Yahoo Tuning Groups Ultimate Backup tuning Virtual Fundamental Period?

--- In tuning@y..., J Gill <JGill99@i...> wrote:
>
>Paul accurately points out that (in order to include sets of pitches
>such as 1/10, 1/9, 1/8, 1/7, 1/6, 1/5, 1/4, 1/3, 1/2), one also has
>to "flip" the perspective (of the 'LCMR' algorithm I proposed), and
>apply the (same) LCM algorithms to the *period* [equal to 1/>
(frequency)] of such an "undertone series" (as described by PE).

Well, maybe that's your interpretation of what I was saying, but
that's not what I meant at all! I was simply pointing out a behavior
of your 'LCMR' algorithm that you may not have been aware of.

>Interestingly, Terhardt's "virtual pitch" (which, unless I am
>missing something here) refers to an (implied fundamental)
>*frequency* (and *not* to an "implied fundamental period") of the >
(generally termed) "implied spectral component of reference" (to
>which the "aural mind", in interpreting our ears' input, may so
>perceptually "tether").

Well, I'm not sure what you're meaning, but the virtual pitch will
correspond to the fundamental of the best-fit harmonic series to the
spectral components you hear.

>How does Paul and/or Terhardt approach this reference "signal
>period" (as opposed to reference "signal frequency") business (in
>relation to the "virtual pitch" hypothesis)?

Again, I have no idea what you mean.

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

>Terhardt states:

><<An explanation of virtual pitch that on first sight is highly
>suggestive is the "time-domain solution", i.e., measurement of the
>period of the tone signal. (Which, of course, requires that the
>signal actually is periodic.) That kind of solution was suggested
>already in the 19th Century, in particular by Seebeck (1841a).
>Schouten (1940c, 1970a) strongly promoted that solution. However,
>already in the 1950s and 1960s new observations had indicated that
>the time-domain model in its original design was not adequate.>>

>FOLLOWED BY SEVERAL RELEVANT STATEMENTS, FOLLOWED BY:

><<In spite of these difficulties with explaining virtual pitch in
>the time domain, a considerable number of attempts of that type have
>been described in the literature until present days. However, I am
>not aware of a satisfactory solution.>>

>How, then, does one "square" (in a rhetorical sense) the "undertone
>series" analytical approaches with Terhardt's "virtual pitch" >
(fundamental *frequency* based) hypothesis?

I think you have two misunderstandings here:

(1) That I find any validity in "undertone series" analytical
approaches like your 'LCMR'.

(2) That time-domain theories of virtual pitch have anything to do
with a period-based RATHER THAN a frequency-based approach.

I'd like to help you clear up your misunderstandings . . . where do
we begin?

🔗unidala <JGill99@imajis.com>

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., J Gill <JGill99@i...> wrote:
> >
> >Paul accurately points out that (in order to include sets of pitches
> >such as 1/10, 1/9, 1/8, 1/7, 1/6, 1/5, 1/4, 1/3, 1/2), one also has
> >to "flip" the perspective (of the 'LCMR' algorithm I proposed), and
> >apply the (same) LCM algorithms to the *period* [equal to 1/>
> (frequency)] of such an "undertone series" (as described by PE).
>
> Well, maybe that's your interpretation of what I was saying, but
> that's not what I meant at all! I was simply pointing out a behavior
> of your 'LCMR' algorithm that you may not have been aware of.
>
> >Interestingly, Terhardt's "virtual pitch" (which, unless I am
> >missing something here) refers to an (implied fundamental)
> >*frequency* (and *not* to an "implied fundamental period") of the >
> (generally termed) "implied spectral component of reference" (to
> >which the "aural mind", in interpreting our ears' input, may so
> >perceptually "tether").
>
> Well, I'm not sure what you're meaning, but the virtual pitch will
> correspond to the fundamental of the best-fit harmonic series to the
> spectral components you hear.
>
> >How does Paul and/or Terhardt approach this reference "signal
> >period" (as opposed to reference "signal frequency") business (in
> >relation to the "virtual pitch" hypothesis)?
>
> Again, I have no idea what you mean.
>
> >At: http://www.mmk.e-technik.tu-
> >muenchen.de/persons/ter/top/virtualp.html
>
> >Terhardt states:
>
> ><<An explanation of virtual pitch that on first sight is highly
> >suggestive is the "time-domain solution", i.e., measurement of the
> >period of the tone signal. (Which, of course, requires that the
> >signal actually is periodic.) That kind of solution was suggested
> >already in the 19th Century, in particular by Seebeck (1841a).
> >Schouten (1940c, 1970a) strongly promoted that solution. However,
> >already in the 1950s and 1960s new observations had indicated that
> >the time-domain model in its original design was not adequate.>>
>
> >FOLLOWED BY SEVERAL RELEVANT STATEMENTS, FOLLOWED BY:
>
> ><<In spite of these difficulties with explaining virtual pitch in
> >the time domain, a considerable number of attempts of that type have
> >been described in the literature until present days. However, I am
> >not aware of a satisfactory solution.>>
>
> >How, then, does one "square" (in a rhetorical sense) the "undertone
> >series" analytical approaches with Terhardt's "virtual pitch" >
> (fundamental *frequency* based) hypothesis?
>
> I think you have two misunderstandings here:
>
> (1) That I find any validity in "undertone series" analytical
> approaches like your 'LCMR'.
>
> (2) That time-domain theories of virtual pitch have anything to do
> with a period-based RATHER THAN a frequency-based approach.
>
> I'd like to help you clear up your misunderstandings . . . where do
> we begin?

