Yahoo Tuning Groups Ultimate Backup tuning subharmonics and combination tones

on 8/12/04 1:41 PM, traktus5 <kj4321@hotmail.com> wrote:

> I read through Monz's dictionary at tonalsoft, but am still
> unclear....How do subharmonics and combinations relate? It's my
> understanding that the former is a harmonic series inverted (going
> downward), whereas combination tones and their primary tones exist in
> a normal harmonic series...but they both involve diffenece tones...so
> I'm confused...

You've got it right, I think. If you flip the intervals in a "harmonic
chord" (also called otonal) top-to-bottom then in the resulting chord
(called utonal) the difference tones don't tend to coincide and reinforce
each other as they do in an otonal chord. But the otonal/utonal distinction
is a little ambiguous and depends on the numbers involved.

For example the major triad 4:5:6 consists of 4:5 below 5:6 and flipped this
becomes the minor triad, with a 4:5 above a 5:6 which is no longer obviously
a harmonic chord. Actually 4:5:6 flipped can be spelled as 1/6:1/5:1/4 [or
abbreviated as 1/(6:5:4)], a utonal spelling, but can also be spelled (by
multiplying by 60) as 10:12:15 which is also a harmonic (otonal) spelling.
But the numbers are higher in the otonal spelling and so the harmonic aspect
is less clear and the chord is more likely to be considered utonal because
the utonal spelling involves lower numbers. And also the minor triad
intervals themselves are individually still 5:6 and 4:5 yet the chord can
not be spelled anymore using the numbers 4,5,6, so this is another
indication that something is quite different from the situation in a
harmonic chord.

There was a thread about this in relation to why minor chords are "sad"
several months back. It makes sense to me that without the harmonic
reinforcement it is much like the individual intervals are "lonely" rather
than really forming a single harmony. Yet the same intervals may be present
as are present in a harmonic chord, so there is still a sense of harmony,
just a different kind.

So for the difference tones, 4:5:6 produces difference tones of 1,1, and 2,
reinforcing the fundamental and the octave. But 10:12:15 produces
difference tones of 2, 3, and 5 and so reinforcement is scattered. This
becomes even clearer in bigger chords: 4:5:6:7 has even stronger
reinforcement whereas 1/(7:6:5:4) again has scattered difference tones.

Probably other people on the list can offer other ways of describing this.
You can also talk about an implied fundamental, which I didn't get into for
lack of time at the moment.

-Kurt

🔗Carl Lumma <ekin@lumma.org>

>> I read through Monz's dictionary at tonalsoft, but am still
>> unclear....How do subharmonics and combinations relate? It's my
>> understanding that the former is a harmonic series inverted (going
>> downward), whereas combination tones and their primary tones exist
>> in a normal harmonic series...but they both involve diffenece
>> tones...so I'm confused...
>
>You've got it right, I think. If you flip the intervals in
>a "harmonic chord" (also called otonal) top-to-bottom then
>in the resulting chord (called utonal) the difference tones
>don't tend to coincide and reinforce each other as they do
>in an otonal chord.

That's only partly true. Here's a classic from Paul Erlich:

>>>The importance and beauty of utonal formations in Partch's
>>>music and elsewhere is undeniable (I heard Catler play
>>>some beautiful 7-limit utonalities on Sunday). The fact that
>>>undertone series as scales can be more easily constructed
>>>by man than overtone series as scales is also undeniable.
>>>The fact that utonal chords have a lower first common
>>>overtone, and a greater rate of occurence of higher
>>>common overtones, and hence are in a sense easier to tune
>>>with beats, that any comparable chords including otonal
>>>ones, is also undeniable.

>But the otonal/utonal distinction
>is a little ambiguous and depends on the numbers involved.

There are borderline cases, such as 1/1:6/5:3/2 vs. 16:19:24.

>For example the major triad 4:5:6 consists of 4:5 below 5:6
>and flipped this becomes the minor triad, with a 4:5 above
>a 5:6 which is no longer obviously a harmonic chord.
>Actually 4:5:6 flipped can be spelled as 1/6:1/5:1/4 [or
>abbreviated as 1/(6:5:4)], a utonal spelling, but can also be
>spelled (by multiplying by 60) as 10:12:15 which is also a
>harmonic (otonal) spelling. But the numbers are higher in
>the otonal spelling and so the harmonic aspect is less clear
>and the chord is more likely to be considered utonal because
>the utonal spelling involves lower numbers. And also the
>minor triad intervals themselves are individually still 5:6
>and 4:5 yet the chord can not be spelled anymore using the
>numbers 4,5,6, so this is another indication that something
>is quite different from the situation in a harmonic chord.

True, and here's a paraphrase of another classic observation
by Paul that is even stronger:

'Comparing otonal and utonal chords beyond the 5-limit, we find
that though such pairs always have the same intervals they do not
sound equally consonant to the vast majority of listeners.
Therefore traditional 'place' models of consonance are incomplete.'

>There was a thread about this in relation to why minor chords
>are "sad" several months back. It makes sense to me that
>without the harmonic reinforcement it is much like the
>individual intervals are "lonely" rather than really forming
>a single harmony. Yet the same intervals may be present as are
>present in a harmonic chord, so there is still a sense of harmony,
>just a different kind.

Exactly. But the missing part doesn't have to do with partials
(in my version of the thesis), but rather the virtual pitch
mechanism.

