Re: The Most Consonant Scale In The Universe; Rigorous Analysis

--- In tuning@y..., "jacky_ligon" <jacky_ligon@y...> wrote:
> Introducing the Unitonic Chant Tuning:
>
> 1/1
> 1/1
> 1/1
> 1/1
> 1/1
> 1/1

The Jacky Hexiligonic Tonic Gamut

"propriety quotient" = 0.000000

"banality factor" = ~infinite

"moment of inertia" = wha?

"implied fundamental pitch"
(and I *don't* mean frequency) = 1/0

"complexity factor" = in stock

"degree of acceptable mis-tuning" = 1 gnat's ass

"convolutional tensor warp" = still under debate ...

Jacky's comedic brilliance = doubtless

JG :)

> From: unidala <JGill99@imajis.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, January 09, 2002 6:00 PM
> Subject: [tuning] Re: The Most Consonant Scale In The Universe; Rigorous
Analysis
>
>
> --- In tuning@y..., "jacky_ligon" <jacky_ligon@y...> wrote:
> > Introducing the Unitonic Chant Tuning:
> >
> > 1/1
> > 1/1
> > 1/1
> > 1/1
> > 1/1
> > 1/1
>
>
> The Jacky Hexiligonic Tonic Gamut
>
> "propriety quotient" = 0.000000
>
> "banality factor" = ~infinite
>
> "moment of inertia" = wha?
>
> "implied fundamental pitch"
> (and I *don't* mean frequency) = 1/0
>
> "complexity factor" = in stock
>
> "degree of acceptable mis-tuning" = 1 gnat's ass
>
> "convolutional tensor warp" = still under debate ...
>
>
> Jacky's comedic brilliance = doubtless

This whole thread has been hilarious.
Nice to see on the tuning list once in a while.

J, regarding that "1 gnat's ass" measurement",
have you ever seen this?
http://www.ixpres.com/interval/dict/wafso.htm

-monz

_________________________________________________________
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Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: unidala <JGill99@i...>
> > To: <tuning@y...>
> > Sent: Wednesday, January 09, 2002 6:00 PM
> > Subject: [tuning] Re: The Most Consonant Scale In The Universe; Rigorous
> Analysis
> >
> >
> > --- In tuning@y..., "jacky_ligon" <jacky_ligon@y...> wrote:
> > > Introducing the Unitonic Chant Tuning:
> > >
> > > 1/1
> > > 1/1
> > > 1/1
> > > 1/1
> > > 1/1
> > > 1/1
> >
> >
> > The Jacky Hexiligonic Tonic Gamut
> >
> > "propriety quotient" = 0.000000
> >
> > "banality factor" = ~infinite
> >
> > "moment of inertia" = wha?
> >
> > "implied fundamental pitch"
> > (and I *don't* mean frequency) = 1/0
> >
> > "complexity factor" = in stock
> >
> > "degree of acceptable mis-tuning" = 1 gnat's ass
> >
> > "convolutional tensor warp" = still under debate ...
> >
> >
> > Jacky's comedic brilliance = doubtless
>
>
> This whole thread has been hilarious.
> Nice to see on the tuning list once in a while.
>
> J, regarding that "1 gnat's ass" measurement",
> have you ever seen this?
> http://www.ixpres.com/interval/dict/wafso.htm
>
>
>
> -monz

J Gill: No, Joe, I had *not* (and I thought that *I* had
acheived immortal status with this "micro-metric" ...)

How about: "1/2 gnat's ass" (a sub-octave equivalency)!

J Gill :)

> From: jacky_ligon <jacky_ligon@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, January 10, 2002 9:23 AM
> Subject: [tuning] Re: The Most Consonant Scale In The Universe; Rigorous
Analysis
>
>
> > > --- In tuning@y..., "jacky_ligon" <jacky_ligon@y...> wrote:
> > > > Introducing the Unitonic Chant Tuning:
> > > >
> > > > 1/1
> > > > 1/1
> > > > 1/1
> > > > 1/1
> > > > 1/1
> > > > 1/1
>
>
> Master Monz,
>
> Thanks!
>
> The discovery of the Unitonic Chant Tuning grows out of my desire to
> remove annoying and painful dissonance caused by using primes higher
> than and including 2.
>
> I found that no matter which intervals I used as generators - either
> RI or ET, there always seemed to be at least one or more dissonant
> intervals left - always some Wolfy intervals remaining somewhere in
> the chain.
>
> This realization led directly to my discovery of the "1/1 Bliss
> Generator". Named so because it completely eliminates all dissonance,
> and gives the best alignment of partials.
>
> One can see from a simple lattice of the tuning why this is so:
>
> 1/1 ; ) 1/1 ; ) 1/1 ; ) 1/1 ; ) 1/1 ; ) 1/1
>
> While the UCT may not provide the usual melodic variety that I
> usually look for, I enjoy this tuning for all the reasons that Haresh
> has pointed out.

Well, Jacky, I think you've hit on the key as to exactly why
this is such a powerful tuning: it is *totally prime-less*!

By definition, "1" is not a prime number, and "2" is the first
prime. So you're excluding *all* primes from your tuning!
Brilliant!

(and I guess that really puts the nail in the coffin of my
"prime-affect" theories! ... be sure to tell McLaren ...)

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: jacky_ligon <jacky_ligon@y...>
> > To: <tuning@y...>
> > Sent: Thursday, January 10, 2002 9:23 AM
> > Subject: [tuning] Re: The Most Consonant Scale In The Universe; Rigorous
> Analysis
> >
> >
> > > > --- In tuning@y..., "jacky_ligon" <jacky_ligon@y...> wrote:
> > > > > Introducing the Unitonic Chant Tuning:
> > > > >
> > > > > 1/1
> > > > > 1/1
> > > > > 1/1
> > > > > 1/1
> > > > > 1/1
> > > > > 1/1
> >
> >
> > Master Monz,
> >
> > Thanks!
> >
> > The discovery of the Unitonic Chant Tuning grows out of my desire to
> > remove annoying and painful dissonance caused by using primes higher
> > than and including 2.
> >
> > I found that no matter which intervals I used as generators - either
> > RI or ET, there always seemed to be at least one or more dissonant
> > intervals left - always some Wolfy intervals remaining somewhere in
> > the chain.
> >
> > This realization led directly to my discovery of the "1/1 Bliss
> > Generator". Named so because it completely eliminates all dissonance,
> > and gives the best alignment of partials.
> >
> > One can see from a simple lattice of the tuning why this is so:
> >
> > 1/1 ; ) 1/1 ; ) 1/1 ; ) 1/1 ; ) 1/1 ; ) 1/1
> >
> > While the UCT may not provide the usual melodic variety that I
> > usually look for, I enjoy this tuning for all the reasons that Haresh
> > has pointed out.
>
>
>
> Well, Jacky, I think you've hit on the key as to exactly why
> this is such a powerful tuning: it is *totally prime-less*!
>
> By definition, "1" is not a prime number, and "2" is the first
> prime. So you're excluding *all* primes from your tuning!
> Brilliant!
>
> (and I guess that really puts the nail in the coffin of my
> "prime-affect" theories! ... be sure to tell McLaren ...)
>
> -monz

Monz,

In a brief moment of seriousness, I would like to point out
that (if one is to doubt the Octavian dogma of equivalence
as absolute), then a difference *would* exist between a prime
number and its even multiples equal to powers of 2. Therefore,
(non-multiplied) primes may, afterall, be said to possess a
certain "uniqueness". However, so would their "non-equivalent"
multiples! So, two-ish-ness and three-ish-ness, etc. would not
exist as underlying core characteristics, but it *could* be
said that *all* (non-equivalent) tones deserve an independent
status. I don't know how intellectually satisfying that is,
but it may well "hold some water", due to the data quoted in:

/tuning/topicId_32256.html#32281

Alas, the freeware "overtone.exe" appears to have
disappeared from the URL http://www.clab.unibe.ch/overtone
(or it's midnight in Czechoslovakia, and the server is for
some reason down).

