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Message: 100 - Contents - Hide Contents

Date: Fri, 01 Jun 2001 22:11:41

Subject: Marc's math

From: Paul Erlich

>I T ' S J U N E ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! >Well congratulations, Tuning List posters, >for far surpassing the Sept '00 record of 1671 posts in a month, >with one post more than QUADRUPLE the 505 posts of May of last year, >giving us a whopping 2221 posts for May '01 ! ! !
Ummm . . . 4*505 + 1 = 2021.
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Message: 102 - Contents - Hide Contents

Date: Fri, 01 Jun 2001 23:14:39

Subject: Re: Temperament program issues

From: Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> So under 7-limit, I presume this refers to my decatonic: > > 2/11, 111.043 cent generator
Yes. So it appears Graham has reverted to including intervals which are purely a multiple of the period, when calculating the MA optimum generator. It needs to include these when giving the actual MA error, but not when finding the optimum generator. Bu this is a minor detail.
> Can you explain where you're getting 8 for smallest MOS? There are > MOSs with 2, 4, 6, 8 notes, but 10 is the first proper one . . . how > is complexity measure defined now?
8 is the smallest MOS that contains a complete otonality (or utonality). "complexity measure" (perhaps not a good name since it is completely unrelated to harmonic complexity) is the max number of notes (not necc MOS) that contain zero complete otonalities. You get an otonality for every note after that. It's the width of the otonality in generators times the number of chains in the octave.
> P.S. If you're using propriety for anything, I'd chuck it in favor of > CS.
Good idea. We now have a simple algorithm that, given the generator and period and width of an otonality, will give us the size of either the smallest MOS containing one, or the smallest strictly-proper MOS containing one. Can you write us a CS recogniser that, when given the generator, period and the number of notes in each chain, will say yea or nay? And we'll slot it in. Here's a wild idea that my mathematical intuition just popped up. Could it be that a MOS is CS if and only if it is either strictly-proper (denom of convergent) or it is either the first or last in a series of improper MOS (denoms of semiconvergents) between two strictly-proper MOS. Is Miracle-11 CS? I guess this idea is wrong because you implied that 8 of paultone in not CS. -- Dave Keenan
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Message: 103 - Contents - Hide Contents

Date: Fri, 01 Jun 2001 23:41:00

Subject: Re: Temperament program issues

From: Paul Erlich

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

>I guess this idea is wrong > because you implied that 8 of paultone in not CS.
I think I implied wrong!
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Message: 104 - Contents - Hide Contents

Date: Sat, 02 Jun 2001 01:19:48

Subject: Re: Temperament program issues

From: Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote: >
>> I guess this idea is wrong >> because you implied that 8 of paultone in not CS. >
> I think I implied wrong!
You'll let us know when you're sure, won't you? Here's part of the MOS-cardinality series for the 380 cent generator approx 6/19 oct (single chain). 3 (4 7 10 13 16) 19 3 and 19 are strictly proper. Can you tell us which of those in between are CS? I'm not very familiar with CS.
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Message: 105 - Contents - Hide Contents

Date: Sat, 02 Jun 2001 01:47:28

Subject: Re: Temperament program issues

From: Dave Keenan

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> Here's part of the MOS-cardinality series for the 380 cent generator > approx 6/19 oct (single chain). > > 3 (4 7 10 13 16) 19 > > 3 and 19 are strictly proper. Can you tell us which of those in > between are CS? I'm not very familiar with CS.
It seems from Definitions of tuning terms: constant structur... * [with cont.] (Wayb.) that Carl Lumma was briefly seduced into thinking that all MOS are CS, or rather that a CS was a family of MOS scales with a particular property (which seems to be simply having the same generator) rather than a single scale with a particular property (i.e. for all intervals available in a scale, all instances of that interval are subtended by the same number of scale steps). Can someone give me an example of a MOS that is not CS?
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Message: 106 - Contents - Hide Contents

Date: Sat, 02 Jun 2001 02:18:19

Subject: Re: Temperament program issues

From: Dave Keenan

Ok. Try this:

All MOS whose generator is an irrational fraction of their period, are 
CS.

If I take the generator to be say

1 +  6*phi
---------- oct ~= 380.8191213102 cents
3 + 19*phi

Then I find MOS of sizes 3 (4 7 10 13 16) 19, all to be CS.

It's only when we set the generator to precisely 6/19 oct that we find 
that 13 and 16 are not CS (but 16 is proper).

Agreed?

That makes Erv's definition of CS (as relayed by Kraig and Monz) 
pretty useless. Are CS only defined for scales embedded in EDPs (Equal 
Divisions of the Period)? Did they leave that bit out? Or are we 
supposed to parameterise the definition of CS with some allowable 
error whereby two slightly different intervals are to be considered as 
instances of the same interval?