>So for the difference tones, 4:5:6 produces difference tones of
>1,1, and 2, reinforcing the fundamental and the octave. But
>10:12:15 produces difference tones of 2, 3, and 5 and so
>reinforcement is scattered. This becomes even clearer in bigger
>chords: 4:5:6:7 has even stronger reinforcement whereas
>1/(7:6:5:4) again has scattered difference tones.

Hon't forget sum tones, which I believe are assumed in many models
to be the same amplitude in every order as difference tones. It is
true that they fall in a register which makes them poor candidates
as roots, and here we already are assuming something special about
roots.

>Probably other people on the list can offer other ways of
>describing this. You can also talk about an implied fundamental,
>which I didn't get into for lack of time at the moment.

If we assume implied fundamentals are based on harmonics (which is
at least plausible given their prevalence over subharmonics in the
spectra of naturally-occurring sounds), then we have a candidate
that explains the difference between major and minor. It's rather
ad hoc, but sometimes ad hoc is good. Throw in some hand waving
about how combination-sensitive neurons could account for an
implied-fundamental perception, and you can even start sounding
mildly affected.

-Carl

🔗Carl Lumma <ekin@lumma.org>

>> I read through Monz's dictionary at tonalsoft, but am still
>> unclear....How do subharmonics and combinations relate? It's my
>> understanding that the former is a harmonic series inverted (going
>> downward), whereas combination tones and their primary tones exist in
>> a normal harmonic series...but they both involve diffenece tones...so
>> I'm confused...
>
>You've got it right, I think. If you flip the intervals in a "harmonic
>chord" (also called otonal) top-to-bottom then in the resulting chord
>(called utonal) the difference tones don't tend to coincide and reinforce
>each other as they do in an otonal chord.

That's only partly true. Here's a classic from Paul Erlich:

>>>The importance and beauty of utonal formations
>>>in Partch's music and elsewhere is undeniable (I heard Catler play some
>>>beautiful 7-limit utonalities on Sunday). The fact that undertone series
>>>as scales can be more easily constructed by man than overtone series as
>>>scales is also undeniable. The fact that utonal chords have a lower
>>>first common overtone, and a greater rate of occurence of higher common
>>>overtones, and hence are in a sense easier to tune with beats, that any
>>>comparable chords including otonal ones, is also undeniable.

>But the otonal/utonal distinction
>is a little ambiguous and depends on the numbers involved.

There are borderline cases, such as 1/1:6/5:3/2 vs. 16:19:24.

>For example the major triad 4:5:6 consists of 4:5 below 5:6 and flipped this
>becomes the minor triad, with a 4:5 above a 5:6 which is no longer obviously
>a harmonic chord. Actually 4:5:6 flipped can be spelled as 1/6:1/5:1/4 [or
>abbreviated as 1/(6:5:4)], a utonal spelling, but can also be spelled (by
>multiplying by 60) as 10:12:15 which is also a harmonic (otonal) spelling.
>But the numbers are higher in the otonal spelling and so the harmonic aspect
>is less clear and the chord is more likely to be considered utonal because
>the utonal spelling involves lower numbers. And also the minor triad
>intervals themselves are individually still 5:6 and 4:5 yet the chord can
>not be spelled anymore using the numbers 4,5,6, so this is another
>indication that something is quite different from the situation in a
>harmonic chord.

True, and here's a paraphrase of another classic from Paul that is
even stronger:

>There was a thread about this in relation to why minor chords are "sad"
>several months back. It makes sense to me that without the harmonic
>reinforcement it is much like the individual intervals are "lonely" rather
>than really forming a single harmony. Yet the same intervals may be present
>as are present in a harmonic chord, so there is still a sense of harmony,
>just a different kind.

Exactly. But the missing part doesn't have to do with partials (in
my version of the thesis), but rather the virtual pitch mechanism.

>So for the difference tones, 4:5:6 produces difference tones of 1,1, and 2,
>reinforcing the fundamental and the octave. But 10:12:15 produces
>difference tones of 2, 3, and 5 and so reinforcement is scattered. This
>becomes even clearer in bigger chords: 4:5:6:7 has even stronger
>reinforcement whereas 1/(7:6:5:4) again has scattered difference tones.

Hon't forget sum tones, which I believe are assumed in many models
to be the same amplitude in every order as difference tones. It is
true that they fall in a register which makes them poor candidates as
roots, and here we already are assuming something special about roots.

>Probably other people on the list can offer other ways of describing this.
>You can also talk about an implied fundamental, which I didn't get into for
>lack of time at the moment.

If we assume implied fundamentals are based on harmonics (which is at
least plausible given their prevalence over subharmonics in the spectra
of naturally-occurring sounds), then we have a candidate that explains
the difference between major and minor. It's rather ad hoc, but sometimes
ad hoc is good. Throw in some hand waving about how combination-
sensitive neurons could account for an implied-fundamental perception,
and you can even start sounding mildly affected.

-Carl

🔗Kurt Bigler <kkb@breathsense.com>

on 8/13/04 12:25 AM, Carl Lumma <ekin@lumma.org> wrote:

>> You've got it right, I think. If you flip the intervals in a "harmonic
>> chord" (also called otonal) top-to-bottom then in the resulting chord
>> (called utonal) the difference tones don't tend to coincide and reinforce
>> each other as they do in an otonal chord.
>
> That's only partly true. Here's a classic from Paul Erlich:
>
>>>> The importance and beauty of utonal formations
>>>> in Partch's music and elsewhere is undeniable (I heard Catler play some
>>>> beautiful 7-limit utonalities on Sunday). The fact that undertone series
>>>> as scales can be more easily constructed by man than overtone series as
>>>> scales is also undeniable. The fact that utonal chords have a lower
>>>> first common overtone,

Of what consequence is a first common overtone? Do you ever hear it?

