This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

- Contents - Hide Contents - Home - Section 5

Previous Next

4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 4500 4550 4600 4650 4700 4750 4800 4850 4900 4950

4400 - 4425 -



top of page bottom of page up down


Message: 4400 - Contents - Hide Contents

Date: Wed, 27 Mar 2002 22:10:34

Subject: Re: Hermite normal form version of "25 best"

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., Herman Miller <hmiller@I...> wrote: >
>> The best 16 would be enough for me; I don't see AMT as particularly >> interesting as a 5-limit temperament, and the last three just don't > seem
>> all that useful for musical purposes as far as I can tell. I > haven't done
>> as much playing around with these scales as I'd like, but it > generally
>> seems to be the case that the less complex scales are also the ones > that
>> are most musically interesting to me. >
> I think we should leave room for various preferences in this > department, and don't see why we can't have a best 20.
that would be fine with me (though it's not really the 'best 20' by any complete criterion, just the 'first 20' in complexity to pass a certain condition).
top of page bottom of page up down


Message: 4401 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 03:27:34

Subject: Re: Pitch Class and Generators

From: genewardsmith

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

> yes, but to claim (as balzano did) that the fundamental importance of > the diatonic scale hinges on this fact is to pull the wool over the > eyes of the numerically inclined reader.
It's an incorrect assessment, I don't suppose it was intentionally misleading. the important properties of the diatonic scale must, i
> feel, be found in the scale itself, in whatever tuning it may be > found (with reasonable allowances for the ear's ability to accept > small errors) -- any 'chromatic totality' considerations should wait > until, and be completely dependent upon, the establishment of the > fundamental 'diatonic' entity upon which the music is to be based.
Is it fundamental to the diatonic scale that 81/80~1, in your view?
top of page bottom of page up down


Message: 4402 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 17:28:45

Subject: Re: Decatonics

From: Carl Lumma

>arranged--that's a valid insight. The number ... * [with cont.] (Wayb.) > >he didn't really get wilson's 'constant structures' right so maybe >he's mistaken on 'Rothenberg's "propriety"' as well?
Well, let me reply to his post:
> The problem as I saw it was that so many not very interesting >looking scales were being generated that stronger criteria were >necessary. Just ensuing that all the notes were parts of major or >minor (or septimal) chords was not a fine enough filter. Melodic >characteristics are also important.
So he has already an embarrassment of riches as far as he's concerned, so erring on the side of fewer scales may not be his worst fear.
> The criterion I've used the most is Rothenberg's "propriety," >which is equivalent essentially to Balzano's "coherence,"
So he agrees with us.
>and Wilson's "constant structure."
This isn't right. Both John and I independently made the mistake that CS was equivalent to *strict* propriety in 1999, forgetting that there are improper scales which are CS.
>A proper scale is one which holds together as a scale, rather than >being perceived as a set of principal and ornamental tones.
This is a gloss, but seems okay. Rothenberg says that in practice, especially without a drone, for scales with very low stability, composition will tend to trace out proper subset(s) of the scale, if they exist, and treat the propriety-breaking scale degrees as ornamental. Playing with the harmonic series segment from 8-16, I found that I tended to use 13 this way, and I later looked, and indeed the scale was proper without it. Yes, he should have said stability.
> A more important question is how are the scales going to be >used. Are they collections of tones (gamuts, like the 12-tone set) >or are they going to be projected as scales and are scalar passages >going to be used thematically. Ah-ha! > I have considerable doubt about JI in nature. Most sounds in >nature are nonharmonic, if only because the oscillators and resonators >are complex 3-D structures instead of ideal 1-D strings and air columns. >It takes considerable effort to train the voice to make harmonic timbres, >and the vocal quality is decidedly un-natural, whether in relation to >speech or the usual singing voice. Most animal cries don't strike me >as especially musical, except perhaps song birds. I suspect that our >perception of JI as something special is an epiphenomenon of our >auditory system. A system evolved to recognize the sounds of predators, >human voices, and especially to decode speech, may find JI easy to >process because of acoustic and/or neurological laws. Hence, we find >it exceptional and special.
He's right. Many authors use the former (natural sounds) as a justification for inharmonic/nonharmonic timbres/music, forgetting the latter (biology).
> It may also be partly a learned phenomenon; JI may not be so >easily perceived or appreciated in cultures whose musical instruments >have mostly nonharmonic timbres (Indonesia, S.E. Asia). There is >some speculation that in pre-columbian Mesoamerican music, melodic >contour was more important than interval size. Perhaps we learn >to perceive JI because it is part of our unnatural environment, >even in its approximated tempered form. It's hard to escape harmonic >timbres from muzak, radio, stereos, etc. But, this is speculation.
Well, I tend to think cultural conditioning is not significant here. I don't buy the predator noises thing -- we have excellent spatial stuff for that (not only timing between ears, but apparently also the the direction sounds enter the outer ear -- somebody's even got an algorithm to simulate this over headphones now). The key thing I think is vowels and inflection in speech, which exist in all cultures, and the biology here is so strong and exposed that we're already beginning to discover it! -Carl
top of page bottom of page up down


