Yahoo Tuning Groups Ultimate Backup tuning Superparticular Spectrums

J Gill's ("2 Cents"): I would venture to say here that Jacky is here proposing that "minimums in spectral entropy" (entropy as a degree of "randomness"), appearing like an oasis to tease the thirsty desert traveler, cannot (in "real life") reliably be characterized, as they (rarely, if ever) might briefly appear amidst the "chaotic tonal miasma" which (in the doing) perennially defies the "conscious mind" (which can only look backwards in time towards a pre-determined outcome)...? Would, then, anyone be able to reliably state that their "personal metaphors" of "tonality" are (at all) transferrable?

J Gill

--- In tuning@y..., "jacky_ligon" <jacky_ligon@y...> wrote:
> Paul Erlich stated:
>
> <<The simpler the
> ratio describing the interval _between_ two pitches, the more
> spectral commonality the two pitches will have, and the simpler the
> periodic pattern they will produce. Therefore, virtually all my work
> in this area has concerned itself with finding scales that have a
> very high proportion of simple ratios (or good approximations
> thereof) _between_ pitches.>>
>
> What???!!!
>
> What ***spectrum*** are you talking about here?
>
>
> Has Paul (or anyone else), in their conceptual labors surrounding
> the functional utility in a "spectral commonality" existing between
> pairs of JI scale pitch-ratios which may be desirable to include
> within the notes of JI scales (due to such "spectral commonality")
> generated any numerical proofs that such *is* the case when
> "superparticular step-sizes" are utilized (to an extent which clearly
> surpasses other approaches in the determination of JI pitch-ratios)?
>
> Well - I don't know about this but it brings up a pervasive question
> for me:
>
> Why does music where the tuning does not allign to the partials of
> one or many instruments of an ensemble of timbres - *still sound
> beautiful* - as in the case of Gamelan? Obviously there's something
> perhaps more, or as, important as "spectral commonality", or else
> music that didn't obey these imaginings would sound utterly horrible
> to *everyone* that hears them.
>
> My personal and humble feeling is that when you are dealing with a
> musical context in which there is a mixed ensemble of different
> timbres with radically different partial structures, that this idea
> of "spectral commonality" is utterly meaningless.
>
> Numerologically Yours,
>
> Jacky Ligon

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "jacky_ligon" <jacky_ligon@y...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:

> > J Gill's ("2 Cents"): I would venture to say here that Jacky is
> here proposing that "minimums in spectral entropy" (entropy as a
> degree of "randomness"), appearing like an oasis to tease the >thirsty
> desert traveler, cannot (in "real life") reliably be characterized,
> as they (rarely, if ever) might briefly appear amidst the "chaotic
> tonal miasma" which (in the doing) perennially defies >the "conscious
> mind" (which can only look backwards in time towards a pre->determined
> outcome)...? Would, then, anyone be able to reliably state that
> their "personal metaphors" of "tonality" are (at all) >transferrable?
> >
> > J Gill
>
> J,
>
> I'd say you are reading me rightly for the most part, because when >we
> begin to speak about our "personal metaphors" of "tonality", and
> there is no mention of timbre, rhythm, space and and musical >context
> under consideration, we are only seeing (and not hearing) an
> incomplete picture.
>
> At a minimum we've got to have timbre involved in any discussion of
> consonance, and recognize that the harmonic series is not the only
> series found in nature - there are many "chords of nature", and
> myriad of them inharmonic, yet just as musically beautiful and >useful
> as any other. Because of this timbral diversity which we see in the
> worlds musical cultures - *no series should be looked at as better
> than any other*.
>
> When we begin to consider the complex interactions of mixed timbres -
> many of which may be inharmonic in nature, a whole new set of rules
> and thought must be applied, which might not readily fit what works
> for our beloved harmonic series and imaginings about the lower >number
> integers.
>
> To be sure its one thing to consider only one "series" or partial
> structure at a time, but altogether another to think about many
> happening together in a real world, living, breathing musical
> context. In the "chaotic tonal miasma" of real world musical >context,
> is where the *ear* reigns supreme.
>
> Jacky Ligon
> President of the American Numerological Society

But Jacky, have you no mercy (or pity) for a poor mathematician like myself who (being rather a hack at playing music - actually, I'm a baaaad motorscooter...) yearns to (before leaving this moronic existence we call "life") leave his fellow "man" (in the rhetorical sense) with just a few "pearls of wisdom" to guide the unwary, just a few "pointers" on how man may "better himself" through anal-retentive and sweeping "global generalizations" as to the "true" nature of the creative process? Have you no "faith in the numbers", my "numerological" brother? How dare you defy the great Oz!

