RE: Improvisation and our musical sounds

> Although improvisation may be a skill helpful to the
> mastery of a particular musical instrument I am not convinced that in
> itself it adds anything to ultimate musical experience. Nor do I feel
> that improvisational skill leads to more creative musical expression. In
> fact it much more readily falls into the realm of noodling than much of
> the so-called Western classical repertoire.

I let this sit for a while, but I find this positively repulsive.

(1) If you can hum what you're composing (a good sign that you're making
music rather than marks on a page), then humming spontaneously (even slowly,
and correcting errors) is improvisational skill. That may not be helpful to
the mastery of a particular instrument, but it sure leads to better
composition!

(2) Many masters of instruments cannot improvise to save their lives. Many
master improvisers create compositions in a realm very different from the
five-line staff -- details will change from performance to performance,
rhythms will transcend rational notation, the musical experience will
reflect the immediate conditions of the moment, communication between
audience and performer will occur -- in all, the music will have a structure
that has more in common with real-time human thought processes.

(3) Noodling on a staff is less musical than noodling on an instrument, and
noodling on an instrument qualifies as poor improvisation at best.

(4) I went to a very prestigious university. The senior composition class
was far beyond my qualifications to enroll in. The performance of final
compositions from this class SUCKED. There was not one note of music in the
whole thing -- I couldn't have IMPROVISED anything worse if I tried.
Meanwhile, rock groups were playing in basements, improvising away, and
sometimes reaching levels beyond pure entertainment. (I should note that a
performance of compositions from the graduate school of the same university
was far more musical and far more challenging -- however, the undergraduate
symphony orchestra positively DEMOLISHES the grad school orchestra!)

(5) The ultimate musical experience. Hmm. Aside from that once-in-a-lifetime
improvisation where it feels like your hands are controlled by a higher
power, this would have to include listening to Jimi, or Duke, or one of Bach
or Chopin's "frozen improvisations" (if only we could capture their actual
performances out of the vibrations in the air). I'll skip Wynton Marsalis'
jazz "compositions."

Blah blah. No one can say it better than Neil already did. Anyway, we need a
lot more composition AND improvisation before the 12-tone monarchy is
overthrown; each element of music-making needs the other to grow and
prosper. Denying that is denying microtonality's very viability.


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PAULE wrote:

> Matt Nathan wrote in TUNING digest 951,
>
> >Punning is purposely using the wrong pitch.
>
> Let's say you had a 6th chord, I'm sure we'd both be happy to tune it
>
> 1/1 5/4 3/2 5/3.
>
> Okay, now add the 9th. Do you add the 9/8 or the 10/9? The former forms a
> yucky 40/27 against the 5/3, and the latter forms the same icky interval
> against the 3/2. The best JI solution is to put the whole thing in
> Pythagorean, but I like meantone even better, so you have a nice major
> third. Anyway, the 9th of this chord is an example of how punning can be
> necessary without getting into the aesthetic arguments over comma shifts,
> scales as the basis of melody, etc.

I've played around with exactly this idea in JI intervals before.
I haven't tried it in the various meantones.

Outside of musical context, just comparing these various pitch sets and say
randomly arpeggiating them or something, I find the sound of the pythagorean
the most entertaining. It's nice and stretched out sounding (adding one more
fifth at B 243/128 starts to "stretch" my tolerence though).

To me, this is still not a "problem". Each of these pitch sets is a separate
sound with its own identity and possible musical uses, in other words, no
punning. To me, it would only be punning if you really wanted one voicing, but
your instrument only had another, so you substituted. That would be at least
conceptually ugly. Maybe I'm using pun differently than Daniel Wolf or others?

The "problem" here may be in trying to transfer what in 12tet is considered a
consonant chord--a Major 6 9 chord--into JI and trying to make it serve the
same purpose. I don't really consider any of the JI approximations of this as
consonant. All of them contain relationships which tug at the ear. If I had
a nice 8:10:12, or even 8:9:10:12 going on, I probably wouldn't want to add
some version of a dissonant 6th, or if I did, it would be for a musical reason
which would automatically suggest which pitch would be correct, for instance,
if the A were left sustaining from a previous F chord as we moved to a C
"add 9" chord then it would make sense as 5/3 and the dissonance against the
9/8 D would highlight its function and meaning.

