Yahoo Tuning Groups Ultimate Backup tuning nutty professor has me confused (JI)

--- In tuning@y..., jpehrson@r... wrote:
> I really can't understand what the "nutty professor" on
> the "impolite" tuning list has against just intonation...
> Isn't it perfectly clear that lower-integer ratios (mistuned) are
> the foundation of our Western music??
> And, isn't there a particular joy in hearing them adaptively tuned
> correctly...?? even if those methods weren't available to the
> composers of earlier eras?
> Even if it's not part of a historical style, I really have to
> applaud John deLaubenfels' incredible computer-assisted efforts in >
> this direction...
> And how about my other just intonation friends, "liars" Joe Monzo
> and David Beardsley...?
> Certainly all these folks can't be off their rockers...
> Exactly what is wrong in striving for just intonation??
> ________ ______ ________
> Joseph Pehrson

I think Brian's objection to just intonation is an overreaction to
theoretical claims that just intonation is somehow primary in music.
In fact, I don't think he really objects to using small number ratios,
but he does get really mad a people whose theories clash with his, and
I believe he thinks small number ratios have almost nothing (maybe
even nothing at all) to do with choosing good tonal resources. Thus
when he goes off on someone who thinks small number ratios are
important somehow, just intonation gets burned in the firefight. I may
be reading him wrong (and if I am, I bet I get set on fire) but that's
my take.

Personally, I think small number ratios are important (for what I
don't know), because they are signposts in the harmonic landscape.
Almost anyone, even without training, can tune two sine wave
oscillators to a small number ratio by slowing the beats to a stop,
but almost no one can tune a non just interval that way. To me, that
says something about the way we hear. I know that very few instruments
have waveforms close to sine waves, but once you learn the tuning
trick with a sine wave, you can do it with a wide variety of waves.

Yes, I think it is perfectly clear that small number ratios are the
foundation of western music. I enjoy listening to them and singing
them to my air compressor drone. I like what JDL's program does, too,
and it is my opinion that most composers he has retuned would like it
too. I have no facts, just pure impression on that count. I don't
think Monz and Beardsley are off their rockers, but I do think 60
million Frenchmen *can* be wrong. But why criticise peoples stylistic
choices? You would have to ask Brian about that.

On the other hand, tuning sine waves is not really music, unless you
are LaMonte Young. Personally, I think the rhythmic character of a
melody, the interpretation, and the fuzzy intervals are more important
than the precise pitches used. Listen to Frank Sinatra. His pitch is
off a lot of the time, but it still sounds right.

🔗Jon Szanto <JSZANTO@ADNC.COM>

🔗monz <joemonz@yahoo.com>

🔗Jon Szanto <JSZANTO@ADNC.COM>

🔗Paul Erlich <paul@stretch-music.com>

🔗John Starrett <jstarret@carbon.cudenver.edu>

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
<snip>
> > Almost anyone, even without training, can tune two sine wave
> > oscillators to a small number ratio by slowing the beats to a
stop,
> > but almost no one can tune a non just interval that way.
>
> This is very true except for one thing -- the sine wave is
> practically the only repeating waveform that will _not_ work for
this
> experiment! Since the sine wave has no harmonics, you'll hear no
> beating stopping at any ratio other than 1:1 (and 2:1 due to
> subjective second-order beats).

HMMM... you are right Paul. I had demonstrated the tuning by beats
phenomenon to physics of music classes with a high precision dual
waveform generator using sine waves and never thought about the
physics of what I was doing! Apparently, I must have been getting
speaker or power amp distortion that let me tune the beats. How
embarrasing! Oh well, they've probably all graduated by now.

This reminds me of a section in the instruction manual for my first
synth, an EMS 101. I paraphrase "After every 200 hours of use you
should run a sine wave through the filters to flush out trapped
overtones."

John Starrett

🔗FreinagelR@netscape.net

--- In tuning@y..., "John Starrett" <jstarret@c...> wrote:
> --- In tuning@y..., jpehrson@r... wrote:
>
> Personally, I think small number ratios are important (for what I
> don't know), because they are signposts in the harmonic landscape.
> Almost anyone, even without training, can tune two sine wave
> oscillators to a small number ratio by slowing the beats to a stop,
> but almost no one can tune a non just interval that way. To me,
that
> says something about the way we hear. I know that very few
instruments
> have waveforms close to sine waves, but once you learn the tuning
> trick with a sine wave, you can do it with a wide variety of waves.

