Yahoo Tuning Groups Ultimate Backup tuning-math Question for Dave Keenan

If a timbre has

2nd partial off by < 10.4 cents
3rd partial off by < 16.5 cents
4th partial off by < 20.8 cents
5th partial off by < 24.1 cents
6th partial off by < 26.9 cents

does it 'hold together' as a single pitch, or does it fall apart into
multiple pitches?

(I'll try to prepare some examples, playing random scales . . .)

If yes:

If I take any inharmonic timbre with one loud partial and some quiet,
unimportant ones (very many fall into this category), and use a
tuning system where

2:1 off by < 10.4 cents
3:1 off by < 16.5 cents
4:1 off by < 20.8 cents
5:1 off by < 24.1 cents
6:1 off by < 26.9 cents

and play a piece with full triadic harmony, doesn't it follow that
the harmony should 'hold together' the way 5-limit triads should?

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> If a timbre has
>
> 2nd partial off by < 10.4 cents
> 3rd partial off by < 16.5 cents
> 4th partial off by < 20.8 cents
> 5th partial off by < 24.1 cents
> 6th partial off by < 26.9 cents
>
> does it 'hold together' as a single pitch, or does it fall apart into
> multiple pitches?

Or is it experienced as somewhere in between, having a single but
poorly-defined pitch. I don't know. I trust your ears more than mine
on such questions, but I'd like to hear it.

Note that my objection to your claimed 5-limit temperaments with very
large errors was not a disagreement between yours and my perception,
but more of an epistemological problem. I sense you are setting up
something to try to catch me in a similar logical mistake. This is
fun. :-)

> (I'll try to prepare some examples, playing random scales . . .)
>
> If yes:

What does "yes" mean here?

> If I take any inharmonic timbre with one loud partial and some quiet,
> unimportant ones (very many fall into this category), and use a
> tuning system where
>
> 2:1 off by < 10.4 cents
> 3:1 off by < 16.5 cents
> 4:1 off by < 20.8 cents
> 5:1 off by < 24.1 cents
> 6:1 off by < 26.9 cents
>
> and play a piece with full triadic harmony, doesn't it follow that
> the harmony should 'hold together' the way 5-limit triads should?

I don't know. What has the single loud partial got to do with it? Is
this partial one of those mentioned above?

We know that with quiet sine waves nothing special happens with any
dyad except a unison, and that loud sine waves work like harmonic
timbres presumably due to harmonics being generated in the
nonlinearities of the ear-brain system. Don't we?

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > If a timbre has
> >
> > 2nd partial off by < 10.4 cents
> > 3rd partial off by < 16.5 cents
> > 4th partial off by < 20.8 cents
> > 5th partial off by < 24.1 cents
> > 6th partial off by < 26.9 cents
> >
> > does it 'hold together' as a single pitch, or does it fall apart
into
> > multiple pitches?
>
> Or is it experienced as somewhere in between, having a single but
> poorly-defined pitch. I don't know. I trust your ears more than mine
> on such questions, but I'd like to hear it.

OK, I'll put something together when I have a chance . . .

> Note that my objection to your claimed 5-limit temperaments with
very
> large errors was not a disagreement between yours and my perception,
> but more of an epistemological problem. I sense you are setting up
> something to try to catch me in a similar logical mistake. This is
> fun. :-)
>
> > (I'll try to prepare some examples, playing random scales . . .)
> >
> > If yes:
>
> What does "yes" mean here?

the sound holds together as a single pitch.

> > If I take any inharmonic timbre with one loud partial and some
quiet,
> > unimportant ones (very many fall into this category), and use a
> > tuning system where
> >
> > 2:1 off by < 10.4 cents
> > 3:1 off by < 16.5 cents
> > 4:1 off by < 20.8 cents
> > 5:1 off by < 24.1 cents
> > 6:1 off by < 26.9 cents
> >
> > and play a piece with full triadic harmony, doesn't it follow
that
> > the harmony should 'hold together' the way 5-limit triads should?
>
> I don't know. What has the single loud partial got to do with it? Is
> this partial one of those mentioned above?

No, it essentially determines the pitch of the timbre.

> We know that with quiet sine waves nothing special happens with any
> dyad except a unison, and that loud sine waves work like harmonic
> timbres presumably due to harmonics

and combinational tones . . .

> being generated in the
> nonlinearities of the ear-brain system.

quiet harmonic timbres don't generate combinational tones, so they
won't "work like" loud sine waves.

> Don't we?

That also ignores virtual pitch. A set of quiet sine waves can evoke
a single pitch which does not agree with any combinational tone . . .
at certain intervals, the pitch evoked will be least ambiguous, which
is certainly 'something special happening' . . .