My question would be, "why try to use a structure which doesn't suggest itself
musically?". If it occurs because of the juxtaposition of melodic fragments,
then it would both be justified by and serve to highlight those melodies. If
we were trying to make the most of the possible combinations of a predetermined
pitch set as some sort of "challenge", then I think we would be making the wrong
approach. (Sorry; I know you didn't want to get into the subject of whether
melody and harmony should determine pitch set or the reverse, but that's what
I'm thinking about these days. I reserve the right to change my position at a
later date. )

Incidentally, if you use the 9/8 and the 5/3, but add a 4/3 to the chord, like

1/1 9/8 5/4 4/3 3/2 5/3

the thing starts to support its own dissonance in a really nice way since
each little otonal 8:9:10 subset resonates on its own even with the 40/27's
and stuff going on between them, at least to my ear.

Matt Nathan

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PAULE writes:
>
>Let's say you had a 6th chord, I'm sure we'd both be happy to tune it:
>
>1/1 5/4 3/2 5/3.
>

Suppose you had a minor triad 10/12/15 - what major sixth would you add to
make a minor 6th chord?

Just curious,
-- Pat.



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From: PAULE

Sorry, I disagree with you about comma shifts. Melody is historically and
aesthetically prior to harmony. If the unaccompanied melody uses equal-sized
whole tones and no comma shifts, then the harmonized melody should too.

>Will you explain this dissonance function with
>figures please? It sounds interesting but I'm
>not sure what you mean by derivative.

"Derivative" means "rate of change" of a function in calculus. What I was
getting at is that if you assume that dissonance has a local minima at
certain simple just ratios (I believe this is certainly true at least for
5-limit intervals smaller than one octave), and make the additional
assumption that the rate of change of the rate of change is finite and not
zero at these minima, then a quadratic function is appropriate: the
dissonance function must be shaped like a parabola if you are close to the
just ratio.

>In JI, I prefer to think of the "complexity" (or something,
>rather than "dissonance") of an interval or sonority, which
>I measure not by multiplying the members of a ratio or
>chord, but by adding them! As an example, 4/1 is about as
>"complex" as 3/2 by my rule of thumb, and 8/5 is about as
>"complex" as 9/4.

This may be a valid issue but is irrelevant here, since we are merely
comparing representations of the same interval(s).

>In systems which deviate from just by small amounts,
>I'm not really sure how to calculate dissonance,
>except that it may have something to do with the
>speed of beating (faster beating, more dissonance).

That is the issue and I assume a quadratic function as justified above.
Many, many other justifications are possible.

>I'm afraid I'd still classify
>weighted mean-tone as an error-spreading and pitch-class-
>confounding system though. Its main value would be
>in the tuning of instruments whose pitch sets are finite
>and not manually adjustable during performance, especially
>for period music.

Very well.

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PAULE wrote:
>
> Melody is historically

probably

> and aesthetically

too general an assertion

> prior to harmony.

Melody, harmony, rhythm, timbre, and location
--the basic parameters of music--have varying relative
priority in various musics. There are also musics
(like some Stochausen [sp?]) which attempt to blur
these categories.

> If the unaccompanied melody uses equal-sized whole tones
> and no comma shifts, then the harmonized melody should too.

That's a big "if" there. Even assuming the most innocuous
and supposedly intuitive-for-Westerners diatonic JI scale:
1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1, as a basis for the various
et and other approximations, you see different-sized whole
steps.

We are the final arbitors of our own music, you and I.
My music uses as many intervals and pitch classes as I
want, melodically and harmonically. Yours uses what you
want. If you want to get into discussing what others
do in practice, I'm sure I can produce examples of "comma
shifts" in melody, accompanied and unaccompanied: Bartok's
field notations; lead singers in pop music who
(consistently, not accidentally) use for certain passages
pitches not available from the fixed-pitch accompanying
instruments; blues singers and blues players, jazz players
like Miles Davis, who purposely use different fingerings
to produce pitches near the "normal" ones; string quartets;
barborshop quartets; etc.

The only times you'll probably hear equal-sized whole
steps and no comma shifts in melody will be on fixed-
pitch instruments or from performers rigorously trained
to match such instruments. (I read, I think in Helmoholtz,
about the comparative difficulty of solfege students to
stay in tune as a group when accompanied by the (et)
church organ and ease when unaccompanied and allowed
to sign in JI.)

Comma shift is probably a misnomer too (like "wandering tonic").
I'd rather think of it not as a "shift" to a different version
of the same pitch class, but a movement to a new pitch class.

Maybe there's a psychological transistion or grey area
between when "shifts" become "new classes". In other
words, maybe historically (following scale-expansion
theories like Yasser's) new pitches were added as
"shifts", but over time became distinct "classes". For
example, if you were composing in a diatonic C Major
scale and were inspired to use a V7 of the relative minor
(E7), the G# might initially be thought of as a "shift"
of the psychological grouping "fifth of scale" rather
than a new class. In fact, our notation system still
shows the fossil remains of such thought in the very name
"G#" (one of 7 letter names plus a "shift" command, rather
than having 12 letter names with no shift commands).

