Sinusoidal Entropy?

Paul,

I believe that you have stated (and please do correct me if I am
incorrect in my memory) that "Harmonic Entropy" is applicable to pure
tone dyads (such as those constructed from simultaneously sounded
sines and cosine waves).

If that is the case, what relation does the term "harmonic" have to
your "Harmonic Entropy" concept?

What effects upon the output function would you expect for the case
of each tone within the dyad having harmonic content at integer
multiples of each of the two fundamental tones, as well?
Have you attempted to calculate the results for such waveforms?

Curiously, J Gill

>I believe that you have stated (and please do correct me if I am
>incorrect in my memory) that "Harmonic Entropy" is applicable to pure
>tone dyads (such as those constructed from simultaneously sounded
>sines and cosine waves).

Yes -- although the standard deviations typically used would only be
applicable to such a case in a certain optimal frequency range, around
3000Hz.

>If that is the case, what relation does the term "harmonic" have to
>your "Harmonic Entropy" concept?

(a) the dyad is compared with intervals in the harmonic series
(b) the dyad is harmonic, not melodic.

>What effects upon the output function would you expect for the case
>of each tone within the dyad having harmonic content at integer
>multiples of each of the two fundamental tones, as well?

That's the case I usually intend harmonic entropy for. That's because
harmonic partials will lead to the same set of ratio-interpretations for the
dyad as the bare dyad, but with tighter standard deviation because

(a) there will normally be some partials in, or at least closer to, the
3000Hz frequency range
(b) the multiplicity of partials will represent several independent sources
of information for the same ratio-interpretations

>Have you attempted to calculate the results for such waveforms?

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.

--- In harmonic_entropy@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

> 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.

> >JG had asked:...what relation does the term "harmonic" have to
> >your "Harmonic Entropy" concept?

> PE:(a) the dyad is compared with intervals in the harmonic series

JG: Of one, the other, or both fundamental frequencies of the two
tones from which the dyad is constructed?

> PE:(b) the dyad is harmonic, not melodic.

JG: Do you mean by this simply that the dyad is (potentially, but not
necessarily) constructed of complex (as opposed to sinusoidal) tones?

>PE:...harmonic partials will lead to the same set of ratio-
>interpretations for the dyad as the bare dyad,
>but with tighter standard deviation because
>
> (a) there will normally be some partials in, or at least closer to,
> the 3000Hz frequency range
>
> (b) the multiplicity of partials will represent several independent
> sources of information for the same ratio-interpretations

JG: Would not (b) directly above imply that - for the equivalence
of "ratio-interpretations" to exist from "several independent sources
of information" (those sources being the individual complex tones
from which the dyad is constructed) - the spectral amplitudes of each
of the overtones of the individual fundamental frequencies of such
individual complex tones must be equivalent. That is - the (steady
state, as well as the transient) frequency spectrums of the (complex)
tones number 1 and number 2 from which the dyad is constructed must
be identical (or nearly identical) in order for your assumption (b)
directly above to be valid?

Curiously, J Gill

--- In harmonic_entropy@y..., "unidala" <JGill99@i...> wrote:
> --- In harmonic_entropy@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
>
> >PE: 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.

> >PE:...harmonic partials will lead to the same set of ratio-
> >interpretations for the dyad as the bare dyad,
> >but with tighter standard deviation because
> >
> > (a) there will normally be some partials in, or at least closer >
> > to, the 3000Hz frequency range

> >
> > (b) the multiplicity of partials will represent several
> > independent
> > sources of information for the same ratio-interpretations
>
>
> JG: Would not (b) directly above imply that - for the equivalence
> of "ratio-interpretations" to exist from "several independent
sources
> of information" (those sources being the individual complex tones
> from which the dyad is constructed) - the spectral amplitudes of
each
> of the overtones of the individual fundamental frequencies of such
> individual complex tones must be equivalent. That is - the (steady
> state, as well as the transient) frequency spectrums of the
(complex)
> tones number 1 and number 2 from which the dyad is constructed must
> be identical (or nearly identical) in order for your assumption (b)
> directly above to be valid?