The fact is that, when using inharmonic timbres of the sort I
described, Western music seems to retain all it meaning: certain
(dissonant) chords resolving to other (consonant) chords, etc., all
sounds quite logical. My sense (and the opinion expressed in
Parncutt's book, for example) is that *harmony* is in fact very
closely related to the virtual pitch phenomenon. We already know,
from our listening tests on the harmonic entropy list, that the
sensory dissonance of a chord isn't a function of the sensory
dissonances of its constituent dyads. Furthermore, you seem to be
defining "something special" in a local sense as a function of
interval size, but in real music you don't get to evaluate each
sonority by detuning various intervals various amounts, which
this "specialness" would seem to require for its detection.

The question I'm asking is, with what other tonal systems, besides
the Western one, is this going to be possible in.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> > What does "yes" mean here?
>
> the sound holds together as a single pitch.

My guess is that it will be experienced as a single pitch, but one
that cannot be accurately determined. The pitch will be fuzzy or vague
in a similar way to that of a harmonic note of very short duration.

> > > If I take any inharmonic timbre with one loud partial and some
> quiet,
> > > unimportant ones (very many fall into this category), and use a
> > > tuning system where
> > >
> > > 2:1 off by < 10.4 cents
> > > 3:1 off by < 16.5 cents
> > > 4:1 off by < 20.8 cents
> > > 5:1 off by < 24.1 cents
> > > 6:1 off by < 26.9 cents
> > >
> > > and play a piece with full triadic harmony, doesn't it follow
> that
> > > the harmony should 'hold together' the way 5-limit triads should?
> >
> > I don't know. What has the single loud partial got to do with it? Is
> > this partial one of those mentioned above?
>
> No, it essentially determines the pitch of the timbre.

So the waveform is essentially sinusoidal? Why not use sinusoidal
waves for this thought experiment?

> > We know that with quiet sine waves nothing special happens with any
> > dyad except a unison, and that loud sine waves work like harmonic
> > timbres presumably due to harmonics
>
> and combinational tones . . .

Good point.

> > being generated in the
> > nonlinearities of the ear-brain system.
>
> quiet harmonic timbres don't generate combinational tones, so they
> won't "work like" loud sine waves.
>
> > Don't we?
>
> That also ignores virtual pitch. A set of quiet sine waves can evoke
> a single pitch which does not agree with any combinational tone . . .
> at certain intervals, the pitch evoked will be least ambiguous, which
> is certainly 'something special happening' . . .

How many sine waves in an approximate harmonic series do you need for
this to be experienced? And what arrangements work? I was only
speaking of dyads.

> The fact is that, when using inharmonic timbres of the sort I
> described, Western music seems to retain all it meaning: certain
> (dissonant) chords resolving to other (consonant) chords, etc., all
> sounds quite logical. My sense (and the opinion expressed in
> Parncutt's book, for example) is that *harmony* is in fact very
> closely related to the virtual pitch phenomenon. We already know,
> from our listening tests on the harmonic entropy list, that the
> sensory dissonance of a chord isn't a function of the sensory
> dissonances of its constituent dyads. Furthermore, you seem to be
> defining "something special" in a local sense as a function of
> interval size, but in real music you don't get to evaluate each
> sonority by detuning various intervals various amounts, which
> this "specialness" would seem to require for its detection.
>
> The question I'm asking is, with what other tonal systems, besides
> the Western one, is this going to be possible in.

If by "Western tonal systems", you mean any based on approximating
small whole number ratios of frequency, and by "something special" you
mean "consonance and dissonance between simultaneous tones when using
only sine waves", then I suppose the answer is "none".

What's your point?

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > > What does "yes" mean here?
> >
> > the sound holds together as a single pitch.
>
> My guess is that it will be experienced as a single pitch, but one
> that cannot be accurately determined. The pitch will be fuzzy or
vague
> in a similar way to that of a harmonic note of very short duration.

Thanks for that. Sounds to come.

> > > > If I take any inharmonic timbre with one loud partial and
some
> > quiet,
> > > > unimportant ones (very many fall into this category), and use
a
> > > > tuning system where
> > > >
> > > > 2:1 off by < 10.4 cents
> > > > 3:1 off by < 16.5 cents
> > > > 4:1 off by < 20.8 cents
> > > > 5:1 off by < 24.1 cents
> > > > 6:1 off by < 26.9 cents
> > > >
> > > > and play a piece with full triadic harmony, doesn't it follow
> > that
> > > > the harmony should 'hold together' the way 5-limit triads
should?
> > >
> > > I don't know. What has the single loud partial got to do with
it? Is
> > > this partial one of those mentioned above?
> >
> > No, it essentially determines the pitch of the timbre.
>
> So the waveform is essentially sinusoidal? Why not use sinusoidal
> waves for this thought experiment?