I don't think so. Two pure sine waves, say at 200 Hz and 300 Hz,
heard together produce _no_ beats - only a primary difference tone at
100 Hz, a sum tone at 500 Hz, and other very faint higher order
products at multiples of 100 Hz. These product tones only arise
because of the non-linear amplitude response of the human ear.
The "beating" of two tones will only be heard if the original tones
have some harmonic overtones present (and therefore are _not_ sine
waves). In the above example, that would be between the 3rd harmonic
of the 200 Hz tone, and the 2nd harmonic of the 300 Hz tone, both
being at 600 Hz. Of course, less than ideal oscillators, amplifiers,
and headphones can introduce harmonic distortion which would account
for the apparent beating of two "sine waves".

Respectfully,

Fred Reinagel

🔗John Starrett <jstarret@carbon.cudenver.edu>

🔗ha.kellner@t-online.de

Dear members, -(I'm back for a few days!)

In view of the appreciation through centuries of mean-tone,
that organ builders were virtually unable to give up, there is
definitely something absolutely significant to pure intonation.

Well-tempered systems do take to various degrees the best they
can have, "make the best of it": what can (or could) be realized and
implemented in place of mesotonic. And the compositions, somehow
like was the case for mean tone, have to follow suit. Even
if well tempered systems admit all 24 keys in their
historical-tonal framework and environment.

Kind regards,
Herbert Anton

John Starrett schrieb:
> --- In tuning@y..., jpehrson@r... wrote:
> > I really can't understand what the "nutty professor" on
> > the "impolite" tuning list has against just intonation...
> > Isn't it perfectly clear that lower-integer ratios (mistuned) are
> > the foundation of our Western music??
> > And, isn't there a particular joy in hearing them adaptively tuned
> > correctly...?? even if those methods weren't available to the
> > composers of earlier eras?
> > Even if it's not part of a historical style, I really have to
> > applaud John deLaubenfels' incredible computer-assisted efforts in >
> > this direction...
> > And how about my other just intonation friends, "liars" Joe Monzo
> > and David Beardsley...?
> > Certainly all these folks can't be off their rockers...
> > Exactly what is wrong in striving for just intonation??
> > ________ ______ ________
> > Joseph Pehrson
>
> I think Brian's objection to just intonation is an overreaction to
> theoretical claims that just intonation is somehow primary in music.
> In fact, I don't think he really objects to using small number ratios,
> but he does get really mad a people whose theories clash with his, and
> I believe he thinks small number ratios have almost nothing (maybe
> even nothing at all) to do with choosing good tonal resources. Thus
> when he goes off on someone who thinks small number ratios are
> important somehow, just intonation gets burned in the firefight. I may
> be reading him wrong (and if I am, I bet I get set on fire) but that's
> my take.
>
> Personally, I think small number ratios are important (for what I
> don't know), because they are signposts in the harmonic landscape.
> Almost anyone, even without training, can tune two sine wave
> oscillators to a small number ratio by slowing the beats to a stop,
> but almost no one can tune a non just interval that way. To me, that
> says something about the way we hear. I know that very few instruments
> have waveforms close to sine waves, but once you learn the tuning
> trick with a sine wave, you can do it with a wide variety of waves.
>
> Yes, I think it is perfectly clear that small number ratios are the
> foundation of western music. I enjoy listening to them and singing
> them to my air compressor drone. I like what JDL's program does, too,
> and it is my opinion that most composers he has retuned would like it
> too. I have no facts, just pure impression on that count. I don't
> think Monz and Beardsley are off their rockers, but I do think 60
> million Frenchmen *can* be wrong. But why criticise peoples stylistic
> choices? You would have to ask Brian about that.
>
> On the other hand, tuning sine waves is not really music, unless you
> are LaMonte Young. Personally, I think the rhythmic character of a
> melody, the interpretation, and the fuzzy intervals are more important
> than the precise pitches used. Listen to Frank Sinatra. His pitch is
> off a lot of the time, but it still sounds right.
>
>
>
>
>
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>
>