JG: Additionally, due to the nonlinear amplitude transfer function of
perceived loudness levels [shown as approximately equal to a value of
( (SPL)b/(SPL)a )^(2/3) at SPL levels greater than 20 PHONS (in SPL),
which is equal to 0.1 SONE (in LOUDNESS UNITS, being 20 dB above the
threshold of human hearing)] [from "Music, Physics, and Engineering",
Olson, 1967, page 252], is it not also true that, in order that "the
spectral amplitudes of each of the overtones of the individual
fundamental frequencies of such individual complex tones" remain
identical (or nearly identical) relative to the amplitudes of those
fundamental frequencies - that the two identical (steady-state as
well as transient) frequency spectrums of the (complex) tones number
1 and number 2 from which the dyad is constructed ALSO be of equal
(or near equal) scalar amplitude throughout the time period during
which each of the individual complex tones are sounded (steady-state
as well as transient)?

NOTE: The phrases "steady-state" as well as "transient" are utilized
above in an inclusive manner to describe all phases of the process of
time-varying amplitude envelopes multiplying each of the values of
the individual complex tones (existing for some given time duration)
from which the dyad is constructed.

Curiously, J Gill

>--- In harmonic_entropy@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

>> 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.

> > >JG had asked:...what relation does the term "harmonic" have to
> > >your "Harmonic Entropy" concept?

> > PE:(a) the dyad is compared with intervals in the harmonic series

> JG: Of one, the other, or both fundamental frequencies of the two
> tones from which the dyad is constructed?

I don't understand this question. What I'm trying to say is that, in the
harmonic series, one finds certain intervals -- for example, if you go up
throught the fourth harmonic, you find the intervals 1:1, 2:1, 3:1, 3:2,
4:1, 4:3.

> > PE:(b) the dyad is harmonic, not melodic.

> JG: Do you mean by this simply that the dyad is (potentially, but not
> necessarily) constructed of complex (as opposed to sinusoidal) tones?

Although that's true, it's not what I meant. What I meant was that the tones
are played simultaneously (i.e., _harmony_) rather than consecutively (i.e.,
_melody_).

> >PE:...harmonic partials will lead to the same set of ratio-
> >interpretations for the dyad as the bare dyad,
> >but with tighter standard deviation because
> >
> > (a) there will normally be some partials in, or at least closer to,
> > the 3000Hz frequency range
> >
> > (b) the multiplicity of partials will represent several independent
> > sources of information for the same ratio-interpretations

> JG: Would not (b) directly above imply that - for the equivalence
> of "ratio-interpretations" to exist from "several independent sources
> of information" (those sources being the individual complex tones
> from which the dyad is constructed)

Actually the sources are, for example, several pairs of overtones, one from
each complex tone.

> That is - the (steady
> state, as well as the transient) frequency spectrums of the (complex)
> tones number 1 and number 2 from which the dyad is constructed must
> be identical (or nearly identical) in order for your assumption (b)
> directly above to be valid?

For the assumption to be valid, both tones would have to have a fairly
complete complement of overtones, present with decent amplitude, in the
frequency range in question. Otherwise, the brain would be forced to rely
more heavily upon the fundamentals, and the s (uncertainty) values we
typically use would be too small to model what happens. For large s values,
of course, one would observe a largely featureless harmonic entropy curve,
with local minima at only the very simplest of ratios, say 1/1 and 2/1.

>JG: Additionally, due to the nonlinear amplitude transfer function of
>perceived loudness levels [shown as approximately equal to a value of
>( (SPL)b/(SPL)a )^(2/3) at SPL levels greater than 20 PHONS (in SPL),
>which is equal to 0.1 SONE (in LOUDNESS UNITS, being 20 dB above the
>threshold of human hearing)] [from "Music, Physics, and Engineering",
>Olson, 1967, page 252], is it not also true that, in order that "the
>spectral amplitudes of each of the overtones of the individual
>fundamental frequencies of such individual complex tones" remain
>identical (or nearly identical) relative to the amplitudes of those
>fundamental frequencies - that the two identical (steady-state as
>well as transient) frequency spectrums of the (complex) tones number
>1 and number 2 from which the dyad is constructed ALSO be of equal
>(or near equal) scalar amplitude throughout the time period during
>which each of the individual complex tones are sounded (steady-state
>as well as transient)?

Again, it would help. If one tone is much louder than the other, then the
brain is more likely to disregard the quieter one, and simply use the louder
one as its source of pitch information.