> >Will you explain this dissonance function with
> >figures please? It sounds interesting but I'm
> >not sure what you mean by derivative.
>
> ...calculus.

Interesting. Do you have a formula?

> >I measure ["complexity"] not by multiplying the members of a
> >ratio or chord, but by adding them!
>
> This may be a valid issue but is irrelevant here, since we are
> merely comparing representations of the same interval(s).

Technically, "we" includes me, so we are talking about
both issues.

Is your dissonance function not applicable to comparing
perfect representations of different intervals? (I guess
it would give equal dissonance, 0, for both?)

Matt Nathan

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From: PAULE

I think the diatonic scale long predated the discovery of 5-limit harmony in
most cultures (India is a possible exception, though only 3-limit
consonances are mentioned in the original texts). It was constructed from
melodic considerations only, which means that ratios beyond the 3-limit are
of negligible importance. Ascribing 5-limit ratios to the diatonic scale is
in most cases, as I have said before, a historico-geographic fallacy, and a
very Eurocentric one at that.

It must be said, though it partially supports Matt's viewpoint, that many
cultures (the West, India, Arabia) seem to have discovered the 5-limit
relations through their SCHISMATIC, not SYNTONIC, approximations. It seems
that Pythagorean tuning was extremely well developed long before any
theoretical understanding of the 5-limit came about.

So something resembling a just diatonic scale did arise in all of these
cultures. In Pythagorean tuning, this is D E Gb G A Cb Db D. But this scale
is already "corrupted" by harmonic considerations.

The paradigm of similar tetrachords seems to be important for melodic
comprehensibility. This in itself can lead to many scales as are found
throughout the world, including some non-octave-repeating middle eastern
scales. The diatonic scale simply takes this concept to its extreme, as
tetrachordality manifests itself in every octave species. This requires a
single size of whole step, though I must admit that the syntonic comma is
too small to make the 5-limit JI diatonic scale very disturbing melodically.
Increase the size of the syntonic comma while keeping excellent 5-limit
approximations (try going from 34-equal to 22-equal to 15-equal) and the
scale does sound melodically incorrect. Melodically, I much prefer the
"Pythagorean" diatonic scale in 22-equal, where the thirds are tuned 7:6 and
9:7, to the 5-limit diatonic scale in the same tuning.

Ptolemy's scales (much like his astronomical models) seem to have been
derived from geometrical and mystical, not scientific (the appropriate
science for music is psychoacoustics), considerations, so they don't count.
I must qualify this by saying that Ptolemy did specifically mention the
psychological desirability of the division of the major third into two equal
parts, but felt that since all musical intervals must be measured
rationally, this could only be satisfied approximately! In this sense, JI is
but the imperfect temperament and something meantone-like is the Platonic
ideal!

The assertion that melody is aesthetically prior to harmony is of course a
subjective one, but something I have learned the hard way. As a child I
taught myself keyboard harmony, and equated composition of and meaning in
music to the harmonic progressions. But unknowingly, I was already obeying
melodic considerations in voice leading, etc. I once composed a progression
on guitar which, I had no idea until I wrote it down, alternated between
chords diatonic to the key and highly chromatic ones.

The diatonic scale defines tonality, and shifts thereof, in Western music.
Harmonic usage is, I believe, governed by harmonic-series or
small-integer-ratio-type considerations, but the melodic aspects are at
least as important for the overall effect of the music. Deriving the
diatonic scale from the tonic, subdominant, and dominant triads is utterly
inaccurate, in my opinion. There are too many pieces in major that focus on
I-ii-I progressions, and pieces in minor (aeolian, to be more academic) that
focus on i-VII-i progressions, for this to be an accurate explanation, for
not even the tuning of the chords will be correctly specified by such an
explanation.

>(I read, I think in Helmoholtz,
>about the comparative difficulty of solfege students to
>stay in tune as a group when accompanied by the (et)
>church organ and ease when unaccompanied and allowed
>to sign in JI.)

If they were not harmonizing but singing in unison, I seriously doubt that
they gravitated towards a scale with unequal whole steps, though Helmholtz
was clearly biased in favor of believing that they did!

>Comma shift is probably a misnomer too (like "wandering tonic").
>I'd rather think of it not as a "shift" to a different version
>of the same pitch class, but a movement to a new pitch class.

I don't think such an analytic framework would hold up if applied to
common-practice Western music. For starters, principles of thematic
development are likely to be utterly destroyed.