They're not especially musical -- you'll have an easier time hearing
chords as sets of separate notes when the timbre is not a pure sine
wave.

> > > We know that with quiet sine waves nothing special happens with
any
> > > dyad except a unison, and that loud sine waves work like
harmonic
> > > timbres presumably due to harmonics
> >
> > and combinational tones . . .
>
> Good point.
>
> > > being generated in the
> > > nonlinearities of the ear-brain system.
> >
> > quiet harmonic timbres don't generate combinational tones, so
they
> > won't "work like" loud sine waves.
> >
> > > Don't we?
> >
> > That also ignores virtual pitch. A set of quiet sine waves can
evoke
> > a single pitch which does not agree with any combinational
tone . . .
> > at certain intervals, the pitch evoked will be least ambiguous,
which
> > is certainly 'something special happening' . . .
>
> How many sine waves in an approximate harmonic series do you need
for
> this to be experienced? And what arrangements work? I was only
> speaking of dyads.

The effect has been observed with dyads, but most of the experiments
concern three (or more) sine waves, since they evoke the phenomenon
more readily.

> > The fact is that, when using inharmonic timbres of the sort I
> > described, Western music seems to retain all it meaning: certain
> > (dissonant) chords resolving to other (consonant) chords, etc.,
all
> > sounds quite logical. My sense (and the opinion expressed in
> > Parncutt's book, for example) is that *harmony* is in fact very
> > closely related to the virtual pitch phenomenon. We already know,
> > from our listening tests on the harmonic entropy list, that the
> > sensory dissonance of a chord isn't a function of the sensory
> > dissonances of its constituent dyads. Furthermore, you seem to be
> > defining "something special" in a local sense as a function of
> > interval size, but in real music you don't get to evaluate each
> > sonority by detuning various intervals various amounts, which
> > this "specialness" would seem to require for its detection.
> >
> > The question I'm asking is, with what other tonal systems,
besides
> > the Western one, is this going to be possible in.
>
> If by "Western tonal systems", you mean any based on approximating
> small whole number ratios of frequency,

No, I meant diatonic/meantone.

> What's your point?

Did the above really not say anything to you?

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> Thanks for that. Sounds to come.

Thanks. I've listened to them. Definitely single pitches. Can you tell
us the relative amplitides of all the partials. And can we hear a
sustained note around middle C.

> > So the waveform is essentially sinusoidal? Why not use sinusoidal
> > waves for this thought experiment?
>
> They're not especially musical -- you'll have an easier time hearing
> chords as sets of separate notes when the timbre is not a pure sine
> wave.
> > > The fact is that, when using inharmonic timbres of the sort I
> > > described, Western music seems to retain all it meaning: certain
> > > (dissonant) chords resolving to other (consonant) chords, etc.,
> all
> > > sounds quite logical. My sense (and the opinion expressed in
> > > Parncutt's book, for example) is that *harmony* is in fact very
> > > closely related to the virtual pitch phenomenon. We already know,
> > > from our listening tests on the harmonic entropy list, that the
> > > sensory dissonance of a chord isn't a function of the sensory
> > > dissonances of its constituent dyads. Furthermore, you seem to be
> > > defining "something special" in a local sense as a function of
> > > interval size, but in real music you don't get to evaluate each
> > > sonority by detuning various intervals various amounts, which
> > > this "specialness" would seem to require for its detection.
> > >
> > > The question I'm asking is, with what other tonal systems,
> besides
> > > the Western one, is this going to be possible in.
> >
> > If by "Western tonal systems", you mean any based on approximating
> > small whole number ratios of frequency,
>
> No, I meant diatonic/meantone.

OK. So is your question, "In what tonal systems other than
diatonic/meantone is it going to be possible to have dissonant chords
resolving to consonant chords?"?

The obvious answer would seem to be systems in which there are
consonant chords, i.e which approximate (or are) JI at least partially.

> > What's your point?
>
> Did the above really not say anything to you?

Certainly not until you clarified the above. And it might still be a
good idea for you to spell out the conclusion you intend.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > Thanks for that. Sounds to come.
>
> Thanks. I've listened to them. Definitely single pitches. Can you
tell
> us the relative amplitides of all the partials.

sqrt(1), sqrt(2), . . . sqrt(6).

> And can we hear a
> sustained note around middle C.