🔗BobWendell@technet-inc.com

Hi, Herbert Anton and all interested parties. In connection with this
discussion, I offer the following excerpt from an Appendix for a
choral training workship I am developing that is mostly aimed at a
cappella ensembles. It is necessarily a bit oversimplified in some
its statements, but true enough to work as a reasonable explanatory
support structure for the EXPERIENTIAL focus of the workshop. The
excerpt follows:

Pleasing harmony has its basis in these three elements of acoustics:

1. The harmonic structure of a single musical tone
2. The difference tones produced when two or more musical tones are
sounded simultaneously
3. The coherent alignment of the relationships that exist between
these two aspects of harmonic structure.

Fortunately, this third element and fundamentally important
requirement for pleasing harmony is satisfied by a single, simple
condition:

Coherent alignment among all the elements of harmonic structure
exists only when the frequencies (number of vibrations per second)
between simultaneously sounded tones form a ratio expressible as
simple whole numbers, such as 3:2, 4:3, 5:4, etc.

It is vitally important for us to realize that this coherent
alignment is INSTRINSICALLY pleasing to the human ear, and is not a
merely subjective or conditioned phenomenon! The self-reinforcing
order, efficiency, and organizing power of the coherence that
characterizes laser light, for example, is even more directly,
deeply, and intimately appreciated in sound by the human ear when it
is emanating from a collection of human voices singing beautiful
music. It reaches deep into the human spirit to integrate our
perception, unify our feelings, inspire our hearts, order our minds,
calm our bodies, and heal our souls. This is why both naïve and
seasoned listeners alike react to accurate intonation as sounding
better, even if completely unaware of what underlies this
superiority, and even if they would not have noticed its absence had
it not been accurate.

Justly yours,

Bob
Music Director
Cantus Angelicus Choral Society
http://www.cangelic.org

P.S. Anyone who pretends that 12t-ET is an arbitrary tonal convention
with no basis in just intervals is not only ignoring the long history
behinds its development and use, but the theoretical principles and
traditional auditory methodology underlying its tuning. I once had a
music theory professor, very erudite in some respects, who argued
that scales and melodic/harmonic systems were arbitrary cultural
conventions, naively pointing to the plethora of systems in actual
use in various cultures. In musical practice, he was a trombone
player who sounded like he was blowing his nose when he played, and
waxed ecstatic over singers who couldn't carry a tune in a bucket!

--- In tuning@y..., ha.kellner@t... wrote:
> Dear members, -(I'm back for a few days!)
>
> In view of the appreciation through centuries of mean-tone,
> that organ builders were virtually unable to give up, there is
> definitely something absolutely significant to pure intonation.
>
> Well-tempered systems do take to various degrees the best they
> can have, "make the best of it": what can (or could) be realized
and
> implemented in place of mesotonic. And the compositions, somehow
> like was the case for mean tone, have to follow suit. Even
> if well tempered systems admit all 24 keys in their
> historical-tonal framework and environment.
>
> Kind regards,
> Herbert Anton
>
>
>
> John Starrett schrieb:
> > --- In tuning@y..., jpehrson@r... wrote:
> > > I really can't understand what the "nutty professor" on
> > > the "impolite" tuning list has against just intonation...
> > > Isn't it perfectly clear that lower-integer ratios (mistuned)
are
> > > the foundation of our Western music??
> > > And, isn't there a particular joy in hearing them adaptively
tuned
> > > correctly...?? even if those methods weren't available to the
> > > composers of earlier eras?
> > > Even if it's not part of a historical style, I really have to
> > > applaud John deLaubenfels' incredible computer-assisted efforts
in >
> > > this direction...
> > > And how about my other just intonation friends, "liars" Joe
Monzo
> > > and David Beardsley...?
> > > Certainly all these folks can't be off their rockers...
> > > Exactly what is wrong in striving for just intonation??
> > > ________ ______ ________
> > > Joseph Pehrson

> >
> >
> >
> >
> >

🔗BobWendell@technet-inc.com

I might add to my last post that extensive experience in working with
vocally produced just harmonies indicates strongly that the
phenomenon of beat frequencies generated between common harmonics
used in setting a temperament, or sometimes in tuning just intervals,
is NOT what musicians with accurate intonation intuitively use to
judge the precision of harmonic intervals. It is much too slow, for
one thing.

