Yahoo Tuning Groups Ultimate Backup tuning Re: Smallness/Primeness

--- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:

> > > Subject: [tuning] Re: Tonal Affinities - Nature or Nurture?
> > >
> > > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > >
> > > > I would argue that the reasons *why* those particular intervals
> > > > have been chosen, all around the earth, time and time again, are
> > > > because of properties of affinity which result from the low
> > > > primeness of these intervals:
> > >
> > > I would argue that it has nothing to do with the "low primeness", but
> > > rather with the fact that these are the simplest possible ratios, the
> > > ratios that involve the smallest numbers.
> >
> >
> > Hmmm... well, I can go along with you to some extent.
> >
> > But 5 is only one more than 4, so why is there so much difference
> > and inconsistency in the way the world's musical cultures have
> > treated 5-limit intervals, when there's so much similarity and
> > consistency with the way those with factors of 2 and 3 have been
> > treated? Clearly, there is some very special difference between
> > 3-limit and 5-limit ratios, more than simply the size of the numbers.
>
> Or look at meter. 3 and 2, and their "octave equivalents, 4, 6, 8 (and
> to a lesser extent 9 and 12) dominate musics which have a metric
> orientation. The more comlicated prime rhythm musics (Balkan and Indian
> musics) tend to heirarchically arrive at those primes by summing 2's, 3's
> and 4's.
>
> Its not just that small numbers are "good", but very small numbers are
> "good". As has been pointed out, due to the definition of prime numbers,
> the population of primes is more dense in the population of small
> numbers.
>
> Bob Valentine

J Gill: It also seems that -

The business of any differences arising out of the "in-divisibility" or "divisibility" of a number (this being the difference between primes and their multiples) still seems murky (despite the proposed "octave-equivalence" which has been applied by some). What about pitch multiplications/divisions of the other prime numbers 3, 5, and 7? And *their* integer multiples (6, 10, 14, etc)?

And what do you think of a concept of "rooted-ness" occuring where powers of the prime 2 appear in the numerator or the denominator (which, if I have interpreted Dave Keenan correctly, may impart a character unique due to "octave-equivalence" effects)? Additionally, are "octave-invariance" phenomena relevant to *melody*, or just to *harmony*? Do "limits of recognizability" exist in the degree of octave stretching/shrinking (the number of octaves shifted) when concepts of "octave-invariance" are applied in "harmony"? In "melody"?

(1) IF it is true that it *is* the integers "smallness" (and not its "prime status") which is the cogent characteristic giving rise (at least in part) to a sort of "recognizability" (and thus, perhaps, distinct tonal "personality") in certain (but likely not *all*) contexts (primarily related, then, to a small numerical vaue, as opposed to whether the value is either divisible or in-divisible); AND

(2) IF it is true that listeners (perhaps even untrained ones, to a certian extent) are, upon hearing "complex" tones sounded in unison with a "complex" tone at 1/1, are able to "recognize and categorize" various levels of the pitch-heights of the secondary "complex" tone of such a "complex" dyad; AND

(3) BEATS (whether they be perceived as benefical, beautiful, or maddening, or familiar - as Paul Erlich pointed out in relation to the "beats" created by 12-tone ET which are a familiar and *desired* spectral characteristic "absent" in JI tunings), being a (necessary) blessing/curse in any practical system, and being a natural and non-controllable (due to random phase between sound sources) result of two separate sound sources at play in *harmonies*; AND

(4) We think of, speak of, and appear to (in some cases) "aurally perceive" various possible combinations of such "complex" tones formed into dyads [dyads as described above [in (2)] as representing a (simultaneously sounded, or "harmonious") tonal "affinity" of sorts, etc; THEN

(5) WHY(?) would one tend to minimize the fact that:

The *lower* in value of the size of the number, the lower the harmonic number (of a 1/1 reference pitch) at which the *first* "harmonic coincidence" takes place, and (correspondingly) the *more* "harmonic coincidences" *occur* (per unit bandwidth) throughout the resulting frequency spectrum...

Thus it would (to me) seem that - while this (historically widely discussed) phenomena may well *not*, in itself, "sum-up" perceptual metaphors such as "consonance", or "con-cordance", or "affinity", etc, is it not (at least, one of) the most tangible and widely applicable (to *multiple* tones, in addition to dyads) descriptor of "physically" (as opposed to "psychically") describable and characterizable tonal inter-relationships between members of groups of "complex" (fundamental plus harmonic content possible) tones in unison.

Of the tangible "starting points" in modelling (relatively) simple sound spectrums composed of multiple "complex" sound sources of number greater than two (triads, tetrads, etc), where the individual "complex" tones making up these combinations may be *linearly* combined and evaluated as to the composite resulting "coincidences" in pitch, this seems to be a fertile ground for the possible characterization of the specific (and not necessarily "con" or "dis", but "cordant") "harmonic signatures" achieved by combining actual multiple "complex" tones sounded in unison (or serially) from within a given N-tone scale (JI or ET).

The ear may not be "second guessed" by such "spectral signatures" generated by the processes described above. Perhaps it is foolish to "make a physical science" of how a given real-world "complex" tone will be perceived when sounded (simultaneously or serially) along with the other available scale pitches in such a "controlled" model.

Therefore, while being no substitute for audio listening tests on the instrument of application, perhaps there are some things to learn from such categorization of the "harmonic coincidence" of various combinations of "complex" tones selected from within the set of available pitches in a particular "tuning under consideration".

Am I here "re-inventing the wheel" (unaware of such very categorizations by others in the past/present)?

Is such information (when gathered) complicated enough to appear without pattern, rhyme, or reason?

Is there another reason (or are there other reasons) why such work appears (to me) not to be applied to the selection of scale-pitches?

Curiously, J Gill

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

Has it not occurred to anyone that if you were to take all the
instruments, including the human voice, that have relatively high
harmonicity of timbre and check how much acoustic energy after
ignoring the fundamental that is represented in harmonics 2, 3, and 4
vis-a-vis all the other higher harmonics, it might go a long way
toward explaining this great mystery?

And let's not forget instruments like the so-called "Jews harp" and
the didjery-doo, in which a substantial part of what's going on
musically is the perception and manipulation of the relative strength
of the harmonics that comprise the timbre. Combine all these things
with the frequency with which drones are used in many folk idioms and
the phenomenon of overlapping melodic material even in the
predominantly non-harmonic history of world musical cultures and
maybe the mystery becomes less than competely mysterious.

--- In tuning@y..., "unidala" <JGill99@i...> wrote:
> --- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:
>
> > > > Subject: [tuning] Re: Tonal Affinities - Nature or Nurture?
> > > >
> > > > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > > >
> > > > > I would argue that the reasons *why* those particular
intervals
> > > > > have been chosen, all around the earth, time and time
again, are
> > > > > because of properties of affinity which result from the low
> > > > > primeness of these intervals:
> > > >
> > > > I would argue that it has nothing to do with the "low
primeness", but
> > > > rather with the fact that these are the simplest possible
ratios, the
> > > > ratios that involve the smallest numbers.
> > >
> > >
> > > Hmmm... well, I can go along with you to some extent.
> > >
> > > But 5 is only one more than 4, so why is there so much
difference
> > > and inconsistency in the way the world's musical cultures have
> > > treated 5-limit intervals, when there's so much similarity and
> > > consistency with the way those with factors of 2 and 3 have been
> > > treated? Clearly, there is some very special difference between
> > > 3-limit and 5-limit ratios, more than simply the size of the
numbers.
> >

🔗paulerlich <paul@stretch-music.com>

🔗jpehrson2 <jpehrson@rcn.com>

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> /tuning/topicId_31805.html#31805
>
> > (5) WHY(?) would one tend to minimize the fact that:
> >
> > The *lower* in value of the size of the number, the lower the
> harmonic number (of a 1/1 reference pitch) at which the
> *first* "harmonic coincidence" takes place, and (correspondingly) the
> *more* "harmonic coincidences" *occur* (per unit bandwidth)
> throughout the resulting frequency spectrum...
> >
>
> Hello J. Gill!
>
> Well, I'm not certain if I'm totally understanding your "Baroque"
> postulate, but I'll try to give it a "Classical" answer! :)
>
> Isn't part of this situation the fact that the lower interger
> harmonics of a sound generating source are, generally speaking, at a
> higher *amplitude* than the higher harmonics??
>
> I think that's generally the case, isn't it... certainly among square
> waves, triangle waves and such like...

JG: Yes. Square, triangle, and ramp waveforms all have
harmonics which "roll-off" (in amplitude over frequency)
at a rate equal to 1/N (where N=harmonic number), or at
-6 dB/Octave over frequency.

> Wouldn't that tend to emphasize the lower partials just by virtue >of
> the *amplitude??* and their associations between simultaneously-
> sounding harmonic series?
>
> Dunno. Something to think about, perhaps??

JG: That (as you state) emphasizes the "lower partials"
relative to *each other* within the spectrum of a given
"square, triangle, or ramp waveform".

However, my point was meant to address the situation
where (*relative to a reference pitch of 1*) the amount
of *harmonic coincidences* which occur between the
overtones of *multiple* notes whose fundamental
frequencies are 2 and 3 (for instance) are more
*numerous* than the amount of "harmonic coincidences"
which occur between the overtones of *multiple* notes
whose fundamental frequencies are located at 3 and 4
(or at 4 and 6, 3 and 5, 5 and 7, etc.)

This is an issue separate from that of the magnitude
of the overtones of a fundamental frequency relative
to the magnitude of that fundamental "complex" tone,
and relates to the "harmonic coincidences" *between*
the overtones of multiple "complex" tones, regardless
of their specific individual spectra as "complex" tones.

Obviously, if there are no overtones to "coincide"
(as in the sinusoidal case), then such coincidences
cannot occur. However, this issue (as I see it) is
not one relating to the various possible magnitudes
of overtones relative to to a fundamental, but instead,
relates only to the possibilities (should such overtones
be present in the individual "complex" tones combined)
of the "coincidences" of such overtones (if present in
two or more of the so combined "complex tones").

Best Regards, J Gill

🔗unidala <JGill99@imajis.com>

> > > > I would argue that it has nothing to do with the "low >>>>primeness", but
> > > > rather with the fact that these are the simplest possible >>>>ratios, the
> > > > ratios that involve the smallest numbers.

> > Its not just that small numbers are "good", but very small >>numbers are
> > "good". As has been pointed out, due to the definition of prime >>numbers,
> > the population of primes is more dense in the population of small
> > numbers.

J Gill: I have read that the chance of a random integer N
being prime is about 1/(log(N)). See "consequence three" at:

http://www.utm.edu/research/primes/howmany.shtml

This indicates that primes are more pervasive when N is small.

There seems to be much agreement that "small is good".
Yet, the issue of whether 4 (as a non-prime multiple
of the prime 2) or 6 (as a non-prime multiple of the
prime 3), etc, is characteristically *different* from
the primes from which they are derived does not appear
to have been addressed in these related threads... :)

And what of multiples by 3 and 5 (as opposed to multiples
of 2 only)?

And, is 2 (as an "octave equivalent") multiplier unique?

Curiously, J Gill

🔗paulerlich <paul@stretch-music.com>

🔗jpehrson2 <jpehrson@rcn.com>

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

/tuning/topicId_31805.html#31992

>
> However, my point was meant to address the situation
> where (*relative to a reference pitch of 1*) the amount
> of *harmonic coincidences* which occur between the
> overtones of *multiple* notes whose fundamental
> frequencies are 2 and 3 (for instance) are more
> *numerous* than the amount of "harmonic coincidences"
> which occur between the overtones of *multiple* notes
> whose fundamental frequencies are located at 3 and 4
> (or at 4 and 6, 3 and 5, 5 and 7, etc.)
>
> This is an issue separate from that of the magnitude
> of the overtones of a fundamental frequency relative
> to the magnitude of that fundamental "complex" tone,
> and relates to the "harmonic coincidences" *between*
> the overtones of multiple "complex" tones, regardless
> of their specific individual spectra as "complex" tones.
>
> Obviously, if there are no overtones to "coincide"
> (as in the sinusoidal case), then such coincidences
> cannot occur. However, this issue (as I see it) is
> not one relating to the various possible magnitudes
> of overtones relative to to a fundamental, but instead,
> relates only to the possibilities (should such overtones
> be present in the individual "complex" tones combined)
> of the "coincidences" of such overtones (if present in
> two or more of the so combined "complex tones").
>
>
> Best Regards, J Gill

Hello J. Gill!

This seemed a little peculiar to me, though, when I first read it.
Is it not the cast that overtones become closer together and more
numerous as they go up the series?

