, 11 Jan 2002 03:05:35

Subject: Re: Dictionary query

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Thursday, January 10, 2002 1:56 PM
> > Subject: [tuning-math] Re: Dictionary query
> >
> >
> > Today, on these lists, we tend to call negative 
systems "meantone" 
> > and positive systems "schismic". The reason 700 cents was chosen 
as 
> > the dividing line between "negative" and "positive" is that when 
the 
> > fifth is below 700 cents, the "meantone" (+4 fifths) 
approximation to 
> > the 5/4 is better than the "schismic" (-8 fifths) approximation 
to 
> > the 5/4. When the fifth is above 700 cents, the "schismic" 
> > approximation to the 5/4 is better than the "meantone" 
approximation 
> > to the 5/4. I might differ, saying that there is a "gray area", 
and 
> > also factoring the 6/5 into consideration . . . but the 
definitions 
> > are well-established and there is no reason to favor ones which 
could 
> > breed potential contradictions.
> > 
> > As for your definition pages, Monz, they definitely give the 
wrong 
> > idea. Positive systems should be characterized by the fraction of 
a 
> > _schisma_ that the fifths differ from just -- this is the 
relevant 
> > measure of them. Knowing what fraction of a syntonic comma a 
positive 
> > system's fifth might have been _increased_ by is irrelevant for 
> > understanding the functioning of the system, and is potentially 
> > misleading.
> 
> 
> Thanks very much for that, Paul.  So how does it look now?
> Definitions of tuning terms: positive system, (c) 2001 by Joe Monzo *
> 
> 
> 
> -monz

Unfortunately, 22 is not a schismic temperament . . . this is my 
fault, of course . . . I later alluded to the correct definition in 
conversation with Gene, as you can see . . . I'm a bit too tired to 
correct this now, but I'm sure Graham or John Chalmers can help you 
if they're available before I can get back to you.

P.S. Monz, why do you like to keep incomplete/incorrect 
definitions/descriptions at the top of your dictionary pages, or even 
in there at all? Why not attempt for the more precise, univerally 
agreed-on definitions/examples first, and then post 
alternate/intermediary-stages-in-someone's-thinking stuff later, 
preferably on entirely separate webpages?


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Message: 3135

Date: Fri, 11 Jan 2002 21:19:18

Subject: Re: More 72-et decatonics

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> We are reaching the end of the 4375/4374 lot; I had 132 of these, 
and these look like the best.
> 
> [0, 5, 12, 19, 24, 35, 42, 49, 54, 61]
> [5, 7, 7, 5, 11, 7, 7, 5, 7, 11]
> edges   16   25   32   33   connectivity   2   3   5   6
> 
> 
> [0, 5, 12, 19, 30, 35, 42, 49, 56, 61]
> [5, 7, 7, 11, 5, 7, 7, 7, 5, 11]
> edges   13   23   30   33   connectivity   0   2   4   5
> 
> 
> [0, 5, 12, 19, 24, 31, 42, 47, 54, 61]
> [5, 7, 7, 5, 7, 11, 5, 7, 7, 11]
> edges   16   24   31   32   connectivity   2   2   4   5
> 
> [0, 7, 12, 19, 30, 35, 42, 49, 56, 61]
> [7, 5, 7, 11, 5, 7, 7, 7, 5, 11]
> edges   15   23   31   32   connectivity   1   3   5   5

Is the advantage of "connectivity" over "edges" is 
that "connectivity" tends to favor triads, tetrads, etc., 
while "edges" merely looks at dyads?


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Message: 3136

Date: Fri, 11 Jan 2002 00:39:19

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> (Midway between 16 and 23?)

(A little closer to 23.)


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Message: 3137

Date: Fri, 11 Jan 2002 21:23:49

Subject: positive/skhismic systems (was: Re: Dictionary query)

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> What has me puzzled is the large size of the "fraction of the 
skhisma"
> that's tempered out in the examples on my webpage.
> 
> You and Gene already discussed 22-EDO, but what about 17- and
> 39-EDO, where the "5th" is +2 and nearly +3 skhismas wider,
> respectively, than a 3:2?  And even 29-EDO's "5th" is nearly
> a whole skhisma wider.