When rather than tuning fixed-pitch instruments, you're tuning
perpetually during performance as do singers and string players, not
much music allows you to sit on a tone long enough for common-
harmonic beat phenomena to be of any use. Rather, the key phenomenon
is the coherent alignment of the subtones generated with the
fundamentals of the sung or played pitches and their harmonic
structures. Anyone who has ever listened to single side-band radio
reception (SSB)has heard the otherworldly quacking quality the radio-
transmitted human voice acquires when mistuning of the receiver's
local oscillator shifts the harmonics so they are no longer whole-
number multiples of the voice's fundamental.

!!!===>>>The important point here is that even the unmusical,
untrained ear is VERY QUICK to detect this! It takes a subtler ear to
notice when the tonal substructure generated by two simultaneous
tones of rich harmonic content is not in such alignment with the
tones generating it and their harmonics, BUT THE PRINCIPLE REMAINS
THE SAME.

Thanks all,

- Bob

🔗Paul Erlich <paul@stretch-music.com>

🔗BobWendell@technet-inc.com

Thank you, Paul. My participation here has been truly inspiring and
enlightening for me, and your contributions have figured very large
in that indeed!

The statement below must be taken in context with the preceding
sentence. I assume that you read that sentence, and so infer that
perhaps you're unfamiliar with single side-band radio(SSB). I'm a
musician with a technical background, both sound and radio
engineering, so I can assume too much at times.

Regular AM radio consists of modulating the amplitude of an otherwise
steady state radio frequency signal with the audio signal to be
transmitted. This generates sum and difference signals symmetrically
above and below the carrier frequency. Two thirds of the transmitter
power is spent in transmitting the carrier frequency, which has no
useful information in it, but only serves as means for the
information to ride "piggyback" on it. All useful information is in
the sidebands.

Since the sidebands are identical except for the inversion of the
audio spectrum with respect to the carrier frequency, they contain
identical information. Eliminating one of them allows narrower tuning
in the receiver, eliminating ambient radio frequency noise, so single
SSB radio puts all the transmitter power into the information
contained in one sideband. The SSB receiver reinserts the carrier
with a small local sine-wave oscillator, which beats against the
sideband information, and the difference signal is the original
audio. The overall effect is about a 9 db gain in signal-to-noise
ratio over conventional AM and vastly increased range for a given
transmitter power.

HOWEVER, the local oscillator seldom tracks perfectly the original
carrier frequency! This means that the speech signal (the only kind
of signal for which this works passably well), is not accurately
reconstructed in the receiver. If the local carrier reinsertion
oscillator is 10 HZ off, this will add or subtract 10 Hz to the
fundamental and every harmonic!

This will yield an overtone structure that is NOT harmonic, NOT exact
whole number multiples of the fundamental since the error does not
multiply with the order of the harmonic, but remains constant
throughout the spectrum. This doesn't occur naturally to anything
like this degree even with piano strings and their inharmonicity, not
to mention the human voice, which as a rule is quite harmonically
coherent.

I'm simply stating that this phenomenon sounds weird and otherworldy
to the even the crudest of ears, and differs not in principle but
only in degree with the misalignment of subtones with the two or more
musical tones that generate them in musical harmonic structures.
Relatively simple whole number ratios between frequencies of musical
fundamentals with reasonably harmonic overtones are the only
conditions that produce good alignment and an overall effect of
coherence among all the harmonic elements.

Hope this was not too arcane!

Yours,

Bob

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., BobWendell@t... wrote:
> >
> > It takes a subtler ear to
> > notice when the tonal substructure generated by two simultaneous
> > tones of rich harmonic content is not in such alignment with the
> > tones generating it and their harmonics, BUT THE PRINCIPLE
REMAINS
> > THE SAME.
>
> You lost me there. Can you explain this sentence?
>
> In any case, I think we're _very_ much in agreement (again). Why do
I
> get the feeling that you are one of the few people who really
> _listens_? Now, I'm going to post something about harmonic entropy,
> and I hope you'll read it.