Message: 4403 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 03:28:58

Subject: Re: Hermite normal form version of "25 best"

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Lets move on to the full 7-limit. When you've got time Gene, could you > set your limits somewhat wide to start with. Say 35 cents rms error. > Weighted complexity of 30 gens, badness to give about 40 temperaments?
I've been thinking of doing 7-limit planars first.
top of page bottom of page up down


Message: 4404 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 19:15:16

Subject: Re: Decatonics

From: Carl Lumma

Sorry to all of you who have already gotten this from the other
list.  I've deleted it there and posted it here.

>> and Wilson's "constant structure." >
>This isn't right. Both John and I independently made the mistake >that CS was equivalent to *strict* propriety in 1999, forgetting >that there are improper scales which are CS.
It's worth noting that CS can be interpreted in the same light as propriety, though. Both can be viewed as measures of interval recognizability -- linking acoustic intervals to scalar ones (ie 3:2 -> 5th). If you assume that the ear tracks acoustic intervals by *relative* size, you get propriety/stablity. If you assume the ear can recognize *particular* *absolute* intervals, you get CS instead. Since I think the latter idea is complete nonsense (with the possible exception of a few strong consonances), I think CS should not be applied to the generalized diatonics problem. However, you, Paul, have shown (and it seems that Wilson intuitively understands) that CS is important with respect to periodicity blocks. I can only remember one facet of this -- that all PBs with unison vectors smaller than their smallest 2nd are CS (was there ever a proof?). -Carl
top of page bottom of page up down


Message: 4405 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 09:36 +0

Subject: Re: Decatonics

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <a7tcea+f5v4@xxxxxxx.xxx>
Mark:
>> This is inconsistent with my rule ii: it contains segments of 3 and > more >> adjacent PCs >> >> 0 1 2, 4 5 6 7, 9 10 11 12, etc >> >> So it will show up as intervallically inchoerent (as defined by > Balzano). Paul:
> this is what we call 'rothenberg improper'. but i don't think that's > a good reason to throw it out. the diatonic scale in pythagorean > tuning is rothenberg improper!
It doesn't look like impropriety to me. The diatonic scale would only fail in the degenerate case of 7-equal. The classic pentatonic would be incoherent in 7-equal, because it's a "pentatonic".
> i sure hope my other responses to you show up, mark. but basically, i > wish there was some text accompanying your decatonic diagram. i have > no idea why you are using 4 (218) and 5 (273) as your 'generators', > for example.
Yes, I'm looking forward to these. How come the only messages that come through from you are ones where you say your messages aren't coming through? Graham
top of page bottom of page up down