Have just been experimenting with "Overtone.exe", and find
(possibly due to my ears, or some particular characteristic
of pitch perception) that - IF the fundamental is tuned to
around 200 Hz or below (particularly around 160-170 Hz),
THEN the 2nd, 4th, and 8th harmonics (when added on an
individual basis, or in combination) really do *not*
(to my tin ears) sound very equivalent to the fundamental!

Past a fundamental frequency of around 200 Hz, this effect
seems to disappear. Tell me what you ears tell you ...

Though the "Overtone.exe" freeware download may be "history"
(and I cannot, of course, post it to the groups "Files"
section), a WAV file could be made by someone in possesion
of "Overtone.exe" which would demonstrate this effect, and
the WAV could then be posted to the "Files" section.

Beware the "Octavians", they are everywhere, and have many
"ears". You just might get a visit from the "men in black"!

The "Prime Believers" may have to become a "secret-sect" ...

Regards, J Gill :)

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

> Monz,
>
> In a brief moment of seriousness, I would like to point out
> that (if one is to doubt the Octavian dogma of equivalence
> as absolute),

Hope you forgive my interjection here, fellas, but I feel inclined to
participate in this moment of seriousness . . .

Who are the promulgators of the Octavian dogma of equivalence as
absolute? I know of not one such individual.

> then a difference *would* exist between a prime
> number and its even multiples equal to powers of 2.

Again, who would doubt this? I'd like to repeat at this point that
the musician's concept of octave-equivalence is mathematically a
mere "similarity relation", which therefore extends to a prime (or
any other number) and its even multiples formed by multiplying by
powers of two -- but it's not a mathematical "equivalence relation".
The "similarity" is only an equivalence relation when viewing the
pitch-chroma helix "head on" and "without perspective" so that it
becomes a circle showing chroma only -- but not at all when one views
the "full picture" of the helix, which includes pitch.

Let me propose to you again, Jeremy, that, when engaging in
discussions on as technical a level as you like to have them, we
speak not of "octave equivalence" -- which is a red herring -- but
rather of some more precise concept -- how about "octave chroma-
equivalence"? This would recognize that the "pitch" aspect is clearly
and greatly altered by multiplying by powers of 2, while the "chroma"
aspect, be it an inborn mode of perception or a culturally-acquired
construct, is not.

Again, my apologies for the interjection . . . now back to your
regularly-scheduled program . . .

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

/tuning/topicId_32480.html#32521

> Let me propose to you again, Jeremy, that, when engaging in
> discussions on as technical a level as you like to have them, we
> speak not of "octave equivalence" -- which is a red herring -- but
> rather of some more precise concept -- how about "octave chroma-
> equivalence"? This would recognize that the "pitch" aspect is
clearly
> and greatly altered by multiplying by powers of 2, while
the "chroma"
> aspect, be it an inborn mode of perception or a culturally-acquired
> construct, is not.
>

This is actually pretty cool... and I think I have a glimmer about
this. Is the "chroma" a set of extended overtones? And, how does
the helix work? Or is it impossible to explain that in "layman's"
terms?? This is really interesting...

JP

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> This is actually pretty cool... and I think I have a glimmer about
> this. Is the "chroma" a set of extended overtones?

Chroma is merely what musicians call "pitch class". It goes around in
a circle: C, C#, D, D# . . . eventually coming back around to C.

> And, how does
> the helix work?

Think of a spiral staircase. As you go up, the pitch gets higher; as
you go down, it gets lower. But every time you go around a "full
circle", you've gone up or down one octave. Hence your angular
position on the spiral staircase corresponds to the "chroma" or
musicians' "pitch class"; points directly above and below your
current position are higher or lower in pitch by one or more octaves,
but have the same "chroma" . . . if that doesn't make sense to you,
I'll try again . . .

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

/tuning/topicId_32480.html#32528

>
> Think of a spiral staircase. As you go up, the pitch gets higher;
as
> you go down, it gets lower. But every time you go around a "full
> circle", you've gone up or down one octave. Hence your angular
> position on the spiral staircase corresponds to the "chroma" or
> musicians' "pitch class"; points directly above and below your
> current position are higher or lower in pitch by one or more
octaves,
> but have the same "chroma" . . . if that doesn't make sense to you,
> I'll try again . . .

****Well, I can certainly see the "positioning." I'm assuming
the "height" of the stair-climber is the actual "pitch."

But then, if you're in the same "position" does that mean that
certain overtones are lining up. *Something* is obviously lining up,
so what is it??

Thanks!

JP

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> ****Well, I can certainly see the "positioning." I'm assuming
> the "height" of the stair-climber is the actual "pitch."

Right . . .

> But then, if you're in the same "position" does that mean that
> certain overtones are lining up.

Sure, assuming you're using a harmonic timbre.

But if you use sine waves, the model still holds good! Similarity, or
chroma-equivalence, still happens for notes separated by an octave.

The funny thing is, for sine waves (no overtones), the interval that
sounds like a "melodically correct chroma match", is not 1200 cents,
but around 1209 cents, and even more in the extreme lower and higher
registers! There's plenty of experimental evidence for this, and you
can try the experiment on yourself if you don't believe it.

See Terhardt's pages for a possible explanation of this "octave
stretch" phenomenon.

Finally, using an _inharmonic spectrum_ with a prominent _off-octave_
partial, one can "bend" the value of the perceived "melodically
correct chroma match" considerable further . . . as I'm sure you
already know from Sethares' examples . . .

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

/tuning/topicId_32480.html#32533

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > ****Well, I can certainly see the "positioning." I'm assuming
> > the "height" of the stair-climber is the actual "pitch."
>
> Right . . .
>
> > But then, if you're in the same "position" does that mean that
> > certain overtones are lining up.
>
> Sure, assuming you're using a harmonic timbre.
>
> But if you use sine waves, the model still holds good! Similarity,
or chroma-equivalence, still happens for notes separated by an octave.
>
> The funny thing is, for sine waves (no overtones), the interval
that sounds like a "melodically correct chroma match", is not 1200
cents, but around 1209 cents, and even more in the extreme lower and
higher registers! There's plenty of experimental evidence for this,
and you can try the experiment on yourself if you don't believe it.

****Well, I assume, then, that this phenominon is what causes piano
tuners to "stretch" octaves, yes??

>
> Finally, using an _inharmonic spectrum_ with a prominent _off-
octave_ partial, one can "bend" the value of the
perceived "melodically
> correct chroma match" considerable further . . . as I'm sure you
> already know from Sethares' examples . . .

Oh sure... his now "famous" minor 9th (sounds like an octave) at the
very beginning of _Tuning, Timbre, Spectrum, Scale_...

JP

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:

> > The funny thing is, for sine waves (no overtones), the interval
> that sounds like a "melodically correct chroma match", is not 1200
> >cents, but around 1209 cents, and even more in the extreme lower
and
> > higher registers! There's plenty of experimental evidence for
this,
> > and you can try the experiment on yourself if you don't believe
it.
>
> ****Well, I assume, then, that this phenominon is what causes piano
> tuners to "stretch" octaves, yes??

Not so much . . . more important is that piano strings have
_inharmonic_ partials -- more specifically, their partials are
stretched. Therefore, the "Sethares-bent-octave" effect mentioned
below takes effect . . .

BTW, can you actually try to play with sine waves? You need good
speakers . . . it would be instructive and you'd learn it better if
you experienced it yourself.

> > Finally, using an _inharmonic spectrum_ with a prominent _off-
> octave_ partial, one can "bend" the value of the
> perceived "melodically
> > correct chroma match" considerable further . . . as I'm sure you
> > already know from Sethares' examples . . .
>
> Oh sure... his now "famous" minor 9th (sounds like an octave) at
the
> very beginning of _Tuning, Timbre, Spectrum, Scale_...