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., BobWendell@t... wrote:
>
> The statement below must be taken in context with the preceding
> sentence. I assume that you read that sentence, and so infer that
> perhaps you're unfamiliar with single side-band radio(SSB).

Actually, I'm familiar with that, but thanks for the refresher!
>
> I'm simply stating that this phenomenon sounds weird and
otherworldy
> to the even the crudest of ears, and differs not in principle but
> only in degree with the misalignment of subtones with the two or
more
> musical tones that generate them in musical harmonic structures.

I'm confused as to what that means, but let's move on.

> Relatively simple whole number ratios between frequencies of
musical
> fundamentals with reasonably harmonic overtones are the only
> conditions that produce good alignment and an overall effect of
> coherence among all the harmonic elements.

I pretty much agree, though you'll find that others (Bill Sethares
and company) believe that one can do equally well using arbitrary
inharmonic timbres, and scales which mimic the intervals in said
timbres' spectra. Inharmonic timbres are fascinating and lots of
great music can be made with them, but they can never match the
immediate sense of "coherence" one gets from harmonic timbres.
Sethares and others base their thinking on the idea that dissonance
results from beating and roughness between partials that occupy one
another's critical bands. While this beating and roughness is real
enough, it explains only a relatively small component of perceived
dissonance. See my "On Harmonic Entropy" post for a little more on
this.

By the way, Bob, you never responded to my assertion that JI may not
always maximize consonance, e.g., for chords like CEGAD. Any response?

🔗BobWendell@technet-inc.com

Bob:
I'm simply stating that this phenomenon sounds weird and otherworldy
> to the even the crudest of ears, and differs not in principle but
> only in degree with the misalignment of subtones with the two or
more
> musical tones that generate them in musical harmonic structures.

Paul:
I'm confused as to what that means, but let's move on.

Bob answers:
Well, respectfully begging your indulgence, I'd rather not move on,
since I consider this fundamental to what I'm attempting to
communicate. The weird phenomenon of unnatural vocal "inharmonic
harmonics" that occurs when SSB is MISTUNED sounds so weird to us
because it is a misalignment of the harmonic structure which,
although it is very orderly, consistent, and mathematically simple,
it is nonetheless a way that one never hears in naturally occurring
overtone structures.

Now if one tunes a just 3:2 P5, the principal subtone generated is a
difference frequency at 1, an octave below the bottom note of the P5.
Assuming harmonic overtones for the source tones, this so-called
differential tone doesn't exist as a pure sine wave, but has an
overtone structure resulting from the difference tones of all the
harmonics in the two tones generating it, as well as the harmonic
structure imposed by the ear's non-linearity responsible for the
difference tones, not to mention neurological phenomena that also
contribute to it.

In this justly tuned P5, the difference tone and all its harmonics
are coherently related, dovetailing perfectly and reinforcing each
other. If we mistune the fifth slightly, the difference tone shifts
and it's harmonic structure is no longer in alignment with either the
fundamentals of the two generating tones or their harmonics.

Although this phenomenon is sadly quite available in nature, most
notably at many concert performances organized by well-meaning human
beings, it is still not pleasing to sensitive, musical ears for
essentially the same reasons that the SSB phenomenon sounds weird
even to the crudest, most untrained and unmusical ears. At bottom, it
is not so much a matter of difference in kind, but rather in degree.
Hope this clarifies the point. If not, please let me know
specifically what aspect is not clear so I know what to address.

Paul had also said:
Inharmonic timbres are fascinating and lots of great music can be
made with them, but they can never match the immediate sense of
"coherence" one gets from harmonic timbres.

Bob answers:
Since I work in practice with a cappella singers, my discussion
assumes a high degree of harmonicity in the overtone structure.

Paul also had said:
By the way, Bob, you never responded to my assertion that JI may not
always maximize consonance, e.g., for chords like CEGAD. Any response?

Bob answers:
Sorry, Paul! CEGAD has quartal structure coexisting with triadic. For
quartal structures, Pythagorean JI works up to a point, but 12 ET
works better in this instance because ET is a more or less reasonable
compromise between quartal and triadic structures, although I
wouldn't argue that it absolutely represents the least harmonically
entropic tuning.