Wouldn't that mean that there would be *more* coincidences among
higher partials than lower, even approaching *infinity?*

Or am I just misunderstanding something...

best,

J. Pehrson

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> /tuning/topicId_31805.html#31992
>
> >
> > However, my point was meant to address the situation
> > where (*relative to a reference pitch of 1*) the amount
> > of *harmonic coincidences* which occur between the
> > overtones of *multiple* notes whose fundamental
> > frequencies are 2 and 3 (for instance) are more
> > *numerous* than the amount of "harmonic coincidences"
> > which occur between the overtones of *multiple* notes
> > whose fundamental frequencies are located at 3 and 4
> > (or at 4 and 6, 3 and 5, 5 and 7, etc.)
> >
> > This is an issue separate from that of the magnitude
> > of the overtones of a fundamental frequency relative
> > to the magnitude of that fundamental "complex" tone,
> > and relates to the "harmonic coincidences" *between*
> > the overtones of multiple "complex" tones, regardless
> > of their specific individual spectra as "complex" tones.
> >
> > Obviously, if there are no overtones to "coincide"
> > (as in the sinusoidal case), then such coincidences
> > cannot occur. However, this issue (as I see it) is
> > not one relating to the various possible magnitudes
> > of overtones relative to to a fundamental, but instead,
> > relates only to the possibilities (should such overtones
> > be present in the individual "complex" tones combined)
> > of the "coincidences" of such overtones (if present in
> > two or more of the so combined "complex tones").
> >
> >
> > Best Regards, J Gill
>
> Hello J. Gill!
>
> This seemed a little peculiar to me, though, when I first read it.
> Is it not the cast that overtones become closer together and more
> numerous as they go up the series?
>
> Wouldn't that mean that there would be *more* coincidences among
> higher partials than lower, even approaching *infinity?*
>
> Or am I just misunderstanding something...
>
> best,
>
> J. Pehrson

Hello, J Pehrson!

If you were counting partials on a per *octave* basis, yes.
If you count partials on a per Hz (CPS) frequency basis, no.

Which matters? Issues of frequency response and perceived
pitch of a single tone are geometric in the "aural mind".
Issues of overtones (they being spaced on a per/Hz linear
basis) are rightly addressed in the "arithmetic" viewpoint
[as the "implied fundamentalists" :) would have you believe].

Best Regards, J Gill

J Gill

🔗paulerlich <paul@stretch-music.com>

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> > > Hello J. Gill!
> > >
> > > This seemed a little peculiar to me, though, when I first read
> it.
> > > Is it not the cast that overtones become closer together and more
> > > numerous as they go up the series?
> > >
> > > Wouldn't that mean that there would be *more* coincidences among
> > > higher partials than lower, even approaching *infinity?*
> > >
> > > Or am I just misunderstanding something...
> > >
> > > best,
> > >
> > > J. Pehrson

JG: Two questions were posed by JP: (1) "Is it not the cast that overtones become closer together and more numerous as they go up the series?", and (2) "Wouldn't that mean that there would be *more* coincidences among higher partials than lower, even approaching
*infinity?*". I responded only to question (1) [sorry, Joe...]

> > JG: Hello, J Pehrson!
> > If you were counting partials on a per *octave* basis, yes.
> PE: No, the answer in this case is no.

JG: The answer to question (1) would be that the tones do not become "closer together" or become "more numerous" (on a per unit frequency basis), but they would appear to be "closer together" and "more numerous" (on a per unit octave basis).

JG: In reference to question (1)
> > JG: If you count partials on a per Hz (CPS) frequency basis, no.
> PE: Correct, the answer is again, "no".

JG: In reference to question (2) above. Coincidences among higher partials (when they exist due to prescence of energy at the partials spoken of here) happen at an initial frequency, and (potentially) at all frequencies that are integer multiples of the initial frequency. Therefore, a fixed number of them occurs per unit bandwidth. If one looked with "per unit octave" perspective, they would count a larger number of such "coincidences" over frequency (in each *octave* unit).

My original point related to differences between - the number of "coinciding" partials (on a per unit frequency basis) when the lowest frequency of "coincidence" appears at a lower frequency (in the case of smaller integers), as opposed to the same appearing at a higher frequency (in the case of larger integers).

> > Which matters? Issues of frequency response and perceived
> > pitch of a single tone are geometric in the "aural mind".
>
> Yes.
>
> > Issues of overtones (they being spaced on a per/Hz linear
> > basis) are rightly addressed in the "arithmetic" viewpoint
> > [as the "implied fundamentalists" :) would have you believe].
>
> Not if you're trying to relate _both_ issues.

JG: Both (the linear as well as logarithmic) issues are, of course, important within their respective contexts of application...

J Gill

🔗robert_wendell <BobWendell@technet-inc.com>

The "mystery" is the one already referred to in the thread to which
I'm responding, namely why octaves, fifths and fourths are so
prevalent as fundamentally important melodic intervals across
cultures and across time.

The point is that there is a hugely greater amount of acoustic energy
in the first three harmonics (meaning 2, 3, and 4) taken globally and
statistically as a general phenomenon than in any of the higher ones
if we take the sum total of all musical instrumments, including the
human voice, throughout history and the world. It's a simple matter
of physics and the relative abundance of acoustic energy in the
fundamental and lower harmonics.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> > Has it not occurred to anyone that if you were to take all the
> > instruments, including the human voice, that have relatively high
> > harmonicity of timbre and check how much acoustic energy after
> > ignoring the fundamental that is represented in harmonics 2, 3,
and
> 4
> > vis-a-vis all the other higher harmonics, it might go a long way
> > toward explaining this great mystery?
>
> I don't know what you mean. Many times a human voice or other
> instrument will happen to have a little more acoustic energy in
> harmonic 5 than in any other harmonic, and it doesn't sound
> particularly unnatural.
>
> > And let's not forget instruments like the so-called "Jews harp"
and
> > the didjery-doo, in which a substantial part of what's going on
> > musically is the perception and manipulation of the relative
> strength
> > of the harmonics that comprise the timbre. Combine all these
things
> > with the frequency with which drones are used in many folk idioms
> and
> > the phenomenon of overlapping melodic material even in the
> > predominantly non-harmonic history of world musical cultures and
> > maybe the mystery becomes less than competely mysterious.
>
> Maybe I'm not clear on what mystery exactly you're thinking of?

🔗monz <joemonz@yahoo.com>

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> > The "mystery" is the one already referred to in the thread to
> which
> > I'm responding, namely why octaves, fifths and fourths are so
> > prevalent as fundamentally important melodic intervals across
> > cultures and across time.
> >
> > The point is that there is a hugely greater amount of acoustic
> energy
> > in the first three harmonics (meaning 2, 3, and 4) taken
> globally and
> > statistically as a general phenomenon than in any of the higher
> ones
> > if we take the sum total of all musical instrumments, including
> the
> > human voice, throughout history and the world. It's a simple
> matter
> > of physics and the relative abundance of acoustic energy in the
> > fundamental and lower harmonics.
>
> As I said, this is way overstating the case, for example look at
the
> human voice -- often a higher harmonic, say the fifth harmonic, is
> loudest of all, and the very identities of vowel sounds are carried
> by formants which work by emphasizing far higher harmonics
> than this. To understand the universality of octaves and fifths, I
> think you would be well served to read, and think carefully about,
> Terhardt's discussion of "affinity".

Bob:
My statements are well-qualified as general and statistical if you
read them with clear awareness of their tone and ALL their content.
Exceptions don't necessarily cancel out general trends.

As a musician, I am first and foremost a singer. I am intimately
familiar with formants and vowel perception, Helmoltz resonator-like
behavior of the vocal tract in structuring formants, etc., not just
conceptually, but also experientially with my own voice. Further,
vowel perceptions involving higher harmonics do not presuppose their
dominance in the energy spectrum.

Overall, even the voice tends to have more energy in the lower three
harmonics than in the higher, the exceptions cited notwithstanding.
More importantly, the heavier concentrations of acoustic energy in
the higher harmonics tends to be associate with the vocal production
of trained professionals with consequently limited significance in
the context of the cross-cultural phenomenon we are discussing here.

🔗robert_wendell <BobWendell@technet-inc.com>

Bob quoting from Terhardt:
"Pitch commonality of harmonic complex tones is quite pronounced for
a 1:2 ratio of oscillation frequencies, i.e., when the tones are in
an octave interval. Pitch commonality is less pronounced for tones in
fifth and fourth intervals. And it is more or less negligible for any
of the other intervals. This elucidates and explains the particular
role played by the octave and the fifth, at least where sensory
affinity is concerned."

Bob reponds to this quote and Paul's comment below:
Yes, I agree that pitch commonality of the harmonics is crucial to
the understanding of pitch affinity. I would think, however, that the
statistical predominance of acoustic energy in the lower harmonics is
a contributor that is not at all neligible in its importance and in
noticing the presence of these intervals withing a single tone.

Any good ear will notice the octave overtone of many kinds of flute
timbres and the fifth is also quite prominent in some. Moreover, the
flute overtones vary substantially with fluctuations in embouchure,
which would tend to make them more noticeable even within a single
tone. Flutes are, after all, among the most culturally universal
instrumental phenomena.

I don't quite understand why he says "more or less negligible for any
of the other intervals", though, unless he refers only to the NUMBER
of common pitches RELATIVE to those in the octaves, fifths, and
fourths, which case he is undoubtedly right on the mark. We couldn't
accurately tune these pitches by timing beats in the setting of
temperaments if the common harmonics were essentially negligible in
any other sense.

🔗unidala <JGill99@imajis.com>

🔗jpehrson2 <jpehrson@rcn.com>

🔗robert_wendell <BobWendell@technet-inc.com>

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> >
> > /tuning/topicId_31805.html#32181
> >
> > > The "mystery" is the one already referred to in the thread to
> which
> > > I'm responding, namely why octaves, fifths and fourths are so
> > > prevalent as fundamentally important melodic intervals across
> > > cultures and across time.
> > >
> > > The point is that there is a hugely greater amount of acoustic
> > energy
> > > in the first three harmonics (meaning 2, 3, and 4) taken globally
> > and
> > > statistically as a general phenomenon than in any of the higher
> > ones
> > > if we take the sum total of all musical instrumments, including
> the
> > > human voice, throughout history and the world. It's a simple
> matter
> > > of physics and the relative abundance of acoustic energy in the
> > > fundamental and lower harmonics.
> > >
> >
> > Hello Bob!
> >
> > By "acoustic energy" I'm assuming you mean *amplitude,* correct??
> >
> > JP
>
> Bob:
> Well, technically I meant acoustic power, which is a function of both
> amplitude and frequency. It is directly proportional to both so the
> same power or energy per unit time is represented by lower amplitudes
> at higher frequencies. This is because energy per unit time for the
> same amplitude is greater at higher frequencies.

J Gill (my "2-cents"): Joe, Bob is (I think) referring to "spectral power bandwidth" [the product of a given voltage/current squared multiplied by the bandwidth over which the "spectral power" is referenced, or the "area under the (power = amplitude^2) curve"].

I would "beg to differ" with Bob's statement:

<< ... energy per unit time for the
same amplitude is greater at higher frequencies. >>

The "energy" (of a time-domain sinusoidal wave, or of the corresponding "spectral line" in the frequency domain) is
independent of frequency. If one were to examine the power
of "geometric bandwidths" (such as octave bandwidths), by
integrating over such (wider and wider in unit *bandwidth*)
frequency ranges, then the "spectral power" of these
(wider and wider) bands *does* increase with increasing
frequency (as the "center frequency" of such a bandwidth
increases).

Clear as mud?

J Gill

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> >
> > /tuning/topicId_31805.html#32181
> >
> > > The "mystery" is the one already referred to in the thread to
> which
> > > I'm responding, namely why octaves, fifths and fourths are so
> > > prevalent as fundamentally important melodic intervals across
> > > cultures and across time.
> > >
> > > The point is that there is a hugely greater amount of acoustic
> > energy
> > > in the first three harmonics (meaning 2, 3, and 4) taken globally
> > and
> > > statistically as a general phenomenon than in any of the higher
> > ones
> > > if we take the sum total of all musical instrumments, including
> the
> > > human voice, throughout history and the world. It's a simple
> matter
> > > of physics and the relative abundance of acoustic energy in the
> > > fundamental and lower harmonics.
> > >
> >
> > Hello Bob!
> >
> > By "acoustic energy" I'm assuming you mean *amplitude,* correct??
> >
> > JP
>
> Bob:
> Well, technically I meant acoustic power, which is a function of both
> amplitude and frequency. It is directly proportional to both so the
> same power or energy per unit time is represented by lower amplitudes
> at higher frequencies. This is because energy per unit time for the
> same amplitude is greater at higher frequencies.

I would "beg to differ" with Bob's statement:

<< ... energy per unit time for the
same amplitude is greater at higher frequencies. >>

Clear as mud?