As for quantifying dissonance, there are many psychoacoustic factors at play
and incorporating them all into a single model does not seem advisable since
the different mathematical definitions of dissonance probably have
different, but equally valid, manifestations in the field of musical
perception. I recommend
The Acoustical Society of America Volume 45 No 6: Cononance Theory Part I+II
from Akio Kameoka and Mamoru Kuryagawa
for understanding the "roughness" aspect of dissonance. Plomp and Levelt and
Bill Sethares are good sources too. As for the "tonalness" aspect, which is
mandated by the phenomena known variously as virtual pitch, residue pitch,
complex tone pitch, fundamental tracking, and possibly periodicity pitch, I
have been working on this for some time. I have been communicating
particularly with James Kukula on this topic (James, are you there?).

>Is your dissonance function not applicable to comparing
>perfect representations of different intervals? (I guess
>it would give equal dissonance, 0, for both?)

I don't believe all just intervals, or even all just intervals within a
certain odd-number limit, have zero dissonance. However, since the exact
form of such a dissonance function is contreversial and certainly depends on
what _kind_ of dissonance you're talking about, I simply made a highly
plausible calculus approximation to only that part of such a function as is
needed to optimize meantone tunings, etc. Other work (see above) is
concerned with more general questions.

>> >Will you explain this dissonance function with
>> >figures please? It sounds interesting but I'm
>> >not sure what you mean by derivative.
>>
>> ...calculus.

>Interesting. Do you have a formula?

The whole point is that it doesn't matter what the exact form dissonance
function is! The formulas that result from this assumption is, as I tried to
make clear in words,
minimize (error of 3:1)^2+(error of 5:1)^2+(error of 5:3)^2 and
minimize (3*error of 3:1)^2+(5*error of 5:1)^2+(5*error of 5:3)^2.

The results of applying these to meantone tunings I have already posted in
both cents and exact formulae. If you want me to work you through the
calculus please try to reach me offline, I don't think we need to scare any
more musicians off this list.

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PAULE wrote the following:

'' It must be said, though it partially supports Matt's viewpoint, that
many
cultures (the West, India, Arabia) seem to have discovered the 5-limit
relations through their SCHISMATIC, not SYNTONIC, approximations. It seems
that Pythagorean tuning was extremely well developed long before any
theoretical understanding of the 5-limit came about.''

and:

'' Ptolemy's scales (much like his astronomical models) seem to have been
derived from geometrical and mystical, not scientific (the appropriate
science for music is psychoacoustics), considerations, so they don't count.
''

Response:

You can't have it both ways. There are alternative readings of the
historical materials, but the most convincing rationale for the use of
schismatic approximations of intervals with five as a factor would be a
doctrinal or mystical commitment to a three limit. (And, in practice, most
Saz players correct their schisma-off thirds to 5-limit ratios). That
Ptolemy would go beyond the three limit in his tunings was contrary to the
prevailing neo-pythagoreanism, and, among the classical harmonicists,
Ptolemy is probably farthest from any such mystical tradition. In fact, his
_Harmonics_ is virtually the only classical source to clearly discern
between theoretical and actual tunings. The remainder of the remark,
criticising Ptolemy on the basis of the superior propriety of a discipline
(Psychoacoustics) which would wait until the late 19th century to be
established (in Leipzig), and whose scope and dimensions were well ouside
of the scientific discourse of Ptolemy's time is, simply, ridiculous.

As to the historical priority of a three-limit in music theory (given
Assyrian and Chinese evidence), I assume this is true, but the advent of
work with a monochord (and the entire discipline of harmonics) is not
characterised by an innocence towards higher prime numbers. I would suggest
that the prior three-limit theories were not constructed with such
innocence but in full comprehension of the fact that additional primes were
not going to create any systemic closure, and that factors of three were
sufficiently complex to generate a large supply of useable tones, thus the
''mystical'' attributes of the number three were a - for us -
comprehensible recogition of real mathematical and musical properties,
while not explicitly condemning the introduction of higher primes - which
may very well have been part and parcel of the actual musical practice.

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While there are a lot of interesting issues that have been raised in this
exchange, I would specifically like to respond to the one about historical
precedence of tuning models for the diatonic scale.