J Gill: Thanks for taking your time to consider my thoughts here.

My thoughts (above) arose out of my interpretation of some of your statements in: /tuning/topicId_31418.html#31552
(quoted from below with comments added):

> >PE: Actually the 'TH' always agrees exactly with the LCM when the
> > frequency is measured in units of _the implied fundamental_
>
> JG: Could you give me a more specific definition of what you mean
> directly above by the phrase "implied fundamental"? :)

The "virtual pitch", or the highest possible "missing root" such that
if there were a tone there, it would have harmonics right were the
tones in question are.

> >PE: -- or
> > when the _wave period_...
>
> JG: You do mean "period" (not it's algebraic inverse
of "frequency")?

PE: Yes.

> >PE (continued):... is measured in units of the _lowest common
> > overtone_.
>
> JG: So you are saying here that - once one has determined the
> numerical (period, and not frequency?) value of the "lowest common
> overtone", ... then (and only then) does one know at what precise
> pitch the value of the "Tenney height" ('HT') is intended to be
> *referenced* to???

PE: No -- there are at least three ways to reference it.

(1) You can reference it to the frequency of the "implied
fundamental", which you therefore call "1" -- and then the LCM of the
frequencies of the two notes equals the 'TH'.

JG: This possibility seems (to me) to be essentially equivalent to what I was getting at in our "lowest common multiple (of) reference" (or, as I coined, 'LCMR') discussions [where you aptly pointed out that my approach to the case of the existence of "irrational" (including "transcendental") numberical values needs some work].
In the case that such a 1/1 "reference" pitch would also be found to be equal to a (somehow selected) "implied fundamental" ('IF'), or represented a numerically "normalized" (to 1/1) form of that 'IF', then it appears (excepting the "irrationality" issue) equivalent.

(2) You can reference it to the period of the lowest common overtone,
which you therefore call "1" -- and then the LCM of the periods of
the two notes equals the 'TH'.

JG: Here [where you verified above that you mean "period" to refer to the quantity = 1/(frequency)], I interpreted this possibilty to amount to a conceptual "(algebraic) inversion" of your possibility '(1)' above. While the terms "implied fundamental" (frequency of reference) and "lowest common overtone" initially appeared to me as being distinct from each other in meaning (and perhaps this is where a conceptual "faux pas" has, on my part, occured), when I imagined such a possibility's application to an "undertone series"...

[Note: your phrase from message #31564, stating, "You'll end up finding that all members of an infinite undertone series under 1/1
(i.e., 1/1, 1/2, 1/3, 1/4, 1/5, etc.) have an 'LCMR' of 1 when taken two, or more, at a time.]

... consisting of frequencies such as 1/2, 1/3, 1/4, 1/5, it occured to me that [in such a case where *both* rational values under consideration have a value equal to less than unity, and, as a result, the location at which the "lowest common overtone" occurs is identical to the location of the (normalized) "implied fundamental" -at a value of 1/1 (in "period" as well as "frequency")], your possibilty '(2)' above appears (to me) to represent the *same thing* as your possibility '(1)' above, with pitch "period" substituted for pitch "frequency".

Do you see your possibility '(2)' differently [when applied to other cases than such an "equivalence" (of the "lowest common overtone" and the (normalized) "implied fundamental" both occuring at a numerical value equal to 1/1]?

I then (perhaps falsely) came to an assumption that your possibility '(2)' above represented your approach to dealing with cases where *both* of the two (rational valued) pitches under consideration each are of a numerical value *less* than unity (1/1).