Message: 4406 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 18:54:36

Subject: Re: A common notation for JI and ETs

From: gdsecor

i--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>> I thought that, in the event somebody *absolutely must* have 23, one >> could allow a little bit of slack if the following were taken into >> account: > ...
>> and, in addition, 21, 25, and 27 are all >> consistent with that. >
> You're right. I'd even like to see if we can push it to 31-limit, > consistent with 311-ET, since we're so close, but this would use > additional flags (and/or additional schismas like 4095:4096 and > 3519:3520) and not affect the existing 23-limit, 217-ET > correspondence.
Considering that the semantics of the notation have already put us past the 19 limit, that 217 is not 23-limit consistent, and that 311 is such an excellent division, I'd say let's go for it!
> You know I went thru the prime factorisation of all the > superparticulars in John Chalmer's list, and you've found the only two > that are useful for this purpose.
I guess I just have a knack for finding useful commas (even before I start looking for them). Are you ready for the next one? It's a honey: 20735:20736 (5*11*13*29:2^8*3^4, ~0.083 cents). And it turns out that we don't need any new flags to get the 29 factor: its defining interval is 256:261 (2^8:3^2*29, ~33.487 cents), and the convex left flag that we already have (715:729) is ~33.571 cents.
> ... I was considering putting a blob on the end of the straight 7 > flag, but no. I agree with you now. Keep the curved flag for the > 7-comma. It is most important to get the 11-limit right. The rest is > just icing on the cake, and a little lateral confusability there can > be tolerated.
As long as we are going for a higher prime limit that will almost certainly require an additional kind of flag, perhaps that will present an opportunity to de-confuse the situation a bit, but that remains to be seen. Here's something to keep in mind as we raise the prime limit. I am sure that there are quite a few people who would think that making a notation as versatile as this one promises to get is overkill. I think that such a criticism is valid only if its complexity makes it more difficult to do the simpler things. Let's try to keep it simple for the ET's under 100 (as I believe we have been able to do so far), keeping the advanced features in reserve for the power-JI composer who wants a lot of prime numbers. If we build everything in from the start and do it right, then there will be no need to revise it later and upset a few people in the process.
>> By the way, something else I figured out over the weekend is how to >> notate 13 through 20 degrees of 217 with single symbols, i.e., how to >> subtract the 1 through 8-degree symbols from the sagittal apotome >> (/||\). The symbol subtraction for notation of apotome complements >> works like this: >> >> For a symbol consisting of: >> 1) a left flag (or blank) >> 2) a single (or triple) stem, and >> 3) a right flag (or blank): >> 4) convert the single stem to a double (or triple to an X); >> 5) replace the left and right flags with their opposites according to >> the following: >> a) a straight flag is the opposite of a blank (and vice versa); >> b) a convex flag is the opposite of a concave flag (and vice versa). >
> You gotta admit this isn't exactly intuitive (particularly 5a). I'm > more interested in the single-stem saggitals used with the standard > sharp-flat symbols, but it's nice that you can do that.
Believe it or not, the logic behind 5a) is pretty solid, while it is 5b) that is a bit contrived. The above is an expansion of what I originally did for the 72-ET notation before any curved flags were introduced. Allow me to elaborate on this. Consider the following: 81:80 upward is a left flag: /| 33:32 upward is both flags: /|\ so 55:54 upward is 33:32 *less* a left flag: |\ Since an apotome upward is two stems with both flags: /||\ then an apotome *minus 81:80* is the apotome symbol *less a left flag*: ||\ which illustrates how we arrive at a symbol for the apotome's complement of 81/80 by changing /| to ||\ according to 4) and 5a) above. Using curved flags in the 72-ET native notation to alleviate lateral confusibility complicates this a little when we wish to notate the apotome's complement (4deg72) of 64/63 (2deg72), a single *convex right* flag. I was doing it with two stems plus a *convex left* flag, but the above rules dictate two stems with *straight left* and *concave right* flags. As it turns out, the symbol having a single stem with *concave left* and *straight right* flags is also 2deg72, and its apotome complement is two stems plus a *convex left* flag (4deg72), which gives me what I was using before for 4 degrees. So with a little bit of creativity I can still get what I had (and really want) in 72; the same thing can be done in 43-ET. This is the only bit of trickery that I have found any need for in divisions below 100. As you noted, it is nice that, given the way that we are developing the symbols, this notation will allow the composer to make the decision whether to use a single-symbol approach or a single-symbols- with-sharp-and-flat approach. And the musical marketplace could eventually make a final decision between the two. So while we can continue to debate this point, we are under no pressure or obligation to come to an agreement on it.
>> I will prepare a diagram illustrating the progression of symbols for >> JI and for various ET's so we can see how all of this is going to >> look. >> >> Stay tuned! >
> Sure. This is fun.
More fun (if more complicated) than I had ever expected! --George
top of page bottom of page up down


