Tuning-Math Archive Section 2: 1225

Message: 1225 - Contents - Hide Contents Date: Mon, 06 Aug 2001 19:40:46 Subject: Re: Another BP linear temperament From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> > But this is just a linear temperament. How is it more specifically a > "generalised meantone"? Is it merely because it seeks to approximate > only two primes (other than the interval-of-equivalence), in this case > 2 and 7? That certainly isn't enough for _me_ to consider it a > "generalised meantone". What do others think?

I'd have to agree with you Dave -- I think "meantone" would have to mean something a bit more specific, even if you "generalize" it. But that's just terminology.

> > So this BP temperament still has an approximate 7:9 generator but it's > typically a narrow one, where the previous BP temperament has a wide > one. The mapping from primes to numbers of generators is: > > Prime No. generators > ----- -------------- > 2 -6 > 3 0 > 5 (don't care, but 2 is best) > 7 -1 > > The MA optimum {1,2,7} generator is 434.0 cents giving a maximum error > of 1.2 cents. This generator is, as you said, a 7:9 narrowed by 1/6 of > the above comma. > > If 5's are included, but not 4's, i.e. {1,2,5.7}, the optimum > generator is 436.0 cents (1/7 comma wide) with errors of 12.3 cents. > > The optimum for the full 7-limit, i.e. {1,2,4,5,7}, is 435.15 cents > (in between the previous two) but with errors of 14.1 cents.

So I take it Dan's suggestion of which commas temper out doesn't look as good? I'm a little lost here.

Message: 1226 - Contents - Hide Contents Date: Mon, 06 Aug 2001 19:44:08 Subject: Re: Another BP linear temperament From: Paul Erlich --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Hi Dave, > > <<Ok. But for this to be a "generalised meantone" in any sense other > than the one which is much better described by the term "linear > temperament", then a chain of _four_ 9/7's would have to be > tritave-equivalent to a 7/6. It needs a chain of five according to > your comma.>> > > I disagree, and I think your looking at this in much too narrow a way. > The process of determining a comma and fractionalizing and > distributing it is what's being generalized...

But no one ever called schismic temperament, which seems to have predated meantone, a "generalized meantone" . . .

> > Again, it's a generalization of meantone because the "tone" and the > "mean" are generalized!

Ah . . . this may be what Dave and I are missing. What are the JI "tones" here? Anyway, let's not let squabbles over terminology (Margo suggested some nice ideas for this) blind us to the fact that all of the others on this list have great pools of insight from which they are drawing.

Message: 1227 - Contents - Hide Contents Date: Mon, 06 Aug 2001 20:01:13 Subject: Re: HyperMOS From: Paul Erlich --- In tuning-math@y..., carl@l... wrote:

> Finally, I'll take the time to say something on this > fascinating thread... > > () Does "HyperMOS" refer to the generalized higher-D > MOS's of Paul's hypothesis, or are they another type?

I don't know . . . it's really a term you coined for something we may not have unambiguously defined yet.

> > () Paul, you said that 'as far as we can tell, CS > and PB are the same thing'. But aren't there many > irrational CS scales that don't have any sensible > PB interp?

I mean the CSs that Kraig has drawn from Wilson's work, all of which are JI scales.

> > () Paul's (hi, Paul!) hypothesis post may be the most > feverishly dense and impressive post I've seen since > his original harmonic entropy post in '97. I don't > think a notion of HyperMOS is nearly as important as > harmonic entropy, but who knows.... I'll quote the > pertinent parts here, and maybe say something more > later when (a) I understand more of it and (b) I don't > have an apartment full of boxes to unpack. For now, > please accept my once-over comments:

OK, but note that none of this relates to "HyperMOS", merely to the Hypothesis which relates PBs to MOS scales. I think it's important because many modern music theorists are obsessed with MOS, which they call well-formed or deep scales, but the Hypothesis _derives_ the concept from basic JI considerations.