>>>> and a greater rate of occurence of higher common
>>>> overtones,

And I thought I'd heard Paul Erlich say elsewhere something to the effect
that the significance of common overtones is not clear. At least I'm pretty
sure he said that the highest common overtone (what would in an utonal chord
be *analogous* to the implied fundamental of an otaonal chord) is not
*heard*. But I can't find the reference, as usual!

>> There was a thread about this in relation to why minor chords are "sad"
>> several months back. It makes sense to me that without the harmonic
>> reinforcement it is much like the individual intervals are "lonely" rather
>> than really forming a single harmony. Yet the same intervals may be present
>> as are present in a harmonic chord, so there is still a sense of harmony,
>> just a different kind.
>
> Exactly. But the missing part doesn't have to do with partials (in
> my version of the thesis), but rather the virtual pitch mechanism.

Yet the virtual pitch mechanism has *something* to do with the implied
fundamental and the implied fundamental in otonal chords is often way too
low to be in the audible range. So the missing part could conceivably be
the audible implied fundamental. However in the case of off-just chords
this may be missing yet the otonal kind of harmony may seem to remain, so
now I am wonderining. But on the other hand what does the virtual pitch
mechanism have to do with an off-just otonal chord?

>> So for the difference tones, 4:5:6 produces difference tones of 1,1, and 2,
>> reinforcing the fundamental and the octave. But 10:12:15 produces
>> difference tones of 2, 3, and 5 and so reinforcement is scattered. This
>> becomes even clearer in bigger chords: 4:5:6:7 has even stronger
>> reinforcement whereas 1/(7:6:5:4) again has scattered difference tones.
>
> Hon't forget sum tones, which I believe are assumed in many models
> to be the same amplitude in every order as difference tones. It is
> true that they fall in a register which makes them poor candidates as
> roots, and here we already are assuming something special about roots.

Yes, yet I don't tend to hear them much, if at all. They don't seem to
enter into my sense of harmony, at least not in a way I have yet become
conscious of.

>> Probably other people on the list can offer other ways of describing this.
>> You can also talk about an implied fundamental, which I didn't get into for
>> lack of time at the moment.
>
> If we assume implied fundamentals are based on harmonics (which is at
> least plausible given their prevalence over subharmonics in the spectra
> of naturally-occurring sounds), then we have a candidate that explains
> the difference between major and minor. It's rather ad hoc, but sometimes
> ad hoc is good. Throw in some hand waving about how combination-
> sensitive neurons could account for an implied-fundamental perception,
> and you can even start sounding mildly affected.

This sounds like it contradicts what you were saying above, that the thing
"missing" doesn't have to do with partials but rather with virtual pitch. ??

-Kurt

🔗Kurt Bigler <kkb@breathsense.com>

on 8/13/04 4:42 PM, Kurt Bigler <kkb@breathsense.com> wrote:

> on 8/13/04 12:25 AM, Carl Lumma <ekin@lumma.org> wrote:
>
>>> You've got it right, I think. If you flip the intervals in a "harmonic
>>> chord" (also called otonal) top-to-bottom then in the resulting chord
>>> (called utonal) the difference tones don't tend to coincide and reinforce
>>> each other as they do in an otonal chord.
>>
>> That's only partly true. Here's a classic from Paul Erlich:
>>
>>>>> The importance and beauty of utonal formations
>>>>> in Partch's music and elsewhere is undeniable (I heard Catler play some
>>>>> beautiful 7-limit utonalities on Sunday). The fact that undertone series
>>>>> as scales can be more easily constructed by man than overtone series as
>>>>> scales is also undeniable. The fact that utonal chords have a lower
>>>>> first common overtone,
>
> Of what consequence is a first common overtone? Do you ever hear it?

Meanwhile I found this other quote from you and from Paul:

on 12/3/03 12:29 PM, Paul Erlich <paul@stretch-music.com> wrote:
> In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>> I've gotten this with utonal 9th chords on my slide guitar
...
> Anyway, guitar is about the only instrument I've gotten
> real "specialness" from such utonalities. It might have something to
> do with sympathetic resonance involving that common overtone.

Interesting. I definitely seem to have "matured" into the beginning of an
otonal phase. As with wine, so with music.

-Kurt

🔗Carl Lumma <ekin@lumma.org>

>>>>> and a greater rate of occurence of higher common
>>>>> overtones,
>
>And I thought I'd heard Paul Erlich say elsewhere something to the
>effect that the significance of common overtones is not clear.

It isn't.

>At least I'm pretty sure he said that the highest common overtone
>(what would in an utonal chord be *analogous* to the implied
>fundamental of an otaonal chord)

How is it analogous?

>>> There was a thread about this in relation to why minor chords
>>> are "sad" several months back. It makes sense to me that without
>>> the harmonic reinforcement it is much like the individual
>>> intervals are "lonely" rather than really forming a single
>>> harmony. Yet the same intervals may be present as are present in
>>> a harmonic chord, so there is still a sense of harmony, just a
>>> different kind.
>>
>> Exactly. But the missing part doesn't have to do with partials
>> (in my version of the thesis), but rather the virtual pitch
>> mechanism.
>
>Yet the virtual pitch mechanism has *something* to do with the
>implied fundamental and the implied fundamental in otonal chords
>is often way too low to be in the audible range.