-- Dave Keenan


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Message: 107 - Contents - Hide Contents

Date: Sat, 02 Jun 2001 02:37:25

Subject: Re: Temperament program issues

From: Dave Keenan

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
I'm having a great time replying to my own posts. Thank goodness you 
guys aren't in the same time zone or I'd never get anything else done. 
:-)

Notice that a chain of 21 Miracle generators is CS when embedded in 
72-EDO or 41-EDO, but not in 31-EDO, although it becomes proper in 
31-EDO.


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Message: 108 - Contents - Hide Contents

Date: Sat, 02 Jun 2001 02:50:45

Subject: Re: Temperament program issues

From: monz

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

Yahoo groups: /tuning-math/message/106 * [with cont.] 

> That makes Erv's definition of CS (as relayed by Kraig and Monz) > pretty useless. Are CS only defined for scales embedded in EDPs > (Equal Divisions of the Period)? Did they leave that bit out? > Or are we supposed to parameterise the definition of CS with > some allowable error whereby two slightly different intervals > are to be considered as instances of the same interval? Hi Dave,
I'm glad you're asking these questions, because I've long been unhappy with my definition of CS, along with many of the other terms that feature prominently in Wilson's theory. Unfortunately, I understand them less than you do, so I'm of no help. Is Daniel Wolf on this list? He might be of some real help here. -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0"
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Message: 109 - Contents - Hide Contents

Date: Sat, 02 Jun 2001 07:05:34

Subject: Re: Temperament program issues

From: carl@l...

>It seems from Definitions of tuning terms: constant structur... * [with cont.] (Wayb.) >that Carl Lumma was briefly seduced into thinking that all MOS >are CS Nope. >or rather that a CS was a family of MOS scales with a particular >property (which seems to be simply having the same generator)
Not a family of scales, but a family of scale cardinalities common to MOSs with the same ie but different generators. This was simply my guess at which one of Erv's ideas (that I knew about) he might call CS, knowing absolutely nothing more about CS than the name itself.
>rather than a single scale with a particular property >(i.e. for all intervals available in a scale, all instances of >that interval are subtended by the same number of scale steps).
The correct definition later furnished by Kraig Grady.
>Can someone give me an example of a MOS that is not CS?
Any MOS with L:s = 2:1 will not be CS. The diatonic scale in 12-equal comes to mind. Monz, that was Erlich, not me, who linked PBs and CS in message 15405. -Carl
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Message: 110 - Contents - Hide Contents

Date: Sat, 2 Jun 2001 08:13 +01

Subject: Re: Temperament program issues

From: graham@m...

Paul wrote:

> The latter. I must have missed the former -- where it it?
<Miracle Temperament Home Page * [with cont.] (Wayb.)>. You complained about it ending with a neutral triad IIRC. Graham
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Message: 111 - Contents - Hide Contents

Date: Sat, 02 Jun 2001 07:21:59

Subject: Re: Temperament program issues

From: Dave Keenan

--- In tuning-math@y..., carl@l... wrote:
>> It seems from Definitions of tuning terms: constant structur... * [with cont.] (Wayb.) >> that Carl Lumma was briefly seduced into thinking that all MOS >> are CS > > Nope. >
>> or rather that a CS was a family of MOS scales with a particular >> property (which seems to be simply having the same generator) >
> Not a family of scales, but a family of scale cardinalities > common to MOSs with the same ie but different generators. > This was simply my guess at which one of Erv's ideas (that I > knew about) he might call CS, knowing absolutely nothing more > about CS than the name itself.
Ok. Monz, you might want to delete that bit from your dictionary.
>> rather than a single scale with a particular property >> (i.e. for all intervals available in a scale, all instances of >> that interval are subtended by the same number of scale steps). >
> The correct definition later furnished by Kraig Grady. >
>> Can someone give me an example of a MOS that is not CS? >
> Any MOS with L:s = 2:1 will not be CS. The diatonic scale > in 12-equal comes to mind.
Thanks Carl. But the above definition means that I can make an infinitesimal change to the generator (in either direction) and the scale suddenly becomes CS. Not a very useful scale property. Or not a very useful definition of it. Anyone want to propose a better definition? One that doesn't have this defect. -- Dave Keenan
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Message: 112 - Contents - Hide Contents

Date: Sat, 02 Jun 2001 15:03:13

Subject: About CS

From: Pierre Lamothe

I am not yet truly available to post on tuning-math but I have to comment
here.