Message: 4407 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 20:08:56

Subject: Re: Decatonics

From: Carl Lumma

>>> >he criterion I've used the most is Rothenberg's "propriety," >>> which is equivalent essentially to Balzano's "coherence," >>
>> So he agrees with us. >
>i thought you objected to calling it *rothenberg* propriety . . . >isn't that the whole point?
No, Rothenberg did coin the term propriety, and a scale is either proper or not, and IIRC is is the same as Balzano coherence. It's just that R. doesn't use it to eliminate scales, and talks most often of stability. I suppose I've been guilty here too. From now on, I'm saying "Rothenberg stability". -Carl
top of page bottom of page up down


Message: 4408 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 12:44:35

Subject: Re: Decatonics

From: Carl Lumma

>> >his is what we call 'rothenberg improper'. but i don't think that's >> a good reason to throw it out. the diatonic scale in pythagorean >> tuning is rothenberg improper! >
>It doesn't look like impropriety to me. The diatonic scale would only >fail in the degenerate case of 7-equal. The classic pentatonic would be >incoherent in 7-equal, because it's a "pentatonic".
When I read Balzano's paper, incoherency was indeed equivalent to propriety. I don't remember anything about "pentatonic" and "diatonic" being a majority or minority of s vs. L, but maybe I'm just repressing it cause it's such a poor idea. -Carl
top of page bottom of page up down


Message: 4409 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 20:54:57

Subject: Re: Decatonics

From: Carl Lumma

>> >t's just that R. doesn't use it to eliminate scales, >
>so what? i don't get what you were yelling at me about! i was just >saying exactly the same thing you're saying here -- that it's the >same as balzano coherence. balzano (and some of his followers) *do* >use it to eliminate scales, and *that's* what i was responding to.
Ok, sorry. I did ask...
>> Who has been throwing out scales for being improper in the way >> that the Pythagorean diatonic is improper?
...and in the past, you've used this argument to say that 'propriety doesn't matter', which is true but incomplete, since it assumes the strictest possible definition for the term "propriety". -Carl
top of page bottom of page up down


Message: 4410 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 21:28:27

Subject: Euler and harmonic entropy

From: genewardsmith

How does Euler's ranking of chords, where chord a:b:c:d would be given
a value lcm(a,b,c,d)/gcd(a,b,c,d), compare to harmonic entropy
rankings of chords? Euler's method gives the same value to otonal as
to utonal chords, so it must have some differences. How great are
they?


top of page bottom of page up down


Message: 4411 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 13:35:29

Subject: Re: Euler and harmonic entropy

From: Carl Lumma

>How does Euler's ranking of chords, where chord a:b:c:d would be >given a value lcm(a,b,c,d)/gcd(a,b,c,d), compare to harmonic entropy >rankings of chords? Euler's method gives the same value to otonal as >to utonal chords, so it must have some differences. How great are >they?
I guess we won't really know until we have chordal harmonic entropy. I remember the totient function, which didn't work at all, even for dyads, but the above measure looks different. -Carl
top of page bottom of page up down


Message: 4412 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 13:41:39

Subject: Re: Decatonics

From: Carl Lumma

>this is what we call 'rothenberg improper'. but i don't think that's >a good reason to throw it out. the diatonic scale in pythagorean >tuning is rothenberg improper!
Paul, how long are you going to continue using this fallacious application of Rothenberg? Who has been throwing out scales for being improper in the way that the Pythagorean diatonic is improper? Certainly not Rothenberg! -Carl
top of page bottom of page up down


Message: 4413 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 13:56:03

Subject: Re: Euler and harmonic entropy

From: Carl Lumma

Doesn't have harmonic entropy, but it does have totient and
gradus suavatatis, and n*d, which *is* similar to harmonic
entropy, at least for dyads:

http://www.uq.net.au/~zzdkeena/Music/HarmonicComplexity.zip - Type Ok * [with cont.]  (Wayb.)