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > ****Well, I can certainly see the "positioning." I'm assuming
> > the "height" of the stair-climber is the actual "pitch."
>
> Right . . .
>
> > But then, if you're in the same "position" does that mean that
> > certain overtones are lining up.
>
> Sure, assuming you're using a harmonic timbre.
>
> But if you use sine waves, the model still holds good! Similarity,
> or
> chroma-equivalence, still happens for notes separated by an octave.
>
> The funny thing is, for sine waves (no overtones), the interval
> that
> sounds like a "melodically correct chroma match", is not 1200
> cents,
> but around 1209 cents, and even more in the extreme lower and
> higher
> registers! There's plenty of experimental evidence for this, and
> you
> can try the experiment on yourself if you don't believe it.
>
> See Terhardt's pages for a possible explanation of this "octave
> stretch" phenomenon.
>
> Finally, using an _inharmonic spectrum_ with a prominent _off-octave_
> partial, one can "bend" the value of the perceived "melodically
> correct chroma match" considerable further . . . as I'm sure you
> already know from Sethares' examples . . .

J Gill: I'm confused by these differentiations. Paul,
in /tuning/topicId_32480.html#32521
you state:

<< I'd like to repeat at this point that
the musician's concept of octave-equivalence is mathematically a
mere "similarity relation",

JG: Which appears to imply that a "similarity relation"
is mathematically "imprecise" ...

... which therefore extends to a prime (or
any other number) and its even multiples formed by multiplying by
powers of two -- but it's not a mathematical "equivalence relation".

JG: Which appears to (further) imply that a "similarity
relation" is mathematically "imprecise", and is not
mathematically "equivalent" ...

The "similarity" is only an equivalence relation when viewing the
pitch-chroma helix "head on" and "without perspective" so that it
becomes a circle showing chroma only -- but not at all when one views
the "full picture" of the helix, which includes pitch. >>

JG: What is this "full picture" you speak of?
Why would a recognition of such "chromas"
appearing at higher/lower "pitches" justify
what you seem to describe as being equivalent
*mathematically*, but (yet) not *musically*.

How, then, can we comfortably utilize such
*mathematical* "equivalences" to describe what
(you state) appear to only be *musical*
"similarities" (1209 Cents, and all)???

In your /tuning/topicId_32256.html#32404
one finds:

> JG: Mathematics is not well suited to such vagaries,
> as is the human spirit. This is why the onus rests
> upon the mathemetician to relate his/her numerical
> relationships to the entire gestalt of the musician,
> and not the other way around!

PE: I agree. In my opinion, any attempt to understand music through
mathematics alone is destined for extremely limited success.
____________________________________________________________

Can we really "have our mathematical cakes",
and "eat them too" (where they fail to account
for what listeners actually perceive with sinusoidal
tones)? Or does the "aural mind" take care of all
this, with "virtual" (as opposed to "spectral") pitches?
Somehow, this seems a bit too "convenient" a retort.

Is not such an onus (of accuracy) upon the mathematician
(and not the listener)??? Or do I misunderstand you?

Curiously, J Gill :)

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

> << I'd like to repeat at this point that
> the musician's concept of octave-equivalence is mathematically a
> mere "similarity relation",
>
> JG: Which appears to imply that a "similarity relation"
> is mathematically "imprecise" ...

Why would you say that? Let's take a different mathematical
definition of similarity . . . similar triangles. Clearly if two
triangles are congruent, they are similar, but if two triangles are
similar, they are not necessarily congruent. Nothing mathematically
imprecise about it!
>
> ... which therefore extends to a prime (or
> any other number) and its even multiples formed by multiplying by
> powers of two -- but it's not a mathematical "equivalence relation".
>
> JG: Which appears to (further) imply that a "similarity
> relation" is mathematically "imprecise", and is not
> mathematically "equivalent" ...

Again, it's a "similarity" relation, and this can be defined, just as
with triangles, in a mathematically precise way.

> The "similarity" is only an equivalence relation when viewing the
> pitch-chroma helix "head on" and "without perspective" so that it
> becomes a circle showing chroma only -- but not at all when one
views
> the "full picture" of the helix, which includes pitch. >>
>
> JG: What is this "full picture" you speak of?

A full three-dimensional representation of the helix.

> Why would a recognition of such "chromas"
> appearing at higher/lower "pitches" justify
> what you seem to describe as being equivalent
> *mathematically*, but (yet) not *musically*.

I don't know what you mean. The helix I refer to is a model of
certain psychological sensations associated with the pitch continuum.
The model can be stated in mathematically very precise terms. The
similarity relation which concerns us corresponds mathematically to
the equality of _angular position_, but not necessarily _height_,
along the helix. The extent to which the model accurately reflects
our sensations a question of experimental psychoacoustics, now and
for the forseeable future. It is _not_ a question of pure
mathematics -- pure mathematics fails to decide between falsifiable
theories in the "hard sciences", let alone the "soft sciences" such
as psychology.

In any case, the notational/linguistic systems which musicians from
virtually all cultures use to describe pitch, show a mathematical
correspondence with this helix model. So either the helix, or some
approximation of it, is "built in" to the way humans hear, or it is
a "frozen accident" of the musical evolution of early human culture,
that has been passed down to virtually all living musical cultures.

> How, then, can we comfortably utilize such
> *mathematical* "equivalences" to describe what
> (you state) appear to only be *musical*
> "similarities" (1209 Cents, and all)???

Excuse my impatience, for I had no sleep last night, but let me
assert to you for the umpteenth time, that what musicians refer to
as "octave equivalence" refers not to a mathematical equivalence at
all, but merely to a mathematical similarity.
>
> In your /tuning/topicId_32256.html#32404
> one finds:
>
> > JG: Mathematics is not well suited to such vagaries,
> > as is the human spirit. This is why the onus rests
> > upon the mathemetician to relate his/her numerical
> > relationships to the entire gestalt of the musician,
> > and not the other way around!
>
> PE: I agree. In my opinion, any attempt to understand music through
> mathematics alone is destined for extremely limited success.
>
> Can we really "have our mathematical cakes",
> and "eat them too" (where they fail to account
> for what listeners actually perceive with sinusoidal
> tones)?

I don't know what you mean by having the cake and eating it too.
Personally, I employ mathematical models for music theory only
insofar as they accurately reflect the characteristics of our
perception. At least, this is true 99% of the time, for occasionally
I will delve into systems such as the Bohlen-Pierce system where the
helix is supposed to spiral every 3:1, rather than every 2:1. If I
were to describe a music theory for melodies in sine waves, I would
certainly take into account the actual interval at which "chroma
similarity" is perceived.

> Or does the "aural mind" take care of all
> this, with "virtual" (as opposed to "spectral") pitches?

I don't know what that would mean.

> Somehow, this seems a bit too "convenient" a retort.

Well, it certainly isn't mine.

> Is not such an onus (of accuracy) upon the mathematician
> (and not the listener)??? Or do I misunderstand you?

Perhaps you do.

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

> Again, who would doubt this? I'd like to repeat at this point that
> the musician's concept of octave-equivalence is mathematically a
> mere "similarity relation", which therefore extends to a prime (or
> any other number) and its even multiples formed by multiplying by
> powers of two -- but it's not a mathematical "equivalence relation".

It's symmetric, transitive and reflexive, and hence is in fact an equivalence relationship. I'm not sure what you mean by a similarity relationship, a term which is sometimes used in fuzzy logic, but this is a completely standard equivalence relationship, and defines equivalence classes which people love to make use of.

In other words, we have

(1) a~a (any pitch is octave equivalent to itself)

(2) a~b <==> b~a (a equivalent to b is the same as b equivalent to a)

(3) a~b & b~c ==> a~c (if a is octave equivalent to b, and b to c, then a is octave equivalent to c.)