I don't know why you would put 39-tET in this category. As for 17 and 
29, have a look at the implied major thirds in these systems . . . 
that should clear up your confusion.


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Message: 3138

Date: Fri, 11 Jan 2002 00:51:51

Subject: Re: All in the spirit of friendship, Gene

From: clumma

>Hmmm...well, I suppose you know people say exactly the same sort
>of things about you. In fact, I think I've heard more about PE's
>arrogance than of GWS's.

Give it time!  :)

Paul has trouble being wrong, but that's different than arrogance.
I think Paul is the antithesis of arrogant.  He's extremely innocent
and honest, though this aspect may not come across the line very
well (I didn't realize it until I met him).  It can be hard to
recognize in any case, but in all my years on this list I've seen
none with motives more pure than Paul Erlich.

I haven't felt any arrogance from you, either, Gene.  Maybe a
little reluctance to read the citations, but I took this to be
either a time management decision, or maybe an attempt to keep
free of past mistakes; to 'play dumb' for the benefit of beginner's
luck, so to speak.  Both seem reasonable to me.  I have certainly
been amazed at the power of real mathematics, which I've never
really been exposed to before, outside of a few brief meetings with
David Rothenberg.

I'm really glad you're both here.

-Carl


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Message: 3139

Date: Fri, 11 Jan 2002 21:27:40

Subject: Re: More 72-et decatonics

From: genewardsmith

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

> Is the advantage of "connectivity" over "edges" is 
> that "connectivity" tends to favor triads, tetrads, etc., 
> while "edges" merely
looks at dyads?

Connectivity does tend to favor chords, but one advantage from my
point of view is that I don't need to write any code to compute it. A
high connectivity means the notes are tied together more closely, with
more paths from one to another, as well.


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Message: 3140

Date: Fri, 11 Jan 2002 21:33:01

Subject: Re: More proposed definitions

From: monz

> From: paulerlich <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Friday, January 11, 2002 8:18 PM
> Subject: [tuning-math] Re: More proposed definitions
>
>

> > Hey J, thanks for your input ... and I think, thanks to your
> > suggestions, that I've already hit on the right way to do this:
> > simply include Gene's defintions "as is" as individual entries
> > in the Dictionary, but then have a "see also" link at the bottom
> > of each, which leads to an elementary tutorial webpage which
> > explains this brand of tuning math.
> > 
> > Feedback?
> 
> I don't think you'd want to do this for "pitch", "interval", etc., 
> though . . .


Well ... right, Paul, this is exactly what's on my mind.
Terms like "pitch" and "interval", which are common currency
among all musicians, certainly belong, but most of Gene's
terms obviously lean much more towards pure mathematics.

For those of you who are more familiar with algebra (and
specifically, multilinear algebra) than I am, I guess what
I'm really driving at with this is: are these terms mainly
quite personal to Gene and his work, or are they more
widely accepted among other mathematically-savvy tuning
theorists?  If the former, then are there terms used by
others which are equivalent to Gene's, and therefore which
also need to be included in the Dictionary as synonyms?

This is really important to me, because I get a sense (and
an eyewitness account from Paul) that the last three months
or so on the tuning-math list (i.e., since Gene's arrival)
have borne some of the most comprehensive and most
widely-applicable concepts and algorithms in the history
of tuning theory, and it's high time that me and the rest
of us join the party.


-monz


 



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Message: 3141

Date: Fri, 11 Jan 2002 03:20:09

Subject: Re: Dictionary query

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> OK, but if they don't post anything here, please do give me
> more info.

I have some of the original Bosanquet papers . . . I'll try to bring 
them in.

> If even one other person here agrees with you that the commatic
> description of positive systems is absolutely useless, then
> they're history.  Let me know.