Dave wrote about CS :

<< Not a very useful scale property >>

Hmmm . . . 

Perhaps it might be better to flush the dirty water rather than the baby :)



The following definition of CS found at

<Definitions of tuning terms: constant structur... * [with cont.]  (Wayb.)>

<< A tuning system where each interval occurs always
   subtended by the same number of steps. 
   (THAT IS ALL, NO OTHER RESTRICTIONS) >>

is not, as expressed, a mathematical one but that corresponds, in my
opinion, to the main property of a serious attempt to modelize
mathematically the tonal paradigmatic structures. The temperament questions
don't address that problematic.

The CS property corresponds to the main axiom in the gammoid theory (the
_congruity_ condition) and subtends the _periodicity_ concept in Z-module.
If you flush CS you loss the mathematical sense of _interval class_ (or do
you have a _fuzzy class_ definition to replace it?) 

I can't see how you could define mathematically such terms as step, degree,
class, mode . . . in a system S without the existence of a primordial
epimorphism S --> Z which gives its consistence.


Pierre


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Message: 113 - Contents - Hide Contents

Date: Sat, 02 Jun 2001 20:49:55

Subject: Re: Temperament program issues

From: Paul Erlich

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> That makes Erv's definition of CS (as relayed by Kraig and Monz) > pretty useless.
It's not useless -- most scales characterized as CS are not MOSs.
> Are CS only defined for scales embedded in EDPs (Equal > Divisions of the Period)? Nope! > Did they leave that bit out? Nope! > Or are we > supposed to parameterise the definition of CS with some allowable > error whereby two slightly different intervals are to be considered as > instances of the same interval? Nope!
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Message: 114 - Contents - Hide Contents

Date: Sat, 02 Jun 2001 20:52:42

Subject: Re: Temperament program issues

From: Paul Erlich

--- In tuning-math@y..., graham@m... wrote:
> Paul wrote: >
>> The latter. I must have missed the former -- where it it? >
> <Miracle Temperament Home Page * [with cont.] (Wayb.)>. You complained about it ending > with a neutral triad IIRC.
Oops -- I forgot to listen to this on good speakers. On the speakers I was listening to it on, there was so much distortion that I didn't even know it was supposed to be a neutral triad.
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Message: 115 - Contents - Hide Contents

Date: Sat, 02 Jun 2001 22:56:15

Subject: Re: Temperament program issues

From: Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote: >
>> That makes Erv's definition of CS (as relayed by Kraig and Monz) >> pretty useless. >
> It's not useless -- most scales characterized as CS are not MOSs. >
>> Are CS only defined for scales embedded in EDPs (Equal >> Divisions of the Period)? > > Nope! >
>> Did they leave that bit out? > > Nope! >
>> Or are we >> supposed to parameterise the definition of CS with some allowable >> error whereby two slightly different intervals are to be
This isn't very helpful Paul. Can you give a definition that we can turn into a recognition algorithm for the case of MOS, such that, when considered as a Boolean function of the generator size, it does not contain spikes of infinitesimal width? considered as
>> instances of the same interval? > > Nope!
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Message: 118 - Contents - Hide Contents

Date: Sun, 03 Jun 2001 04:04:15

Subject: CS (was: Re: Temperament program issues)

From: carl@l...

> Thanks Carl. But the above definition means that I can make an > infinitesimal change to the generator (in either direction) and the > scale suddenly becomes CS. Not a very useful scale property. Or not > a very useful definition of it.
Hate to say it Dave, but I don't think CS is that useful a property anyway. Or, maybe I should say, I can almost hear Wilson crying out, 'Of course! It's only a guide, an idea. Why would you expect it to be so precise?'.
> Anyone want to propose a better definition? One that doesn't have > this defect.
A treatment along the lines of what I did for propriety and stability would be one approach. But by that time, I'd just be using those measures anyway. Now, Erlich's consonant constant structures I can live with. One could imagine... 1. Listing the target consonances. Say, for easy of what follows, there is only one target consonance. 2. In h.e. fashion, find for each interval in the scale the probability it will be heard as the target interval. 3. Sum these probabilities for each scale degree. 4. Subtract from the largest sum all the other sums. Or maybe take the difference between the two largest sums. Or maybe the difference between the largest sum and the mean sum. Or something like that. ...hmmm... if one insisted on normal CS and not consonant CS (CCS?), he might use the statistical approach above, but include all the intervals in the scale in #1, and use all the intervals instead of a farey series in #2 (and midpoints between these instead of mediants). -Carl
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Message: 119 - Contents - Hide Contents

Date: Sun, 03 Jun 2001 04:15:38

Subject: Re: Constant Structure (was: Temperament program issues)

From: carl@l...