I do believe that is why some of the richer jazz structures one hears
in "comping" on the piano has evolved into stacked fourths that
contain all the elements of extended triads. For example, Eb A D G C
F in ET works harmonically very nicely in place of the more
conventional F A C EB G D (a minor 7th chord extended with a Major
9th and a Major 13th).

I'm not sure this is the kind of answer you were looking for, Paul,
but
it's my best take on it.

Thanks for asking!

- Bob

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> --- In tuning@y..., BobWendell@t... wrote:
> >
> > The statement below must be taken in context with the preceding
> > sentence. I assume that you read that sentence, and so infer that
> > perhaps you're unfamiliar with single side-band radio(SSB).
>
> Actually, I'm familiar with that, but thanks for the refresher!
> >
> > I'm simply stating that this phenomenon sounds weird and
> otherworldy
> > to the even the crudest of ears, and differs not in principle but
> > only in degree with the misalignment of subtones with the two or
> more
> > musical tones that generate them in musical harmonic structures.
>
> I'm confused as to what that means, but let's move on.
>
> > Relatively simple whole number ratios between frequencies of
> musical
> > fundamentals with reasonably harmonic overtones are the only
> > conditions that produce good alignment and an overall effect of
> > coherence among all the harmonic elements.
>
> I pretty much agree, though you'll find that others (Bill Sethares
> and company) believe that one can do equally well using arbitrary
> inharmonic timbres, and scales which mimic the intervals in said
> timbres' spectra. Inharmonic timbres are fascinating and lots of
> great music can be made with them, but they can never match the
> immediate sense of "coherence" one gets from harmonic timbres.
> Sethares and others base their thinking on the idea that dissonance
> results from beating and roughness between partials that occupy one
> another's critical bands. While this beating and roughness is real
> enough, it explains only a relatively small component of perceived
> dissonance. See my "On Harmonic Entropy" post for a little more on
> this.
>
> By the way, Bob, you never responded to my assertion that JI may
not
> always maximize consonance, e.g., for chords like CEGAD. Any
response?

🔗jpehrson@rcn.com

--- In tuning@y..., BobWendell@t... wrote:

/tuning/topicId_26421.html#26895

> I might add to my last post that extensive experience in working
with
> vocally produced just harmonies indicates strongly that the
> phenomenon of beat frequencies generated between common harmonics
> used in setting a temperament, or sometimes in tuning just
intervals,
> is NOT what musicians with accurate intonation intuitively use to
> judge the precision of harmonic intervals. It is much too slow, for
> one thing.
>
> When rather than tuning fixed-pitch instruments, you're tuning
> perpetually during performance as do singers and string players,
not
> much music allows you to sit on a tone long enough for common-
> harmonic beat phenomena to be of any use. Rather, the key
phenomenon
> is the coherent alignment of the subtones generated with the
> fundamentals of the sung or played pitches and their harmonic
> structures. Anyone who has ever listened to single side-band radio
> reception (SSB)has heard the otherworldly quacking quality the
radio-
> transmitted human voice acquires when mistuning of the receiver's
> local oscillator shifts the harmonics so they are no longer whole-
> number multiples of the voice's fundamental.
>
> !!!===>>>The important point here is that even the unmusical,
> untrained ear is VERY QUICK to detect this! It takes a subtler ear
to
> notice when the tonal substructure generated by two simultaneous
> tones of rich harmonic content is not in such alignment with the
> tones generating it and their harmonics, BUT THE PRINCIPLE REMAINS
> THE SAME.
>
> Thanks all,
>
> - Bob
>

Gee, this is an interesting post, Bob! Sorry I can't say more about
it, but I *can* say "gee, this is an interesting post..." :)

________ _______ ________
Joseph Pehrson

🔗BobWendell@technet-inc.com

Paul had said:
By the way, Bob, you never responded to my assertion that JI may not
> always maximize consonance, e.g., for chords like CEGAD. Any
response?

Bob elaborates previous answer:
Paul, it occurred to me that I should have been more explicit about
one point. C E G A D can be arrange as consecutive perfect fifths
from C (C G D A E). These are the self-same core pitches that are
used to set the temperament in quarter-comma meantone. Tuning the C
and E as a 5:4 just Major 3rd, then shaving a quarter syntonic comma
off of each fifth sets this temperament. The rest can be derived by
tuning just Major 3rds to these reference pitches.