>> Take an n-dimensional lattice, and pick n independent unison >> vectors. Use these to divide the lattice into parallelograms or >> parallelepipeds or hyperparallelepipeds, Fokker style. Each one >> contains an identical copy of a single scale (the PB) with N notes. >

> Paul, did you have to use the letter "n" twice? You seem to want > to use case to distinguish them... I'll hold you to that. OK. Sorry. >

>> Any vector in the lattice now corresponds to a single generic >> interval in this scale no matter where the vector is placed (if >> the PB is CS, which it normally should be). > > Check. >

>> Now suppose all but one of the unison vectors are tempered out. >> The "wolves" now divide the lattice into parallel strips, or >> layers, or hyper-layers. The "width" of each of these, along the >> direction of the chromatic unison vector (the one that remains >> untempered), is equal to the length of exactly one of this >> chromatic unison vector. > > Check. >

>> Now let's go back to "any vector in the lattice". This vector, >> added to itself over and over, will land one back at a pitch in the >> same equivalence class as the pitch one started with, after N >> iterations (and more often if the vector represents a generic >> interval whose cardinality is not relatively prime with N). > > Okay. >

>> In general, the vector will have a length that is some fraction >> M/N of the width of one strip/layer/hyperlayer, measured in the >> direction of this vector (NOT in the direction of the chromatic >> unison vector). M must be an integer, since after N iterations, >> you're guaranteed to be in a point in the same equivalence class as >> where you started, hence you must be an exact integer M >> strips/layers/hyperlayers away. As a special example, the generator >> has length 1/N of the width of one strip/layer/hyperlayer, measured >> in the direction of the generator. >

> I don't see why M/N yet, but I can see what's happening.

M is just the number of times you cross a "wolf" or period boundary before you get back to the note you started with. N is the number of steps before you get back to the note you started with. Now do you see why the length is M/N units in the relevant direction?

>> Anyhow, each occurence of the vector will cross either floor(M/N) >> or ceiling(M/N) boundaries between strips/layers/hyperlayers. Now, >> each time one crosses one of these boundaries in a given direction, >> one shifts by a chromatic unison vector. Hence each specific >> occurence of the generic interval in question will be shifted by >> either floor(M/N) or ceiling(M/N) chromatic unison vectors. Thus >> there will be only two specific sizes of the interval in question, >> and their difference will be exactly 1 of the chromatic unison >> vector. And since the vectors in the chain are equally spaced and >> the boundaries are equally spaced, the pattern of these two sizes >> will be an MOS pattern. >

> Spot on, for the 1-D case I can vidi!!!

I'll have to make an illustration for the 2-D case, as I promised Monz. But this will still be a (1-D) MOS. Whatever n is, if you temper out n-1 unison vectors, you're left with a (1-D) MOS.

Message: 1228 - Contents - Hide Contents Date: Mon, 06 Aug 2001 20:06:55 Subject: Re: Harmonics of Pentachord From: Paul Erlich --- In tuning-math@y..., J Gill <JGill99@i...> wrote:

> The dramatic decrease of the overtones of the 1 Hz voice in the pentachord > as frequency increases appears (to me) to indicate that the commonality in > frequency of the fundamentals (and their respective overtones) of

the 3 Hz,

> 5Hz, and 7 Hz voices within the pentachord may well possess *more* > significance (in terms of any interactions of frequency components > appearing at common frequencies) than interactions between

overtones of the

> 1 Hz fundamental and the individual 3 Hz, 5 Hz, and 7 Hz voices. What do > you think? I would be interested in any comments or feedback.

I wish I understood the intuition behind what you were asking. Can you describe any situations where you feel the comparison is clear- cut, so we can better understand how you would gauge "signifance" here?

Message: 1230 - Contents - Hide Contents Date: Mon, 06 Aug 2001 23:04:57 Subject: Re: HyperMOS From: carl@xxxxx.xxx

>> >) Does "HyperMOS" refer to the generalized higher-D >> MOS's of Paul's hypothesis, or are they another type? >

>I don't know . . . it's really a term you coined for something we >may not have unambiguously defined yet.

Wow! I don't even remember seeing it before Dave's post. All those years of sniffing glue must be starting to catch up with me...

>> () Paul, you said that 'as far as we can tell, CS >> and PB are the same thing'. But aren't there many >> irrational CS scales that don't have any sensible >> PB interp? >

> I mean the CSs that Kraig has drawn from Wilson's work, all of > which are JI scales.

Thanks (just checking).