/tuning-math/files/Erlich/dave1.wav

> > > > The fact is that, when using inharmonic timbres of the sort I
> > > > described, Western music seems to retain all it meaning:
certain
> > > > (dissonant) chords resolving to other (consonant) chords,
etc.,
> > all
> > > > sounds quite logical. My sense (and the opinion expressed in
> > > > Parncutt's book, for example) is that *harmony* is in fact
very
> > > > closely related to the virtual pitch phenomenon. We already
know,
> > > > from our listening tests on the harmonic entropy list, that
the
> > > > sensory dissonance of a chord isn't a function of the sensory
> > > > dissonances of its constituent dyads. Furthermore, you seem
to be
> > > > defining "something special" in a local sense as a function
of
> > > > interval size, but in real music you don't get to evaluate
each
> > > > sonority by detuning various intervals various amounts, which
> > > > this "specialness" would seem to require for its detection.
> > > >
> > > > The question I'm asking is, with what other tonal systems,
> > besides
> > > > the Western one, is this going to be possible in.
> > >
> > > If by "Western tonal systems", you mean any based on
approximating
> > > small whole number ratios of frequency,
> >
> > No, I meant diatonic/meantone.
>
> OK. So is your question, "In what tonal systems other than
> diatonic/meantone is it going to be possible to have dissonant
chords
> resolving to consonant chords?"?

Yes, in a sense, and with timbres that are primarily sine-wave but
have spectral and envelopular aids to being individually heard out.

> The obvious answer would seem to be systems in which there are
> consonant chords, i.e which approximate (or are) JI at least
>partially.

Right -- now is Top pelogic such a system?

> > > What's your point?
> >
> > Did the above really not say anything to you?
>
> Certainly not until you clarified the above. And it might still be a
> good idea for you to spell out the conclusion you intend.

That the phenomenon responsible for central (aka 'virtual') pitch
allows x amount of Tenney-weighted error in the partials of a single
pitch, and should the phenomenon be at least partially responsible
also for the consonance of triads, temperaments with x amount of
Tenney-weighted error stand a chance of exhibiting this triadic
consonance, especially if roughness-inducing harmonic overtones are
absent from each of the pitches.

So it's more than matching tuning and timbre to acheive 'sensory
consonance' -- defined as a local minimum of roughness -- but it's
less than the multiply-caused (central pitch, combinational tones,
and low roughness) consonance that occurs with very-low-error
temperaments or JI when using harmonic timbres.

🔗Carl Lumma <ekin@lumma.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > And can we hear a
> > sustained note around middle C.
>
> /tuning-math/files/Erlich/dave1.wav

I didn't find a sustained note there.

> > OK. So is your question, "In what tonal systems other than
> > diatonic/meantone is it going to be possible to have dissonant
> chords
> > resolving to consonant chords?"?
>
> Yes, in a sense, and with timbres that are primarily sine-wave but
> have spectral and envelopular aids to being individually heard out.
>
> > The obvious answer would seem to be systems in which there are
> > consonant chords, i.e which approximate (or are) JI at least
> >partially.
>
> Right -- now is Top pelogic such a system?

Possibly. But if so, it's marginal. This is of course what I've been
saying for some time. I don't see that you've given me any reason to
change this position.

> > And it might still be a
> > good idea for you to spell out the conclusion you intend.
>
> That the phenomenon responsible for central (aka 'virtual') pitch
> allows x amount of Tenney-weighted error in the partials of a single
> pitch, and should the phenomenon be at least partially responsible
> also for the consonance of triads, temperaments with x amount of
> Tenney-weighted error stand a chance of exhibiting this triadic
> consonance, especially if roughness-inducing harmonic overtones are
> absent from each of the pitches.
>
> So it's more than matching tuning and timbre to acheive 'sensory
> consonance' -- defined as a local minimum of roughness -- but it's
> less than the multiply-caused (central pitch, combinational tones,
> and low roughness) consonance that occurs with very-low-error
> temperaments or JI when using harmonic timbres.

But remember that the disagreement on things like "beep" and "father"
is not whether they contain consonances but whether those consonances
have anything to do with their supposed 5-limit mappings.

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > But remember that the disagreement on things like "beep"
> and "father"
> > is not whether they contain consonances but whether those
> consonances
> > have anything to do with their supposed 5-limit mappings.
>
> What would you call a virtual-pitch-based phenomenon where chord
> tones are assigned by the brain to specific partials within a
> subsuming 'harmony' -- to take typical examples, 3:4:5, 4:5:6, 5:6:8,
> or hypothetically, 10:12:15 (or still more hypothetically -- remember
> George Kahrimanis? -- 1/6:1/5:1/4)?