J Gill

🔗jpehrson2 <jpehrson@rcn.com>

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> > > The "mystery" is the one already referred to in the thread to
> > which
> > > I'm responding, namely why octaves, fifths and fourths are so
> > > prevalent as fundamentally important melodic intervals across
> > > cultures and across time.
> > >
> > > The point is that there is a hugely greater amount of acoustic
> > energy
> > > in the first three harmonics (meaning 2, 3, and 4) taken
> > globally and
> > > statistically as a general phenomenon than in any of the higher
> > ones
> > > if we take the sum total of all musical instrumments, including
> > the
> > > human voice, throughout history and the world. It's a simple
> > matter
> > > of physics and the relative abundance of acoustic energy in the
> > > fundamental and lower harmonics.
> >
> > As I said, this is way overstating the case, for example look at
> the
> > human voice -- often a higher harmonic, say the fifth harmonic,
is
> > loudest of all, and the very identities of vowel sounds are
carried
> > by formants which work by emphasizing far higher harmonics
> > than this. To understand the universality of octaves and fifths,
I
> > think you would be well served to read, and think carefully
about,
> > Terhardt's discussion of "affinity".
>
>
> Bob:
> My statements are well-qualified as general and statistical if you
> read them with clear awareness of their tone and ALL their
content.
> Exceptions don't necessarily cancel out general trends.
>
> As a musician, I am first and foremost a singer. I am intimately
> familiar with formants and vowel perception, Helmoltz resonator-
like
> behavior of the vocal tract in structuring formants, etc., not just
> conceptually, but also experientially with my own voice. Further,
> vowel perceptions involving higher harmonics do not presuppose
their
> dominance in the energy spectrum.
>
> Overall, even the voice tends to have more energy in the lower
three
> harmonics than in the higher, the exceptions cited notwithstanding.
> More importantly, the heavier concentrations of acoustic energy in
> the higher harmonics tends to be associate with the vocal
production
> of trained professionals with consequently limited significance in
> the context of the cross-cultural phenomenon we are discussing here.

My point is that IMO the phenomenon is a result of greater affinity,
as Terhardt describes it, and would function just fine if all vocal
sounds had equal partial amplitudes up to the 10th partial or
whatever.

🔗paulerlich <paul@stretch-music.com>

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> Bob quoting from Terhardt:
> "Pitch commonality of harmonic complex tones is quite pronounced
for
> a 1:2 ratio of oscillation frequencies, i.e., when the tones are in
> an octave interval. Pitch commonality is less pronounced for tones
in
> fifth and fourth intervals. And it is more or less negligible for
any
> of the other intervals. This elucidates and explains the particular
> role played by the octave and the fifth, at least where sensory
> affinity is concerned."
>
> Bob reponds to this quote and Paul's comment below:
> Yes, I agree that pitch commonality of the harmonics is crucial to
> the understanding of pitch affinity. I would think, however, that
the
> statistical predominance of acoustic energy in the lower harmonics
is
> a contributor that is not at all neligible in its importance and in
> noticing the presence of these intervals withing a single tone.
>
> Any good ear will notice the octave overtone of many kinds of flute
> timbres and the fifth is also quite prominent in some.

Any good ear? You and I, yes, as we've trained our ears to hear
overtones. Most of the musicians I know with really good ears?
Totally oblivious to overtones.
>
> I don't quite understand why he says "more or less negligible for
any
> of the other intervals", though, unless he refers only to the
NUMBER
> of common pitches RELATIVE to those in the octaves, fifths, and
> fourths, which case he is undoubtedly right on the mark.

You may have to read a few of the other articles on his website too.
His understanding of pitch is a bit more complex, but once you grasp
it, it's very rewarding in its power.

> We couldn't
> accurately tune these pitches by timing beats in the setting of
> temperaments if the common harmonics were essentially negligible in
> any other sense.

We're talking about the "affinity" or "similarity" relationship here,
not the "consonance" relationship.

🔗paulerlich <paul@stretch-music.com>

🔗robert_wendell <BobWendell@technet-inc.com>

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
>
> /tuning/topicId_31805.html#32269
>
> > Bob:
> > Well, technically I meant acoustic power, which is a function of
> both
> > amplitude and frequency. It is directly proportional to both so
the
> > same power or energy per unit time is represented by lower
> amplitudes
> > at higher frequencies. This is because energy per unit time for
the
> > same amplitude is greater at higher frequencies.
>
> Hello Bob!
>
> Well, then, you mean that the higher frequencies would have more
> power if they were at the same amplitude, but in acoustic sounds
the
> higher ones are at lesser amplitudes, yes?
>
> The lower partials are louder... but since they have less energy
per
> unit time, wouldn't they "level out" with the higher, lower
amplitude
> partials??
>
> Signed, confused...
>
> Thanks, Bob!
>
> JP

Bob:
Well, I think Jeremy is thinking electrically and I'm thinking
acoustically. To the best of my rusty memory, I'm fairly confident
(but not absolutely so) that ACOUSTIC power is directly proportional
to both amplitude and frequency. I believe the relationship of
amplitude and power is not the same acoustically as electrically
because of the way they are defined.

The bottom line for our discussion of the relative acoustic power of
lower vs. higher harmonics is that despite this, more acoustic energy
is generally concentrated individually in one or more of the first
few lower harmonics than in any individual harmonics higher than the
first four. There are notable exceptions to this, as Paul Erlich has
pointed out, citing the human voice on certain vowels as an example.
I pointed out in a response that this is truer in trained voices than
in traditional folk singing, the latter of which is more germaine to
this discussion of the universality of octaves, fifths, and fourths.

I think, also, that statistically the phenomenon is sufficiently
significant, especially in the case of the culturally ubiquitous
flute, to add quite a bit of weight to the high rate of common
harmonics that occur between fundamentals representing the same
relationships (octaves, fourths, and fifths) to each other. Taken
together, I believe these physical realities eliminate the mystery
concerning why these intervals would prevail in the melodic systems
of most cultures in any era.

🔗robert_wendell <BobWendell@technet-inc.com>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> > Bob quoting from Terhardt:
> > "Pitch commonality of harmonic complex tones is quite pronounced
> for
> > a 1:2 ratio of oscillation frequencies, i.e., when the tones are
in
> > an octave interval. Pitch commonality is less pronounced for
tones
> in
> > fifth and fourth intervals. And it is more or less negligible for
> any
> > of the other intervals. This elucidates and explains the
particular
> > role played by the octave and the fifth, at least where sensory
> > affinity is concerned."
> >
> > Bob reponds to this quote and Paul's comment below:
> > Yes, I agree that pitch commonality of the harmonics is crucial
to
> > the understanding of pitch affinity. I would think, however, that
> the
> > statistical predominance of acoustic energy in the lower
harmonics
> is
> > a contributor that is not at all neligible in its importance and
in
> > noticing the presence of these intervals withing a single tone.
> >
> > Any good ear will notice the octave overtone of many kinds of
flute
> > timbres and the fifth is also quite prominent in some.
>
> Any good ear? You and I, yes, as we've trained our ears to hear
> overtones. Most of the musicians I know with really good ears?
> Totally oblivious to overtones.
> >
> > I don't quite understand why he says "more or less negligible for
> any
> > of the other intervals", though, unless he refers only to the
> NUMBER
> > of common pitches RELATIVE to those in the octaves, fifths, and
> > fourths, which case he is undoubtedly right on the mark.
>
> You may have to read a few of the other articles on his website
too.
> His understanding of pitch is a bit more complex, but once you
grasp
> it, it's very rewarding in its power.
>
> > We couldn't
> > accurately tune these pitches by timing beats in the setting of
> > temperaments if the common harmonics were essentially negligible
in
> > any other sense.
>
> We're talking about the "affinity" or "similarity" relationship
here,
> not the "consonance" relationship.

Nothing in my last statement above implies that I'm switching the
subject to consonance. I'm just saying the common harmonics have to
be there in those intervals or we couldn't tune them. How does this
imply such a switch to you?

Puzzled,

Bob

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> >
> > /tuning/topicId_31805.html#32269
> >
> > > Bob:
> > > Well, technically I meant acoustic power, which is a function of
> > both
> > > amplitude and frequency. It is directly proportional to both so
> the
> > > same power or energy per unit time is represented by lower
> > amplitudes
> > > at higher frequencies. This is because energy per unit time for
> the
> > > same amplitude is greater at higher frequencies.
> >
> > Hello Bob!
> >
> > Well, then, you mean that the higher frequencies would have more
> > power if they were at the same amplitude, but in acoustic sounds
> the
> > higher ones are at lesser amplitudes, yes?
> >
> > The lower partials are louder... but since they have less energy
> per
> > unit time, wouldn't they "level out" with the higher, lower
> amplitude
> > partials??
> >
> > Signed, confused...
> >
> > Thanks, Bob!
> >
> > JP
>
> Bob:
> Well, I think Jeremy is thinking electrically and I'm thinking
> acoustically. To the best of my rusty memory, I'm fairly confident
> (but not absolutely so) that ACOUSTIC power is directly proportional
> to both amplitude and frequency. I believe the relationship of
> amplitude and power is not the same acoustically as electrically
> because of the way they are defined.
>
> The bottom line for our discussion of the relative acoustic power of
> lower vs. higher harmonics is that despite this, more acoustic energy
> is generally concentrated individually in one or more of the first
> few lower harmonics than in any individual harmonics higher than the
> first four. There are notable exceptions to this, as Paul Erlich has
> pointed out, citing the human voice on certain vowels as an example.
> I pointed out in a response that this is truer in trained voices than
> in traditional folk singing, the latter of which is more germaine to
> this discussion of the universality of octaves, fifths, and fourths.
>
> I think, also, that statistically the phenomenon is sufficiently
> significant, especially in the case of the culturally ubiquitous
> flute, to add quite a bit of weight to the high rate of common
> harmonics that occur between fundamentals representing the same
> relationships (octaves, fourths, and fifths) to each other. Taken
> together, I believe these physical realities eliminate the mystery
> concerning why these intervals would prevail in the melodic systems
> of most cultures in any era.

Bob and Joe,

I've done some digging into Olson's "Music, Physics,
and Engineering", 2nd Edition, Dover, 1967, and here
is what I find:

The units of *sound pressure* are those of "force".
The unit is the "dyne" [acceleration of 1 gram mass
at 1 centimeter per(second^2)]. "Sound pressure" is
measured in units of "dynes/(cm)^2" (force in dynes
per square centimeter). "Sound pressure level" ('SPL').

The units of *power* are in energy transferred per second.
The unit is the "erg" [the work done when a 1 dyne force
displaces that 1 gram of mass by 1 centimeter of distance].

Whether a sound pressure wave is a plane or spherical shape,
the same relationships apply (relative to the issue at hand).

"effective sound pressure" ('ESP') is the root mean square
('RMS') value over 1 cycle of that wave [in dynes/(cm)^2].

At a constant distance from the sound source, and
in order to *maintain a CONSTANT amplitude* (of air
particle displacement), the "sound pressure" must be
*increased* in a manner directly proportional to
the frequency of the sound wave.

At a constant distance from the sound source, and
in order to *maintain a CONSTANT amplitude* (of air
particle displacement), the "acoustic power" must be
*increased* in a manner directly proportional to
the *square* of the frequency of the sound wave.

The "loudness" of a sound (equal to the magnitude
of auditory sensation) is defined in units of "Sones"
(which are related to "Phons" by measured non-linear
relationships for sinusoidal tones). 1 Phon is equivalent
to 40 Sones (an 'SPL' of 2 x 10^4 dynes/cm^2).

So, "loudness" is related to "sound pressure level"
('SPL'), and not directly related to "acoustic power".

While Olson does not cover the "whys" and "wherefores"
of the decrease in the amplitude of the SPL of harmonic
overtones of resonators as the harmonic number increases,
it makes intuitive sense that (since a doubling of the
SPL - and a quadrupling of the "acoustic power" - would
be necessary to *maintain a CONSTANT amplitude* of air
particle displacement each time the frequency doubles),
resonators (such as voice, strings, etc.) do not behave
in this manner. In fact, they tend to exhibit *decreasing*
SPLs as the harmonic number *increases* over frequency.

From measurements by Sivian, Dunn, and White, Olson
(on page 205, Figure 6.3) shows curves depicting the
measured (over a long time period of many cycles)
magnitudes of "average pressure/cycle" over frequency
copared to the "average total pressure" (of the entire
frequency spectrum of that sound source).

Male speech "rolls-off" at around -3db/octave between
100 Hz and 1000 Hz, and at around -6dB/octave between
1000 Hz and 10,000 Hz. Female speech is similar, but
with the first formant frequency being somewhat higher.

A 75-piece orchestra "rolls-off" in a manner rather
similar to that of "male speech" above, as well.

Therefore, the (on average) musical spectrums of
such sources "roll-off" at between -3 dB / octave
below about 1000 Hz (which is equal to "pink noise"),
and -6 dB /octave between about 1000 Hz and 10,000 Hz
(which is equivalent to the "roll-off" of harmonics
as frequency increases of a "square" and "ramp" wave).