The virtual pitch mechanism helps define "audible". Is is one
of the sources of the sensation of pitch. If your calculations
tell you that the implied fundamental is inaudible, your
calculations are probably wrong.

>So the missing part could conceivably be the audible implied
>fundamental. However in the case of off-just chords this may
>be missing

Lost you here...

>yet the otonal kind of harmony may seem to remain, so now I am
>wonderining. But on the other hand what does the virtual pitch
>mechanism have to do with an off-just otonal chord?

The virtual pitch mechanism is very forgiving. It tries, even
forces everything into a harmonic series. You should read up on
harmonic entropy. I thought you had already done so.

>>> So for the difference tones, 4:5:6 produces difference tones of 1,1,
>>> and 2, reinforcing the fundamental and the octave. But 10:12:15
>>> produces difference tones of 2, 3, and 5 and so reinforcement is
>>> scattered. This becomes even clearer in bigger chords: 4:5:6:7
>>> has even stronger reinforcement whereas 1/(7:6:5:4) again has
>>> scattered difference tones.

Have you actually listened to these?

>> Don't forget sum tones, which I believe are assumed in many models
>> to be the same amplitude in every order as difference tones. It is
>> true that they fall in a register which makes them poor candidates as
>> roots, and here we already are assuming something special about roots.
>
>Yes, yet I don't tend to hear them much, if at all. They don't seem to
>enter into my sense of harmony, at least not in a way I have yet become
>conscious of.

But you hear difference tones? My experience is that both are
audible, yet neither are primary in musical consonance or harmony,
in typical Western-music environs.

>> If we assume implied fundamentals are based on harmonics (which is
>> at least plausible given their prevalence over subharmonics in the
>> spectra of naturally-occurring sounds), then we have a candidate
>> that explains the difference between major and minor. It's rather
>> ad hoc, but sometimes ad hoc is good. Throw in some hand waving
>> about how combination-sensitive neurons could account for an
>> implied-fundamental perception, and you can even start sounding
>> mildly affected.
>
>This sounds like it contradicts what you were saying above, that the
>thing "missing" doesn't have to do with partials but rather with
>virtual pitch. ??

I probably should have said, "based no the harmonic series". And
remember, everything in the virtual pitch mechanism happens after
spectral decomposition in the cochlea.

-Carl

🔗Kurt Bigler <kkb@breathsense.com>

on 8/13/04 11:04 PM, Carl Lumma <ekin@lumma.org> wrote:

>>>>>> and a greater rate of occurence of higher common overtones,
>> At least I'm pretty sure he said that the highest common overtone
>> (what would in an utonal chord be *analogous* to the implied
>> fundamental of an otaonal chord)
> How is it analogous?

By mathematical inversion (or whatever you want to call it) only.

Actually "highest common overtone" is maybe misleading. I was looking for
the true mathematical analogy to the "1" which may not be (usually isn't)
present in the otonal chord. This would be the "1" which is not present in
the utonal chord. So that is the highest common harmonic of the
*fundamentals* of the notes appearing in the chord.

Or perhaps it will make it more obvious if I name a chord. Take the 4:5:6:7
as the otonal example. The implied fundamental is the 1. You could add
that to the chord and have 1:4:5:6:7. On the utonal side take 1/(7:6:5:4).
The "analog" to what I called the implied fundamental is again the 1. You
could add it to the chord and have 1/(7:6:5:4:1). Curious how hard it was
to name that "1" in the otonal case. There is no name for the 1 in the
utonal case, right?

> The virtual pitch mechanism helps define "audible". Is is one
> of the sources of the sensation of pitch. If your calculations
> tell you that the implied fundamental is inaudible, your
> calculations are probably wrong.

If the implied fundamental is calculably sub-audible, then there is probably
no implied fundamental at all, as far as perception goes.

>> So the missing part could conceivably be the audible implied
>> fundamental. However in the case of off-just chords this may
>> be missing
> Lost you here...

Detune the just chord until the implied fundamental (or whatever you want to
call it) is no longer there. Yet the harmony is not *gone*. This makes
descriptions of harmony that depend on the implied fundamental questionable,
is all I was saying.
>
>> yet the otonal kind of harmony may seem to remain, so now I am
>> wonderining. But on the other hand what does the virtual pitch
>> mechanism have to do with an off-just otonal chord?
> The virtual pitch mechanism is very forgiving. It tries, even
> forces everything into a harmonic series. You should read up on
> harmonic entropy. I thought you had already done so.

Maybe I was mistaking "virtual pitch" for another term. No I've only
scratched the surface of harmonic entropy, although I thought that I had a
good intuition of it.

>>>> So for the difference tones, 4:5:6 produces difference tones of 1,1,
>>>> and 2, reinforcing the fundamental and the octave. But 10:12:15
>>>> produces difference tones of 2, 3, and 5 and so reinforcement is
>>>> scattered. This becomes even clearer in bigger chords: 4:5:6:7
>>>> has even stronger reinforcement whereas 1/(7:6:5:4) again has
>>>> scattered difference tones.
> Have you actually listened to these?

Yes, and bigger chords. I definitely hear nothing like an implied
fundamental in big otonal chords. But I need to listen more!