Dave, it looks like all of the pointers are broken.  Maybe it has
to do with me running Excel 2000 now?

-Carl


top of page bottom of page up down


Message: 4414 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 22:00:51

Subject: Re: Pitch Class and Generators

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote: >
>> yes, but to claim (as balzano did) that the fundamental importance > of
>> the diatonic scale hinges on this fact is to pull the wool over the >> eyes of the numerically inclined reader. >
> It's an incorrect assessment, I don't suppose it was intentionally > misleading.
well, no, i think balzano managed to fool himself, and much of the academic community in the process.
>> the important properties of the diatonic scale must, i >> feel, be found in the scale itself, in whatever tuning it may be >> found (with reasonable allowances for the ear's ability to accept >> small errors) -- any 'chromatic totality' considerations should > wait
>> until, and be completely dependent upon, the establishment of the >> fundamental 'diatonic' entity upon which the music is to be based. >
> Is it fundamental to the diatonic scale that 81/80~1, in your view?
well, the diatonic scale is pretty strong already in the 3-limit, where 80 of course doesn't even come into play. but yes, in the 5- limit, whether the diatonic scale is in ji or tempered, 81/80 is one of its defining unison vectors. as joe monzo might put it, the smallness of 81/80 helps determine the 'finity' of the diatonic scale -- if 81/80 were large (pardon the arithmetical counterfactual, and don't take it too seriously), you'd tend to keep adding more notes until you did hit up against a small unison vector.
top of page bottom of page up down


Message: 4415 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 22:01:44

Subject: Re: Hermite normal form version of "25 best"

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote: >
>> Lets move on to the full 7-limit. When you've got time Gene, could > you
>> set your limits somewhat wide to start with. Say 35 cents rms > error.
>> Weighted complexity of 30 gens, badness to give about 40 > temperaments? >
> I've been thinking of doing 7-limit planars first.
wow -- that could be a long list. the nice thing is that my heuristic should work well for these, since there's only one unison vector.
top of page bottom of page up down


Message: 4416 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 22:06:02

Subject: Re: Decatonics

From: paulerlich

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
>>> this is what we call 'rothenberg improper'. but i don't think that's >>> a good reason to throw it out. the diatonic scale in pythagorean >>> tuning is rothenberg improper! >>
>> It doesn't look like impropriety to me. The diatonic scale would only >> fail in the degenerate case of 7-equal. The classic pentatonic would be >> incoherent in 7-equal, because it's a "pentatonic". >
> When I read Balzano's paper, incoherency was indeed equivalent to > propriety.
me too. so graham, what gives?
top of page bottom of page up down


Message: 4417 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 00:03:09

Subject: Re: Hermite normal form version of "25 best"

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
>> I think we should leave room for various preferences in this >> department, and don't see why we can't have a best 20. >
> that would be fine with me (though it's not really the 'best 20' by > any complete criterion, just the 'first 20' in complexity to pass a > certain condition).
Sure. If any one of us wants the 20, I'm happy to go with it. Lets move on to the full 7-limit. When you've got time Gene, could you set your limits somewhat wide to start with. Say 35 cents rms error. Weighted complexity of 30 gens, badness to give about 40 temperaments?
top of page bottom of page up down


Message: 4418 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 22:08:22

Subject: Re: Decatonics

From: paulerlich

--- In tuning-math@y..., Carl Lumma <carl@l...> wrote:
>> this is what we call 'rothenberg improper'. but i don't think that's >> a good reason to throw it out. the diatonic scale in pythagorean >> tuning is rothenberg improper! >
> Paul, how long are you going to continue using this fallacious > application of Rothenberg? Who has been throwing out scales > for being improper in the way that the Pythagorean diatonic is > improper? Certainly not Rothenberg!
sorry -- just force of habit (acquired from john chalmers, i believe). so who did introduce the terms 'proper' and 'improper' in this context, if not rothenberg?
top of page bottom of page up down