J Gill

🔗genewardsmith <genewardsmith@juno.com>

🔗jpehrson2 <jpehrson@rcn.com>

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

/tuning/topicId_31805.html#32313

> --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> > --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > > --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> > >
> > > /tuning/topicId_31805.html#32269
> > >
> > > > Bob:
> > > > Well, technically I meant acoustic power, which is a function
of
> > > both
> > > > amplitude and frequency. It is directly proportional to both
so
> > the
> > > > same power or energy per unit time is represented by lower
> > > amplitudes
> > > > at higher frequencies. This is because energy per unit time
for
> > the
> > > > same amplitude is greater at higher frequencies.
> > >
> > > Hello Bob!
> > >
> > > Well, then, you mean that the higher frequencies would have
more
> > > power if they were at the same amplitude, but in acoustic
sounds
> > the
> > > higher ones are at lesser amplitudes, yes?
> > >
> > > The lower partials are louder... but since they have less
energy
> > per
> > > unit time, wouldn't they "level out" with the higher, lower
> > amplitude
> > > partials??
> > >
> > > Signed, confused...
> > >
> > > Thanks, Bob!
> > >
> > > JP
> >
> > Bob:
> > Well, I think Jeremy is thinking electrically and I'm thinking
> > acoustically. To the best of my rusty memory, I'm fairly
confident
> > (but not absolutely so) that ACOUSTIC power is directly
proportional
> > to both amplitude and frequency. I believe the relationship of
> > amplitude and power is not the same acoustically as electrically
> > because of the way they are defined.
> >
> > The bottom line for our discussion of the relative acoustic power
of
> > lower vs. higher harmonics is that despite this, more acoustic
energy
> > is generally concentrated individually in one or more of the
first
> > few lower harmonics than in any individual harmonics higher than
the
> > first four. There are notable exceptions to this, as Paul Erlich
has
> > pointed out, citing the human voice on certain vowels as an
example.
> > I pointed out in a response that this is truer in trained voices
than
> > in traditional folk singing, the latter of which is more germaine
to
> > this discussion of the universality of octaves, fifths, and
fourths.
> >
> > I think, also, that statistically the phenomenon is sufficiently
> > significant, especially in the case of the culturally ubiquitous
> > flute, to add quite a bit of weight to the high rate of common
> > harmonics that occur between fundamentals representing the same
> > relationships (octaves, fourths, and fifths) to each other. Taken
> > together, I believe these physical realities eliminate the
mystery
> > concerning why these intervals would prevail in the melodic
systems
> > of most cultures in any era.
>
>
> Bob and Joe,
>
> I've done some digging into Olson's "Music, Physics,
> and Engineering", 2nd Edition, Dover, 1967, and here
> is what I find:
>
> The units of *sound pressure* are those of "force".
> The unit is the "dyne" [acceleration of 1 gram mass
> at 1 centimeter per(second^2)]. "Sound pressure" is
> measured in units of "dynes/(cm)^2" (force in dynes
> per square centimeter). "Sound pressure level" ('SPL').
>
> The units of *power* are in energy transferred per second.
> The unit is the "erg" [the work done when a 1 dyne force
> displaces that 1 gram of mass by 1 centimeter of distance].
>
> Whether a sound pressure wave is a plane or spherical shape,
> the same relationships apply (relative to the issue at hand).
>
> "effective sound pressure" ('ESP') is the root mean square
> ('RMS') value over 1 cycle of that wave [in dynes/(cm)^2].
>
> At a constant distance from the sound source, and
> in order to *maintain a CONSTANT amplitude* (of air
> particle displacement), the "sound pressure" must be
> *increased* in a manner directly proportional to
> the frequency of the sound wave.
>
> At a constant distance from the sound source, and
> in order to *maintain a CONSTANT amplitude* (of air
> particle displacement), the "acoustic power" must be
> *increased* in a manner directly proportional to
> the *square* of the frequency of the sound wave.
>
> The "loudness" of a sound (equal to the magnitude
> of auditory sensation) is defined in units of "Sones"
> (which are related to "Phons" by measured non-linear
> relationships for sinusoidal tones). 1 Phon is equivalent
> to 40 Sones (an 'SPL' of 2 x 10^4 dynes/cm^2).
>
> So, "loudness" is related to "sound pressure level"
> ('SPL'), and not directly related to "acoustic power".
>
> While Olson does not cover the "whys" and "wherefores"
> of the decrease in the amplitude of the SPL of harmonic
> overtones of resonators as the harmonic number increases,
> it makes intuitive sense that (since a doubling of the
> SPL - and a quadrupling of the "acoustic power" - would
> be necessary to *maintain a CONSTANT amplitude* of air
> particle displacement each time the frequency doubles),
> resonators (such as voice, strings, etc.) do not behave
> in this manner. In fact, they tend to exhibit *decreasing*
> SPLs as the harmonic number *increases* over frequency.
>
> From measurements by Sivian, Dunn, and White, Olson
> (on page 205, Figure 6.3) shows curves depicting the
> measured (over a long time period of many cycles)
> magnitudes of "average pressure/cycle" over frequency
> copared to the "average total pressure" (of the entire
> frequency spectrum of that sound source).
>
> Male speech "rolls-off" at around -3db/octave between
> 100 Hz and 1000 Hz, and at around -6dB/octave between
> 1000 Hz and 10,000 Hz. Female speech is similar, but
> with the first formant frequency being somewhat higher.
>
> A 75-piece orchestra "rolls-off" in a manner rather
> similar to that of "male speech" above, as well.
>
> Therefore, the (on average) musical spectrums of
> such sources "roll-off" at between -3 dB / octave
> below about 1000 Hz (which is equal to "pink noise"),
> and -6 dB /octave between about 1000 Hz and 10,000 Hz
> (which is equivalent to the "roll-off" of harmonics
> as frequency increases of a "square" and "ramp" wave).
>
>
> J Gill

Hi J!

So, if I'm reading this correctly, the higher partials *do* have a
greater "sound pressure" or "acoustic power" to maintain a given
amplitude, as Robert Wendell originally suggested... ??

J. Pehrson

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> /tuning/topicId_31805.html#32313
>
>
> > --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> > > --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > > > --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> > > >
> > > > /tuning/topicId_31805.html#32269
> > > >
> > > > > Bob:
> > > > > Well, technically I meant acoustic power, which is a function
> of
> > > > both
> > > > > amplitude and frequency. It is directly proportional to both
> so
> > > the
> > > > > same power or energy per unit time is represented by lower
> > > > amplitudes
> > > > > at higher frequencies. This is because energy per unit time
> for
> > > the
> > > > > same amplitude is greater at higher frequencies.
> > > >
> > > > Hello Bob!
> > > >
> > > > Well, then, you mean that the higher frequencies would have
> more
> > > > power if they were at the same amplitude, but in acoustic
> sounds
> > > the
> > > > higher ones are at lesser amplitudes, yes?
> > > >
> > > > The lower partials are louder... but since they have less
> energy
> > > per
> > > > unit time, wouldn't they "level out" with the higher, lower
> > > amplitude
> > > > partials??
> > > >
> > > > Signed, confused...
> > > >
> > > > Thanks, Bob!
> > > >
> > > > JP
> > >
> > > Bob:
> > > Well, I think Jeremy is thinking electrically and I'm thinking
> > > acoustically. To the best of my rusty memory, I'm fairly
> confident
> > > (but not absolutely so) that ACOUSTIC power is directly
> proportional
> > > to both amplitude and frequency. I believe the relationship of
> > > amplitude and power is not the same acoustically as electrically
> > > because of the way they are defined.
> > >
> > > The bottom line for our discussion of the relative acoustic power
> of
> > > lower vs. higher harmonics is that despite this, more acoustic
> energy
> > > is generally concentrated individually in one or more of the
> first
> > > few lower harmonics than in any individual harmonics higher than
> the
> > > first four. There are notable exceptions to this, as Paul Erlich
> has
> > > pointed out, citing the human voice on certain vowels as an
> example.
> > > I pointed out in a response that this is truer in trained voices
> than
> > > in traditional folk singing, the latter of which is more germaine
> to
> > > this discussion of the universality of octaves, fifths, and
> fourths.
> > >
> > > I think, also, that statistically the phenomenon is sufficiently
> > > significant, especially in the case of the culturally ubiquitous
> > > flute, to add quite a bit of weight to the high rate of common
> > > harmonics that occur between fundamentals representing the same
> > > relationships (octaves, fourths, and fifths) to each other. Taken
> > > together, I believe these physical realities eliminate the
> mystery
> > > concerning why these intervals would prevail in the melodic
> systems
> > > of most cultures in any era.
> >
> >
> > Bob and Joe,
> >
> > I've done some digging into Olson's "Music, Physics,
> > and Engineering", 2nd Edition, Dover, 1967, and here
> > is what I find:

[[[ The units of *sound pressure* are those of "force" per
[[[ unit area. The unit is the "dyne" [acceleration of
[[[ 1 gram of mass at 1 centimeter per(second^2)].
[[[ "Sound pressure" is measured in units of "dynes/(cm)^2"
[[[ (force in dynes per square centimeter).
[[[ "Sound pressure level" is often abbreviated as 'SPL'.

> > The units of *power* are in energy transferred per second.

[[[ The units are in "ergs" of energy transferred per second
[[[ per square centimeter. The "erg" itself is a unit of work
[[[ equal to the work done when a 1 dyne force displaces a mass
[[[ of 1 gram by 1 centimeter of distance]

> > Whether a sound pressure wave is a plane or spherical shape,
> > the same relationships apply (relative to the issue at hand).
> >
[[[ "effective sound pressure" is the root mean square
> > ('RMS') value over 1 cycle of that wave [in dynes/(cm)^2].
> >
> > At a constant distance from the sound source, and
> > in order to *maintain a CONSTANT amplitude* (of air
> > particle displacement), the "sound pressure" must be
> > *increased* in a manner directly proportional to
> > the frequency of the sound wave.
> >
> > At a constant distance from the sound source, and
> > in order to *maintain a CONSTANT amplitude* (of air
> > particle displacement), the "acoustic power" must be
> > *increased* in a manner directly proportional to
> > the *square* of the frequency of the sound wave.
> >
> > The "loudness" of a sound (equal to the magnitude
> > of auditory sensation) is defined in units of "Sones"
> > (which are related to "Phons" by measured non-linear
[[[ relationships for sinusoidal tones). 1 Sone is equivalent
[[[ to 40 Phons (an 'SPL' of 2 x 10^4 dynes/cm^2).
> >
> > So, "loudness" is related to "sound pressure level"
> > ('SPL'), and not directly related to "acoustic power".
> >
> > While Olson does not cover the "whys" and "wherefores"
> > of the decrease in the amplitude of the SPL of harmonic
> > overtones of resonators as the harmonic number increases,
> > it makes intuitive sense that (since a doubling of the
> > SPL - and a quadrupling of the "acoustic power" - would
> > be necessary to *maintain a CONSTANT amplitude* of air
> > particle displacement each time the frequency doubles),
> > resonators (such as voice, strings, etc.) do not behave
> > in this manner. In fact, they tend to exhibit *decreasing*
> > SPLs as the harmonic number *increases* over frequency.
> >
> > From measurements by Sivian, Dunn, and White, Olson
> > (on page 205, Figure 6.3) shows curves depicting the
> > measured (over a long time period of many cycles)
> > magnitudes of "average pressure/cycle" over frequency
> > copared to the "average total pressure" (of the entire
> > frequency spectrum of that sound source).
> >
> > Male speech "rolls-off" at around -3db/octave between
> > 100 Hz and 1000 Hz, and at around -6dB/octave between
> > 1000 Hz and 10,000 Hz. Female speech is similar, but
> > with the first formant frequency being somewhat higher.
> >
> > A 75-piece orchestra "rolls-off" in a manner rather
> > similar to that of "male speech" above, as well.
> >
> > Therefore, the (on average) musical spectrums of
> > such sources "roll-off" at between -3 dB / octave
> > below about 1000 Hz (which is equal to "pink noise"),
> > and -6 dB /octave between about 1000 Hz and 10,000 Hz
> > (which is equivalent to the "roll-off" of harmonics
> > as frequency increases of a "square" and "ramp" wave).
> >
> >
> > J Gill
>
> Hi J!
>
> So, if I'm reading this correctly, the higher partials *do* have a
> greater "sound pressure" or "acoustic power" to maintain a given
> amplitude, as Robert Wendell originally suggested... ??
>
> J. Pehrson

J Gill: Joe, I had a couple errors in my previous text,
which Gene was kind to point out. I have edited my post
in the copy above (you will find '[[[' to the left of the
modified text). Hopefully, that will help a bit. What I
mean to say in the above post, and it's implications, are:

(1) "Acoustic power" (which is in units of "ergs" of *energy*
transferred per square centimeter per second ) is *not* the same
thing as "loudness" (which is a measure of *sound pressure*,
and is in units of "dynes" of *force* per square centimeter).

(2) For a "flat" spectrum (equal in value at all frequencies)
of "sound pressure level" (which is closest to what we actually
"hear"), the "acoustic power" of (either a plane or a spherical)
sound wave will *decrease* in a manner proportional to the value
of frequency considered (at -6 dB / octave). Therefore, the
measured "acoustic power" is the *integral* of the measured
"sound pressure level", and is, then, fundamentally different.

(3) The "sound pressure level" spectrum which a microphone
picks up and we might display on a spectrum analyzer produced
by musical resonators with harmonically related overtones
[such as the averaged human voice, or a 75-piece orchestra
where, indeed, some of the sources have non-harmonic energy,
themselves], *decreases* with increasing harmonic number above
the lowest fundamental frequency of that/those source(s).
Male speech "rolls-off" at around -3db/octave between
100 Hz and 1000 Hz, and at around -6dB/octave between
1000 Hz and 10,000 Hz. Female speech is similar, but
with the first formant frequency being somewhat higher.
A 75-piece orchestra "rolls-off" in a manner rather
similar to that of "male speech" above, as well.
By "roll-off", I mean, "decreasing amplitude of sound pressure
level" as frequency (and as harmonic number) increases.