Without this or some kind of temperament, the syntonic comma will
simply get shifted around in any just tuning, but won't go away no
matter what you do. Even strings with only four open string pitches
have this problem. Traditionally, string players tune the open string
fifths justly. This will always produce an octave and a Major sixth
interval between the top and bottom strings that is sharp by the
syntonic comma to a just Major sixth. Its inversion, the minor 3rd,
will be flat by the same amount.

Interestingly, the chord in the spelling you gave is a common final
chord in some jazz and other earlier pop styles. I do not believe
this
usage would have ever evolved musically without a tempered scale
environment.

Thanks for the interesting (and perhaps, chuckle, deliberately
loaded?) question, Paul!

Sincerely,

Bob

P.S. Concerning the derivation of the name "meantone" for this
temperament, please note this interesting observation for the most
elegantly simple version, the quarter-comma. Since the comma division
is symmetrical around D in this tuning (assuming C-based tuning), the
temperament for D should be set first. After tuning C and E as a just
5:4 M3, you can set the temperament for D by:

1) tuning A a Perfect 4th above E (and consequently a just M6 above C
if you want to double check)

2) tuning G a Perfect 5th above C

3) tuning D a Perfect 5th below the just A

4) play the D against the just G (it will be flat to the G by a
synontic comma)

5) Observe the beat rate

6) Raise the D till the beat rate is half the initial rate

This sets the tempered D. Now the two remaining tempered core
pitches, G and A, are tuned as mean pitches between C and D, then D
and E respectively, by an analogous procedure.

*!*!*!*!*!*! NOTE THE FOLLOWING: All three notes are tempered as
MEANS between two pitches, and not just the D. All other pitches from
Eb to G#, or wherever else you want to place the "wolf" fifth
(actually a diminished 6th or its inversion, an augmented 3rd), can
be tuned as just 5:4 Major 3rds to these core pitches.

This simplest and most elegantly symmetrical mother of all the other
meantone temperaments has two difficulites beyond the limits it
imposes on harmonic modulation:

1) Unlike 12-EDO, it does not order the degree of pitch error to
align with the order of the intervalic primes. In ET, the fifths are
almost just, the Major 3rds next (as bad as they may be), etc., so
the most dissonant intervals are also the least accurate. Quarter-
comma meantone has the fifths more than twice as mistuned as ET, but
Major 3rds are perfect, inverting the order of primacy for the
intervals.

2) Further, quarter-comma has diatonic half-steps 5.5 cents wider
than the already fat (to modern ears) diatonic half-steps of JI. The
narrow chromatic half-steps of JI are further narrowed by 5.5 cents.
This has mostly a melodic significance, but not a completely
negligible one.

These weaknesses seem to have bothered some quarter-comma users
enough to seek other variants of meantone, many of which did little
or nothing to extend the range of clean modulations, but rather
shifted some of the pitch error of the perfect 5ths and minor 3rds to
the Major 3rds, narrowing the diatonic half-steps and widening the
chromatic half-steps in the bargain.

🔗Paul Erlich <paul@stretch-music.com>

Hi Bob Wendell, I understood your clarificiation perfectly, and agree.

You then wrote,
>
> Paul had also said:
> Inharmonic timbres are fascinating and lots of great music can be
> made with them, but they can never match the immediate sense of
> "coherence" one gets from harmonic timbres.
>
> Bob answers:
> Since I work in practice with a cappella singers, my discussion
> assumes a high degree of harmonicity in the overtone structure.

Well, perfect harmonicity, to the extent that that is well-defined
for samples of finite length.
>
> Paul also had said:
> By the way, Bob, you never responded to my assertion that JI may
not
> always maximize consonance, e.g., for chords like CEGAD. Any
response?
>
> Bob answers:
> Sorry, Paul! CEGAD has quartal structure coexisting with triadic.
For
> quartal structures, Pythagorean JI works up to a point, but 12 ET
> works better in this instance because ET is a more or less
reasonable
> compromise between quartal and triadic structures, although I
> wouldn't argue that it absolutely represents the least harmonically
> entropic tuning.