Maybe Pythagorus never wrote anything down (that we know of) because he *didn't* want to be misinterpreted posthumously? Perhaps he did not mean (at all) that "all is number", but was misquoted by an over-zealous disciple who (erroneously) recorded the statement (instead) that "all is number-ed". Sort of like in the Monty Python movie entitled "The Life of Brian" where (upon having a bad seat at the "sermon on the mount") the faithful heard "blessed are the cheese-makers"... The rest was history. From then on, all Swiss cheese *had* to have holes - no "ands", "ifs", or "buts" about it!

Cheers, J Gill, Consultant to the American Numerological Society :)

🔗paulerlich <paul@stretch-music.com>

Clearly a harmonic spectrum, which is the context Jeremy raised and
in which this statement was made.

> Well - I don't know about this but it brings up a pervasive
question
> for me:
>
> Why does music where the tuning does not allign to the partials of
> one or many instruments of an ensemble of timbres - *still sound
> beautiful* - as in the case of Gamelan? Obviously there's something
> perhaps more, or as, important as "spectral commonality", or else
> music that didn't obey these imaginings would sound utterly
horrible
> to *everyone* that hears them.

Indeed! However, Jeremy wanted to _start_ with the assumption of
spectral commonality and proceed from there -- perhaps you missed the
earlier messages in this thread. Although I've discussed with him the
limitations of this assumption both here and on the harmonic entropy
list, and you no doubt have even more to contribute along these
lines, I also wanted (and continue to want) to help Jeremy see what
kinds of scale structures these assumptions would lead to.

> My personal and humble feeling is that when you are dealing with a
> musical context in which there is a mixed ensemble of different
> timbres with radically different partial structures, that this idea
> of "spectral commonality" is utterly meaningless.

Why "utterly meaningless"? Because it's a difficult problem? Do you
find Sethares' work in this area to be "utterly meaningless"? This
seems an odd statement coming from you, who has done so much
interesting work, and participated in so many interesting
discussions, in the area of "spectral commonality" between like
inharmonic spectra. When the spectra become different from one
another, does the idea suddenly become "utterly meaningless"?

🔗paulerlich <paul@stretch-music.com>

🔗jpehrson2 <jpehrson@rcn.com>

--- In tuning@y..., "unidala" <JGill99@i...> wrote:

/tuning/topicId_31641.html#31646

> J Gill's ("2 Cents"): I would venture to say here that Jacky is
here proposing that "minimums in spectral entropy" (entropy as a
degree of "randomness"), appearing like an oasis to tease the thirsty
desert traveler, cannot (in "real life") reliably be characterized,
as they (rarely, if ever) might briefly appear amidst the "chaotic
tonal miasma" which (in the doing) perennially defies the "conscious
mind" (which can only look backwards in time towards a pre-determined
outcome)...? Would, then, anyone be able to reliably state that
their "personal metaphors" of "tonality" are (at all) transferrable?
>

Hi J!

Well, we're getting a little into the realm of the "nutty professor"
here (whom I'm reminded is not a *real* professor, although the prior
appelation remains)... and also into the realm of classical
philosophy, Plato, etc., etc., where it finally gets down to the
point that nobody believes in any kind of commonality of experience
at all between one person and another. (It's all "forms" in the
brain...)

That may be a bit "counter scientific" and, also, counter to the
thinking of several people on this list... It seems around *these*
parts people believe there are certain verifiable perceptions related
to low integer ratios...

It seems many of the scientific studies post Helmholtz are rather
contradictory on this matter... All I can say, for certain, is that
low integer ratios appear to create an "affective" or "emotional"
reaction in certain listeners. I can see this all the time when
people listen to such ratios as contrasted with more complex ones or
with "atonal" pitch complexes...