> It's sounds to me as if you're trying to define a property other > than Wilson's CS. As I understand it, Wilson uses CS to describe > a pattern shared by a group of tunings that can be mapped onto a > single scale tree pattern,even if the generator size is only a > average value. For example, he maps the 3(1,3,7,9,11,15 Eikosany > (plus two "pigtails"), a scale based on an Indian Sruti model, and > 22tet onto a 22-tone keyboard and notation. The subsets described > by the scale tree then becomes useful paths for orientation in > īthe larger system.
Amazing, Daniel! This is just what I guessed CS was, before the correspondence from Kraig. See monz's web site reffed earlier in this thread.
> (Writing the above, I've got the strong suspicion that > constructing a formula that will predict whether and where in > the scale tree a given scale will find a CS is probably very hard > to construct.)
On the contrary, unless I mis-understood you, the scale tree shows the generator ranges itself. I know you know this, so maybe I _did_ mis-read you. Can I get you to check monz's web page? Definitions of tuning terms: constant structur... * [with cont.] (Wayb.) -Carl
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Message: 120 - Contents - Hide Contents

Date: Sun, 03 Jun 2001 04:21:21

Subject: CS (was: Re: Temperament program issues)

From: carl@l...

>Anyone want to propose a better definition? One that doesn't have >this defect.
Actually, Dave, if you're only interested in CS, I think you can just use the size of L/s. The stuff I gave in my last message is good for any scale (not just MOS). -Carl
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Message: 121 - Contents - Hide Contents

Date: Sun, 03 Jun 2001 14:29:47

Subject: Re: Fwd: optimizing octaves in MIRACLE scale..

From: jpehrson@r...

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

Yahoo groups: /tuning-math/message/21 * [with cont.] 

> > Mahler wrote to Schoenberg in 1909 that "I have the score of > your [Schoenberg's 2nd] Quartet with me here [in New York] > and study it from time to time, but it's difficult for me." >
Can you imagine this? And, it's one of his "easier" works, in the overview... ________ ______ _______ Joseph Pehrson
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Message: 122 - Contents - Hide Contents

Date: Sun, 03 Jun 2001 16:04:13

Subject: Re: Fwd: optimizing octaves in MIRACLE scale..

From: jpehrson@r...

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

Yahoo groups: /tuning-math/message/44 * [with cont.] 


> Schoenberg then presents a diagram of the overtones and the > resulting scale, which I have adaptated, adding the partial-numbers > which relate all the overtones together as a single set: > > b-45 > g-36 > e-30 > d-27 > c-24 > a-20 > g-18 g-18 > f-16 > c-12 c-12 > f-8 > > > f c g a d e b > 8 12 18 20 27 30 45 > <cut> etc.
This is an extremely interesting post, Monz, and I would recommend that it be reposted to the "big" list as well... _________ _______ ______ Joseph Pehrson
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Message: 123 - Contents - Hide Contents

Date: Sun, 03 Jun 2001 16:14:35

Subject: Re: Temperament program issues

From: jpehrson@r...

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

Yahoo groups: /tuning-math/message/68 * [with cont.] 

> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >> >> Yahoo groups: /tuning-math/message/56 * [with cont.] >>
>>> ... By the way, your Blackjack chord progression totally >>> blew away and inspired Joseph Pehrson. >> >>
>> Me too!!!! I could tell by looking at it that I simply >> *had* to hear it... that's exactly why *I* made the audio file! >
> You deserve a lot of credit also for the voice-leading (which Joseph > and I meticulously reproduced on the keyboard) and for such a good > choice of timbre.
The first time I heard that progression, I was literally paralyzed with shock/pleasure/anticipation...etc. The fact that it seemed like a "mobius" strip and never ended fascinated me, too. It ended up that it was just confusing me by being 7 chords rather than the "usual" 8 chords that are used in such exercises (see, there WERE a couple of numbers in this post!) ___________ ________ _________ Joseph Pehrson
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Message: 124 - Contents - Hide Contents

Date: Mon, 04 Jun 2001 03:35:39

Subject: Re: Temperament program issues

From: Paul Erlich

--- In tuning-math@y..., "Dave 
Keenan" <D.KEENAN@U...> wrote:

> This isn't very helpful Paul. Can you give a definition that we can > turn into a recognition algorithm for the case of MOS, such that, when > considered as a Boolean function of the generator size, it does not > contain spikes of infinitesimal width?
Why? I think you guys are right that almost all generators lead to MOSs that are CS. So the non-MOS scales for which the term CS is used, which don't have a single generator but are rather periodicity blocks in JI, won't concern us here.
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