>> Yes, yet I don't tend to hear them much, if at all. They don't seem to
>> enter into my sense of harmony, at least not in a way I have yet become
>> conscious of.
> But you hear difference tones? My experience is that both are
> audible, yet neither are primary in musical consonance or harmony,
> in typical Western-music environs.

Maybe I need to listen to utonal chords some more and it will come to me. I
almost have a hint of it in my audible reccollection and maybe I just wasn't
listening for the right thing.

>>> If we assume implied fundamentals are based on harmonics (which is
>>> at least plausible given their prevalence over subharmonics in the
>>> spectra of naturally-occurring sounds), then we have a candidate
>>> that explains the difference between major and minor.
>> This sounds like it contradicts what you were saying above, that the
>> thing "missing" doesn't have to do with partials but rather with
>> virtual pitch. ??
> I probably should have said, "based no the harmonic series". And
> remember, everything in the virtual pitch mechanism happens after
> spectral decomposition in the cochlea.

Well it isn't really "decomposition" in an absolute theroetical sense (i.e.
not like an FFT) because the filtering (like any physical resonance) isn't
ideal.

In any case I need to look into this more before I can say more. It seems
to me that separating frequencies into places helps a nonlinear system to
retain more information, makes it more likely that things like periodicity
pitch and/or the relational aspects (e.g. timbre/chord interactions) of
difference tones could be perceived.

But I think you are connecting the harmonic series to the "place" mechanism
(is that what you meant?), whereas I guess I was thinking the place
mechanism was incidental. I'm not expert enough to comment further.

-Kurt

🔗Carl Lumma <ekin@lumma.org>

>>> the highest common overtone
>>> (what would in an utonal chord be *analogous* to the implied
>>> fundamental of an otaonal chord)
>>
>> How is it analogous?
>
>By mathematical inversion (or whatever you want to call it) only.

I won't know exactly what you mean without an example. But it
may be worth pointing out that related-by-inversion doesn't
imply related-by-psychoacoustics. And, for a variety of reasons,
the most prominent difference tone does not always coincide with
the implied fundamental even in the case of otonal chords.

>Or perhaps it will make it more obvious if I name a chord. Take
>the 4:5:6:7 as the otonal example. The implied fundamental is
>the 1.

Not necessarily.

>You could add that to the chord and have 1:4:5:6:7. On the utonal
>side take 1/(7:6:5:4). The "analog" to what I called the implied
>fundamental is again the 1.

That chord may also be written 1/1:7/6:7/5:7/4. If you listen to
this, what do you hear? Try different registers. Then try
different inversions!

>Curious how hard it was to name that "1" in the
>otonal case. There is no name for the 1 in the utonal case, right?

Not sure what you're asking -- both pitches may be called 1.

>> The virtual pitch mechanism helps define "audible". Is is one
>> of the sources of the sensation of pitch. If your calculations
>> tell you that the implied fundamental is inaudible, your
>> calculations are probably wrong.
>
>If the implied fundamental is calculably sub-audible, then there
>is probably no implied fundamental at all, as far as perception goes.

How did you arrive at this method of calculation?

>>> So the missing part could conceivably be the audible implied
>>> fundamental. However in the case of off-just chords this may
>>> be missing
>> Lost you here...
>
>Detune the just chord until the implied fundamental (or whatever you
>want to call it) is no longer there. Yet the harmony is not *gone*.
>This makes descriptions of harmony that depend on the implied
>fundamental questionable, is all I was saying.

Have you tried this?

>>> yet the otonal kind of harmony may seem to remain, so now I am
>>> wonderining. But on the other hand what does the virtual pitch
>>> mechanism have to do with an off-just otonal chord?
>>
>> The virtual pitch mechanism is very forgiving. It tries, even
>> forces everything into a harmonic series. You should read up on
>> harmonic entropy. I thought you had already done so.
>
>Maybe I was mistaking "virtual pitch" for another term. No I've only
>scratched the surface of harmonic entropy, although I thought that I
>had a good intuition of it.

I just forwarded you some introductory material on it.

>I definitely hear nothing like an implied fundamental in big otonal
>chords.

This is an atypical observation.

>>>> If we assume implied fundamentals are based on harmonics (which is
>>>> at least plausible given their prevalence over subharmonics in the
>>>> spectra of naturally-occurring sounds), then we have a candidate
>>>> that explains the difference between major and minor.
>>>
>>> This sounds like it contradicts what you were saying above, that the
>>> thing "missing" doesn't have to do with partials but rather with
>>> virtual pitch. ??

I don't know what you mean by "missing". Try replacing my first
sentence with: ""Virtual pitch assumes a harmonic series.""

>But I think you are connecting the harmonic series to the "place"
>mechanism (is that what you meant?),

That's roughly the opposite of what I meant.

>> remember, everything in the virtual pitch mechanism happens after
>> spectral decomposition in the cochlea.
>
>Well it isn't really "decomposition" in an absolute theroetical
>sense (i.e. not like an FFT) because the filtering (like any physical
>resonance) isn't ideal.

It certainly is decomposition, and for the things humans expect
to hear a constant-Q transform is more appropriate than an FFT.
The human auditory system achieves more realtime temporal and
spectral resolution than any man-made system I'm aware of.

-Carl

🔗traktus5 <kj4321@hotmail.com>

thanks for the informative reply!