Since octave-equivalence satisfies all of these, it is an equivalence relationship. Another way of seeing it, for an algebraist, is that octaves are the kernel of a mapping, and coset equivalence is an equivalence relationship.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > Again, who would doubt this? I'd like to repeat at this point
that
> > the musician's concept of octave-equivalence is mathematically a
> > mere "similarity relation", which therefore extends to a prime
(or
> > any other number) and its even multiples formed by multiplying by
> > powers of two -- but it's not a mathematical "equivalence
relation".
>
> It's symmetric, transitive and reflexive, and hence is in fact an
>equivalence relationship. I'm not sure what you mean by a similarity
>relationship, a term which is sometimes used in fuzzy logic, but
>this is a completely standard equivalence relationship, and defines
>equivalence classes which people love to make use of.
>
> In other words, we have
>
> (1) a~a (any pitch is octave equivalent to itself)
>
> (2) a~b <==> b~a (a equivalent to b is the same as b equivalent to
a)
>
> (3) a~b & b~c ==> a~c (if a is octave equivalent to b, and b to c,
then a is octave equivalent to c.)
>
> Since octave-equivalence satisfies all of these, it is an
>equivalence relationship. Another way of seeing it, for an
>algebraist, is that octaves are the kernel of a mapping, and coset
>equivalence is an equivalence relationship.

Hi Gene . . .

Well, the point I'm trying to get across to Jeremy is that just
because it's called "equivalence" doesn't mean it connotes "sameness"
the way, say, a vanishing unison vector in a temperament could
connote the sameness of two interval vectors. It's more like "similar
triangles" than "congruent triangles", if we make an analogy between
the space of pitches and the space of triangles.

How would you get this idea across to Jeremy? He seems bothered that
mathematical theories of music use an "equivalence relation" to
connect objects which are in fact distinguishable from one
another . . . I reply that the relation is not one of complete
equivalence . . . and he gets the idea that the mathematics is
therefore "imperfect" or something. I could certainly see how the
term "equivalence" could contribute to this false impresssion . . .
so please, help us out here.

Thanks!

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

> Well, the point I'm trying to get across to Jeremy is that just
> because it's called "equivalence" doesn't mean it connotes "sameness"
> the way, say, a vanishing unison vector in a temperament could
> connote the sameness of two interval vectors. It's more like "similar
> triangles" than "congruent triangles", if we make an analogy between
> the space of pitches and the space of triangles.

Mathematically speaking, both similarity and congeuence are equivalence relationships. If ABC is similar to DEF and DEF is similar to GHI, then ABC is similar to GHI, and so forth.

> How would you get this idea across to Jeremy? He seems bothered that
> mathematical theories of music use an "equivalence relation" to
> connect objects which are in fact distinguishable from one
> another . . .

It's done all the time in math--I don't know if that helps or not. One example would be fractions--we put 3/2, 6/4, -10/-5 and so forth together into an equivalence class, and call it a rational number. They are not the same fraction, but they denote the same number.

I reply that the relation is not one of complete
> equivalence . . . and he gets the idea that the mathematics is
> therefore "imperfect" or something. I could certainly see how the
> term "equivalence" could contribute to this false impresssion . . .
> so please, help us out here.

"Equivalence" for a mathematician is just a word you use to say a relation is symmetric, reflexive and transitive, and does not mean the same as equality.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > Well, the point I'm trying to get across to Jeremy is that just
> > because it's called "equivalence" doesn't mean it
connotes "sameness"
> > the way, say, a vanishing unison vector in a temperament could
> > connote the sameness of two interval vectors. It's more
like "similar
> > triangles" than "congruent triangles", if we make an analogy
between
> > the space of pitches and the space of triangles.
>
>Mathematically speaking, both similarity and congeuence are
>equivalence relationships. If ABC is similar to DEF and DEF is
>similar to GHI, then ABC is similar to GHI, and so forth.

Yes Gene. I was wrong about "equivalence" not being the appropriate
mathematical term. But I'm trying to help Jeremy understand that
octave-equivalence is an equivalence relationship more analogous to
similarity of triangles, rather than congruence of triangles. Better
yet, I've tried to reference the helix model of pitch/chroma, and
explain that octave-equivalence refers to an identity in the angular,
but not necessary the linear, coordinate in the helix. Jeremy seems
to feel that an equivalence relationship that does not imply perfect
identical-ness will not be conducive to logical rigor. Please help me
show him that this is not the case.

> > How would you get this idea across to Jeremy? He seems bothered
that
> > mathematical theories of music use an "equivalence relation" to
> > connect objects which are in fact distinguishable from one
> > another . . .
>
>It's done all the time in math--I don't know if that helps or not.
>One example would be fractions--we put 3/2, 6/4, -10/-5 and so
>forth together into an equivalence class, and call it a rational
>number. They are not the same fraction, but they denote the same
>number.

OK . . . though the use of ratios can create some unfortunate
confusion in the present context. Thanks, though -- you're doing
better than I am.

> > I reply that the relation is not one of complete
> > equivalence . . . and he gets the idea that the mathematics is
> > therefore "imperfect" or something. I could certainly see how the
> > term "equivalence" could contribute to this false
impresssion . . .
> > so please, help us out here.
>
>"Equivalence" for a mathematician is just a word you use to say a
>relation is symmetric, reflexive and transitive, and does not mean
>the same as equality.

OK, thanks Gene. This is terrific. What I'm afraid of, though, is
that *mathematically* one can define plenty of other non-trivial
equivalence relations in pitch space (of course one can). It's
important, I feel, to emphasize that of these, only octave-
equivalence corresponds to an actual perceived similarity, rather
universallly accepted by musicians, and whose
biological/psychological origin may (or may not) have been accounted
for by Terhardt. This "special" equivalence relation then leads to
the helical model of pitch/chroma that I described. Finally, this
construct allows us, where it is appropriate (say if we are
interested in octave-repeating tuning systems), to abstract the
chroma concept away from pitch (something that cannot be acheived
with actual tones, except the rather artificial Shepard tones), and
to construct theoretical "castles" such as periodicity blocks, linear
temperaments, evoking "odd-limit" consonance measures, etc. . . . I'm
trying to share with Jeremy my confidence that we are not making any
illegal logical "leaps" in so doing, that we are not over-relying on
an imperfect mathematical model (in particular, the "imperfect
equivalence" that bothers Jeremy) and incorporating unseen implicit
assumptions whose validity may be questionable (or whatever it is
Jeremy is afraid of) . . . I hope this has all made some sense, and
perhaps you can help me explain it better . . .

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

/tuning/topicId_32480.html#32535

>
> BTW, can you actually try to play with sine waves? You need good
> speakers . . . it would be instructive and you'd learn it better if
> you experienced it yourself.
>

I just tried that with "overtone.exe" which I, fortunately, still
have a copy of...

It did seem that some of the higher sine tone octave comparisons
sounded "flat" compared with the lower ones. I noticed it at about
500hz - 1000hz. I kept wishing that the higher pitch was just a
*tad* higher...

JP

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> /tuning/topicId_32480.html#32535
>
>
> >
> > BTW, can you actually try to play with sine waves? You need good
> > speakers . . . it would be instructive and you'd learn it better
if
> > you experienced it yourself.
> >
>
> I just tried that with "overtone.exe" which I, fortunately, still
> have a copy of...
>
> It did seem that some of the higher sine tone octave comparisons
> sounded "flat" compared with the lower ones. I noticed it at about
> 500hz - 1000hz. I kept wishing that the higher pitch was just a
> *tad* higher...

There ya go!

Are you using a good sound system? If so, try low-frequency sine
waves too . . . if you don't have a good sound system, you won't be
able to hear these without the system also producing harmonic
overtones of the sine waves -- which would of course ruin the effect.

JG: Have read Paul and Gene's communications where I find:

> GWS: "Equivalence" for a mathematician is just a word you use to
> say a relation is symmetric, reflexive and transitive, and does
> not mean the same as equality.