Message: 4419 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 22:12:15

Subject: Re: Euler and harmonic entropy

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

>How does Euler's ranking of chords, where chord a:b:c:d would be >given a value lcm(a,b,c,d)/gcd(a,b,c,d), compare to harmonic entropy >rankings of chords? Euler's method gives the same value to otonal as >to utonal chords, so it must have some differences. How great are >they?
they could be very great, in many cases. for example, harmonic entropy is a *continuous* function of the input intervals, defined for irrational as well as rational intervals. i wish this were euler's actual ranking method, but of course he went even further (off the deep end) with his final formulae for GS. Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) Euler and music, by Patrice Bailhache, transla... * [with cont.] (Wayb.)
top of page bottom of page up down


Message: 4420 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 22:55:43

Subject: Re: Hermite normal form version of "25 best"

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

let's see how the heuristic for complexity matches up with the g_w 
measure. i'll leave out the heuristic for error, since we appear to 
have settled on an unweighted error measure.


> 135/128 (3)^3*(5)/(2)^7 > > g_w 2.558772839
log(135)/2.558772839 = 1.9170
> 256/243 (2)^8/(3)^5 > > g_w 3.472662942
log(243)/3.472662942 = 1.5818
> 25/24 (5)^2/(2)^3/(3) > > g_w 1.597771402
log(25)/1.597771402 = 2.0146
> 648/625 (2)^3*(3)^4/(5)^4 > > g_w 3.484393186
log(625)/3.484393186 = 1.8476
> 16875/16384 (3)^3*(5)^4/(2)^14 > > g_w 4.719203505
log(16875)/4.719203505 = 2.0625
> 250/243 (2)*(5)^3/(3)^5 > > g_w 3.413658644
log(250)/3.413658644 = 1.6175
> 128/125 (2)^7/(5)^3 > > g_w 2.613294890
log(125)/2.613294890 = 1.8476
> 3125/3072 (5)^5/(2)^10/(3) > > g_w 4.128050871
log(3125)/4.128050871 = 1.9494
> 20000/19683 (2)^5*(5)^4/(3)^9 > > g_w 5.817894303
log(19683)/5.817894303 = 1.6995
> 81/80 (3)^4/(2)^4/(5) > > g_w 2.558772839
log(81)/2.558772839 = 1.7174
> 2048/2025 (2)^11/(3)^4/(5)^2 > > g_w 3.822598772
log(2025)/3.822598772 = 1.9917
> 78732/78125 (2)^2*(3)^9/(5)^7 > > g_w 6.772337791
log(78125)/6.772337791 = 1.6635
> 393216/390625 (2)^17*(3)/(5)^8 > > g_w 6.722154036
log(390625)/6.722154036 = 1.9154
> 2109375/2097152 (3)^3*(5)^7/(2)^21 > > g_w 7.187006703
log(2109375)/7.187006703 = 2.0261
> 4294967296/4271484375 (2)^32/(3)^7/(5)^9 > > g_w 10.74662038
log(4271484375)/10.74662038 = 2.0635
> 15625/15552 (5)^6/(2)^6/(3)^5 > > g_w 4.990527341
log(15625)/4.990527341 = 1.9350
> 1600000/1594323 (2)^9*(5)^5/(3)^13 > > g_w 8.314887839
log(1594323)/8.314887839 = 1.7176
> 1224440064/1220703125 (2)^8*(3)^14/(5)^13 > > g_w 11.61862841
log(1220703125)/11.61862841 = 1.8008
> 10485760000/10460353203 (2)^24*(5)^4/(3)^21 > > g_w 13.57752022
log(10460353203)/13.57752022 = 1.6992
> 6115295232/6103515625 (2)^23*(3)^6/(5)^14 > > g_w 11.20594372
log(6103515625)/11.20594372 = 2.0107
> 19073486328125/19042491875328 (5)^19/(2)^14/(3)^19 > > g_w 16.55086763
log(19073486328125)/16.55086763 = 1.8476
> 32805/32768 (3)^8*(5)/(2)^15 > > g_w 5.957335766
log(32805)/5.957335766 = 1.7455
> 274877906944/274658203125 (2)^38/(3)^2/(5)^15 > > g_w 13.67967551
log(274658203125)/13.67967551 = 1.9254
> 7629394531250/7625597484987 (2)*(5)^18/(3)^27 > > g_w 19.05445924
log(7625597484987)/19.05445924 = 1.5567
> 9010162353515625/9007199254740992 (3)^10*(5)^16/(2)^53 > > g_w 17.87941745
log(9010162353515625)/17.87941745 = 2.0547 the ratios fall between 1.5567 and 2.0635. so it's good as a rough guesstimate . . .
top of page bottom of page up down