J.P.

🔗unidala <JGill99@imajis.com>

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

> >Has Paul (or anyone else), in their conceptual labors surrounding
> >the functional utility in a "spectral commonality" existing between
> >pairs of JI scale pitch-ratios which may be desirable to include
> >within the notes of JI scales (due to such "spectral commonality")
> >generated any numerical proofs that such *is* the case when
> >"superparticular step-sizes" are utilized (to an extent which
> >clearly
> >surpasses other approaches in the determination of JI pitch->>ratios)?

> PE: I now have a different answer to this question. Assume one is >not
> interested in the melodic suitability of the scale, or in the
> periodicity block property. Then one may simply choose the set of
> ratios with the most spectral commonality with a central 1/1,
> confident that these pitches will have a high degree of spectral
> commonality with one another. Now it is my contention that such a >set
> of ratios _will_ have superparticular step sizes. One example is >the
> Farey series.

JG: At: http://www.cut-the-knot.com/ctk/Farey.html one finds the property of what I have been terming "Farey adjacence" described (perhaps more accurately) as "Farey successive"

<< Let m1/n1 and m2/n2 be two successive terms of FN. Then
<<(1) m2n1 - m1n2 = 1

<<For three successive terms m1/n1, m2/n2, and m3/n3, the middle term <<is the mediant of the other two

<<(2) m2/n2 = (m1 + m3)/(n1 + n3)

As a result of the "step-size" ratio beween any (successively related) pair of ratios in the "Farey series" being "superparticular" ratios,

can it be said that:

IF all of the elements of a set of "step-sizes" (from which one plans to derive scale pitch-ratios) are "superparticular" ratios

THEN the scale so derived from that "superparticular" set of of step-size ratios will contain ONLY "Farey successive" scale pitch-ratios?

>PE: Another example is the set of Stern-Brocot ratios up to
> a given level.

JG: Where I have observed that *some* "nodal ratios" (which I have evaluated as being scale "pitch-ratios") along certain branched pathways will be "superparticular", while *some* will *not* be "superparticular". Nevertheless, all "nodal ratios" are "Farey successive" to their neighbors (to the North and South on the diagram as customarily displayed), and thus are represented by what are "superparticular" step-size ratios between those nodal neighbors.

>PE: Another example is the set of ratios within a given
> Tenney Harmonic Distance. In all these cases, and many other >similar
> ones, the set of ratios will have superparticular step sizes. The
> case for the Farey series is proved in Hardy and Wright's textbook >on
> number theory. Should you wish to pursue this further, I refer you >to
>
> tuning-math@y...

Curiously, J Gill

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "jacky_ligon" <jacky_ligon@y...> wrote:
> > Paul Erlich stated:
> >
> > <<The simpler the
> > ratio describing the interval _between_ two pitches, the more
> > spectral commonality the two pitches will have, and the simpler the
> > periodic pattern they will produce. Therefore, virtually all my
> work
> > in this area has concerned itself with finding scales that have a
> > very high proportion of simple ratios (or good approximations
> > thereof) _between_ pitches.>>
> >
> > What???!!!
> >
> > What ***spectrum*** are you talking about here?
>
> Clearly a harmonic spectrum, which is the context Jeremy raised and
> in which this statement was made.
>
> > Well - I don't know about this but it brings up a pervasive
> question
> > for me:
> >
> > Why does music where the tuning does not allign to the partials >>of
> > one or many instruments of an ensemble of timbres - *still sound
> > beautiful* - as in the case of Gamelan? Obviously there's >>something
> > perhaps more, or as, important as "spectral commonality", or else
> > music that didn't obey these imaginings would sound utterly
>> horrible
> > to *everyone* that hears them.
>
> Indeed! However, Jeremy wanted to _start_ with the assumption of
> spectral commonality and proceed from there -- perhaps you missed >the
> earlier messages in this thread. Although I've discussed with him >the
> limitations of this assumption both here and on the harmonic >entropy
> list, and you no doubt have even more to contribute along these
> lines, I also wanted (and continue to want) to help Jeremy see what
> kinds of scale structures these assumptions would lead to.
>
> > My personal and humble feeling is that when you are dealing with >>a
> > musical context in which there is a mixed ensemble of different
> > timbres with radically different partial structures, that this >>idea
> > of "spectral commonality" is utterly meaningless.
>
> Why "utterly meaningless"? Because it's a difficult problem? Do you
> find Sethares' work in this area to be "utterly meaningless"? This
> seems an odd statement coming from you, who has done so much
> interesting work, and participated in so many interesting
> discussions, in the area of "spectral commonality" between like
> inharmonic spectra. When the spectra become different from one
> another, does the idea suddenly become "utterly meaningless"?