(5) "Loudness" itself (the actual units of the perceived
"intensity" of a tone) is, additionally, related in a
non-linear manner to *sound pressure*. At values of less
than 40 Phons of sound pressure (2 x 10^4 dynes/cm^2 at
at 1000 Hz), our "ears" act like a "downward expander",
and at values greater than 40 Phons of sound pressure,
our "ears" act like a "compressor". This effect varies
with frequency both above as well as below 1000 Hz.

Sorry about the previous typos! I think I got them all.

Best Regards, J Gill

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> > > Bob quoting from Terhardt:
> > > "Pitch commonality of harmonic complex tones is quite pronounced
> > for
> > > a 1:2 ratio of oscillation frequencies, i.e., when the tones are
> in
> > > an octave interval. Pitch commonality is less pronounced for
> tones
> > in
> > > fifth and fourth intervals. And it is more or less negligible for
> > any
> > > of the other intervals. This elucidates and explains the
> particular
> > > role played by the octave and the fifth, at least where sensory
> > > affinity is concerned."
> > >
> > > Bob reponds to this quote and Paul's comment below:
> > > Yes, I agree that pitch commonality of the harmonics is crucial
> to
> > > the understanding of pitch affinity. I would think, however, that
> > the
> > > statistical predominance of acoustic energy in the lower
> harmonics
> > is
> > > a contributor that is not at all neligible in its importance and
> in
> > > noticing the presence of these intervals withing a single tone.
> > >
> > > Any good ear will notice the octave overtone of many kinds of
> flute
> > > timbres and the fifth is also quite prominent in some.
> >
> > Any good ear? You and I, yes, as we've trained our ears to hear
> > overtones. Most of the musicians I know with really good ears?
> > Totally oblivious to overtones.
> > >
> > > I don't quite understand why he says "more or less negligible for
> > any
> > > of the other intervals", though, unless he refers only to the
> > NUMBER
> > > of common pitches RELATIVE to those in the octaves, fifths, and
> > > fourths, which case he is undoubtedly right on the mark.
> >
> > You may have to read a few of the other articles on his website
> too.
> > His understanding of pitch is a bit more complex, but once you
> grasp
> > it, it's very rewarding in its power.
> >
> > > We couldn't
> > > accurately tune these pitches by timing beats in the setting of
> > > temperaments if the common harmonics were essentially negligible
> in
> > > any other sense.
> >
> > We're talking about the "affinity" or "similarity" relationship
> here,
> > not the "consonance" relationship.
>
>
> Nothing in my last statement above implies that I'm switching the
> subject to consonance. I'm just saying the common harmonics have to
> be there in those intervals or we couldn't tune them. How does this
> imply such a switch to you?
>
> Puzzled,
>
> Bob

J Gill: I think I see the significance of differentiating
the absence of "roughness" (from "beats") and "dissonance"
(from Plomp and Levelt tonal consonance), allright, but at:

http://www.mmk.e-technik.tu-muenchen.de/persons/ter/top/affinity.html

Terhardt states:

<< Affinity means that tones may be perceived as similar in certain aspects; that in some respect a tone may be replaced by another one; and that one tone may even be confused with another one. These criteria apply practically only to two musical intervals, namely, the octave, and, to a lesser degree, the fifth. >>

While beats seem a separate issue, allright, the "affinity"
concept of Terhardt's *does* relate to the two intervals
invariably (also) described as the most *consonant* by
(it seems) "dissonance curves", and other propositions as
to the (generalized) "compatability" of simultaneous tones.

It seems that a "similarity relationship" between two or
more tones does not preclude those tones from also being
described (in various general terms) as "compatible".

Without knowing the "end of such a journey" [being the
successful "atomization" of the *total* effect into (this
or that) conceptually described "phenomenon"], what real
psychoacoustic assurances can we present ourselves with
which would establish that an "affinity/similarity"
relationship between two or more tones (sinusoidal, or
complex with harmonics) *is* a phenomenon separable from
other (humanly perceived) characteristics of such tonal
combinations which result in (humanly judged) descriptors
of tonal "consonance", "compatability", or "coincidence"?

Of "sensory affinity", Terhardt states:

<< sensory affinity, emerges from the fact that the pitch of any complex tone is multiple (see topic definition of pitch). As a consequence of the multiplicity of pitch, and of the particular intervals that exist between the simultaneous pitches of any single harmonic complex tone, there occurs commonality of pitches when the oscillation frequencies of two successive harmonic complex tones are in a 1:2 or 2:3 ratio. Commonality means that one of the tones has some pitches in common with the other, and this immediately accounts for an enhanced similarity of the tones as compared to other frequency ratios, i.e., for sensory affinity. >>

The above sure appears (to me) to describe a (single) phenomenon
related to "concidence" of partials of complex (harmonic) tones.

Of an "auditory sense", Terhardt states:

<< Octave- and fifth-matching can be done with practically any type of tone, in particular, both with harmonic complex tones and with sine tones. Any listener who can do that experiment with consistent results, thereby reveals that he/she possesses an internal auditory template for the pitch interval that corresponds to an octave and fifth, respectively. >>

The sinusoidal case of the above described effect would seem
to preclude "coincidence" of "partials" (none being present),
*except* for the real possibilty of some (albeit possibly small)
"perceived overtones" (at the 2nd, 3rd, 4th, and 5th harmonics
of a given sinusoidal tone) at SPLs greater than 40 Phons. If
that is the case, "coincidence" of "partials" *may* be a factor.

The non-sinusoidal case of the above described effect would
*not* seem to preclude "coincidence" of "partials", and *may*
be enhanced by the presence of additional harmonic energies.

I would bet that what goes on "inside our heads" does *not*
amount to an astute mathematician watching a frequency-counter
and noting the possibility of expressing the instrument's
reading in terms of rational (thus separable) numerators and
denominators, keenly noting the "prime" status of some.
The ear is not relaxing for a "good evening" in mathematical
conceptual visualization, neurons interconnected in 3rds and
5ths, with "taxicab distances" flattering our sensibilities.

Only the sinusoidal case of Terhardt's "auditory sense"
(described above) would preclude the involvement of the
"coincidence" of the harmonic overtones of complex tones
(and the measured perception of "perceptual partials"
occuring over the upper 60 dB of the (maybe, under very
ideal conditions) 100 dB dynamic range of the human ear.

As (no doubt) many know (in some vernacular):

If two separately sounded harmonic resonators are sounded
with either: (1) a small difference in pitch; and/or
(2) an implicitly non-controllable random phase relationship -
effects from the resultant slowly varying amplitude envelopes
may be troubling, or perhaps (in some cases) judged pleasant.
If the differences in the frequency of the pitches widens,
we might say that it "beats". Further still, and it is "rough".

Past that - for conceptual models which construct relativly
simple spectrums comprised of a few complex (harmonic) tones,
it seems (to me) that one is hard-pressed to find structural
characteristics of the sound waves being described *other than*
"coincident" partials between such complex tones whose pitch
relationships are described by ratios of small valued integers.

Note that (due to the -3 dB to -6 dB per octave rollof rates
of the sound pressure levels of the harmonic overtones of
many musical sound sources), such "coincidences" are, by
the nature of such "roll-off", strongly favored to occur
primarily at the lower (2nd, 3rd, 4th, and, 5th) harmonic
multiples of the fundamental frequency of such a resonator.
Or (put more critically), limited in potential for activity
in all but lower harmonic overtones of such complex tones.

The above is *not* to say that real-world sound sources [with
non-harmonic (as well as harmonically related) overtones, as
well as "continuous spectral energy" introduced into the spectrum
as a result of the amplitude "envelope" of musical notes over time]
present anything so spectrally "simple" as such a model(s).

It is to note that it seems *not* to be the numbers involved
in the ratios which describe the interval between two pitches
sounded (whether they be fundamentals or overtones of such),
but the (expected, and describable) effects of the acoustical
summation of nearby frequencies present into a coherent whole.

Does it not seem that a "harmonic theory of tonal combinations"
would appear to be applicable in the case of *harmonic* tones,
but might well be *inapplicable* to combinations of small numbers
of *sinusoidal* tones? Can the case be made that combinations of
small numbers of *sinusoidal* tones can predict the effects of
combinations of similarly tuned *complex* tones (with harmonics)?

Curiously, J Gill

🔗paulerlich <paul@stretch-music.com>

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> > --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > > --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
>
> > > > We couldn't
> > > > accurately tune these pitches by timing beats in the setting of
> > > > temperaments if the common harmonics were essentially
> negligible
> > in
> > > > any other sense.
> > >
> > > We're talking about the "affinity" or "similarity" relationship
> > here,
> > > not the "consonance" relationship.
> >
> >
> > Nothing in my last statement above implies that I'm switching the
> > subject to consonance.
>
> Well, you're switching the subject to "tunability", which is even
> farther from "affinity" than "consonance" is.
>
> > I'm just saying the common harmonics have to
> > be there in those intervals or we couldn't tune them. How does this
> > imply such a switch to you?
>
> Affinity has little to do with common harmonics. It works fine with
> sine waves, for example. Please read the Terhardt stuff again.

J Gill:

http://www.mmk.e-technik.tu-muenchen.de/persons/ter/top/affinity.html

Terhardt states:

<< Affinity means that tones may be perceived as similar in certain aspects;
that in some respect a tone may be replaced by another one; and that one tone
may even be confused with another one. These criteria apply practically only to
two musical intervals, namely, the octave, and, to a lesser degree, the fifth.
>>

It seems that a "similarity relationship" between two or
more tones does not preclude those tones from also being
described (in various general terms) as "compatible".

Of "sensory affinity", Terhardt states:

<< sensory affinity, emerges from the fact that the pitch of any complex tone
is multiple (see topic definition of pitch). As a consequence of the
multiplicity of pitch, and of the particular intervals that exist between the
simultaneous pitches of any single harmonic complex tone, there occurs
commonality of pitches when the oscillation frequencies of two successive
harmonic complex tones are in a 1:2 or 2:3 ratio. Commonality means that one of
the tones has some pitches in common with the other, and this immediately
accounts for an enhanced similarity of the tones as compared to other frequency
ratios, i.e., for sensory affinity. >>

The above sure appears (to me) to describe a (single) phenomenon
related to "concidence" of partials of complex (harmonic) tones.

Of an "auditory sense", Terhardt states:

<< Octave- and fifth-matching can be done with practically any type of tone, in
particular, both with harmonic complex tones and with sine tones. Any listener
who can do that experiment with consistent results, thereby reveals that he/she
possesses an internal auditory template for the pitch interval that corresponds
to an octave and fifth, respectively. >>

The non-sinusoidal case of the above described effect would
*not* seem to preclude "coincidence" of "partials", and *may*
be enhanced by the presence of additional harmonic energies.

🔗paulerlich <paul@stretch-music.com>

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

> Of "sensory affinity", Terhardt states:
>
> << sensory affinity, emerges from the fact that the pitch of any
complex tone is multiple (see topic definition of pitch). As a
consequence of the multiplicity of pitch, and of the particular
intervals that exist between the simultaneous pitches of any single
harmonic complex tone, there occurs commonality of pitches when the
oscillation frequencies of two successive harmonic complex tones are
in a 1:2 or 2:3 ratio. Commonality means that one of the tones has
some pitches in common with the other, and this immediately accounts
for an enhanced similarity of the tones as compared to other
frequency ratios, i.e., for sensory affinity. >>
>
>
> The above sure appears (to me) to describe a (single) phenomenon
> related to "concidence" of partials of complex (harmonic) tones.
>

Not quite that. Do you understand what he means by "multiplicity of
pitch"?

>
> Of an "auditory sense", Terhardt states:
>
> << Octave- and fifth-matching can be done with practically any type
of tone, in particular, both with harmonic complex tones and with
sine tones. Any listener who can do that experiment with consistent
results, thereby reveals that he/she possesses an internal auditory
template for the pitch interval that corresponds to an octave and
fifth, respectively. >>
>
> The sinusoidal case of the above described effect would seem
> to preclude "coincidence" of "partials" (none being present),
> *except* for the real possibilty of some (albeit possibly small)
> "perceived overtones" (at the 2nd, 3rd, 4th, and 5th harmonics
> of a given sinusoidal tone) at SPLs greater than 40 Phons. If
> that is the case, "coincidence" of "partials" *may* be a factor.

But he's talking about multiplicity of pitch, above, beyond, and
containing within it coincidence of partials or lack thereof.
>
> I would bet that what goes on "inside our heads" does *not*
> amount to an astute mathematician watching a frequency-counter
> and noting the possibility of expressing the instrument's
> reading in terms of rational (thus separable) numerators and
> denominators, keenly noting the "prime" status of some.

Agreed!

> The ear is not relaxing for a "good evening" in mathematical
> conceptual visualization, neurons interconnected in 3rds and
> 5ths, with "taxicab distances" flattering our sensibilities.

No . . . but, it's really cool that you can get some taxicab-distance
metrics from very few assumptions about what the ear does.

> As (no doubt) many know (in some vernacular):
>
> If two separately sounded harmonic resonators are sounded
> with either: (1) a small difference in pitch; and/or
> (2) an implicitly non-controllable random phase relationship -
> effects from the resultant slowly varying amplitude envelopes
> may be troubling, or perhaps (in some cases) judged pleasant.
> If the differences in the frequency of the pitches widens,
> we might say that it "beats". Further still, and it is "rough".
>
>
> Past that - for conceptual models which construct relativly
> simple spectrums comprised of a few complex (harmonic) tones,
> it seems (to me) that one is hard-pressed to find structural
> characteristics of the sound waves being described *other than*
> "coincident" partials between such complex tones whose pitch
> relationships are described by ratios of small valued integers.