Well, let me put it this way. For meantone systems, the meaning of 
_which_ comma you're talking about is clear from the way meantone 
works. If you're describing a non-meantone system as "x/y-comma" 
whatever, then it's ambiguous. At least specify _which_ comma you're 
talking about, that will at least make the specification 
mathematically precise. But functionally, schismic temperaments are 
best described by the fraction of the schimsa that's tempered 
out . . . 1/8 schisma is Helmholtian, 1/9 schism is Sabat-Garibaldi's 
Dinarra tuning . . . am I making any sense?


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Message: 3142

Date: Fri, 11 Jan 2002 21:28:11

Subject: Re: For Joe--proposed definitions

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> Thanks, Gene!  This is great.  One quibble...
> 
> 
> > From: genewardsmith <genewardsmith@j...>
> > To: <tuning-math@y...>
> > Sent: Thursday, January 10, 2002 8:11 PM
> > Subject: [tuning-math] For Joe--proposed definitions
> >
> > 
> > Scale 
> > 
> > A discrete set of real numbers, containing 1, and such
> > that the distance between sucessive elements of the scale
> > is bounded both below and above by positive real numbers.
> > The least upper bound of the intervals between successive
> > elements of the scale is the maximum scale step, and the
> > greatest lower bound is the minimum scale step. The element
> > of the scale obtained by counting up n scale steps is the
> > nth degree, by counting down is the -nth degree; 1 is the
> > 0th degree.
> 
> 
> These definitions are necessary for helping a reader to
> understand your tuning-math posts.  

Are they? I would think that they would be useful mainly for a 
mathematician who may not know anything about music but who still may 
wish to understand Gene's research.


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Message: 3143

Date: Fri, 11 Jan 2002 03:21:10

Subject: Re: Distinct p-limit intervals and ets

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> Anyway, my request still stands: can you please explain these
> tables in more detail?  I don't quite understand what's in them.

I'll leave that to Manuel for now . . . good night.


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Message: 3144

Date: Fri, 11 Jan 2002 21:30:08

Subject: Re: Dictionary query

From: paulerlich

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
> 
> Paul, Gene, Joe,
> 
> You've missed or ignored my answer to Joe's question,
> which was the most concise I could give.
> The borderline is the point where the Pythagorean comma
> vanishes: 700 cents. This choice is not 12-tET centric
> in my view.
> 
> > Thanks very much for that, Paul.  So how does it look now?
> > Definitions of tuning terms: positive system, (c) 2001 by Joe Monzo *
> 
> You could add that systems with p=1 (scale steps) are
> called singly positive, with p=2 doubly positive,
> p=-1 singly negative, etc.
> 
> Manuel

Where p is the Pythagorean comma.


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Message: 3145

Date: Fri, 11 Jan 2002 03:40:52

Subject: More 72-et decatonics

From: genewardsmith

We are reaching the end of the 4375/4374 lot; I had 132 of these, and theselook like the best.

[0, 5, 12, 19, 24, 35, 42, 49, 54, 61]
[5, 7, 7, 5, 11, 7, 7, 5, 7, 11]
edges   16   25   32   33   connectivity   2   3   5   6


[0, 5, 12, 19, 30, 35, 42, 49, 56, 61]
[5, 7, 7, 11, 5, 7, 7, 7, 5, 11]
edges   13   23   30   33   connectivity   0   2   4   5


[0, 5, 12, 19, 24, 31, 42, 47, 54, 61]
[5, 7, 7, 5, 7, 11, 5, 7, 7, 11]
edges   16   24   31   32   connectivity   2   2   4   5

[0, 7, 12, 19, 30, 35, 42, 49, 56, 61]
[7, 5, 7, 11, 5, 7, 7, 7, 5, 11]
edges   15   23   31   32   connectivity   1   3   5   5


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Message: 3146

Date: Fri, 11 Jan 2002 21:32:47

Subject: Re: badly tuned remote overtones

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> How does this compare with the other 12-tone periodicity-block
> you calculated for Schoenberg?  Can you please give a listing
> of the pitches inside *this* PB?