I suppose you'd like to call it 5-limit consonance, and I would have
no great objection to that. But what I want to know is, how can we
tell whether it's happening or not? Let's assume we agree on what
sounds consonant. How would we tell if the consonance is due to one of
these approximate 5-limit alignments or some other more complex but
more accurate alignment.

e.g. with TOP Beep, you were asking us to believe that the 260 c
generator could be "experienced as" an approximate 5:6 even though it
is only 7 cents away from 6:7, and 55 cents away from 5:6.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> > wrote:
> >
> > > But remember that the disagreement on things like "beep"
> > and "father"
> > > is not whether they contain consonances but whether those
> > consonances
> > > have anything to do with their supposed 5-limit mappings.
> >
> > What would you call a virtual-pitch-based phenomenon where chord
> > tones are assigned by the brain to specific partials within a
> > subsuming 'harmony' -- to take typical examples, 3:4:5, 4:5:6,
5:6:8,
> > or hypothetically, 10:12:15 (or still more hypothetically --
remember
> > George Kahrimanis? -- 1/6:1/5:1/4)?
>
> I suppose you'd like to call it 5-limit consonance, and I would have
> no great objection to that. But what I want to know is, how can we
> tell whether it's happening or not?

A psychological experiment concerning 'roots', perhaps? Ultimately it
will come down to perception and the reporting of perception, which
as we know, are not amenable to the exact sciences.

> Let's assume we agree on what
> sounds consonant. How would we tell if the consonance is due to one
of
> these approximate 5-limit alignments or some other more complex but
> more accurate alignment.

Well the perceived position of the 'root' would determine that.

> e.g. with TOP Beep, you were asking us to believe that the 260 c
> generator could be "experienced as" an approximate 5:6 even though
it
> is only 7 cents away from 6:7, and 55 cents away from 5:6.

Not really, because though the optimality of TOP can be seen as
concerning *all intervals*, most of which are even more complex than
5:6 and have worse errors, its optimization property still holds for
any, say, "product limit" n*d not smaller than the largest prime. So
you could use a "product limit" of 5 or even 20 (thus allowing
voicings -- favored anyway -- like 2:3:4:5) without running into this
difficulty.

Also, I'm not convinced what you say would necessarily be impossible
in the right context, such as a full 1:2:3:4:5:6:8 chord, or even,
perhaps, a 4:5:6:8 chord. After all, Gene and others would have us
believe that meantone dominant seventh chords were "experienced as"
4:5:6:7 chords, even though the 6:7 interval would typically be tuned
far closer to 5:6.

Anyway, when I yielded to you on 'father', the same was pretty much
implied for 'beep' . . .

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> e.g. with TOP Beep, you were asking us to believe that the 260 c
> generator could be "experienced as" an approximate 5:6 even though it
> is only 7 cents away from 6:7, and 55 cents away from 5:6.

A feeble objection. The comma for equating 7/6 and 6/5 is 36/35, and
the wedgie we get with that and 27/25 is <<2 3 1 0 -4 -6||. Now
compare TOP tunings:

TOP 27/25: <1200 1879.486 2819.230|

TOP <<2 3 1 0 -4 -6||: <1200 1879.486 2819.230 3329.029|

In other words, 7-limit beep equates 7/6 and 6/5 anyway.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > After all, Gene and others would have us
> > believe that meantone dominant seventh chords were "experienced
as"
> > 4:5:6:7 chords, even though the 6:7 interval would typically be
tuned
> > far closer to 5:6.
>
> Would harmonic entropy suggest something different?

It might, depending on the value of 's' or hearing resolution assumed
(this is essentially the only free parameter in harmonic entropy,
which subsumes considerations of timbre, register, etc.). The
meantone dominant seventh could land outside the low-entropy region
surrounding 4:5:6:7 in the 3-dimensional space of tetrads (where the
axes corresponding to these harmonics lie at 60-degree angles).

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> It might, depending on the value of 's' or hearing resolution assumed
> (this is essentially the only free parameter in harmonic entropy,
> which subsumes considerations of timbre, register, etc.). The
> meantone dominant seventh could land outside the low-entropy region
> surrounding 4:5:6:7 in the 3-dimensional space of tetrads (where the
> axes corresponding to these harmonics lie at 60-degree angles).

If we analyze 1--5/4--3/2--9/5 as a chord of septimal meantone, using
the approximations native to that, we get that 9/5 is a 9-limit
consonance and 5/4--9/5 is equivalent to 10/7 by 126/125, so this
would be a 9-limit magic chord for septimal meantone.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > It might, depending on the value of 's' or hearing resolution
assumed
> > (this is essentially the only free parameter in harmonic entropy,
> > which subsumes considerations of timbre, register, etc.). The
> > meantone dominant seventh could land outside the low-entropy
region
> > surrounding 4:5:6:7 in the 3-dimensional space of tetrads (where
the
> > axes corresponding to these harmonics lie at 60-degree angles).
>
> Can you calculate this for various s?