> OK, but note that none of this relates to "HyperMOS", merely to the > Hypothesis which relates PBs to MOS scales. I think it's important > because many modern music theorists are obsessed with MOS, which > they call well-formed or deep scales, but the Hypothesis _derives_ > the concept from basic JI considerations.

Yeah- that's what makes it so exciting! I've never been a big fan of MOS myself, until now...

>>> In general, the vector will have a length that is some fraction >>> M/N of the width of one strip/layer/hyperlayer, measured in the >>> direction of this vector (NOT in the direction of the chromatic >>> unison vector). M must be an integer, since after N iterations, >>> you're guaranteed to be in a point in the same equivalence class >>> as where you started, hence you must be an exact integer M >>> strips/layers/hyperlayers away. As a special example, the >>> generator has length 1/N of the width of one >>> strip/layer/hyperlayer, measured in the direction of the >>> generator. >>

>> I don't see why M/N yet, but I can see what's happening. >

> M is just the number of times you cross a "wolf" or period boundary > before you get back to the note you started with. N is the number > of steps before you get back to the note you started with. Now do > you see why the length is M/N units in the relevant direction?

Yup. I've got some other questions though. I'll: () Wait for the vis you promised Monz. () Think about 'em on my own for a while.

> I'll have to make an illustration for the 2-D case, as I promised > Monz. But this will still be a (1-D) MOS. Whatever n is, if you > temper out n-1 unison vectors, you're left with a (1-D) MOS.

Right. What's killing me is what happens when you don't... -Carl

Message: 1231 - Contents - Hide Contents Date: Mon, 06 Aug 2001 23:52:45 Subject: Re: Another BP linear temperament From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

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

>> The MA optimum {1,2,7} generator is 434.0 cents giving a maximum > error

>> of 1.2 cents. This generator is, as you said, a 7:9 narrowed by 1/6 > of

>> the above comma. >> >> If 5's are included, but not 4's, i.e. {1,2,5.7}, the optimum >> generator is 436.0 cents (1/7 comma wide) with errors of 12.3 cents. >> >> The optimum for the full 7-limit, i.e. {1,2,4,5,7}, is 435.15 cents >> (in between the previous two) but with errors of 14.1 cents. >

> So I take it Dan's suggestion of which commas temper out doesn't look > as good? I'm a little lost here.

No. Dan's statements were spot-on. If you only want to approximate ratios of {1,2,7}*3^n then a generator which is a 7:9 reduced by 1/6 of the comma he gave (2^1 * 3^10 * 7^-6 = 6.7 cents), is superb (max 1.2c errors and only 6 generators per triad). But if you wanted to include any other non-multiples of 3 (e.g. 4 or 5) then it's very ordinary.

Message: 1232 - Contents - Hide Contents Date: Mon, 06 Aug 2001 23:55:55 Subject: Re: Another BP linear temperament From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> > No. Dan's statements were spot-on. If you only want to approximate > ratios of {1,2,7}*3^n then a generator which is a 7:9 reduced by 1/6 > of the comma he gave (2^1 * 3^10 * 7^-6 = 6.7 cents), is superb (max > 1.2c errors and only 6 generators per triad). But if you wanted to > include any other non-multiples of 3 (e.g. 4 or 5) then it's very > ordinary.

I see. So if you've only got two dimensions in the lattice, Dan's the man!!

Message: 1233 - Contents - Hide Contents Date: Mon, 06 Aug 2001 22:28:59 Subject: Re: Harmonics of Pentachord From: Herman Miller On Mon, 06 Aug 2001 04:26:36 -0700, J Gill <JGill99@xxxxxx.xxx> wrote:

>I've attached (and uploaded to J Gill folder in Files) a diagram which I >came up with recently which attempts to construct a model of a pentachord >made up of complex waveforms "voices" (fundamental + overtones) tuned to >the frequencies 1 Hz, 3 Hz, 5 Hz, Hz, and 9 Hz.

I'm sure this is interesting, but I don't like 3000-line messages suddenly showing up in my inbox. Can we agree to limit this list to reasonably sized messages?