There are at least two phenomena in between the characteristics of
the sound waves described and the apprehension of the musical
sensation they evoke that must be taken into account:

(1) the nonlinearity of the ear (with which you're quite familiar);

(2) the action of the central pitch processor, which seeks out
harmonic relationships between partials in order to assign one or
more (virtual) _pitches_ to the mass of frequencies it is receiving.
This action is of paramount importance for understanding what
Terhardt means by "multiplicity of pitch", and is also central to the
harmonic entropy concept.
>
> Note that (due to the -3 dB to -6 dB per octave rollof rates
> of the sound pressure levels of the harmonic overtones of
> many musical sound sources), such "coincidences" are, by
> the nature of such "roll-off", strongly favored to occur
> primarily at the lower (2nd, 3rd, 4th, and, 5th) harmonic
> multiples of the fundamental frequency of such a resonator.
> Or (put more critically), limited in potential for activity
> in all but lower harmonic overtones of such complex tones.

With low-volume sawtooth waves, I can tune a 17:13 interval (or any
simpler one) by ear, through coincidences of overtones. That isn't
necessarily a musically relevant fact, however.

> "continuous spectral energy" introduced into the spectrum
> as a result of the amplitude "envelope" of musical notes over time

I think this is called "bandwidth".
>
> It is to note that it seems *not* to be the numbers involved
> in the ratios which describe the interval between two pitches
> sounded (whether they be fundamentals or overtones of such),
> but the (expected, and describable) effects of the acoustical
> summation of nearby frequencies present into a coherent whole.

The latter often implies the former.

> Does it not seem that a "harmonic theory of tonal combinations"
> would appear to be applicable in the case of *harmonic* tones,
> but might well be *inapplicable* to combinations of small numbers
> of *sinusoidal* tones?

> Can the case be made that combinations of
> small numbers of *sinusoidal* tones can predict the effects of
> combinations of similarly tuned *complex* tones (with harmonics)?

The two situations could be made absolutely identical and
indistinguishable.

🔗paulerlich <paul@stretch-music.com>

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> > As (no doubt) many know (in some vernacular):
> >
> > If two separately sounded harmonic resonators are sounded
> > with either: (1) a small difference in pitch; and/or
> > (2) an implicitly non-controllable random phase relationship -
> > effects from the resultant slowly varying amplitude envelopes
> > may be troubling, or perhaps (in some cases) judged pleasant.
> > If the differences in the frequency of the pitches widens,
> > we might say that it "beats". Further still, and it is "rough".
> >
> >
> > Past that - for conceptual models which construct relativly
> > simple spectrums comprised of a few complex (harmonic) tones,
> > it seems (to me) that one is hard-pressed to find structural
> > characteristics of the sound waves being described *other than*
> > "coincident" partials between such complex tones whose pitch
> > relationships are described by ratios of small valued integers.
>
> There are at least two phenomena in between the characteristics of
> the sound waves described and the apprehension of the musical
> sensation they evoke that must be taken into account:
>
> (1) the nonlinearity of the ear (with which you're quite familiar);
>
> (2) the action of the central pitch processor, which seeks out
> harmonic relationships between partials in order to assign one or
> more (virtual) _pitches_ to the mass of frequencies it is receiving.
> This action is of paramount importance for understanding what
> Terhardt means by "multiplicity of pitch", and is also central to the
> harmonic entropy concept.
> >
> > Note that (due to the -3 dB to -6 dB per octave rollof rates
> > of the sound pressure levels of the harmonic overtones of
> > many musical sound sources), such "coincidences" are, by
> > the nature of such "roll-off", strongly favored to occur
> > primarily at the lower (2nd, 3rd, 4th, and, 5th) harmonic
> > multiples of the fundamental frequency of such a resonator.
> > Or (put more critically), limited in potential for activity
> > in all but lower harmonic overtones of such complex tones.
>
> With low-volume sawtooth waves, I can tune a 17:13 interval (or any
> simpler one) by ear, through coincidences of overtones. That isn't
> necessarily a musically relevant fact, however.

JG: Why do you think that:

(1) you utilized "coincidences of overtones" in order to do
this [since this would involve hearing a 17th harmonic of one
of the two (sawtooth) tones]; and

(2) Such (if that was the case) "isn't
necessarily a musically relevant fact"?

> > "continuous spectral energy" introduced into the spectrum
> > as a result of the amplitude "envelope" of musical notes over time
>
> I think this is called "bandwidth".

JG: "Bandwidth" is a very general (and vaguely used) term
which simply implies a domain of frequency between some
minimum and some maximum numerical value of frequency.
I would call it something like "continuous frequency
spectral energy introduced by spectral modifications
introduced by the time-domain windowing of audio signals".

> > It is to note that it seems *not* to be the numbers involved
> > in the ratios which describe the interval between two pitches
> > sounded (whether they be fundamentals or overtones of such),
> > but the (expected, and describable) effects of the acoustical
> > summation of nearby frequencies present into a coherent whole.
>
> The latter often implies the former.

JG: If only such implications amounted to explanations...
>
> > Does it not seem that a "harmonic theory of tonal combinations"
> > would appear to be applicable in the case of *harmonic* tones,
> > but might well be *inapplicable* to combinations of small numbers
> > of *sinusoidal* tones?
>
> > Can the case be made that combinations of
> > small numbers of *sinusoidal* tones can predict the effects of
> > combinations of similarly tuned *complex* tones (with harmonics)?
>
> The two situations could be made absolutely identical and
> indistinguishable.

JG: That's a rather tall order to fill, it seems. I guess
that you would necessarily assume that if it were to be the
case that "harmonic entropy" is independent of waveform ...

J Gill

🔗paulerlich <paul@stretch-music.com>

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

> > With low-volume sawtooth waves, I can tune a 17:13 interval (or
any
> > simpler one) by ear, through coincidences of overtones. That
isn't
> > necessarily a musically relevant fact, however.
>
> JG: Why do you think that:
>
> (1) you utilized "coincidences of overtones" in order to do
> this [since this would involve hearing a 17th harmonic of one
> of the two (sawtooth) tones];

I adjusted the size of the interval, little by little (without
looking), until I heard a certain high-pitched beating come to a stop
(well, as close as I could manage) -- so it _had_ to be the
coinciding overtones. Look at the dB decay of sawtooth waves -- the
17th harmonic is only about 12 dB (I think) quieter than the
fundamental.

> and
>
> (2) Such (if that was the case) "isn't
> necessarily a musically relevant fact"?

Well, let me turn that around and ask you why you think that _would_
be a musically relevant fact? If you think it is, and you think that
such coincidences are the only musically relevant facts, I'm more
than happy to work out the consequences of these assumptions with
you . . . but I'll also be able to show you that they don't explain
everything.

> > > "continuous spectral energy" introduced into the spectrum
> > > as a result of the amplitude "envelope" of musical notes over
time
> >
> > I think this is called "bandwidth".
>
> JG: "Bandwidth" is a very general (and vaguely used) term
> which simply implies a domain of frequency between some
> minimum and some maximum numerical value of frequency.
> I would call it something like "continuous frequency
> spectral energy introduced by spectral modifications
> introduced by the time-domain windowing of audio signals".

Well, this sort of assumes that our ears do one big Fourier transform
on the whole "signal", after the "end" of a signal is detected, which
is of course not how our ears work.

> > > It is to note that it seems *not* to be the numbers involved
> > > in the ratios which describe the interval between two pitches
> > > sounded (whether they be fundamentals or overtones of such),
> > > but the (expected, and describable) effects of the acoustical
> > > summation of nearby frequencies present into a coherent whole.
> >
> > The latter often implies the former.
>
> JG: If only such implications amounted to explanations...

The simpler the ratio between two frequencies (assuming it's simpler
than, say, 11:9), the more easily the two pitches will be heard to
possess _commonality_, since, as we've seen, the pitches evoked by a
frequency are all in simple harmonic/subharmonic relationships with
one another. That's because the whole virtual pitch mechanism seems
to be _designed_ to deal with harmonic overtone series, perhaps an
evolutionary advantage, related to the fact that human (and other)
voices have harmonic overtones, or perhaps something acquired
prenatally via exposure to the mother's voice . . . in any case, the
reality and properties of the virtual pitch phenomenon have been
demonstrated in experiment after experiment, and remains a thriving
area of research today, as some list members can attest . . .

BTW, it might help our discussion if we kept the terms
straight . . . "pitch" is not the same thing as "spectral component",
and you meant ""spectral components" rather than "pitches" above.

> > > Does it not seem that a "harmonic theory of tonal combinations"
> > > would appear to be applicable in the case of *harmonic* tones,
> > > but might well be *inapplicable* to combinations of small
numbers
> > > of *sinusoidal* tones?
> >
> > > Can the case be made that combinations of
> > > small numbers of *sinusoidal* tones can predict the effects of
> > > combinations of similarly tuned *complex* tones (with
harmonics)?
> >
> > The two situations could be made absolutely identical and
> > indistinguishable.
>
> JG: That's a rather tall order to fill, it seems. I guess
> that you would necessarily assume that if it were to be the
> case that "harmonic entropy" is independent of waveform ...

It has nothing do do with harmonic entropy. I'm merely referring to
the mathematical equivalence of a description in terms of complex
tones and a description in terms of sine waves. Certainly I'm not
claiming that these effects are timbre-independent!

🔗paulerlich <paul@stretch-music.com>

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> > > With low-volume sawtooth waves, I can tune a 17:13 interval (or
> any
> > > simpler one) by ear, through coincidences of overtones. That
> isn't
> > > necessarily a musically relevant fact, however.
> >
> > JG: Why do you think that:
> >
> > (1) you utilized "coincidences of overtones" in order to do
> > this [since this would involve hearing a 17th harmonic of one
> > of the two (sawtooth) tones];
>
> I adjusted the size of the interval, little by little (without
> looking), until I heard a certain high-pitched beating come to
> a stop
> (well, as close as I could manage) -- so it _had_ to be the
> coinciding overtones. Look at the dB decay of sawtooth waves -- the
> 17th harmonic is only about 12 dB (I think) quieter than the
> fundamental.

JG: In /tuning/topicId_31805.html#32046
it appears that you use "sawtooth" to describe a repeating
"ramp" waveform (as opposed to describing a "triangle" wave.
In that post (where you pointed out my error regarding the
rate of "roll-off" with increasing frequency in the amplitude
of the partials of a "triangle" wave), we seem to agree that
the "sawtooth" waveform "rolls-off" at rate of -6 dB / octave.

With decibels (scaled to describe current, voltage, or sound
pressure levels, as opposed to sound *power* levels), I get

Amplitude of 17th harmonic = 20*(LOG_10 (1/17)) = -24.61 dB

The identity 10*(LOG_10 (1/17)) would be appropriate only
if the *power* (equal to the square of SPL, current, voltage,
etc) of spectral components (the harmonics) decreased in a
manner inversely proportional to the change in frequency.
>
> > and
> >
> > (2) Such (if that was the case) "isn't
> > necessarily a musically relevant fact"?
>
> Well, let me turn that around and ask you why you think that >_would_
> be a musically relevant fact? If you think it is, and you think >that
> such coincidences are the only musically relevant facts, I'm more
> than happy to work out the consequences of these assumptions with
> you

JG: Well, no, I do not see these "coincidences" as necessarily
constituting "the only musically relevant facts" at all, but I
*am* exploring with curiosity a number of other descriptors ...

>. . . but I'll also be able to show you that they don't explain
> everything.

JG: Cogent facts are always welcome.

> > > > "continuous spectral energy" introduced into the spectrum
> > > > as a result of the amplitude "envelope" of musical notes over
> time
> > >
> > > I think this is called "bandwidth".
> >
> > JG: "Bandwidth" is a very general (and vaguely used) term
> > which simply implies a domain of frequency between some
> > minimum and some maximum numerical value of frequency.
> > I would call it something like "continuous frequency
> > spectral energy introduced by spectral modifications
> > introduced by the time-domain windowing of audio signals".
>
> Well, this sort of assumes that our ears do one big Fourier >transform
> on the whole "signal", after the "end" of a signal is detected, >which
> is of course not how our ears work.

JG: I've looked a bit at Terhardt's texts relating to the
Laplace/Fourier stuff, and his concepts of applying the
Fourier integral more appropriately for "shorter-term"
algorithms approaching a (sort of) "real-time" status.

Since you seem to have covered Terhardt's material so
comprehensively - what texts on his website do you feel
best address your ideas of "how our ears work" (if that
may be where some of these ideas are found)?

> > > > It is to note that it seems *not* to be the numbers involved
> > > > in the ratios which describe the interval between two pitches
> > > > sounded (whether they be fundamentals or overtones of such),
> > > > but the (expected, and describable) effects of the acoustical
> > > > summation of nearby frequencies present into a coherent whole.
> > >
> > > The latter often implies the former.
> >
> > JG: If only such implications amounted to explanations...
>
> The simpler the ratio between two frequencies (assuming it's >simpler
> than, say, 11:9), the more easily the two pitches will be heard to
> possess _commonality_, since, as we've seen, the pitches evoked by >a
> frequency are all in simple harmonic/subharmonic relationships with
> one another.