It hasn't been done yet, but I promise to do it post haste, if you
first give me a sincere effort to improve the computational
efficiency of the calculation (starting, of course, with the dyadic
case). I'd prefer to discuss this on the harmonic entropy list, for
the convenience of those who wish to follow its development.

🔗Paul Erlich <perlich@aya.yale.edu>

Yup.

🔗Carl Lumma <ekin@lumma.org>

>> I suppose you'd like to call it 5-limit consonance, and I would have
>> no great objection to that. But what I want to know is, how can we
>> tell whether it's happening or not?
>
>A psychological experiment concerning 'roots', perhaps? Ultimately it
>will come down to perception and the reporting of perception, which
>as we know, are not amenable to the exact sciences.

No, you just ask people to walk over to the piano and pick out the
notes they've heard. Rinse, lather, repeat. You'll find massive
agreement.

>> Let's assume we agree on what
>> sounds consonant. How would we tell if the consonance is due to one
>of
>> these approximate 5-limit alignments or some other more complex but
>> more accurate alignment.
>
>Well the perceived position of the 'root' would determine that.

Exactly.

-Carl

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Carl Lumma <ekin@lumma.org>

>> >> I'm assuming the predicted fundamentals will be based on the beep
>> >> mapping -- that the spectra are set up with the nearest beep-
>> >> mapped approx. to the harmonics. That is what your test
>> >> tones (Paul) are, right?
>> >>
>> >> -Carl
>> >
>> >No, those were the top pelogic rounded to 0.1 cent ones, since
>> >that's what I asked Dave about in the original message.
>>
>> Ok, but what I meant was, you have some way to make the predicted
>> fundamental come out differently if the 2ndary approximations are
>> at work?
>
>You lost me.

If the partials are individually at their nearest pelogic-mapped
values and people consistently identify the 'fundamental' as
being the pitch of the tone, could it be claimed that the extra-map
consonances Dave is pointing out were at work?

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> I'm assuming the predicted fundamentals will be based on the
beep
> >> >> mapping -- that the spectra are set up with the nearest beep-
> >> >> mapped approx. to the harmonics. That is what your test
> >> >> tones (Paul) are, right?
> >> >>
> >> >> -Carl
> >> >
> >> >No, those were the top pelogic rounded to 0.1 cent ones, since
> >> >that's what I asked Dave about in the original message.
> >>
> >> Ok, but what I meant was, you have some way to make the predicted
> >> fundamental come out differently if the 2ndary approximations are
> >> at work?
> >
> >You lost me.
>
> If the partials are individually at their nearest pelogic-mapped
> values and people consistently identify the 'fundamental' as
> being the pitch of the tone, could it be claimed that the extra-map
> consonances Dave is pointing out were at work?

How could they possibly come into play? The map (from primes --
clearly generators are irrelevant here) is only carried out as far as
a single big major triad, no further. The melodic intervals, should
it matter, were not pelogic ones.

🔗Carl Lumma <ekin@lumma.org>

>> >> >> I'm assuming the predicted fundamentals will be based on the
>> >> >> beep mapping -- that the spectra are set up with the nearest
>> >> >> beep-mapped approx. to the harmonics. That is what your test
>> >> >> tones (Paul) are, right?
>> >> >>
>> >> >> -Carl
>> >> >
>> >> >No, those were the top pelogic rounded to 0.1 cent ones, since
>> >> >that's what I asked Dave about in the original message.
>> >>
>> >> Ok, but what I meant was, you have some way to make the predicted
>> >> fundamental come out differently if the 2ndary approximations are
>> >> at work?
>> >
>> >You lost me.
>>
>> If the partials are individually at their nearest pelogic-mapped
>> values and people consistently identify the 'fundamental' as
>> being the pitch of the tone, could it be claimed that the extra-map
>> consonances Dave is pointing out were at work?
>
>How could they possibly come into play? The map (from primes --
>clearly generators are irrelevant here) is only carried out as far as
>a single big major triad, no further.

Just clarifying. So one could conduct this experiment and if the
results are as I suggest it would allow one to reasonably claim that
the 5-limit (if one didn't use any other partials in the synthesis)
approximations of pelogic are indeed 5-limit approximations.

-Carl

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> It might, depending on the value of 's' or hearing resolution
assumed
> >> (this is essentially the only free parameter in harmonic
entropy,
> >> which subsumes considerations of timbre, register,
>
> Not register, I thought.

Yes register. S is clearly larger lower down.

> The neigborhood around 5/2 probably looks
> a lot like that around 5/4 but the actual entropy values will be
> different, right?