I agree that ET is better than Pythagorean or 5-limit JI in this
case. Some form of meantone is probably even better.
>
> I do believe that is why some of the richer jazz structures one
hears
> in "comping" on the piano has evolved into stacked fourths that
> contain all the elements of extended triads. For example, Eb A D G
C
> F in ET works harmonically very nicely in place of the more
> conventional F A C EB G D (a minor 7th chord extended with a Major
> 9th and a Major 13th).

Umm . . . you mean a dominant seventh chord? Actually, we had quite
an interesting discussion on the tuning of a dominant seventh chord
with added 9th, 13th, and sharp 11th a while back. Monz made a
webpage with some of the suggested tunings.
>
> I'm not sure this is the kind of answer you were looking for, Paul,
> but
> it's my best take on it.

Sounds like we're in agreement, yet again!

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., BobWendell@t... wrote:

> Interestingly, the chord in the spelling you gave is a common final
> chord in some jazz and other earlier pop styles. I do not believe
> this
> usage would have ever evolved musically without a tempered scale
> environment.

Agreed yet again!
>
> 6) Raise the D till the beat rate is half the initial rate

Almost half, but not exactly (if you want exact 1/4-comma meantone).

> *!*!*!*!*!*! NOTE THE FOLLOWING: All three notes are tempered as
> MEANS between two pitches, and not just the D. All other pitches
from
> Eb to G#, or wherever else you want to place the "wolf" fifth
> (actually a diminished 6th or its inversion, an augmented 3rd), can
> be tuned as just 5:4 Major 3rds to these core pitches.

It took me about a week to tune my piano to meantone. If I didn't
know this, it would have taken a month.
>
> 1) Unlike 12-EDO, it does not order the degree of pitch error to
> align with the order of the intervalic primes. In ET, the fifths
are
> almost just, the Major 3rds next (as bad as they may be), etc., so
> the most dissonant intervals are also the least accurate. Quarter-
> comma meantone has the fifths more than twice as mistuned as ET,
but
> Major 3rds are perfect, inverting the order of primacy for the
> intervals.

That may be desirable. In fact, 2/7-comma meantone was favored at one
point (when meantone was first mathematically described), primacy
being given to both major and minor thirds over fifths. However, by
the late Baroque era and through most of the Classical, 55-tET and
1/6-comma meantone were deemed more desirable.
>
> 2) Further, quarter-comma has diatonic half-steps 5.5 cents wider
> than the already fat (to modern ears) diatonic half-steps of JI.

"To modern ears" is the key phrase. Most likely in the Renaissance,
the importance of incisive leading tones (both ascending and
descending) was rather lessened due to the non-tonal, often
panconsonant harmonic language. Then, after the advent of tonality,
incisive leading tones became the grammatical key to tonal meaning in
the music -- thus the move to 1/6-comma meantone.

🔗BobWendell@technet-inc.com

Bob had said:
*!*!*!*!*!*! NOTE THE FOLLOWING: All three notes are tempered as
MEANS between two pitches, and not just the D. All other pitches
from
Eb to G#, or wherever else you want to place the "wolf" fifth
(actually a diminished 6th or its inversion, an augmented 3rd), can
be tuned as just 5:4 Major 3rds to these core pitches.

Paul replied:
It took me about a week to tune my piano to meantone. If I didn't
know this, it would have taken a month.

Bob Answers:
Yes. We do seem to be pretty much justly tuned to each other on
almost every point, Paul! The point here was not to insult your
intelligence by pretending to teach you how to tune quarter-comma
meantone, but to point out its beautiful SYMMETRY in way that would
be clear to ALL interested readers.

In turn, the point of THAT was to show how the D is not the only
pitch tuned to a mean between two other pitches, but that the G and A
are also means symetrically disposed around the D in the core group
of fifths from C to E and tuned as means between just tunings to the
adjacent fifths on either side, namely the meantone D and the C or E
respectively.

On the "not quite half" comment, if you mean that theoretically the
mean is geometrical and not a linear arithmetic mean, you're right,
of course, but for these very small pitch ranges (splitting ~22 cents
for D, ~11 cents for G and A), the difference is much smaller than
the accuracy with which ordinary mortals can tune.