J Gill writes:

Jacky, thank you (and I mean that!) for "goring" my "sacred cow". Helps keep calcium deposits to a minimum to let go of one's own "seriousness", noting that "Nature (in its indifference) favors not the lowly mathematician"...

Nevertheless, it seems (to me) that Paul (in the paragraph quoted directly above this paragraph) asks some good questions.

A (theory, or knowledge-claim) of spectral "un-commonality" or "deep complexity of interelatedness" may well adequately express that perhaps some are "not seeing the forest for the trees" (where it comes to direct practicable applicability in the *act* of making sounds). And, such a tangible (musical) product *is* the subject.

I do *not* utilize my conceptual ideas about music when *playing* music, and I do agree that the vast complexity of the spectral components presents very daunting as well as humbling limitations upon one's abilities to generalize in making any assumptions.

However, there seem to be some "guidlines" which you, yourself follow (as to certain choices of scales and scale "pitch-ratios", etc.). I'll bet that you would not refer to those as "useless" (because they constitute valuable tools with which to present your musical ideas).

While humility is (without debate) a quality, and not a liability in this world - it seems (to me) that the (possibly meaningless, but who is to say?) world of mathematical expression (the hallmark of conceptual reductionists, theorizers, and speculators) should (as a form of human expression), be afforded a similar respect as you (I would think) would, as a practicing musician engaged in the act of *making sounds* (and very interesting ones, at that!). :)

Some difference do exist between these parallel endeavors. You would not (I presume) dream of telling me that, after listening to your upcoming CD, I should expect mathematical "solutions" to suddenly, effervescently just ... jump out of the page at me, giving me the insight of Euler (pouring out pages of revolutionary mathematical breakthroughs on a daily basis).

The experience of listening to your upcoming CD makes a direct "connection" to the matter of your "making sounds". While such is no small wonder, that activity does *not* purport to say more about the nature of things (or claim to be instructive in the art of numerical analysis).

However (and please do correct me if I am off-base here), perhaps you are feeling (?) that when mathematicians set about determining (or at least, in some manner, tending to enjoy "recommending") certain alignments (and successions and combinations thereof) in pitch, they should (at least) *attempt* to tackle the "entire enchilada" (as it stands, warts, non-harmonic energy, and all)?

Somewhere between sinusoidal and cosinusoidal "twinky sounds", and the overwhelming spectral forest of a symphony, one (at least hopes, perhaps futily), that some conceptual structures may prove to be useful in aiding the processes of tuning and tonal selections in time by persons actually engaging in the action of making interesting sounds.

Few are surprized when few (if any) of a collection of persons necessarily agree as to the musical "value" (to them as listeners) of a musical piece performed by you, me, or anyone else. We recognize (hopefully) the futility of attempting to impose an "aural template" upon another (as if they could be thou). I have no expectations or hopes that others would necessarily share my love for blues shuffles and loathing of polka music, etc.

So, perhaps it comes as little of a surprize that it seems that the business of mathematical conceptions surrounding a human activity which, itself (in the absence of any mathematical gadgets at play) cannot be tethered (by description utilizing language) in a manner which does justice to what is a (necessarily subjective) human perceptual process - the act of listening to one's own (or another's) "musical sounds".

Nevertheless, I (personally) am very interested in concepts which *do* (at least to an extent surpassing sin/cos sound sources) possibly lend some insight into a more comprehensive (albeit not necessarily total) "viewpoint" (you might say "earpoint") of the "elephant" of complicated composite musical spectrums. They are not presented (I hope) in what seems a dictatorial fashion, but may bring you some joy (perhaps only whimsical or visually aesthetic) at some point (you being a self-professed lover of pattern and symmetry, yourself).