JG: So you *do* consider "subharmonic" relationships to be
significant in the perception of a "harmonic" spectra? You
had seemed (to me) to restrict its significance in prior
communication to applications involving (3 or more tone)
"utonal" chords. Would you then look at a harmonic series
from the perspective of *other* spectral components than
the fundamental "pitch" (used here to attempt to address
your sense of a "pitch" arising out of *one* "spectral
component" out of a group of spectral components)?

>That's because the whole virtual pitch mechanism seems
> to be _designed_ to deal with harmonic overtone series, perhaps an
> evolutionary advantage, related to the fact that human (and other)
> voices have harmonic overtones, or perhaps something acquired
> prenatally via exposure to the mother's voice . . . in any case, the
> reality and properties of the virtual pitch phenomenon have been
> demonstrated in experiment after experiment, and remains a thriving
> area of research today, as some list members can attest . . .

JG: Where else (other than Terhardt) have you seen published
references to "virtual pitch"? Any authors come to mind? It
might be instructive if you could post a "bibliography" of
such papers.

> > > > Does it not seem that a "harmonic theory of tonal >>>>combinations"
> > > > would appear to be applicable in the case of *harmonic* tones,
> > > > but might well be *inapplicable* to combinations of small
>>>> numbers
> > > > of *sinusoidal* tones?
> > > > Can the case be made that combinations of
> > > > small numbers of *sinusoidal* tones can predict the effects of
> > > > combinations of similarly tuned *complex* tones (with
>>>>harmonics)?

> > > The two situations could be made absolutely identical and
> > > indistinguishable.

JG: Sounds like "magic" to me! :)

> > JG: That's a rather tall order to fill, it seems. I guess
> > that you would necessarily assume that if it were to be the
> > case that "harmonic entropy" is independent of waveform ...
>
> It has nothing do do with harmonic entropy. I'm merely referring to
> the mathematical equivalence of a description in terms of complex
> tones and a description in terms of sine waves. Certainly I'm not
> claiming that these effects are timbre-independent!

In "harmonic_entropy" message #531, you said:

<< 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. >>

It would *seem* that the question of whether
or not the case can be made that "combinations
of small numbers of *sinusoidal* tones can predict the effects of
combinations of similarly tuned *complex* tones (with harmonics)"
would be highly relevant to whether "harmonic entropy" can be
utilized in the characterization of "harmonic timbres"!...

J Gill

🔗jpehrson2 <jpehrson@rcn.com>

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

/tuning/topicId_31805.html#32328

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > --- In tuning@y..., "unidala" <JGill99@i...> wrote:
> >
> > /tuning/topicId_31805.html#32313
> >
> >
> > > --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> > > > --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > > > > --- In tuning@y..., "robert_wendell" <BobWendell@t...>
wrote:
> > > > >
> > > > > /tuning/topicId_31805.html#32269
> > > > >
> > > > > > Bob:
> > > > > > Well, technically I meant acoustic power, which is a
function
> > of
> > > > > both
> > > > > > amplitude and frequency. It is directly proportional to
both
> > so
> > > > the
> > > > > > same power or energy per unit time is represented by
lower
> > > > > amplitudes
> > > > > > at higher frequencies. This is because energy per unit
time
> > for
> > > > the
> > > > > > same amplitude is greater at higher frequencies.
> > > > >
> > > > > Hello Bob!
> > > > >
> > > > > Well, then, you mean that the higher frequencies would have
> > more
> > > > > power if they were at the same amplitude, but in acoustic
> > sounds
> > > > the
> > > > > higher ones are at lesser amplitudes, yes?
> > > > >
> > > > > The lower partials are louder... but since they have less
> > energy
> > > > per
> > > > > unit time, wouldn't they "level out" with the higher, lower
> > > > amplitude
> > > > > partials??
> > > > >
> > > > > Signed, confused...
> > > > >
> > > > > Thanks, Bob!
> > > > >
> > > > > JP
> > > >
> > > > Bob:
> > > > Well, I think Jeremy is thinking electrically and I'm
thinking
> > > > acoustically. To the best of my rusty memory, I'm fairly
> > confident
> > > > (but not absolutely so) that ACOUSTIC power is directly
> > proportional
> > > > to both amplitude and frequency. I believe the relationship
of
> > > > amplitude and power is not the same acoustically as
electrically
> > > > because of the way they are defined.
> > > >
> > > > The bottom line for our discussion of the relative acoustic
power
> > of
> > > > lower vs. higher harmonics is that despite this, more
acoustic
> > energy
> > > > is generally concentrated individually in one or more of the
> > first
> > > > few lower harmonics than in any individual harmonics higher
than
> > the
> > > > first four. There are notable exceptions to this, as Paul
Erlich
> > has
> > > > pointed out, citing the human voice on certain vowels as an
> > example.
> > > > I pointed out in a response that this is truer in trained
voices
> > than
> > > > in traditional folk singing, the latter of which is more
germaine
> > to
> > > > this discussion of the universality of octaves, fifths, and
> > fourths.
> > > >
> > > > I think, also, that statistically the phenomenon is
sufficiently
> > > > significant, especially in the case of the culturally
ubiquitous
> > > > flute, to add quite a bit of weight to the high rate of
common
> > > > harmonics that occur between fundamentals representing the
same
> > > > relationships (octaves, fourths, and fifths) to each other.
Taken
> > > > together, I believe these physical realities eliminate the
> > mystery
> > > > concerning why these intervals would prevail in the melodic
> > systems
> > > > of most cultures in any era.
> > >
> > >
> > > Bob and Joe,
> > >
> > > I've done some digging into Olson's "Music, Physics,
> > > and Engineering", 2nd Edition, Dover, 1967, and here
> > > is what I find:
>
> [[[ The units of *sound pressure* are those of "force" per
> [[[ unit area. The unit is the "dyne" [acceleration of
> [[[ 1 gram of mass at 1 centimeter per(second^2)].
> [[[ "Sound pressure" is measured in units of "dynes/(cm)^2"
> [[[ (force in dynes per square centimeter).
> [[[ "Sound pressure level" is often abbreviated as 'SPL'.
>
> > > The units of *power* are in energy transferred per second.
>
> [[[ The units are in "ergs" of energy transferred per second
> [[[ per square centimeter. The "erg" itself is a unit of work
> [[[ equal to the work done when a 1 dyne force displaces a mass
> [[[ of 1 gram by 1 centimeter of distance]
>
> > > Whether a sound pressure wave is a plane or spherical shape,
> > > the same relationships apply (relative to the issue at hand).
> > >
> [[[ "effective sound pressure" is the root mean square
> > > ('RMS') value over 1 cycle of that wave [in dynes/(cm)^2].
> > >
> > > At a constant distance from the sound source, and
> > > in order to *maintain a CONSTANT amplitude* (of air
> > > particle displacement), the "sound pressure" must be
> > > *increased* in a manner directly proportional to
> > > the frequency of the sound wave.
> > >
> > > At a constant distance from the sound source, and
> > > in order to *maintain a CONSTANT amplitude* (of air
> > > particle displacement), the "acoustic power" must be
> > > *increased* in a manner directly proportional to
> > > the *square* of the frequency of the sound wave.
> > >
> > > The "loudness" of a sound (equal to the magnitude
> > > of auditory sensation) is defined in units of "Sones"
> > > (which are related to "Phons" by measured non-linear
> [[[ relationships for sinusoidal tones). 1 Sone is equivalent
> [[[ to 40 Phons (an 'SPL' of 2 x 10^4 dynes/cm^2).
> > >
> > > So, "loudness" is related to "sound pressure level"
> > > ('SPL'), and not directly related to "acoustic power".
> > >
> > > While Olson does not cover the "whys" and "wherefores"
> > > of the decrease in the amplitude of the SPL of harmonic
> > > overtones of resonators as the harmonic number increases,
> > > it makes intuitive sense that (since a doubling of the
> > > SPL - and a quadrupling of the "acoustic power" - would
> > > be necessary to *maintain a CONSTANT amplitude* of air
> > > particle displacement each time the frequency doubles),
> > > resonators (such as voice, strings, etc.) do not behave
> > > in this manner. In fact, they tend to exhibit *decreasing*
> > > SPLs as the harmonic number *increases* over frequency.
> > >
> > > From measurements by Sivian, Dunn, and White, Olson
> > > (on page 205, Figure 6.3) shows curves depicting the
> > > measured (over a long time period of many cycles)
> > > magnitudes of "average pressure/cycle" over frequency
> > > copared to the "average total pressure" (of the entire
> > > frequency spectrum of that sound source).
> > >
> > > Male speech "rolls-off" at around -3db/octave between
> > > 100 Hz and 1000 Hz, and at around -6dB/octave between
> > > 1000 Hz and 10,000 Hz. Female speech is similar, but
> > > with the first formant frequency being somewhat higher.
> > >
> > > A 75-piece orchestra "rolls-off" in a manner rather
> > > similar to that of "male speech" above, as well.
> > >
> > > Therefore, the (on average) musical spectrums of
> > > such sources "roll-off" at between -3 dB / octave
> > > below about 1000 Hz (which is equal to "pink noise"),
> > > and -6 dB /octave between about 1000 Hz and 10,000 Hz
> > > (which is equivalent to the "roll-off" of harmonics
> > > as frequency increases of a "square" and "ramp" wave).
> > >
> > >
> > > J Gill
> >
> > Hi J!
> >
> > So, if I'm reading this correctly, the higher partials *do* have
a
> > greater "sound pressure" or "acoustic power" to maintain a given
> > amplitude, as Robert Wendell originally suggested... ??
> >
> > J. Pehrson
>
>
> J Gill: Joe, I had a couple errors in my previous text,
> which Gene was kind to point out. I have edited my post
> in the copy above (you will find '[[[' to the left of the
> modified text). Hopefully, that will help a bit. What I
> mean to say in the above post, and it's implications, are:
>
> (1) "Acoustic power" (which is in units of "ergs" of *energy*
> transferred per square centimeter per second ) is *not* the same
> thing as "loudness" (which is a measure of *sound pressure*,
> and is in units of "dynes" of *force* per square centimeter).
>
> (2) For a "flat" spectrum (equal in value at all frequencies)
> of "sound pressure level" (which is closest to what we actually
> "hear"), the "acoustic power" of (either a plane or a spherical)
> sound wave will *decrease* in a manner proportional to the value
> of frequency considered (at -6 dB / octave). Therefore, the
> measured "acoustic power" is the *integral* of the measured
> "sound pressure level", and is, then, fundamentally different.
>
> (3) The "sound pressure level" spectrum which a microphone
> picks up and we might display on a spectrum analyzer produced
> by musical resonators with harmonically related overtones
> [such as the averaged human voice, or a 75-piece orchestra
> where, indeed, some of the sources have non-harmonic energy,
> themselves], *decreases* with increasing harmonic number above
> the lowest fundamental frequency of that/those source(s).
> Male speech "rolls-off" at around -3db/octave between
> 100 Hz and 1000 Hz, and at around -6dB/octave between
> 1000 Hz and 10,000 Hz. Female speech is similar, but
> with the first formant frequency being somewhat higher.
> A 75-piece orchestra "rolls-off" in a manner rather
> similar to that of "male speech" above, as well.
> By "roll-off", I mean, "decreasing amplitude of sound pressure
> level" as frequency (and as harmonic number) increases.
>
> (5) "Loudness" itself (the actual units of the perceived
> "intensity" of a tone) is, additionally, related in a
> non-linear manner to *sound pressure*. At values of less
> than 40 Phons of sound pressure (2 x 10^4 dynes/cm^2 at
> at 1000 Hz), our "ears" act like a "downward expander",
> and at values greater than 40 Phons of sound pressure,
> our "ears" act like a "compressor". This effect varies
> with frequency both above as well as below 1000 Hz.
>
>
> Sorry about the previous typos! I think I got them all.
>
>
> Best Regards, J Gill

****Thanks a lot, J.G. for the clarifications. Now that we have them
straight, can you please explain to me what these results have to do
with either music or hearing? I'm not picking that up, so any
explanation would be greatly appreciated.

best,

Joe

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> /tuning/topicId_31805.html#32328
>
> > --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > > --- In tuning@y..., "unidala" <JGill99@i...> wrote:
> > >
> > > /tuning/topicId_31805.html#32313
> > >
> > >
> > > > --- In tuning@y..., "robert_wendell" <BobWendell@t...> wrote:
> > > > > --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > > > > > --- In tuning@y..., "robert_wendell" <BobWendell@t...>
> wrote:
> > > > > >
> > > > > > /tuning/topicId_31805.html#32269
> > > > > >
> > > > > > > Bob:
> > > > > > > Well, technically I meant acoustic power, which is a
> function
> > > of
> > > > > > both
> > > > > > > amplitude and frequency. It is directly proportional to
> both
> > > so
> > > > > the
> > > > > > > same power or energy per unit time is represented by
> lower
> > > > > > amplitudes
> > > > > > > at higher frequencies. This is because energy per unit
> time
> > > for
> > > > > the
> > > > > > > same amplitude is greater at higher frequencies.
> > > > > >
> > > > > Bob:
> > > > > Well, I think Jeremy is thinking electrically and I'm
> thinking
> > > > > acoustically. To the best of my rusty memory, I'm fairly
> > > confident
> > > > > (but not absolutely so) that ACOUSTIC power is directly
> > > proportional
> > > > > to both amplitude and frequency. I believe the relationship
> of
> > > > > amplitude and power is not the same acoustically as
> electrically
> > > > > because of the way they are defined.
> > > > >
> > > > > The bottom line for our discussion of the relative acoustic
> power
> > > of
> > > > > lower vs. higher harmonics is that despite this, more
> acoustic
> > > energy
> > > > > is generally concentrated individually in one or more of the
> > > first
> > > > > few lower harmonics than in any individual harmonics higher
> than
> > > the
> > > > > first four> >

J Gill: Joe, my involvement in this thread arose out of
my reading about "acoustic power" as it is related to
"sound pressure level" in my Olson text, and offering
my viewpoint as to the *difference* between the two
ways of measuring the magnitude of the frequency spectrum.