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 8/12/04 1:41 PM, traktus5 <kj4321@h...> wrote:
>
> > I read through Monz's dictionary at tonalsoft, but am still
> > unclear....How do subharmonics and combinations relate? It's my
> > understanding that the former is a harmonic series inverted (going
> > downward), whereas combination tones and their primary tones
exist in
> > a normal harmonic series...but they both involve diffenece
tones...so
> > I'm confused...
>
> You've got it right, I think. If you flip the intervals in
a "harmonic
> chord" (also called otonal) top-to-bottom then in the resulting
chord
> (called utonal) the difference tones don't tend to coincide and
reinforce
> each other as they do in an otonal chord. But the otonal/utonal
distinction
> is a little ambiguous and depends on the numbers involved.
>
> For example the major triad 4:5:6 consists of 4:5 below 5:6 and
flipped this
> becomes the minor triad, with a 4:5 above a 5:6 which is no longer
obviously
> a harmonic chord. Actually 4:5:6 flipped can be spelled as
1/6:1/5:1/4 [or
> abbreviated as 1/(6:5:4)], a utonal spelling, but can also be
spelled (by
> multiplying by 60) as 10:12:15 which is also a harmonic (otonal)
spelling.
> But the numbers are higher in the otonal spelling and so the
harmonic aspect
> is less clear and the chord is more likely to be considered utonal
because
> the utonal spelling involves lower numbers. And also the minor
triad
> intervals themselves are individually still 5:6 and 4:5 yet the
chord can
> not be spelled anymore using the numbers 4,5,6, so this is another
> indication that something is quite different from the situation in a
> harmonic chord.
>
> There was a thread about this in relation to why minor chords
are "sad"
> several months back. It makes sense to me that without the harmonic
> reinforcement it is much like the individual intervals are "lonely"
rather
> than really forming a single harmony. Yet the same intervals may
be present
> as are present in a harmonic chord, so there is still a sense of
harmony,
> just a different kind.
>
> So for the difference tones, 4:5:6 produces difference tones of
1,1, and 2,
> reinforcing the fundamental and the octave. But 10:12:15 produces
> difference tones of 2, 3, and 5 and so reinforcement is scattered.
This
> becomes even clearer in bigger chords: 4:5:6:7 has even stronger
> reinforcement whereas 1/(7:6:5:4) again has scattered difference
tones.
>
> Probably other people on the list can offer other ways of
describing this.
> You can also talk about an implied fundamental, which I didn't get
into for
> lack of time at the moment.
>
> -Kurt

🔗jjensen142000 <jjensen14@hotmail.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Kurt Bigler <kkb@breathsense.com>

on 8/15/04 12:25 PM, jjensen142000 <jjensen14@hotmail.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>> Detune the just chord until the implied fundamental (or whatever
> you want to
>> call it) is no longer there. Yet the harmony is not *gone*. This
> makes
>> descriptions of harmony that depend on the implied fundamental
> questionable,
>> is all I was saying.
>
>
> I would say that if you de-tune a chord to the point where it
> sound horrible, that is where you have utterly lost any implied
> fundamental. That is what harmony is!

I could be a little more specific here. Major chords in the good keys of a
well-temperament are still harmonious to me, yet I pretty much never
*physically* heard an implied fundamental in a well-temperament. Maybe it
is still there but it is so much softer that I never heard it.

On the other hand I've always had *impressions* of implied fundamentals in
non-just tunings. But I can't say I ever actually *heard* them, and so I
think it is really a different phenomenon.

-Kurt

🔗Carl Lumma <ekin@lumma.org>

🔗Kurt Bigler <kkb@breathsense.com>

Carl,

on 8/15/04 6:15 PM, Carl Lumma <ekin@lumma.org> wrote:

>> I could be a little more specific here. Major chords in the good
>> keys of a well-temperament are still harmonious to me, yet I
>> pretty much never *physically* heard an implied fundamental in
>> a well-temperament. Maybe it is still there but it is so much
>> softer that I never heard it.
>
> I don't think you're listening to the right thing. But then, I
> have no way to be sure that I am.

Well I'm not really talking about anything that requires much in the way of
deep listening. In a just tuning it comes out and knocks me on the head.
In a well-temperament it ain't there. Ain't there means it didn't come out
and knock me on the head. I could try to listen more closely but to me
that's already a statement about the strength of the audible implied
fundamental--if I have to listen that hard then it isn't very strong.

We can argue about whether I am hearing this or that phenomenon but I am
talking about something that has the correct pitch to be the fundamental of
a just chord. Maybe its the implied fundamental, maybe its difference
tones. Whatever it is, it works.

On the other hand in Paul Bailey's "modified meantone" tuning on my piano I
can hear similar resultant tones in the C major chord. The key C gets
pretty close to just in that tuning, but I don't have the data handy. In
any case I think the resultant tones don't even occur strongly in any key
besides C so it is an indication of just how fussy the threshold is for me
to here them. On the other hand the piano is another beastie and makes it
harder to judge tunings in an absolute way. But the tuning on my piano is a
narrow tuning, meaning the nearby consonances are emphasized, so I think
this is still a fairly accurate statement about the tuning itself when
playing chords within an octave. I recall the same was true of that tuning
on the organ, but I'm not 100% sure.

When you're over hear some time we can listen together for all the things we
have discussed in this category recently:

* implied fundamental
* sum tones
* common overtones in utonal chords

and then we can report our results back to the list.