Only Jacky will know whether Jeremy's (occaisionally, perhaps) interesting diagrams of pitch-ratios have direct meaning to the joy which Jacky finds in making sounds. And I would not have it any other way! I'm really glad that you are there doing what you are doing, and hope to see and hear more of your ideas about music, and what you find to be your personal likings, dislikings, and style!

Pontifications and intolerance in general are poisionous to a creative environment... Viva la difference, and I hope that you (might) enjoy some of the fruits of my "nerding-out", my friend!

Sincerely, J Gill :)

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "jacky_ligon" <jacky_ligon@y...> wrote:

> Gamelan is one of the best examples of this I know for
> the context I was intending (although there are others - too many
to
> type about at once). This music does not necessarily deal with
> concepts of "spectral commonality",

Well, I might argue that it does, but rather than trying to eliminate
beating, Gamelan tuning-timbre combinations seek to acheive prominent
beating within a certain range of beats per second, independently of
register. This effect, impossible to achieve in a tuning-timbre
combination that repeats itself exactly every octave or every ______,
may be a way of getting "expressiveness" out of the metallophones of
the Gamelan, which are incapable of vibrato. The beating rates in
question are not too different from the vibrato rates we find in our
coloratura singers and violinists in the West.

The concept of a just octave, in which second-order beating slows to
a standstill, is well known to Gamelan musicians, who call
it "pleng". Different degrees of stretching and contracting of the
octave, which will vary by register to achieve the desired rate of
second-order beating, are applied quite consciously and intentionally
by Gamelan tuners, if Daniel Wolf is to be believed.

> The implications of why this is so - I feel - should not be to
> quickly overlooked, and personally I see it as yet another rich and
> fertile field to explore. It seems on some level to be an "art of
the
> ear" - music tuned by ear, and I've heard it said that there are as
> many Gamelan tunings as there are Gamelan orchestras.

There are definitely identifiable _regional_ tuning tendencies,
though.

> In Gamelan
> music, it is usually not the goal to tune out the beats, but to
> actually and deliberately have beating in the music - it is a
> characteristic feature of this music. Again - *it beats*,

Let it beat!

> and it is
> not necessarily about RI/JI, Spectrum Tuning, ETs or whatever -
yet -
> it sounds gorgeous. That it is as ancient as it is,

Gamelan tuning as practiced today may have very little to do with
Gamelan tuning a century ago.

> and as beautiful
> to the ear as it is, to me, implies a very wide open set of
> possibilities for tuning.

Yes it does. And I think Jeremy is also seeking to open the set of
possibilities for tuning, relative to the "norm" in the English-
speaking world (and much more) of 12-tET. 12-tET *beats* too, and the
majority of professional musicians in our culture are quite attached
to that beating, such that JI sounds "insipid" to them. Jeremy is, it
seems to me, interested in laying a foundation for a new (or perhaps
old) system of music, based on a concept which some people in our
subculture _do_ find a musically desirable direction, whether from
the experience of hearing a barbershop quartet singing perfectly in
tune, or playing with analog synthesizers, or what have you . . .

> As I was jokingly showing yesterday, 22 EDO does not actually fit
> into the harmonic series template,

If I were to take this seriously, I'd say that 12-tET was arrived at
in our culture partly because it does give an "acceptable" (after
some painful historical struggles) fit to the first 6 harmonics, and
22-tET is something I like because, among other things, it gives an
equally "acceptable" fit to the first 8, or even 12, harmonics.

> Suppose in a musical setting you have multiple instruments playing
> together, and each of these has radically different partial
> structures, say for instance you are using a combinanation of tuned
> gongs, and strings - just for an imaginary scenario. Well - you
> perform and FFT on the gongs, and you find that from gong to gong
> each one has an entirely different partial structure from the next,
> some having what is nearest to the 2nd partial is in the range of
> 1155-1240 cents, and the partial nearest to the 3rd harmonic of the
> strings is between a range of 1855-1940 (these are typically what
one
> might see looking at gong spectra),

I'm shocked to hear that gongs have their 2nd and 3rd partials so
close to harmonic series values. Are you sure about this?