Carrying that further, it seemed (to me) that (whether or not either one, or both, or none of the pitch-ratios under consideration possess a numerical value of less than unity), a "period based" analysis should (in addition to a "frequency based" analysis) be included in my consideration of "coincident multiples" [the phrase now generalized, in order to encompass "multiples of a *period* of reference" in addition to (as formerly) including "multiples of a *frequency* of reference" (or "harmonics")]. Thus (it seems), one could escape the "integer-limit" world which you have spoken of [by accounting for "coincidences" in the "period" (which are associated with the values of the *denominators* of the rational numbers considered), in addition to accounting for "coincidences" in the "frequency" (which are associated with the values of the *numerators* of the rational numbers considered)].

The whole business of references to the term "undertones" (which do *not*, in *linear* systems, *actually exist*) has always (for that very reason) troubled me (though I can see how a series of values such as 1/1, 1/2, 1/3, 1/4, 1/5, etc, might be described by such a term in conversation, and I see what is implied by the term's use).

So (perhaps failing to comprehensively consider the full meaning and/or implications of your statements, and their applicability or non-applicability to the concepts of which my mind has been engaged), it occured to me that - this "undertone series" business, and (what appears to be the associated) "coincidence of periods" business *should* (at least, "could") have a corollary in the world of numerical analysis of musical tones... So (I thought), "where is it"?

Upon reflection, it may be that I (alone) have brought an expectation (of a kind of "related physical evidence" existing in the realm of "research surrounding aural perceptions") into my desires that a given numerical property (topology, pattern, "periodicity", etc) *could* (at least in part) "be found" to objectively describe demonstrable physical phenomena which *might* appear to "correlate" with my own (I'll speak only for myself) subjective sensory impressions.

None of this precludes the possibility that: certain numerical operations are (by their nature) required in order to numerically process certain relationships surrounding pitch and (with harmonic/non-harmonic spectral components also considered) timbre,

while no such corollary may exist in the world of sensory research (or, for that matter, the human "aural mind" as it would be subjectively conceived by the same listener who conceives it).

Nature (in its indifference) favors not the lowly "mathematician"...

J Gill

🔗paulerlich <paul@stretch-music.com>

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

> J Gill: Thanks for taking your time to consider my thoughts here.
>
> My thoughts (above) arose out of my interpretation of some of your
statements in: /tuning/topicId_31418.html#31552
> (quoted from below with comments added):
>
> > >PE: Actually the 'TH' always agrees exactly with the LCM when
the
> > > frequency is measured in units of _the implied fundamental_
> >
> > JG: Could you give me a more specific definition of what you mean
> > directly above by the phrase "implied fundamental"? :)
>
> The "virtual pitch", or the highest possible "missing root" such
that
> if there were a tone there, it would have harmonics right were the
> tones in question are.
>
> > >PE: -- or
> > > when the _wave period_...
> >
> > JG: You do mean "period" (not it's algebraic inverse
> of "frequency")?
>
> PE: Yes.
>
> > >PE (continued):... is measured in units of the _lowest common
> > > overtone_.
> >
> > JG: So you are saying here that - once one has determined the
> > numerical (period, and not frequency?) value of the "lowest
common
> > overtone", ... then (and only then) does one know at what precise
> > pitch the value of the "Tenney height" ('HT') is intended to be
> > *referenced* to???
>
> PE: No -- there are at least three ways to reference it.
>
> (1) You can reference it to the frequency of the "implied
> fundamental", which you therefore call "1" -- and then the LCM of
the
> frequencies of the two notes equals the 'TH'.
>
> JG: This possibility seems (to me) to be essentially equivalent to
>what I was getting at in our "lowest common multiple (of) reference"
>(or, as I coined, 'LCMR') discussions [where you aptly pointed out
>that my approach to the case of the existence of "irrational" >
(including "transcendental") numberical values needs some work].
> In the case that such a 1/1 "reference" pitch would also be found
>to be equal to a (somehow selected) "implied fundamental" ('IF'), or
>represented a numerically "normalized" (to 1/1) form of that 'IF',
>then it appears (excepting the "irrationality" issue) equivalent.

Right, but your measure appeared problematic to me because the only
case that the 'IF' would equal your 1/1 reference pitch is if both of
the pitches in the dyad being evaluated happened to be of the form
n/1, n an integer -- a rather special case indeed!