What does that have to do with it? The right comparison would be a
given interval with that same interval transposed lower down, not
between two different intervals.

For octave-equivalent harmonic entropy, s still depends on register,
but *voicing* and *inversion* of course become irrelevant to the
actual entropy values.

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > It hasn't been done yet, but I promise to do it post haste, if
you
> > first give me a sincere effort to improve the computational
> > efficiency of the calculation (starting, of course, with the
dyadic
> > case). I'd prefer to discuss this on the harmonic entropy list,
for
> > the convenience of those who wish to follow its development.
>
> It's your definition; I've thought about trying to compute it, but
it
> looks nasty.

The idea is to find an easier way to express it or compute it, but
the direct computation is pretty easy for dyads. Let's use means
instead of mediants, which will allow us to use voronoi cells in the
generalization to higher dimensions. The harmonic entropy of a dyad i
is simply

-SUM (p(j,i) log(p(j,i)))
j

where j runs in order of cents size over all lowest-term ratios where
n*d < 10000 or 65536 or some large number, and p(j,i) is proportional
to

(cents(j+1)-cents(j-1))*exp( -(cents(j)-cents(i))^2 / (2s^2) )

and the constant of proportionality is such that

-SUM (p(j,i)) = 1
j

or, if you prefer, you could define p(j,i) as 1/(s*sqrt(2*pi)) times
the integral from
(cents(j-1) + cents(j))/2
to
(cents(j+1) + cents(j))/2
of
exp( -(cents(t)-cents(i))^2 / (2s^2) ) dt

> Are you suggesting trying to find something similar, but
> easier to compute?

Yes -- similar or even equal.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
>
> > After all, Gene and others would have us
> > believe that meantone dominant seventh chords were "experienced as"
> > 4:5:6:7 chords, even though the 6:7 interval would typically be tuned
> > far closer to 5:6.
>
> Would harmonic entropy suggest something different?

I assume you mean here the _minimisation_ of harmonic entropy.

There's also the posssibility that the dominant seventh chord
functions best when its harmonic entropy (or maybe only the HE of one
of its dyads) is locally _maximised_.

George Secor alluded to this recently (on the tuning list I think).

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > e.g. with TOP Beep, you were asking us to believe that the 260 c
> > generator could be "experienced as" an approximate 5:6 even though it
> > is only 7 cents away from 6:7, and 55 cents away from 5:6.
>
> A feeble objection.
...
> In other words, 7-limit beep equates 7/6 and 6/5 anyway.

Mad if it didn't. However I don't see that that affects anything I
said about 5-limit TOP Beep.

Paul's objection was better, namely that if you provide a large enough
context of other pitches in an approximate harmonic-series segment,
then maybe even a 55 cent error can be pulled into line, as it were.

But even if that were true, there is a very significant difference
between a normal 5-limit temperament where all the 5-limit harmonies
work (including bare dyads and utonalities), and one in which only
otonal tetrads and larger otonalities "work".

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

It seems to me a Stieltjes integral with respect to the Minkowski ?
function is worth exploring.

http://mathworld.wolfram.com/StieltjesIntegral.html

http://mathworld.wolfram.com/MinkowskisQuestionMarkFunction.html

The ? function has the property that

?((p1+p2)/(q1+q2)) = (p1/q1 + p2/q2)/2

for adjacent Farey fractions.

We can ask for Integral exp(-((x-c)/s)^2) d?

for various values of s, and use it as one definition for the
harmonic entropy of c. ?(x) is continuous and it doesn't look
terribly difficult to compute, though I've never tried.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> We can ask for Integral exp(-((x-c)/s)^2) d?
>
> for various values of s, and use it as one definition for the
> harmonic entropy of c. ?(x) is continuous and it doesn't look
> terribly difficult to compute, though I've never tried.

It may be worth noting that the above is a convolution of ? with the
Gaussian distribution.

http://mathworld.wolfram.com/Convolution.html

This is a smoothing operation, and while the derivaitve ?'(x) does
not exist as a function, it is a well-defined generalized function
like the Dirac delta. The above definition can be thought of as
Gaussian smoothings of ?' into actual functions.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >There's also the posssibility that the dominant seventh chord
> >functions best when its harmonic entropy (or maybe only the HE of one
> >of its dyads) is locally _maximised_.
>
> What would this look like?

Pretty much like a dominant seventh chord in 12-tET.

Using noble-mediants to estimate them, a max entropy minor seventh
should be around 1002 cents, a max entropy diminished fifth should be
around 607 cents, and a max entropy minor third should be around 284
cents.

> The dominant seventh chord is defined
> as a local minimum of entropy.

Who defined it as such, when, and why?