> so in this imaginary scenario,
> are we looking for a tuning that features spectral commonality, or
is
> it best to try to tune by the fundamentals?

I'm not sure what "tuning by the fundamentals" means, but certainly
one could, if it suited one's musical goals, look for a Sethares-like
solution where the maximum possible amount of spectral commonality is
acheived. You could easily tune a bunch of gongs so that they all
shared a common partial, for example.

> Tuning by the
> fundamentals on tuned idiophones may not always be the best
solution,
> as the 2nd and 3rd harmonics may be prominent partials and may be -
> as in this scenario - totally different from the accompanying
string
> harmonics.

In this scenario, they are suprisingly close to those from the
accompaying string harmonics, but that may not be relevant since the
strings may not be playing the same fundamental pitches as the gongs.

> So - where we may be designing tunings for harmonic series timbres:
>
> Where they are being used with a complex blend of inharmonic
timbres,
> the tuning *is not going to display spectral commonality*.

Hopefully everyone understands this -- if not, it's time they did!

> I think more times than not there is a "style" and
> personal subjective views which come through our scale
constructions,
> which either do - or perhaps should reflect what we are trying to
> achieve with, or in, our music.

Absolutely.

> And here, what works for one, might
> not work for all.

Here, hear.
>
> Well - that you are exploring this, and have the good nature that
you
> show, will insure that I will be following along with what you
show,
> because spectral tuning represents one of the most important facets
> tuning to me. Especially interesting to me will be to see when we
> approach the issues of complex timbral interactions.

In our *Western* symphony orchestra, we have a wide variety of
timbres. Yet the strings have perfectly harmonic spectra, the reeds
have perfectly harmonic spectra, and the brass have perfectly
harmonic spectra. Outside the orchestra, the piano and guitar are
surely our most important instruments (out of the ones that we
_tune_), and their spectra are only very slightly inharmonic.
Although such sounds may fall flat to your ears, surely you can
understand why a Westerner might begin his or her theoretical
explorations by assuming harmonic spectra!

🔗jpehrson2 <jpehrson@rcn.com>

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> /tuning/topicId_31641.html#31663
>
>
> > >Curiously, J Gill
> >
> > No -- but what seems to be the case very often, is that when one
> > comes up with such a scale in the form of a periodicity block,
one
> > has quite a few arbitrary choices to make as to which version of
a
> > particular scale degree one wants (the different versions
differing
> > by a unison vector), and then _one such set_ of arbitrary choices
> > does lead to a scale with superparticular step sizes.
>
> Hi Paul...
>
> Well, that's pretty *mysterious* isn't it? Why does that happen
that
> the superparticular step sizes result?

Usually they don't "result", but very often you can arbitrarily
choose to use them.

> Is it just the way the system
> is set up.

Well, superparticulars are not favored _by design_, if that's what
you mean. You should ask Kraig Grady about superparticular step sizes
too.

> Spooky stuff! (If we can't believe in "magic primes"
> that surely is something a little weird... yes?)

One of the themes on the tuning-math list (busier than ever) is
superparticulars . . . for example, we found that the graph of
ET "goodness" has "waves" in it, and the most prominent visible wave
by far for 7-limit rises and falls every 1664 ETs . . . and the
superparticular ratio that Graham made famous in his Blackjack
progression, 2401:2400, fits 1663.9 times in an octave.

Superparticular Spectrums

🔗J Gill <JGill99@imajis.com>

🔗unidala <JGill99@imajis.com>

🔗unidala <JGill99@imajis.com>

🔗paulerlich <paul@stretch-music.com>

🔗paulerlich <paul@stretch-music.com>

🔗paulerlich <paul@stretch-music.com>

🔗paulerlich <paul@stretch-music.com>

🔗jpehrson2 <jpehrson@rcn.com>

🔗unidala <JGill99@imajis.com>

🔗unidala <JGill99@imajis.com>

🔗unidala <JGill99@imajis.com>

🔗paulerlich <paul@stretch-music.com>

🔗paulerlich <paul@stretch-music.com>

🔗jpehrson2 <jpehrson@rcn.com>

🔗paulerlich <paul@stretch-music.com>