Message: 4421 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 14:58:30

Subject: Re: Decatonics

From: Carl Lumma

>> >aul, how long are you going to continue using this fallacious >> application of Rothenberg? Who has been throwing out scales >> for being improper in the way that the Pythagorean diatonic is >> improper? Certainly not Rothenberg! >
>sorry -- just force of habit (acquired from john chalmers, i >believe). so who did introduce the terms 'proper' and 'improper' >in this context, if not rothenberg?
I'm not sure. I've seen it around the list(s) on occasion, and complained bitterly every time. :) To recap: Rothenberg defines his "ideal measure" as one that works on all the subsets of a scale. However, due mainly to computational constraints, he uses stability, without a cutoff -- he lists all the scales in 12-et and ranks them by stability. I've proposed a varient of stability based on the actual log-frequency amount of overlap, and this is available in Scala. Even for low stability scales, Rothenberg does not throw them out -- he simply predicts that a different kind of composition suits them better. I don't endorse all of Rothenberg's model -- in fact, I only understand a small part of it. I do agree that he tries to go too far without considering harmony -- I suspect his derrivation of the diatonic, pentatonic scale, etc, using only stability and efficiency blows up in a tuning other than 12 where almost everything is consonant. He may also have jumped to conclusions with the ethno-musical data, though I've found that trying to transpose themes over the modes of low-stability scales doesn't seem to work. -Carl
top of page bottom of page up down


Message: 4422 - Contents - Hide Contents

Date: Thu, 28 Mar 2002 16:47:29

Subject: generalizing diatonicity

From: Carl Lumma

Hello Mark, Jeff, All,

I've just updated my diatonicity shopping list:

 * [with cont.]  (Wayb.)

Does it make any more sense now?

-Carl


top of page bottom of page up down


Message: 4423 - Contents - Hide Contents

Date: Fri, 29 Mar 2002 09:03:14

Subject: Re: Digest Number 331

From: John Chalmers

Manuel: Thanks for the counter-example to CS equalling strict propriety.
I stand corrected

As for harmonic and inharmonic vocal timbres. I was apparently mistaken.
What confused me was the fact that outside of the European culture area,
vocal timbres are usually nasal and/or strident and their use may be
correlated with non-JI (or close approximations) tunings and intervals.
For example, how harmonic is the spectrum of the Indonesian singing
voice or that of American Indians? For that matter, how harmonically
related are the formants of speech in many languages (Khoisan, North
Caucasian, etc.). It seemed to me that to produce the clear harmonic
tone of European singing (primarily Church and Italianate styles) takes
a lot of training. Untrained voices often sound less harmonic to me, but
I could be wrong. 

How in tune are the harmonics and are the usual pitches of the vowel
formants for most speakers actually close to harmonics? I don't know. Is
this information in the literature (Sundberg, perhaps?)? 

--John


top of page bottom of page up down


Message: 4424 - Contents - Hide Contents

Date: Fri, 29 Mar 2002 09:48:39

Subject: Re: Decatonics

From: Carl Lumma

>> >n 1/3-comma meantone is strictly proper and CS >> in 12-et is proper but not strictly so, not CS >> in Pythagorean tuning is improper and CS >
>All of the above are epimorphic. > >I think of scales in terms of the properties epimorphic, >convex, and connected.
If I'm right about the terms convex and connected, they only apply to periodicity blocks. What about non-just scales? Rothenberg claims some of the stuff of melody doesn't have anything to do with harmony. -Carl
top of page bottom of page up

Previous Next

4000 4050 4100 4150 4200 4250 4300 4350 4400 4450 4500 4550 4600 4650 4700 4750 4800 4850 4900 4950

4400 - 4425 -

top of page