PE: OK, thanks Gene. This is terrific. What I'm afraid of, though,
is that *mathematically* one can define plenty of other non-trivial
equivalence relations in pitch space (of course one can). It's
important, I feel, to emphasize that of these, only octave-
equivalence corresponds to an actual perceived similarity, rather
universallly accepted by musicians, and whose
biological/psychological origin may (or may not) have been accounted
for by Terhardt. This "special" equivalence relation then leads to
the helical model of pitch/chroma that I described. Finally, this
construct allows us, where it is appropriate (say if we are
interested in octave-repeating tuning systems), to abstract the
chroma concept away from pitch (something that cannot be acheived
with actual tones, except the rather artificial Shepard tones), and
to construct theoretical "castles" such as periodicity blocks, linear
temperaments, evoking "odd-limit" consonance measures, etc. . . . I'm
trying to share with Jeremy my confidence that we are not making any
illegal logical "leaps" in so doing, that we are not over-relying on
an imperfect mathematical model (in particular, the "imperfect
equivalence" that bothers Jeremy) and incorporating unseen implicit
assumptions whose validity may be questionable (or whatever it is
Jeremy is afraid of) . . . I hope this has all made some sense, and
perhaps you can help me explain it better . . .
________________________________________________

JG: Thank you both for your consideration of my questions,
but I must say that (alas) it *is* "equality" (of the numerical
substitutions we make in the process of systemically classifying
scale-pitches, together with the actual (tunable) pitches at
which our "aural" (as opposed to "conceptual") "minds" would
make a similar "classification" (such as at 1209 cents in the
sinusoidal case Paul mentioned). Is that so misguided. I just
would like to know whether a valid basis exists upon which to
rely in grouping scale-pitches related by 2^N in a similar class.

I understand the "slinky" bit, and the stretching of "pitch"
independent of "chroma". Such may serve as a model to attempt
to accomodate listener *perceptions*, but it establishes a
*non-equality* (between "chroma" and "pitch") which plainly
(to me) raises (rather than quells) doubts as to the veracity
of making common classifications of scale-pitches which are
scaled by 2^N ...

So, my questions below remain:

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> > << I'd like to repeat at this point that
> > the musician's concept of octave-equivalence is mathematically a
> > mere "similarity relation",
> >
> > JG: Which appears to imply that a "similarity relation"
> > is mathematically "imprecise" ...
>
> Why would you say that? Let's take a different mathematical
> definition of similarity . . . similar triangles. Clearly if two
> triangles are congruent, they are similar, but if two triangles are
> similar, they are not necessarily congruent. Nothing mathematically
> imprecise about it!

JG: What does that have to do with the differences between
the frequencies at which your "octave chroma-equivalence"
(which you suggest in your message #32521), and numerically
applied "octave pitch-equivalence" in numerous mathematical
conceptual sytems?

> > ... which therefore extends to a prime (or
> > any other number) and its even multiples formed by multiplying by
> > powers of two -- but it's not a mathematical "equivalence relation".
> >
> > JG: Which appears to (further) imply that a "similarity
> > relation" is mathematically "imprecise", and is not
> > mathematically "equivalent" ...
>
> Again, it's a "similarity" relation, and this can be defined, just > as
> with triangles, in a mathematically precise way.

JG: You mean as a numerical "approximation" (and not location)?

> > The "similarity" is only an equivalence relation when viewing the
> > pitch-chroma helix "head on" and "without perspective" so that it
> > becomes a circle showing chroma only -- but not at all when one
> views
> > the "full picture" of the helix, which includes pitch. >>
> >
> > JG: What is this "full picture" you speak of?
>
> A full three-dimensional representation of the helix.

JG: How does it's existence as a 3D model impart special meaning?

> > Why would a recognition of such "chromas"
> > appearing at higher/lower "pitches" justify
> > what you seem to describe as being equivalent
> > *mathematically*, but (yet) not *musically*.
>
> I don't know what you mean. The helix I refer to is a model of
> certain psychological sensations associated with the pitch
> continuum.
> The model can be stated in mathematically very precise terms.

JG: But in terms which does not comport with 2^N, right?

> The
> similarity relation which concerns us corresponds mathematically to
> the equality of _angular position_, but not necessarily _height_,
> along the helix.

JG: So your "slinky" is "stretch-able" (in "height along the
helix) depending upon the various measured phenomena to be
explained? Is such deviation (of "pitch" from "chroma")
consistent? Is is *multiplicative* (increasing geometrically
as pitch increases/decreases)?

> The extent to which the model accurately reflects
> our sensations a question of experimental psychoacoustics, now and
> for the forseeable future. It is _not_ a question of pure
> mathematics -- pure mathematics fails to decide between falsifiable
> theories in the "hard sciences", let alone the "soft sciences" such
> as psychology.

JG: My original interest and concern has been (and remains)
the degree of precision with which conceptual models of musical
*systems* comport with what listeners report as "similarities".
If *perceptual* "chroma-octaves" are not equivalent to *numerical-
octaves* described by poweres of 2^N, so be it. That's all. :)

> In any case, the notational/linguistic systems which musicians from
> virtually all cultures use to describe pitch, show a mathematical
> correspondence with this helix model.

JG: Viewed from *which* perspective? "Chroma" *only* (regardless
of numerical "pitch stretching/shrinking" necessary to comport)?

> So either the helix, or some
> approximation of it, is "built in" to the way humans hear, or it is
> a "frozen accident" of the musical evolution of early human
> culture,
> that has been passed down to virtually all living musical cultures.

JG: Once again "viewed from *which* perspective"? Chroma *only*?

> > How, then, can we comfortably utilize such
> > *mathematical* "equivalences" to describe what
> > (you state) appear to only be *musical*
> > "similarities" (1209 Cents, and all)???
>
> Excuse my impatience, for I had no sleep last night, but let me
> assert to you for the umpteenth time, that what musicians refer to
> as "octave equivalence" refers not to a mathematical equivalence at
> all, but merely to a mathematical similarity.

JG: So (then), mathematical multiples of 2^N do not describe
"what musicians refer to as 'octave equivalence'"?

> > In your /tuning/topicId_32256.html#32404
> > one finds:
> >
> > > JG: Mathematics is not well suited to such vagaries,
> > > as is the human spirit. This is why the onus rests
> > > upon the mathemetician to relate his/her numerical
> > > relationships to the entire gestalt of the musician,
> > > and not the other way around!
> >
> > PE: I agree. In my opinion, any attempt to understand music
> > through
> > mathematics alone is destined for extremely limited success.

> > Can we really "have our mathematical cakes",
> > and "eat them too" (where they fail to account
> > for what listeners actually perceive with sinusoidal
> > tones)?
>
> I don't know what you mean by having the cake and eating it too.

JG: Simply, utilizing powers of 2^N to conceptually model
listener responses to music without the differences being
a significant limitation built within such assumptions.

> Personally, I employ mathematical models for music theory only
> insofar as they accurately reflect the characteristics of our
> perception.

JG: Well (then) what about multipications by powers of 2^N
as they relate to the "pitch/chroma" phenomena you describe?

> At least, this is true 99% of the time, for
> occasionally
> I will delve into systems such as the Bohlen-Pierce system where
> the
> helix is supposed to spiral every 3:1, rather than every 2:1.

JG: If the helix were to spiral "every 2:1" (in *pitch*),
then why would we be discussing 1209 Cent "perceived octaves"
(in *chroma*)?

> If I
> were to describe a music theory for melodies in sine waves, I would
> certainly take into account the actual interval at which "chroma
> similarity" is perceived.

JG: How would the presence of "harmonic" (as opposed to
"sinusoidal") tones serve to simplify the situation?

> > Or does the "aural mind" take care of all
> > this, with "virtual" (as opposed to "spectral") pitches?
>
> I don't know what that would mean.

JG: Nor do I. That is why I inquired.

> > Somehow, this seems a bit too "convenient" a retort.
>
> Well, it certainly isn't mine.

JG: Fair enough. So this is not about "virtual" pitches,
and while it *is* about "chroma" pitches, it is *not*
about "spectral" pitches (as might increase by 2^N)?

> > Is not such an onus (of accuracy) upon the mathematician
> > (and not the listener)??? Or do I misunderstand you?
>
> Perhaps you do.

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

> OK, thanks Gene. This is terrific. What I'm afraid of, though, is
> that *mathematically* one can define plenty of other non-trivial
> equivalence relations in pitch space (of course one can). It's
> important, I feel, to emphasize that of these, only octave-
> equivalence corresponds to an actual perceived similarity, rather
> universallly accepted by musicians, and whose
> biological/psychological origin may (or may not) have been accounted
> for by Terhardt.