-Kurt

🔗jjensen142000 <jjensen14@hotmail.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

/tuning/topicId_55432.html#55489

> on 8/13/04 11:04 PM, Carl Lumma <ekin@l...> wrote:
>
> >>>>>> and a greater rate of occurence of higher common overtones,
> >> At least I'm pretty sure he said that the highest common overtone
> >> (what would in an utonal chord be *analogous* to the implied
> >> fundamental of an otaonal chord)
> > How is it analogous?
>
> By mathematical inversion (or whatever you want to call it) only.
>
> Actually "highest common overtone" is maybe misleading. I was
looking for
> the true mathematical analogy to the "1" which may not be (usually
isn't)
> present in the otonal chord. This would be the "1" which is not
present in
> the utonal chord. So that is the highest common harmonic of the
> *fundamentals* of the notes appearing in the chord.
>
> Or perhaps it will make it more obvious if I name a chord. Take
the 4:5:6:7
> as the otonal example. The implied fundamental is the 1. You
could add
> that to the chord and have 1:4:5:6:7. On the utonal side take 1/
(7:6:5:4).
> The "analog" to what I called the implied fundamental is again the
1. You
> could add it to the chord and have 1/(7:6:5:4:1). Curious how hard
it was
> to name that "1" in the otonal case. There is no name for the 1 in
the
> utonal case, right?
>

***Wow... I win bingo. Something I can answer. It's traditionally
called the "guide tone" Kurt...

best,

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗traktus5 <kj4321@hotmail.com>

By the way, this is the best description I've ever seen (and I'm the
one who started the 'why is the minor chord sad thread') of the minor
chord and its character --and with clarification of utonal/otonal to
boot--which for music theory (not so much tuning, but harmonic
entropy yes) guy like me, has been very unclear reading hear in the
past...thanks! This is something I could actually explain to the
average classical music theory student...

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 8/13/04 12:25 AM, Carl Lumma <ekin@l...> wrote:
>
> >> You've got it right, I think. If you flip the intervals in
a "harmonic
> >> chord" (also called otonal) top-to-bottom then in the resulting
chord
> >> (called utonal) the difference tones don't tend to coincide and
reinforce
> >> each other as they do in an otonal chord.
> >
> > That's only partly true. Here's a classic from Paul Erlich:
> >
> >>>> The importance and beauty of utonal formations
> >>>> in Partch's music and elsewhere is undeniable (I heard Catler
play some
> >>>> beautiful 7-limit utonalities on Sunday). The fact that
undertone series
> >>>> as scales can be more easily constructed by man than overtone
series as
> >>>> scales is also undeniable. The fact that utonal chords have a
lower
> >>>> first common overtone,
>
> Of what consequence is a first common overtone? Do you ever hear
it?
>
> >>>> and a greater rate of occurence of higher common
> >>>> overtones,
>
> And I thought I'd heard Paul Erlich say elsewhere something to the
effect
> that the significance of common overtones is not clear. At least
I'm pretty
> sure he said that the highest common overtone (what would in an
utonal chord
> be *analogous* to the implied fundamental of an otaonal chord) is
not
> *heard*. But I can't find the reference, as usual!
>
> >> There was a thread about this in relation to why minor chords
are "sad"
> >> several months back. It makes sense to me that without the
harmonic
> >> reinforcement it is much like the individual intervals
are "lonely" rather
> >> than really forming a single harmony. Yet the same intervals
may be present
> >> as are present in a harmonic chord, so there is still a sense of
harmony,
> >> just a different kind.
> >
> > Exactly. But the missing part doesn't have to do with partials
(in
> > my version of the thesis), but rather the virtual pitch mechanism.
>
> Yet the virtual pitch mechanism has *something* to do with the
implied
> fundamental and the implied fundamental in otonal chords is often
way too
> low to be in the audible range. So the missing part could
conceivably be
> the audible implied fundamental. However in the case of off-just
chords
> this may be missing yet the otonal kind of harmony may seem to
remain, so
> now I am wonderining. But on the other hand what does the virtual
pitch
> mechanism have to do with an off-just otonal chord?
>
> >> So for the difference tones, 4:5:6 produces difference tones of
1,1, and 2,
> >> reinforcing the fundamental and the octave. But 10:12:15
produces
> >> difference tones of 2, 3, and 5 and so reinforcement is
scattered. This
> >> becomes even clearer in bigger chords: 4:5:6:7 has even stronger
> >> reinforcement whereas 1/(7:6:5:4) again has scattered difference
tones.
> >
> > Hon't forget sum tones, which I believe are assumed in many models
> > to be the same amplitude in every order as difference tones. It
is
> > true that they fall in a register which makes them poor
candidates as
> > roots, and here we already are assuming something special about
roots.
>
> Yes, yet I don't tend to hear them much, if at all. They don't
seem to
> enter into my sense of harmony, at least not in a way I have yet
become
> conscious of.
>
> >> Probably other people on the list can offer other ways of
describing this.
> >> You can also talk about an implied fundamental, which I didn't
get into for
> >> lack of time at the moment.
> >
> > If we assume implied fundamentals are based on harmonics (which
is at
> > least plausible given their prevalence over subharmonics in the
spectra
> > of naturally-occurring sounds), then we have a candidate that
explains
> > the difference between major and minor. It's rather ad hoc, but
sometimes
> > ad hoc is good. Throw in some hand waving about how combination-
> > sensitive neurons could account for an implied-fundamental
perception,
> > and you can even start sounding mildly affected.
>
> This sounds like it contradicts what you were saying above, that
the thing
> "missing" doesn't have to do with partials but rather with virtual
pitch. ??
>
> -Kurt

🔗Kurt Bigler <kkb@breathsense.com>

on 8/18/04 1:22 AM, traktus5 <kj4321@hotmail.com> wrote:

> By the way, this is the best description I've ever seen (and I'm the
> one who started the 'why is the minor chord sad thread') of the minor
> chord and its character --and with clarification of utonal/otonal to
> boot--which for music theory (not so much tuning, but harmonic
> entropy yes) guy like me, has been very unclear reading hear in the
> past...thanks! This is something I could actually explain to the
> average classical music theory student...