The set of pitches inside any non-torsional 12-tone periodicity block 
with all the unison vectors tempered out is simply 12-tET.


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Message: 3147

Date: Fri, 11 Jan 2002 21:42:04

Subject: Re: For Joe--proposed definitions

From: genewardsmith

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> Can I have both?  

Scale 

A discrete set of real numbers, containing 0, and regarded as defining tone=

s in a logarithmic measure, such as cents or octaves,
and such that the distance between sucessive elements of the scale is bound=

ed both below and above by positive real numbers. The least upper bound of t=

he intervals between successive elements of the scale is the maximum scale s=

tep, and the greatest lower bound is the minimum scale step. The element of=

the scale obtained by counting up n scale steps is the nth degree, by counti=

ng down is the –nth degree; 0 is the 0th degree. The set of positive realnu=

mbers which are the tones so represented is also regarded as the scale.

Periodic scale

If s(n) denotes the nth degree of a scale, it is *periodic* with period L>0=

 if there is an integer k>0 such that s(n+k) = s(n) + L.
The most common value of L is the octave, which defines octave periodic sca=

les.

Period reduced scale

A period reduced scale with period L is the unique set of representatives i=

n the range 0<=s<L of a periodic scale with period L.

Connected scale

A scale is connected with respect to a set of intevals S if for any
two scale degrees a and b, there is a path a=a_0, a_1, ... a_n = b
such that |a_i - a_{i+1}| is an element of S.


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Message: 3148

Date: Fri, 11 Jan 2002 04:11:29

Subject: For Joe--proposed definitions

From: genewardsmith

Scale 

A discrete set of real numbers, containing 1, and such that the distance be=

tween sucessive elements of the scale  is bounded both below and above by po=

sitive real numbers. The least upper bound of the intervals between successi=

ve elements of the scale is the maximum scale step, and the greatest lower b=

ound is the minimum scale step. The element of the scale obtained by countin=

g up n scale steps is the nth degree, by counting down is the –nth degree; 1=

 is the 0th degree.

Tone group

A set of positive real numbers closed under multiplication and inversion (s=

o that if x is in the set, so is 1/x), and regarded as a set of intervals or=

 pitches.

Rational tone group

A tone group whose elements are rational numbers.

Val

A map from a rational tone group to the integers, which respects multiplica=

tion. If h is a val, then h(a*b) = h(a) + h(b); h(1) = 0; and h(1/a) = -h(a)=

. If we write the rational number "a"as 
a = 2^e2 * 3^e3 * ... * p^ep, we may denote it by a row vector 
[e2, e3, ...., ep]. In that case we denote a val by a column vector of inte=

gers of the same dimension. In the language of abstract algebra, a val is a=

homomorphism from the tone group to the integers.

Canonical val

For any positive integer n, the canonical val hn is the val such that 
hn(p) = round(n * log2(p)), where p is prime and where "round" means round =

to the nearest integer. The restriction of hn to a particular rational tone=

group is also denoted hn.

P-adic val

For any prime p, the p-adic val vp is the val which sends q = 2^e2 3^e3 … p=

^ep … to vp(q) = ep. This is called the "p-adic valuation" in number theory.=



Epimorphic

A scale has the epimorphic property, or is epimorphic, if there is a val h =

such that if qn is the nth scale degree, then h(qn)=n. The val h is the char=

acterizing val of the scale.


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Message: 3149

Date: Fri, 11 Jan 2002 21:44:08

Subject: Re: For Joe--proposed definitions

From: genewardsmith

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

> Are they? I would think that they would be useful mainly for a 
> mathematician who may not know anything about music but who still may 
> wish to understand Gene's research.

They are useful for anyone (eg, me) who might want to state and prove theorems. You can't do that without definitions.


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