I have the impression that the terminology was taken from mathematics, and if you want another term for the same thing also taken from mathematics, "congruence" would be good. To say that
5/4 and 5/2 are congruent modulo octaves might lead to less confusion than saying they are equivalent modulo octaves, and the two mean the same thing.

If you want a special word, not taken from mathematics, you'll have to invent one, but others may be inclined to stick with terms that already have an established meaning, such as "congruence". What about "octagruence" for congruence modulo octaves? Can anyone stand to say that 5/4 is octagruent to 5/2?

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

/tuning/topicId_32480.html#32544
>
> Are you using a good sound system? If so, try low-frequency sine
> waves too . . . if you don't have a good sound system, you won't be
> able to hear these without the system also producing harmonic
> overtones of the sine waves -- which would of course ruin the
effect.

Well, these are fairly decent headphones... not thousands of dollars
or anything...

I'm getting rather strange results, though on the "low end..."

40HZ - 80HZ sounds "flat"... which I guess is the "desired" result
(anything lower really doesn't come out well)

50HZ - 100HZ sounds "correct..." no flatting effect

60HZ - 120HZ is strange. Here the interval actually sounds *too
wide*... I wonder why *that* would be... ??

and from 70HZ- 140HZ to the middle register, everything is "normal..."

So I guess the 60HZ one is a "peculiar" case for some reason....

JP

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

> If you want a special word, not taken from mathematics, you'll have to invent one, but others may be inclined to stick with terms that already have an established meaning, such as "congruence". What about "octagruence" for congruence modulo octaves? Can anyone stand to say that 5/4 is octagruent to 5/2?

J Gill: How about "octageneric",
"octung, baby!" ...

JG :)

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

/tuning/topicId_32480.html#32546

>
> If you want a special word, not taken from mathematics, you'll have
to invent one, but others may be inclined to stick with terms that
already have an established meaning, such as "congruence". What
about "octagruence" for congruence modulo octaves? Can anyone stand
to say that 5/4 is octagruent to 5/2?

Well that's pretty clever, Gene. Actually, *personally* I prefer it
to octave "equivalence." Octaves are not *musically* equivalent or
it would put orchestrators out of business, as well as composers like
Scriabin, and many other *more* moderns (Arvo Part, others??) if it
weren't for the particular and sometimes "peculiar" spacing of their
pitch classes...

JP

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> /tuning/topicId_32480.html#32544
> >
> > Are you using a good sound system? If so, try low-frequency sine
> > waves too . . . if you don't have a good sound system, you won't be
> > able to hear these without the system also producing harmonic
> > overtones of the sine waves -- which would of course ruin the
> effect.
>
> Well, these are fairly decent headphones... not thousands of dollars
> or anything...
>
> I'm getting rather strange results, though on the "low end..."
>
> 40HZ - 80HZ sounds "flat"... which I guess is the "desired" result
> (anything lower really doesn't come out well)
>
> 50HZ - 100HZ sounds "correct..." no flatting effect
>
> 60HZ - 120HZ is strange. Here the interval actually sounds *too
> wide*... I wonder why *that* would be... ??
>
> and from 70HZ- 140HZ to the middle register, everything is "normal..."
>
> So I guess the 60HZ one is a "peculiar" case for some reason....
>
> JP

JG: (without knowing the particulars), but regardless
of what the headphone manufacturer may claim, I would
guess that neither the headphones or (any of our) ears
can resolve/perceive much of *anything* below around
70 Hz. Try listening to 15 inch sub-woofer *only* which
has a steep roll-off equaliztion filter. I (personally)
can only "feel" frequencies below 70 Hz (more as a bodily
sensation than as a tonal entity, in my perception).

How would you describe the "audibility" below 70 Hz
in the case of your perception, Joe?

JG

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

/tuning/topicId_32480.html#32551

>
> JG: (without knowing the particulars), but regardless
> of what the headphone manufacturer may claim, I would
> guess that neither the headphones or (any of our) ears
> can resolve/perceive much of *anything* below around
> 70 Hz. Try listening to 15 inch sub-woofer *only* which
> has a steep roll-off equaliztion filter. I (personally)
> can only "feel" frequencies below 70 Hz (more as a bodily
> sensation than as a tonal entity, in my perception).
>
> How would you describe the "audibility" below 70 Hz
> in the case of your perception, Joe?
>
>
> JG

Well, my *own* woofers are not too woofy... since they're only 8
inchers... (not much "bite" to that "woof..")

But, I'm getting results somewhat similar to the earphones... except
now the 40HZ - 80HZ sounds OK and not flat...

60HZ-120HZ is still *way* too wide, for some strange reason...
possibly the way "overtone.exe" is generating the sines? That seems
strange, though...

But, I agree, we're in the basso profondo here... and it's hard to
hear what we hear here...

JP

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> /tuning/topicId_32480.html#32551
>
> >
> > JG: (without knowing the particulars), but regardless
> > of what the headphone manufacturer may claim, I would
> > guess that neither the headphones or (any of our) ears
> > can resolve/perceive much of *anything* below around
> > 70 Hz. Try listening to 15 inch sub-woofer *only* which
> > has a steep roll-off equaliztion filter. I (personally)
> > can only "feel" frequencies below 70 Hz (more as a bodily
> > sensation than as a tonal entity, in my perception).
> >
> > How would you describe the "audibility" below 70 Hz
> > in the case of your perception, Joe?
> >
> >
> > JG
>
> Well, my *own* woofers are not too woofy... since they're only 8
> inchers... (not much "bite" to that "woof..")
>
> But, I'm getting results somewhat similar to the earphones... except
> now the 40HZ - 80HZ sounds OK and not flat...
>
> 60HZ-120HZ is still *way* too wide, for some strange reason...
> possibly the way "overtone.exe" is generating the sines? That seems
> strange, though...
>
> But, I agree, we're in the basso profondo here... and it's hard to
> hear what we hear here...
>
> JP

J Gill: Joe, bless your (experimenter's) spirit!
I've been running 20-80 Hz single sine waves (from "Overtone")
through my "Altec/Lansing" semi-joke computer speakers.
In that case 50 Hz is inaudible, 60 Hz is a very low "buzz".

In the case of all ("open-baffle" or "sealed-enclosure")
speakers, the (voice-coil) excursion decreases rapidly
below the "free-air-resonance" (about 90 Hz for my 10-inch
Celestion). The "free-air-resonance" of a *15-inch* woofer
is (at best, meaning at the lowest) around 70 Hz or so.

The moral of this story is that in order to reproduce
such low frequencies (down to 60 Hz or so) with any
audibility *whatsoever*, one must push their voice
coil assembly in the speaker to (probably) distorted
extremes (hence, unintended harmonics generated).

Unless you have a "crossed-over" ported (bass-reflex)
speaker system (or sub-woofer system), I would be
cautious about "pouring a lot of power" (high signal
levels) into it, for fear of possibly over-heating
your voice-coil assembly. Then, the melted insulation
around the coil itself can "bubble-up", and "catch"
against the assembly within which the "piston" is
magnetically suspended, causing intermittent "seizing-up"
of your voice coil assembly. This happened to me with
a JBL 15-inch speaker. Would not want it to happen to
a great guy like you (in the course of scientific research)!

Remember, the *longer* the low frequency signals are
sent to the speaker, the more heat will be dissapated,
increasing the chances of such overtemp failures...

Sincerely, J Gill :)

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

/tuning/topicId_32480.html#32553

> >
> The moral of this story is that in order to reproduce
> such low frequencies (down to 60 Hz or so) with any
> audibility *whatsoever*, one must push their voice
> coil assembly in the speaker to (probably) distorted
> extremes (hence, unintended harmonics generated).
> > Unless you have a "crossed-over" ported (bass-reflex)
> speaker system (or sub-woofer system), I would be
> cautious about "pouring a lot of power" (high signal
> levels) into it, for fear of possibly over-heating
> your voice-coil assembly. Then, the melted insulation
> around the coil itself can "bubble-up", and "catch"
> against the assembly within which the "piston" is
> magnetically suspended, causing intermittent "seizing-up"
> of your voice coil assembly. This happened to me with
> a JBL 15-inch speaker. Would not want it to happen to
> a great guy like you (in the course of scientific research)!
>

Ohmygod!