Message: 1234 - Contents - Hide Contents Date: Tue, 07 Aug 2001 17:18:21 Subject: Re: HyperMOS From: Paul Erlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> >> From: <carl@l...> >> To: <tuning-math@y...>

>> Sent: Monday, August 06, 2001 4:04 PM >> Subject: [tuning-math] Re: HyperMOS >> >>

>>> OK, but note that none of this relates to "HyperMOS", merely to the >>> Hypothesis which relates PBs to MOS scales. I think it's important >>> because many modern music theorists are obsessed with MOS, which >>> they call well-formed or deep scales, but the Hypothesis _derives_ >>> the concept from basic JI considerations. >>

>> Yeah- that's what makes it so exciting! I've never been a big >> fan of MOS myself, until now... >

> Paul, I'm *deeply* interested in this too! > > Can you lay the whole thing out (what you have so far... I realize > it's "in progress") simply and clearly, like a "Gentle Introduction"? > With *lots* and lots of visuals. I'll be happy to host it somewhere > at Sonic Arts... I think a Dictionary entry for "HyperMOS" would be > a good gateway to it. Guys?

The problem is, at present, we don't know exactly what HyperMOS will end up meaning, or if we'll even end up agreeing on a meaning. And again, the Hypothesis doesn't concern HyperMOS, just MOS. As soon as I have time, I'll get cracking on those visuals.

> Is a (1-D) MOS what Erv referred to as "extended linear mapping"?

By (1-D) MOS, I just mean MOS (as opposed to HyperMOS). An MOS is a linear tuning (since there is a single generator along with an associated interval of repetition), carried out to some number of notes such that there are only two step sizes.

Message: 1236 - Contents - Hide Contents Date: Tue, 7 Aug 2001 02:09:53 Subject: Re: HyperMOS From: monz

> From: <carl@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, August 06, 2001 4:04 PM > Subject: [tuning-math] Re: HyperMOS > >

>> OK, but note that none of this relates to "HyperMOS", merely to the >> Hypothesis which relates PBs to MOS scales. I think it's important >> because many modern music theorists are obsessed with MOS, which >> they call well-formed or deep scales, but the Hypothesis _derives_ >> the concept from basic JI considerations. >

> Yeah- that's what makes it so exciting! I've never been a big > fan of MOS myself, until now...

Paul, I'm *deeply* interested in this too! Can you lay the whole thing out (what you have so far... I realize it's "in progress") simply and clearly, like a "Gentle Introduction"? With *lots* and lots of visuals. I'll be happy to host it somewhere at Sonic Arts... I think a Dictionary entry for "HyperMOS" would be a good gateway to it. Guys?

> Yup. I've got some other questions though. I'll: > > () Wait for the vis you promised Monz. > () Think about 'em on my own for a while. >

>> I'll have to make an illustration for the 2-D case, as I promised >> Monz. But this will still be a (1-D) MOS. Whatever n is, if you >> temper out n-1 unison vectors, you're left with a (1-D) MOS.

Is a (1-D) MOS what Erv referred to as "extended linear mapping"? And I don't remember you promising this to me! (years of hanging out with Herbert catching up with me)

> Right. What's killing me is what happens when you don't... Me too!

love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 1237 - Contents - Hide Contents Date: Thu, 9 Aug 2001 09:52:24 Subject: Re: [cm] X.J. Scott and Jacky Ligon get Froggy... From: monz

> From: <jacky_ligon@xxxxx.xxx> > To: <crazy_music@xxxxxxxxxxx.xxx> > Sent: Tuesday, August 07, 2001 8:20 PM > Subject: [cm] X.J. Scott and Jacky Ligon get Froggy... > > <Yahoo groups: /crazy_music/files/Jacky Ligon/F... * [with cont.] 3> Hi Jacky,

This is awesome! (and the post describing it was hilarious!)

> Cents Note Names > 0 Throaty > 369.398 Croaker > 564.028 Jeep Jeep > 1509.372 Screech > 1688.114 Jeepers > 2307.751 High Screech > 2895.611 Extra High Jeepers > > I would like to request for Master Monz to chart this out on a > Frog Lattice (if possible and convenient to do).