As far as I know, the only widely accepted definition of the dominant
seventh chord is a chord containing the diatonic scale degrees V, VII,
II, IV. It may or may not be a local harmonic entropy minimum
depending how the scale is tuned. It seems to me that the more
dissonant it is, the more relief is likely to be felt when it
"resolves" to a consonance.

🔗Carl Lumma <ekin@lumma.org>

>> The dominant seventh chord is defined
>> as a local minimum of entropy.
>
>Who defined it as such, when, and why?
>
>As far as I know, the only widely accepted definition of the dominant
>seventh chord is a chord containing the diatonic scale degrees V, VII,
>II, IV. It may or may not be a local harmonic entropy minimum
>depending how the scale is tuned.

Ah; I thought you were referring to '4:5:6:7', when in fact you
want bins to restrict the pitches of the chord. Finding the highest-
entropy chord that fits in the bins makes sense.

>It seems to me that the more
>dissonant it is, the more relief is likely to be felt when it
>"resolves" to a consonance.

I don't personally agree with this, but it is a rather popular
assertion.

>> >There's also the posssibility that the dominant seventh chord
>> >functions best when its harmonic entropy (or maybe only the HE of one
>> >of its dyads) is locally _maximised_.
>>
>> What would this look like?
>
>Pretty much like a dominant seventh chord in 12-tET.
>
>Using noble-mediants to estimate them, a max entropy minor seventh
>should be around 1002 cents, a max entropy diminished fifth should be
>around 607 cents, and a max entropy minor third should be around 284
>cents.

If you're trying to maximize pairwise entropy you'll need to consider
the P-5th and M-3rd too. But tetradic entropy seems more to the
point.

-Carl

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> >
> > > After all, Gene and others would have us
> > > believe that meantone dominant seventh chords were "experienced
as"
> > > 4:5:6:7 chords, even though the 6:7 interval would typically be
tuned
> > > far closer to 5:6.
> >
> > Would harmonic entropy suggest something different?
>
> I assume you mean here the _minimisation_ of harmonic entropy.
>
> There's also the posssibility that the dominant seventh chord
> functions best when its harmonic entropy (or maybe only the HE of
one
> of its dyads) is locally _maximised_.
>
> George Secor alluded to this recently (on the tuning list I think).

The most recent thread related to this is "Just diminished 7th chords
(Was: Chord names in Scala)", which began here:
/tuning/topicId_51743.html#51743
but it concerns diminished rather than dominant 7th chords, and I
emphasize the *melodic* properties of the intervals more than the
harmonic. But I did attempt to relate the two in the last sentence
of this message, which is probably what you had in mind:
/tuning/topicId_51743.html#51769

There is an interesting problem I ran into when trying to incorporate
all of my ideas simultaneously in attempting to resolve a dissonant
dominant 7th chord to a tonic. Starting with 4:5:6 on the dominant
with some sort of minor seventh added, it makes sense to lower the
seventh to a 7:4 with the (dominant) root, so that the interval of
resolution to the 3rd of the tonic triad is smaller. However, when
you raise the leading tone a little bit (to make a smaller interval
of resolution to the tonic root, and also to make the 7th chord more
dissonant), you risk making the tritone so small that it ceases to
sound like a tritone. The interval between a pythagorean major 3rd
and harmonic 7th is ~561c -- only 10 cents larger than 8:11, and very
close to 13:18. So what you end up with is something that functions
very much like a dominant 7th chord, yet sounds somewhat different.

An extreme example of this occurs in 19-ET: Alter both the 3rd and
7th of G B D F by 1deg, to G B# D Fb, to resolve to a C major triad.
The interval between B# and Fb is equivalent to a perfect fifth --
not a tritone by anyone's stretch of the imagination! Still, I think
it's musically useful in that it provides a harmonization:
G-D-F-B to G-D-Fb-B# to C-C-E C
that uses all of the tones of the Greek enharmonic genus as occurs in
19 (with the small intervals at the top of the tetrachord):
C E Fb F G B B# C

--George

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > > >There's also the posssibility that the dominant seventh chord
> > > >functions best when its harmonic entropy (or maybe only the HE
> of one
> > > >of its dyads) is locally _maximised_.
> > >
> > > What would this look like?
> >
> > Pretty much like a dominant seventh chord in 12-tET.
> >
> > Using noble-mediants to estimate them, a max entropy minor seventh
> > should be around 1002 cents, a max entropy diminished fifth should
> be
> > around 607 cents, and a max entropy minor third should be around 284
> > cents.
>
> Why look at these intervals and not the major third, etc.?

Well you _could_ try to locally maximise their entropy too, but I
assumed the dominant triad (without the seventh) normally functions as
a consonance.