It seems that my findings offered might alter some of
Bob's original contentions (as I see it, anyway ...). :)

So it really has to do with technical details offered, and
(while I do think that Bob is onto something with ideas
regarding interactions of the first few harmonics in
*general*), my posts on this thread relate to technical
clarifications. I've copied my conclusions posted below:

> > J Gill: Joe, I had a couple errors in my previous text,
> > which Gene was kind to point out. I have edited my post
> > in the copy above (you will find '[[[' to the left of the
> > modified text). Hopefully, that will help a bit. What I
> > mean to say in the above post, and it's implications, are:
> >
> > (1) "Acoustic power" (which is in units of "ergs" of *energy*
> > transferred per square centimeter per second ) is *not* the same
> > thing as "loudness" (which is a measure of *sound pressure*,
> > and is in units of "dynes" of *force* per square centimeter).
> >
> > (2) For a "flat" spectrum (equal in value at all frequencies)
> > of "sound pressure level" (which is closest to what we actually
> > "hear"), the "acoustic power" of (either a plane or a spherical)
> > sound wave will *decrease* in a manner proportional to the value
> > of frequency considered (at -6 dB / octave). Therefore, the
> > measured "acoustic power" is the *integral* of the measured
> > "sound pressure level", and is, then, fundamentally different.
> >
> > (3) The "sound pressure level" spectrum which a microphone
> > picks up and we might display on a spectrum analyzer produced
> > by musical resonators with harmonically related overtones
> > [such as the averaged human voice, or a 75-piece orchestra
> > where, indeed, some of the sources have non-harmonic energy,
> > themselves], *decreases* with increasing harmonic number above
> > the lowest fundamental frequency of that/those source(s).
> > Male speech "rolls-off" at around -3db/octave between
> > 100 Hz and 1000 Hz, and at around -6dB/octave between
> > 1000 Hz and 10,000 Hz. Female speech is similar, but
> > with the first formant frequency being somewhat higher.
> > A 75-piece orchestra "rolls-off" in a manner rather
> > similar to that of "male speech" above, as well.
> > By "roll-off", I mean, "decreasing amplitude of sound pressure
> > level" as frequency (and as harmonic number) increases.
> >
> > (5) "Loudness" itself (the actual units of the perceived
> > "intensity" of a tone) is, additionally, related in a
> > non-linear manner to *sound pressure*. At values of less
> > than 40 Phons of sound pressure (2 x 10^4 dynes/cm^2 at
> > at 1000 Hz), our "ears" act like a "downward expander",
> > and at values greater than 40 Phons of sound pressure,
> > our "ears" act like a "compressor". This effect varies
> > with frequency both above as well as below 1000 Hz.
> >
> >
> > Sorry about the previous typos! I think I got them all.
> >
> >
> > Best Regards, J Gill
>
>
> ****Thanks a lot, J.G. for the clarifications. Now that we have them
> straight, can you please explain to me what these results have to do
> with either music or hearing? I'm not picking that up, so any
> explanation would be greatly appreciated.
>
> best,
>
> Joe

🔗robert_wendell <BobWendell@technet-inc.com>

--- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> J Gill: Joe, my involvement in this thread arose out of
> my reading about "acoustic power" as it is related to
> "sound pressure level" in my Olson text, and offering
> my viewpoint as to the *difference* between the two
> ways of measuring the magnitude of the frequency spectrum.
>
> It seems that my findings offered might alter some of
> Bob's original contentions (as I see it, anyway ...). :)
>
.
> >
> > best,
> >
> > Joe

Bob:
I really appreciate your technical clarifications, Jeremy! Although
the relationships of loudness to distance is not germaine to our
original discussion and could serve to confuse some into thinking it
is, the sound pressure level being closer to subjective loudness than
acoustic power is an important distinction about which I was not
clear. Thanks again!

My original thoughts were based on the idea that there is a
statistically high probability that one or more of the first four
harmonics will be typically more prominent to the ear than any
individual harmonics higher than that in harmonic timbres from most
common sources, most notably the culturally almost universal flute in
one variation or other. I used the term acoustic energy to express
this, which turns out to have been a bit off.

However, it seems to me that Jeremy's clarifications have only
reinforced rather than weakened this conjecture. I have read
Terhardt's information on the rate of coincident harmonics as we go
to higher harmonics, and I agree that this explains the phenomenon
under discussion, namely the almost universal existence of fifths,
fourths, and octaves as melodic intervals both cross-temporally and
cross-culturally, better than the prominence in terms of loudness
that I postulated.

Nonetheless, I see no reason to regard this as having simply replaced
it as an explanation. There is nothing mutually exclusive about these
two phenonema, and I believe Terhardt's explanation is the primary
contributor and that the very significant prominence in terms of
loudness of these lower harmonics is a secondary but still quite
significant contributor as well. I sure hear them much more loudly
than I do other harmonics in most of the harmonic timbres with which
I'm familiar.

These first few harmonics are also easier to pick out because of the
greater pitch distance between them, since the pitch distance between
successive harmonics decreases logarithmically with the order of the
harmonic. This, I would conjecture, is a third important contributor.

🔗paulerlich <paul@stretch-music.com>

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

> > I adjusted the size of the interval, little by little (without
> > looking), until I heard a certain high-pitched beating come to
> > a stop
> > (well, as close as I could manage) -- so it _had_ to be the
> > coinciding overtones. Look at the dB decay of sawtooth waves --
the
> > 17th harmonic is only about 12 dB (I think) quieter than the
> > fundamental.
>
> JG: In /tuning/topicId_31805.html#32046
> it appears that you use "sawtooth" to describe a repeating
> "ramp" waveform (as opposed to describing a "triangle" wave.
> In that post (where you pointed out my error regarding the
> rate of "roll-off" with increasing frequency in the amplitude
> of the partials of a "triangle" wave), we seem to agree that
> the "sawtooth" waveform "rolls-off" at rate of -6 dB / octave.
>
> With decibels (scaled to describe current, voltage, or sound
> pressure levels, as opposed to sound *power* levels), I get
>
> Amplitude of 17th harmonic = 20*(LOG_10 (1/17)) = -24.61 dB

Right.
>
> The identity 10*(LOG_10 (1/17)) would be appropriate only
> if the *power* (equal to the square of SPL, current, voltage,
> etc) of spectral components (the harmonics) decreased in a
> manner inversely proportional to the change in frequency.

Well, the point is that the beating between the 17th harmonic of one
sawtooth and the 13th harmonic of the other _is_ audible, even at a
moderate volume. David Canright did similar investigations and got
similar results. Try it yourself, if you can!

>
> >. . . but I'll also be able to show you that they don't explain
> > everything.
>
> JG: Cogent facts are always welcome.

For one thing, you might want to examine the early archives of the
harmonic entropy list.
>
> JG: I've looked a bit at Terhardt's texts relating to the
> Laplace/Fourier stuff, and his concepts of applying the
> Fourier integral more appropriately for "shorter-term"
> algorithms approaching a (sort of) "real-time" status.

Cool. I don't know much about this stuff, but I think it's important
to grasp the general gist.

> Since you seem to have covered Terhardt's material so
> comprehensively - what texts on his website do you feel
> best address your ideas of "how our ears work" (if that
> may be where some of these ideas are found)?

On his website? Well, ignorning that bit, besides the Hall and
Roederer books I already mentioned, you should read the Parncutt
book _Harmony: A Psychoacoustical Approach_ and try to extrapolate
beyond Parncutt's strict 12-tET presentation.

>
>
> > > > > It is to note that it seems *not* to be the numbers
involved
> > > > > in the ratios which describe the interval between two
pitches
> > > > > sounded (whether they be fundamentals or overtones of
such),
> > > > > but the (expected, and describable) effects of the
acoustical
> > > > > summation of nearby frequencies present into a coherent
whole.
> > > >
> > > > The latter often implies the former.
> > >
> > > JG: If only such implications amounted to explanations...
> >
> > The simpler the ratio between two frequencies (assuming it's
>simpler
> > than, say, 11:9), the more easily the two pitches will be heard
to
> > possess _commonality_, since, as we've seen, the pitches evoked
by >a
> > frequency are all in simple harmonic/subharmonic relationships
with
> > one another.
>
> JG: So you *do* consider "subharmonic" relationships to be
> significant in the perception of a "harmonic" spectra? You
> had seemed (to me) to restrict its significance in prior
> communication to applications involving (3 or more tone)
> "utonal" chords.

On the contrary, I see subharmonic relationships as becoming
_limited_ in their value in chords of 3 or more tones.

> Would you then look at a harmonic series
> from the perspective of *other* spectral components than
> the fundamental "pitch" (used here to attempt to address
> your sense of a "pitch" arising out of *one* "spectral
> component" out of a group of spectral components)?

If you think I or Terhardt are trying to imply that the sense of
pitch arises from *one* spectral component out of a group of
spectral components, you're sorely mistaken. Did you really read the
Terhardt stuff, for example, the "strike note of bells" page? Also,
go to your library and read the relevant chapter or two or three in
the Hall book.

> >That's because the whole virtual pitch mechanism seems
> > to be _designed_ to deal with harmonic overtone series, perhaps
an
> > evolutionary advantage, related to the fact that human (and
other)
> > voices have harmonic overtones, or perhaps something acquired
> > prenatally via exposure to the mother's voice . . . in any case,
the
> > reality and properties of the virtual pitch phenomenon have been
> > demonstrated in experiment after experiment, and remains a
thriving
> > area of research today, as some list members can attest . . .
>
> JG: Where else (other than Terhardt) have you seen published
> references to "virtual pitch"? Any authors come to mind? It
> might be instructive if you could post a "bibliography" of
> such papers.

Every reputable account of psychoacoustics talks about it!! Just
start with some good intros like Roederer and Hall and follow the
references from there!

> > > > > Does it not seem that a "harmonic theory of tonal
>>>>combinations"
> > > > > would appear to be applicable in the case of *harmonic*
tones,
> > > > > but might well be *inapplicable* to combinations of small
> >>>> numbers
> > > > > of *sinusoidal* tones?
> > > > > Can the case be made that combinations of
> > > > > small numbers of *sinusoidal* tones can predict the
effects of
> > > > > combinations of similarly tuned *complex* tones (with
> >>>>harmonics)?
>
> > > > The two situations could be made absolutely identical and
> > > > indistinguishable.
>
> JG: Sounds like "magic" to me! :)

It's simply "Fourier synthesis", or whatever you want to call it!

> > > JG: That's a rather tall order to fill, it seems. I guess
> > > that you would necessarily assume that if it were to be the
> > > case that "harmonic entropy" is independent of waveform ...
> >
> > It has nothing do do with harmonic entropy. I'm merely referring
to
> > the mathematical equivalence of a description in terms of
complex
> > tones and a description in terms of sine waves. Certainly I'm
not
> > claiming that these effects are timbre-independent!
>
> In "harmonic_entropy" message #531, you said:
>
> << 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. >>

That was about harmonic entropy. Here, we're not talking about
harmonic entropy at all! This is a blatant case of taking things out
of context! Can we forget about harmonic entropy for the purposes of
this discussion? If you can't, at least treat harmonic entropy as an
embryonic plaything that attempts to capture one of Terhardt's two
components of dissonance -- that's all it is. Read my next paragraph
before replying to this one.

> It would *seem* that the question of whether
> or not the case can be made that "combinations
> of small numbers of *sinusoidal* tones can predict the effects of
> combinations of similarly tuned *complex* tones (with harmonics)"
> would be highly relevant to whether "harmonic entropy" can be
> utilized in the characterization of "harmonic timbres"!...

Jeremy, I thought you were talking about something different. If you
simply remove all the upper partials of the complex tones, clearly
you are not faced with the same sound you had before. In harmonic
entropy I would use a much larger s value for the no-upper-partial
case, as I've explained before. But I thought you were talking about
the equivalence of a representation using _many_ sine-wave
components and a representation using _few_ complex tones. I was
merely saying that yes, these are equivalent views (per Fourier).
Sorry for the confusion!