Well keep in mind the corrections other people have made.

In particular I have probably been using the term "implied fundamental"
incorrectly. I used the term to fill an "intuitive" slot for a concept that
in my mind was described by exactly those two words. But existing usage was
more specific. To me "implied fundamental" meant just what the words say
(to me), a pitch that I hear that matches a fundamental pitch that is not
present but which is implied by a chord, i.e. the "1" that is not present in
a 4:5:6:7 chord. This heard phenomenon might be due to difference tones, it
might be due to virtual pitch, whatever. But it is a pitch that is not
present that is heard, and matches the fundamental "implied" by the chord.
But this is not necessarily the correct usage.

So I will try to be clearer and use the term "implied fundamental" only with
its original meaning which I intend to clarify for myself before I resort to
using it again.

-Kurt

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> on 8/13/04 12:25 AM, Carl Lumma <ekin@l...> wrote:
>>
>>>> You've got it right, I think. If you flip the intervals in
> a "harmonic
>>>> chord" (also called otonal) top-to-bottom then in the resulting
> chord
>>>> (called utonal) the difference tones don't tend to coincide and
> reinforce
>>>> each other as they do in an otonal chord.
>>>
>>> That's only partly true. Here's a classic from Paul Erlich:
>>>
>>>>>> The importance and beauty of utonal formations
>>>>>> in Partch's music and elsewhere is undeniable (I heard Catler
> play some
>>>>>> beautiful 7-limit utonalities on Sunday). The fact that
> undertone series
>>>>>> as scales can be more easily constructed by man than overtone
> series as
>>>>>> scales is also undeniable. The fact that utonal chords have a
> lower
>>>>>> first common overtone,
>>
>> Of what consequence is a first common overtone? Do you ever hear
> it?
>>
>>>>>> and a greater rate of occurence of higher common
>>>>>> overtones,
>>
>> And I thought I'd heard Paul Erlich say elsewhere something to the
> effect
>> that the significance of common overtones is not clear. At least
> I'm pretty
>> sure he said that the highest common overtone (what would in an
> utonal chord
>> be *analogous* to the implied fundamental of an otaonal chord) is
> not
>> *heard*. But I can't find the reference, as usual!
>>
>>>> There was a thread about this in relation to why minor chords
> are "sad"
>>>> several months back. It makes sense to me that without the
> harmonic
>>>> reinforcement it is much like the individual intervals
> are "lonely" rather
>>>> than really forming a single harmony. Yet the same intervals
> may be present
>>>> as are present in a harmonic chord, so there is still a sense of
> harmony,
>>>> just a different kind.
>>>
>>> Exactly. But the missing part doesn't have to do with partials
> (in
>>> my version of the thesis), but rather the virtual pitch mechanism.
>>
>> Yet the virtual pitch mechanism has *something* to do with the
> implied
>> fundamental and the implied fundamental in otonal chords is often
> way too
>> low to be in the audible range. So the missing part could
> conceivably be
>> the audible implied fundamental. However in the case of off-just
> chords
>> this may be missing yet the otonal kind of harmony may seem to
> remain, so
>> now I am wonderining. But on the other hand what does the virtual
> pitch
>> mechanism have to do with an off-just otonal chord?
>>
>>>> So for the difference tones, 4:5:6 produces difference tones of
> 1,1, and 2,
>>>> reinforcing the fundamental and the octave. But 10:12:15
> produces
>>>> difference tones of 2, 3, and 5 and so reinforcement is
> scattered. This
>>>> becomes even clearer in bigger chords: 4:5:6:7 has even stronger
>>>> reinforcement whereas 1/(7:6:5:4) again has scattered difference
> tones.
>>>
>>> Hon't forget sum tones, which I believe are assumed in many models
>>> to be the same amplitude in every order as difference tones. It
> is
>>> true that they fall in a register which makes them poor
> candidates as
>>> roots, and here we already are assuming something special about
> roots.
>>
>> Yes, yet I don't tend to hear them much, if at all. They don't
> seem to
>> enter into my sense of harmony, at least not in a way I have yet
> become
>> conscious of.
>>
>>>> Probably other people on the list can offer other ways of
> describing this.
>>>> You can also talk about an implied fundamental, which I didn't
> get into for
>>>> lack of time at the moment.
>>>
>>> If we assume implied fundamentals are based on harmonics (which
> is at
>>> least plausible given their prevalence over subharmonics in the
> spectra
>>> of naturally-occurring sounds), then we have a candidate that
> explains
>>> the difference between major and minor. It's rather ad hoc, but
> sometimes
>>> ad hoc is good. Throw in some hand waving about how combination-
>>> sensitive neurons could account for an implied-fundamental
> perception,
>>> and you can even start sounding mildly affected.
>>
>> This sounds like it contradicts what you were saying above, that
> the thing
>> "missing" doesn't have to do with partials but rather with virtual
> pitch. ??
>>
>> -Kurt