Actually, that happened to me once. I blew out a rather large and
expensive speaker. It was in the course of my "multi media"
presentations in Ann Arbor in my early 20's.

We were *really* driving that sound, and I had long hair and small
beads around my neck, but you needn't know any more about that.

There were also slide projections filling the performance space, and
a friend of mine decided to "enhance" the experience by spraying a
specially crafted "perfume" from a perfume spray bottle by squeezing
the bulb and walking around the audience. (Who were also not
sitting, free to roam).

However, in addition to blowing out the speaker, I eventually got a
hernia from carrying equipment!

Ah, those were the days...

JP

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> /tuning/topicId_32480.html#32553
>
> > >
> > The moral of this story is that in order to reproduce
> > such low frequencies (down to 60 Hz or so) with any
> > audibility *whatsoever*, one must push their voice
> > coil assembly in the speaker to (probably) distorted
> > extremes (hence, unintended harmonics generated).
> > > Unless you have a "crossed-over" ported (bass-reflex)
> > speaker system (or sub-woofer system), I would be
> > cautious about "pouring a lot of power" (high signal
> > levels) into it, for fear of possibly over-heating
> > your voice-coil assembly. Then, the melted insulation
> > around the coil itself can "bubble-up", and "catch"
> > against the assembly within which the "piston" is
> > magnetically suspended, causing intermittent "seizing-up"
> > of your voice coil assembly. This happened to me with
> > a JBL 15-inch speaker. Would not want it to happen to
> > a great guy like you (in the course of scientific research)!
> >
>
> Ohmygod!

JG: I hope that all is well with your stereo system speakers
following your scientific endeavors yesterday!

> Actually, that happened to me once. I blew out a rather large and
> expensive speaker. It was in the course of my "multi media"
> presentations in Ann Arbor in my early 20's.

JG: Free John Sinclair! The Detroit Free Press. The MC-5.
Kick out the Jams!

> We were *really* driving that sound, and I had long hair and small
> beads around my neck, but you needn't know any more about that.

JG: That's certainly *better* than having had "small hair" and
"long beads" around one's neck, Joe! I shall refrain from
referring to you as "dude", brother. Ah, those halcyon days
of innocence, and limitless possibility... "Farm out"! :)

> There were also slide projections filling the performance space, and
> a friend of mine decided to "enhance" the experience by spraying a
> specially crafted "perfume" from a perfume spray bottle by squeezing
> the bulb and walking around the audience. (Who were also not
> sitting, free to roam).

JG: We were starlight, we were golden ... :)

> However, in addition to blowing out the speaker, I eventually got a
> hernia from carrying equipment!

JG: War stories. I (somehow, with one other fellow) carried *all*
of "The Who's" *hundreds* of massive speaker cabinets from a truck
down a steep, icy loading ramp, and into the Rainbow Theatre in
London in the middle of the night (for a benefit performance of
"Tommy" including The Who, and a bevy of "rock stars" participating). It's *amazing* what a (wide-eyed) kid will do for "rock and roll"!
>
> Ah, those were the days...
>
> JP

JG: Ah, youth!

J Gill

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

> JG: Thank you both for your consideration of my questions,
> but I must say that (alas) it *is* "equality" (of the numerical
> substitutions we make in the process of systemically classifying
> scale-pitches, together with the actual (tunable) pitches at
> which our "aural" (as opposed to "conceptual") "minds" would
> make a similar "classification" (such as at 1209 cents in the
> sinusoidal case Paul mentioned).

What do you mean when you say it *is* "equality"? What *is* equality?

> Is that so misguided. I just
> would like to know whether a valid basis exists upon which to
> rely in grouping scale-pitches related by 2^N in a similar class.

One possible basis would be found in psychoacoustics / musical
practice, as we've discussed. Another possible basis would be if one
decided beforehand that one wishes to have an octave-repeating scale
or tuning system. Either of these are perfectly good justifications
for making use of an octave equivalence relation, and dealing with
pitch classes instead of pitches. If one does not wish to do so,
that's no big deal, one can proceed and build all kinds of systems
(as Gene has hinted on the tuning-math list) without making any use
whatsoever of octave-equivalence.

> I understand the "slinky" bit, and the stretching of "pitch"
> independent of "chroma".

I don't think it's independent at all -- it's highly _dependent_.

> Such may serve as a model to attempt
> to accomodate listener *perceptions*, but it establishes a
> *non-equality* (between "chroma" and "pitch")

The two were never equal to begin with!

> which plainly
> (to me) raises (rather than quells) doubts as to the veracity
> of making common classifications of scale-pitches which are
> scaled by 2^N ...

If one were making single-voice melodies in sine waves, one would not
use 2^N, but instead a slightly different equivalence relation.

There will be no end to this conversation, it seems to me, since you
may in the future bring up the dependence of psychoacoustical pitch
on not only frequency but also on amplitude, etc. etc. . . . but just
as Newtonian physics works fine for understanding the earth's tides,
so a "classical" pitch-chroma model works fine for understanding, for
example, human voices and the music that's been made with/for them.
The small imperfections in the mathematical model are far less
important than the fact that music has linguistic, emotional, and
other associations that are (for the forseeable future) far beyond
the capacity of any mathematical model, or any consciously expressed
intellectual framework, to capture.

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

> JG: (without knowing the particulars), but regardless
> of what the headphone manufacturer may claim, I would
> guess that neither the headphones or (any of our) ears
> can resolve/perceive much of *anything* below around
> 70 Hz.

I thought our ears were supposed to be able to hear pitches as low as
20-25 Hz.

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> 60HZ-120HZ is still *way* too wide, for some strange reason...
> possibly the way "overtone.exe" is generating the sines?

Try using a different sound source . . . perhaps your synth?

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

/tuning/topicId_32480.html#32572

> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> > JG: (without knowing the particulars), but regardless
> > of what the headphone manufacturer may claim, I would
> > guess that neither the headphones or (any of our) ears
> > can resolve/perceive much of *anything* below around
> > 70 Hz.
>
> I thought our ears were supposed to be able to hear pitches as low
as 20-25 Hz.

I remember learning that, too... a long time ago...

JP

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

/tuning/topicId_32480.html#32570

> There will be no end to this conversation, it seems to me, since
you may in the future bring up the dependence of psychoacoustical
pitch on not only frequency but also on amplitude, etc. etc. . . .
but just as Newtonian physics works fine for understanding the
earth's tides, so a "classical" pitch-chroma model works fine for
understanding, for example, human voices and the music that's been
made with/for them.
> The small imperfections in the mathematical model are far less
> important than the fact that music has linguistic, emotional, and
> other associations that are (for the forseeable future) far beyond
> the capacity of any mathematical model, or any consciously
expressed intellectual framework, to capture.

This is timed interestingly, with a quote I just posted by Roger
Sessions....

JP

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> > JG: Thank you both for your consideration of my questions,
> > but I must say that (alas) it *is* "equality" (of the numerical
> > substitutions we make in the process of systemically classifying
> > scale-pitches, together with the actual (tunable) pitches at
> > which our "aural" (as opposed to "conceptual") "minds" would
> > make a similar "classification" (such as at 1209 cents in the
> > sinusoidal case Paul mentioned).
>
> What do you mean when you say it *is* "equality"? What *is* equality?

JG: Gee, my dictionary defines "equality" as - "of the same
quantity, quality, value, number, or status as another, one
that is equal". An example of an equality would be "2+2=4".

> > Is that so misguided. I just
> > would like to know whether a valid basis exists upon which to
> > rely in grouping scale-pitches related by 2^N in a similar class.
>
> One possible basis would be found in psychoacoustics / musical
> practice, as we've discussed. Another possible basis would be if >one
> decided beforehand that one wishes to have an octave-repeating >scale
> or tuning system. Either of these are perfectly good justifications
> for making use of an octave equivalence relation, and dealing with
> pitch classes instead of pitches. If one does not wish to do so,
> that's no big deal, one can proceed and build all kinds of systems
> (as Gene has hinted on the tuning-math list) without making any use
> whatsoever of octave-equivalence.