> (2) You can reference it to the period of the lowest common
overtone,
> which you therefore call "1" -- and then the LCM of the periods of
> the two notes equals the 'TH'.
>
> JG: Here [where you verified above that you mean "period" to refer
>to the quantity = 1/(frequency)], I interpreted this possibilty to
>amount to a conceptual "(algebraic) inversion" of your
>possibility '(1)' above. While the terms "implied fundamental" >
(frequency of reference) and "lowest common overtone" initially
>appeared to me as being distinct from each other in meaning (and
>perhaps this is where a conceptual "faux pas" has, on my part,
>occured),

They _are_ distinct from each other in meaning.

>when I imagined such a possibility's application to an "undertone
>series"...

Not following you.

> [Note: your phrase from message #31564, stating, "You'll end up
>finding that all members of an infinite undertone series under 1/1
> (i.e., 1/1, 1/2, 1/3, 1/4, 1/5, etc.) have an 'LCMR' of 1 when
>taken two, or more, at a time.]

Again, just pointing out the shortcomings of 'LCMR'.

> ... consisting of frequencies such as 1/2, 1/3, 1/4, 1/5, it
>occured to me that [in such a case where *both* rational values
>under consideration have a value equal to less than unity, and, as a
>result, the location at which the "lowest common overtone" occurs is
>identical to the location of the (normalized) "implied fundamental"

It certainly isn't! For example, if you take 1/2 and 1/3, the lowest
common overtone is 1/1, but the implied fundamental is 1/6.

> Do you see your possibility '(2)' differently [when applied to
>other cases than such an "equivalence" (of the "lowest common
>overtone" and the (normalized) "implied fundamental" both occuring
>at a numerical value equal to 1/1]?

The only time they'll both occur at 1/1 is when both of the pitches
are themselves equal to 1/1!

> I then (perhaps falsely) came to an assumption that your
>possibility '(2)' above represented your approach to dealing with
>cases where *both* of the two (rational valued) pitches under
>consideration each are of a numerical value *less* than unity (1/1).

No -- 1/1 has nothing to do with it. (1), (2), and (3) _always_ work.

>Carrying that further, it seemed (to me) that (whether or not
>either one, or both, or none of the pitch-ratios under consideration
>possess a numerical value of less than unity), a "period based"
>analysis should (in addition to a "frequency based" analysis) be
>included in my consideration of "coincident multiples" [the phrase
>now generalized, in order to encompass "multiples of a *period* of
>reference" in addition to (as formerly) including "multiples of a
>*frequency* of reference" (or "harmonics")]. Thus (it seems), one
>could escape the "integer-limit" world which you have spoken of [by
>accounting for "coincidences" in the "period" (which are associated
>with the values of the *denominators* of the rational numbers
>considered), in addition to accounting for "coincidences" in
>the "frequency" (which are associated with the values of the
>*numerators* of the rational numbers considered)].

I'm confused, but maybe you should think about this again in light of
the above. And again, (1), (2), and (3) give you ways of
understanding the 'TH', not the "integer limit".

> So (perhaps failing to comprehensively consider the full meaning
>and/or implications of your statements, and their applicability or
>non-applicability to the concepts of which my mind has been
>engaged), it occured to me that - this "undertone series" business,
>and (what appears to be the associated) "coincidence of periods"
>business *should* (at least, "could") have a corollary in the world
>of numerical analysis of musical tones... So (I thought), "where is
>it"?

The "coincidence of periods" business comes from a rather old view of
psychoacoustics, mentioned by Terhardt, which holds that the shorter
the wave period of the combined waveforms of the tones, the
more "consonant" the sensation. Even Partch at one point seems to
hold this view.

>Upon reflection, it may be that I (alone) have brought an
>expectation (of a kind of "related physical evidence" existing in
>the realm of "research surrounding aural perceptions") into my
>desires that a given numerical property (topology,
>pattern, "periodicity", etc) *could* (at least in part) "be found"
>to objectively describe demonstrable physical phenomena which
>*might* appear to "correlate" with my own (I'll speak only for
>myself) subjective sensory impressions.

Well, I think I'm with you on that, but a simple rule like the ones
we've been considering obviously breaks down for ratios such as
3001:2001, which is very consonant, despite the extreme height of the
lowest common overtone, the extreme length of the wave period,
etc. . . . This is why I developed the harmonic entropy theory, so
that a very simple conceptual/mathematical model could still account
for my sensory impressions quite well.

Virtual Fundamental Period?

🔗J Gill <JGill99@imajis.com>

🔗paulerlich <paul@stretch-music.com>

🔗unidala <JGill99@imajis.com>

🔗paulerlich <paul@stretch-music.com>

🔗Ray Tomes <rtomes@kcbbs.gen.nz>