Ask and ye shall receive... (partially, anyway...) I did a search thru a 13-limit ratio database delimited by 3^(-10...10) * 5^(-2...2) * (7,11,13)^(-1...1), (that's 2,835 different 8ve-invariant pitches), adjusted the exponents of 2 to match Jacky's 8ve-specific list, and came up with the following: ratio Jacky's delta note 2 3 5 7 11 13 decimal fraction cents cents cents names | 0 0 0 0 0 0| 1.0 1/1 0.000 0.000 0 Throaty | 8 0 -2 -1 1 -1| 1.237802198 2816/2275 369.337 369.398 -0.061 Croaker | 6 -1 1 -1 -1 0| 1.385281385 320/231 564.215 564.028 +0.187 Jeep Jeep | 8 -7 0 -1 1 1| 2.391273107 36608/15309 1509.335 1509.372 -0.037 Screech |-1 -1 2 1 -1 0| 2.651515152 175/66 1688.180 1688.114 +0.066 Jeepers | 9 -3 -1 0 0 0| 3.792592593 512/135 2307.821 2307.751 +0.070 High Screech | 7 -7 0 1 0 1| 5.326017375 11648/2187 2895.669 2895.611 +0.058 Extra High Jeepers The greatest error between my ratios and Jacky's cents values is less than 1/5-cent (in the case of "jeep jeep"). The errors for the other notes are all less than 1/10-cent, mostly in the range of about 1/15 to 1/30 of a cent. I've been working on the Frog Lattice for several hours, but it's quite complicated and will take much longer to finish. When it's done I'll make a webpage of it, with this data. It would be nice to have an algorithm that can convert from cents values into prime-factorized format, to varying degrees of accuracy, for general use. Any takers? (Dan Stearns once showed me an algorithm for converting from cents values to ratios, but prime-factoring irrational numbers seems a bit more difficult as the possibilities are endless.) love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 1239 - Contents - Hide Contents Date: Fri, 10 Aug 2001 03:58:23 Subject: Re: Hi gang. From: Paul Erlich --- In tuning-math@y..., "John Starrett" <jstarret@c...> wrote:

> Just a short not to let you know I am lurking. > > John Starrett Hi John!

Dear Graham and Dave, What we need is a really user-friendly, _practical_ guide to a bunch of the new temperaments and their MOSs (and ideally, tetrachordal alterations of those MOSs in cases like 10-of-22 and 22-of-46). If one of you does this, you'll be a real hero. My only other plead would be to please optimize for all saturated chords (especially the 9-limit ones) and not just for all complete chords, through some limit. I want to see Margo's Wonder scale (generator = 2*MIRACLE generator) in there. Hey, my ~22-tET decatonics made the rankings for 7-limit, didn't they? How about 9-limit? -Paul

Message: 1240 - Contents - Hide Contents Date: Fri, 10 Aug 2001 04:53:55 Subject: On harmonic entropy From: Paul Erlich Harmonic entropy is the simplest possible model of consonance. Nothing simpler is possible. It asks the question, "how confused is my brain when it hears an interval?" It assumes only one parameter in answering this question. Our _brain_ determines what pitch we'll hear when we listen to a sound. It does so by trying to match the frequencies in the sound's spectrum (timbre) with a harmonic series. The pitch we hear is high or low depending on whether the frequency of the fundamental of the best-fit harmonic series is high or low. The pitch corresponding to the fundamental itself need not be physically present in the sound. Sometimes, the meaning of "best-fit" will not be clear and we'll hear more than one pitch. This happens when several tones are playing together, or when the spectrum of the instrument is highly inharmonic. Entropy is a mathematical measure of disorder, or confusion. For a dyad, consisting of two tones which are sine waves or have harmonic spectra, one can immediately understand the behavior of the harmonic entropy function. The brain's attempt to fit the stimulus to a harmonic series is quite unambiguous when the ratio between the frequencies is a simple one, such as 2:1 or 3:2. More complex ratios, or irrational ones far enough from any simple one, and the limited resolution with which the brain receives frequency information makes it harder for it to be sure about how to fit the stimulus into a harmonic series. The resolution mentioned is parameterized by the variable s. A computer program is used to calculate the entropy for every possible interval (in, say, 1¢ increments). The set of potential "fitting" ratios is chosen to be large enough (by going high enough in the harmonic series) so that further enlargements of the set cease to affect the basic shape of the harmonic entropy curve. You can see some examples at Yahoo groups: /harmonic_entropy/files/dyadic/h... * [with cont.] pg Yahoo groups: /harmonic_entropy/files/dyadic/t... * [with cont.] 2877.jpg (the "weighting" referred to here is not a weighting of anything in the model but merely refers to a computational shortcut used). Considering ratios to be different if they are not in lowest terms (appropriate, for example, if we assume 6:3 might be interpreted as the sixth and third harmonics, rather than simply as a 2:1 ratio) leads to this slightly different appearance: Yahoo groups: /harmonic_entropy/files/dyadic/t... * [with cont.] 877.jpg Certain chords of three or more notes blend so well that it sounds like fewer notes are playing than there actually are. We hear a "root" which is kind of the overall pitch of the chord, and the most stable bass note for it. The harmonic entropy of these chords (which is not a function of the harmonic entropies of the intervals in the chord) is low. Our non-laboratory experiments on the harmonic entropy list seem to conclusively show that the dissonance of a chord can't be even close to a function of the dissonances of the constituent intervals. For example, everyone put the 4:5:6:7 chord near the top of their ranking of 36 recorded tetrads from least to most dissonant, while everyone put 1/7:1/6:1/5:1/4 much lower. These two chords have the same intervals. Therefore, it seems to be the case that dissonance measures which are functions of dyadic (intervallic) dissonance account for, at best, a relatively small portion of the dissonance of chords. Such measures include those of Plomp and Levelt, Kameoka and Kuriyagawa, and Sethares. If you're interested in discussing further, please join the harmonic entropy group: Yahoo groups: /harmonic_entropy * [with cont.] ______________________________________________________________________