JG: The simple original question was: does 2^N accurately describe
our perception of equivalent pitch classes, or does it not?

> > I understand the "slinky" bit, and the stretching of "pitch"
> > independent of "chroma".
>
> I don't think it's independent at all -- it's highly _dependent_.

JG: Yet I do not hear you elucidating any known systematic way
in which this is so. It's all so "mystical", and non-coherent ...

> > Such may serve as a model to attempt
> > to accomodate listener *perceptions*, but it establishes a
> > *non-equality* (between "chroma" and "pitch")
>
> The two were never equal to begin with!

JG: Well then, that explains it! It's all *so* clear ...

> > which plainly
> > (to me) raises (rather than quells) doubts as to the veracity
> > of making common classifications of scale-pitches which are
> > scaled by 2^N ...
>
> If one were making single-voice melodies in sine waves, one would >not
> use 2^N, but instead a slightly different equivalence relation.
>
> There will be no end to this conversation,

JG: Don't worry, there is an end to this conversation.

>it seems to me, since >you
> may in the future bring up the dependence of psychoacoustical pitch
> on not only frequency but also on amplitude, etc. etc. . . . but >just
> as Newtonian physics works fine for understanding the earth's tides,
> so a "classical" pitch-chroma model works fine for understanding, >for
> example, human voices and the music that's been made with/for them.

> The small imperfections in the mathematical model

JG: Thanks for here providing a straightforward answer.

>are far less
> important than the fact that music has linguistic, emotional, and
> other associations that are (for the forseeable future) far beyond
> the capacity of any mathematical model, or any consciously >expressed
> intellectual framework, to capture.

JG: So, in "harmonic_entropy" message #512, where you stated,
"Waveform has nothing to do with it. Harmonic entropy is the
simplest possible model of consonance and cannot be regarded
as specific to any waveform.", you actually meant it?

J Gill

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> > JG: (without knowing the particulars), but regardless
> > of what the headphone manufacturer may claim, I would
> > guess that neither the headphones or (any of our) ears
> > can resolve/perceive much of *anything* below around
> > 70 Hz.
>
> I thought our ears were supposed to be able to hear pitches as low as
> 20-25 Hz.

J Gill: So Paul, tell me, have *you* ever "heard" a 20-25 Hz
tone, or have you ever met anyone who has "heard" a 20-25 Hz
tone?

J Gill

--- In tuning@y..., "unidala" <JGill99@i...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning@y..., "unidala" <JGill99@i...> wrote:
> >
> > > JG: Thank you both for your consideration of my questions,
> > > but I must say that (alas) it *is* "equality" (of the numerical
> > > substitutions we make in the process of systemically classifying
> > > scale-pitches, together with the actual (tunable) pitches at
> > > which our "aural" (as opposed to "conceptual") "minds" would
> > > make a similar "classification" (such as at 1209 cents in the
> > > sinusoidal case Paul mentioned).
> >
> > What do you mean when you say it *is* "equality"? What *is*
equality?
>
> JG: Gee, my dictionary defines "equality" as - "of the same
> quantity, quality, value, number, or status as another, one
> that is equal". An example of an equality would be "2+2=4".

Oh, thanks J, you're being really helpful. Can't you do any better
than that in explaining what you meant above?

> > > Is that so misguided. I just
> > > would like to know whether a valid basis exists upon which to
> > > rely in grouping scale-pitches related by 2^N in a similar
class.
> >
> > One possible basis would be found in psychoacoustics / musical
> > practice, as we've discussed. Another possible basis would be if
>one
> > decided beforehand that one wishes to have an octave-repeating
>scale
> > or tuning system. Either of these are perfectly good
justifications
> > for making use of an octave equivalence relation, and dealing
with
> > pitch classes instead of pitches. If one does not wish to do so,
> > that's no big deal, one can proceed and build all kinds of
systems
> > (as Gene has hinted on the tuning-math list) without making any
use
> > whatsoever of octave-equivalence.
>
> JG: The simple original question was: does 2^N accurately describe
> our perception of equivalent pitch classes, or does it not?

It seems to come very close when the tones have a decent complement
of harmonic partials. Otherwise, it may depart from that -- I
detailed some cases to Joseph here, yesterday I think.

> > > I understand the "slinky" bit, and the stretching of "pitch"
> > > independent of "chroma".
> >
> > I don't think it's independent at all -- it's highly _dependent_.
>
> JG: Yet I do not hear you elucidating any known systematic way
> in which this is so. It's all so "mystical", and non-coherent ...

The stretching of pitch that you refer to would seem to mean the
stretching of octaves? So that causes the chromas to be a little more
spread out in pitch, no? Seems coherent to me . . .

> > > Such may serve as a model to attempt
> > > to accomodate listener *perceptions*, but it establishes a
> > > *non-equality* (between "chroma" and "pitch")
> >
> > The two were never equal to begin with!
>
> JG: Well then, that explains it! It's all *so* clear ...

I don't know what to say to you, Jeremy. What exactly are you getting
at with this "non-equality"?

> >are far less
> > important than the fact that music has linguistic, emotional, and
> > other associations that are (for the forseeable future) far
beyond
> > the capacity of any mathematical model, or any consciously
>expressed
> > intellectual framework, to capture.
>
> JG: So, in "harmonic_entropy" message #512, where you stated,
> "Waveform has nothing to do with it. Harmonic entropy is the
> simplest possible model of consonance and cannot be regarded
> as specific to any waveform.", you actually meant it?

Yes, although

(a) I don't see what relevance harmonic entropy has to anything we're
talking about here; and

(b) I do tend to use different values of harmonic entropy's free
parameter, s, depending on timbre. This is far from arbitrary -- both
mathematics and experience suggest that ratio-intepretation of
intervals in timbres with weak, absent, or inharmonic partials are
more "blurred" ("pastelized" is one of Margo's terms for it) than in
timbres with a strong complement of harmonic partials.

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

> J Gill: So Paul, tell me, have *you* ever "heard" a 20-25 Hz
> tone, or have you ever met anyone who has "heard" a 20-25 Hz
> tone?

I don't know, but considering the vast amount of psychoacoustic data
that you cite, I felt it appropriate to note that your lower bound of
70 Hz definitely seemed out of line with the published evidence.

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

/tuning/topicId_unknown.html#32597

> >
> > J Gill: So Paul, tell me, have *you* ever "heard" a 20-25 Hz
> > tone, or have you ever met anyone who has "heard" a 20-25 Hz
> > tone?
> >
> >
> > J Gill
>
>
> Well - you both have now. Me! And my old ears can still hear up to
18-19 khz too - insect, bird frequencies!
>
> What's the big deal anyway? Is there some dispute that this can't
be heard?
>
> J:L

Hi Jacky!

Yes, there was....

JP

>>>>> "jpehrson2" == jpehrson2 <jpehrson@rcn.com> writes:

jpehrson2> We were *really* driving that sound, and I had long hair and small
jpehrson2> beads around my neck, but you needn't know any more about that.

Somehow that puts me in my place -- i still have long hair and beads
round my neck as I type this. The beads only date from the mid 60s,
but the long hair was in place at (high) school.

==John ffitch

--- In tuning@y..., <jpff@c...> wrote:

/tuning/topicId_32480.html#32676

> >>>>> "jpehrson2" == jpehrson2 <jpehrson@r...> writes:
>
> jpehrson2> We were *really* driving that sound, and I had long
hair and small
> jpehrson2> beads around my neck, but you needn't know any more
about that.
>
> Somehow that puts me in my place -- i still have long hair and beads
> round my neck as I type this. The beads only date from the mid 60s,
> but the long hair was in place at (high) school.
>
> ==John ffitch

Hello John FFFFFitch! CSOUND Maestro!

Well, John, at least you've found something of "interest" to read on
this list! :)

Soon, I'm hoping to try again to "wrap my head around" CSOUND. Again
next week, or perhaps the following week... maybe the week after
that... :)

best wishes,

J. Pehrson