Message: 1241 - Contents - Hide Contents Date: Fri, 10 Aug 2001 05:46:46 Subject: Re: Hi gang. From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Dear Graham and Dave, > > What we need is a really user-friendly, _practical_ guide to a bunch > of the new temperaments and their MOSs (and ideally, tetrachordal > alterations of those MOSs in cases like 10-of-22 and 22-of-46). > > If one of you does this, you'll be a real hero. > > My only other plead would be to please optimize for all saturated > chords (especially the 9-limit ones) and not just for all complete > chords, through some limit. I want to see Margo's Wonder scale > (generator = 2*MIRACLE generator) in there. > > Hey, my ~22-tET decatonics made the rankings for 7-limit, didn't > they? How about 9-limit? > > -Paul

It's all yours Graham.

Message: 1243 - Contents - Hide Contents Date: Fri, 17 Aug 2001 18:22:14 Subject: Re: Hi! Seeking advice From: Paul Erlich --- In tuning-math@y..., BobWendell@t... wrote:

> So I would love to know of any recommendations you might offer as a > good first primer on lattices, unison vectors, periodicity blocks, > linear temperaments, etc. and their relationships to each other. > > Thanks tons in advance! > > Sincerely, > > Bob

As far as I know, we here are on the cutting edge of this stuff, and it really isn't dealt with anywhere else. My advice would be to go through the archives of this list, and of the tuning list before this list split off from it. Ask us lots of questions.

Message: 1245 - Contents - Hide Contents Date: Fri, 17 Aug 2001 19:31:07 Subject: Re: Hi! Seeking advice From: Paul Erlich --- In tuning-math@y..., BobWendell@t... wrote:

> Thanks so much, Paul! I feel very fortunate indeed to have run across > you all here. I was hoping for some kind of more coherent > presentation format, but I guess that doesn't exist yet. I will take > your generous advice and hope that I don't pester you too much. You > all seem to have a high tolerance for my naivete in such matters and > I deeply appreciate that! > > Sincerely, > > Bob

Bob, I feel particularly close to you since we seem to share a very similar set of opinions and ideals, in both of our cases acquired through genuine musical experience. I'm curious whether my last response to you on the tuning list made sense to you, and if you have any outstanding questions. -Paul

Message: 1246 - Contents - Hide Contents Date: Fri, 17 Aug 2001 19:56:20 Subject: Re: Hi! Seeking advice From: Paul Erlich --- In tuning-math@y..., BobWendell@t... wrote:

> Thanks so much, Paul! I feel very fortunate indeed to have run across > you all here. I was hoping for some kind of more coherent > presentation format

I trust you've read the _Gentle Introduction to Fokker Periodicity Blocks_ (A gentle introduction to Fokker periodicity bl... * [with cont.] (Wayb.))?