PAULE wrote:

>I think the diatonic scale long predated the discovery of 5-limit harmony in
>most cultures (India is a possible exception, though only 3-limit
>consonances are mentioned in the original texts). It was constructed from
>melodic considerations only, which means that ratios beyond the 3-limit are
>of negligible importance. Ascribing 5-limit ratios to the diatonic scale is
>in most cases, as I have said before, a historico-geographic fallacy, and a
>very Eurocentric one at that.
>
>It must be said, though it partially supports Matt's viewpoint, that many
>cultures (the West, India, Arabia) seem to have discovered the 5-limit
>relations through their SCHISMATIC, not SYNTONIC, approximations. It seems
>that Pythagorean tuning was extremely well developed long before any
>theoretical understanding of the 5-limit came about.
>
Dan Wolf talked some about the distinction between theory and practice, and
I think it's a crucial distinction. First, when does a culture NEED a
precisely defined, standardized tuning system? Several possible answers
occur to me (though there may be others): 1) when it is desirable for
several fixed-pitch instruments to play together (the definition of
"fixed-pitch" may be a gray area, I know), 2) when there are instruments
with frets or a single sounding body per pitch (even then they don't
necessarily have to be standardized), 3) when there is an important
numerological justification.

Many cultures don't meet any of these criteria. Even those for whom #3 is
important, such as India, China, Persia, and (sometimes) ancient Greece,
there's often no evidence that these mathematical constructs had any
relationship to practice. Lacking that evidence, the possibility arises
that so-called Pythagorean tuning was invented as a way of describing
musical systems already in practice. While it's possible to derive scales
purely mathematically, I am skeptical that most scales, including the
diatonic, were initiated this way.

Perhaps scales evolve intuitively, though that intuition may be based on
mathematical models that may not be verbalized by the musician. (Thus the
octave is virtually universal, but musicians may not be aware that its
significance derives from its closeness to 2:1). Overblown fifths, small
number ratios, Pythagorean cycles, tendency towards equal step sizes, these
are all possible models for an (originally) unspoken basis for the diatonic
scale. If that's the case, I don't think it's possible to say for sure
which is correct from a historical point of view.

When dealing with non-fixed-pitch instruments, we are in even more trouble.
The deviations from a theoretical standard tuning system of a melodic line
sung by a voice alone or played on a bamboo flute alone can easily approach
the syntonic comma. Oh, I know, some of this may just be bad musicianship,
but I still think it's very problematic to derive a precise tuning system
from such music. Indian sources, it's true, give elaborate tuning
instructions for the vina, but nearly all of them, back to the vedas,
ascribe primacy to the human voice.

Bill

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)621-8360 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^




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> PAULE wrote:
>
> I think the diatonic scale long predated the discovery of 5-limit harmony in
> most cultures (India is a possible exception, though only 3-limit
> consonances are mentioned in the original texts). It was constructed from
> melodic considerations only, which means that ratios beyond the 3-limit are
> of negligible importance. Ascribing 5-limit ratios to the diatonic scale is
> in most cases, as I have said before, a historico-geographic fallacy, and a
> very Eurocentric one at that.

A few other list participants have responded to the historical
issues, and I will defer to them, not being an historian. I do
think that just as practice differs from theory today, the two
have probably differed through history. Even if your 3-limit
history is correct regarding practice, we live today, and 5
is important now and has been for hundreds of years.

I will hopefully live tomorrow as well, and 7, 11, 13, 17,
and 19 are important to me too, so my music will reflect
this. My ear doesn't seem to want to do anything with 23 or
above right now, in fact, it kind of ignores 13 and 17 and
jumps right from 11 to 19. This may be because I keep
hearing 12tET minor thirds which are within 3 cents of 19/16.

[snipped interesting stuff which I don't really have a response to]

> The paradigm of similar tetrachords seems to be important
> for melodic comprehensibility.

I don't know where you get the data for statements as
general as this. All I can say is that I can find examples
of comprehensible melodies without simliar tetrachords.

> The diatonic scale simply takes this concept to its extreme,
> as tetrachordality manifests itself in every octave species.

Neither ascending melodic minor nor harmonic minor have
similar tetrachords. Each of these has modes which have
also been used. Maybe you wouldn't considered these diatonic?

[snipped more interesting stuff which I don't really have a response to]

> ...Deriving the
> diatonic scale from the tonic, subdominant, and dominant triads is utterly
> inaccurate, in my opinion. There are too many pieces in major that focus on
> I-ii-I progressions, and pieces in minor (aeolian, to be more academic) that
> focus on i-VII-i progressions, for this to be an accurate explanation, for
> not even the tuning of the chords will be correctly specified by such an
> explanation.

Here would be cases, like I-ii-I, which I would say are
implying more than 7 pitches, or at least a different
7 pitches than say I-IV-I, which is what I meant when
I said that much diatonic music implies more than 7
pitch classes and will be played that way when the
ear is given sway.

> >(I read, I think in Helmoholtz,
> >about the comparative difficulty of solfege students to
> >stay in tune as a group when accompanied by the (ET)
> >church organ and ease when unaccompanied and allowed
> >to sing in JI.)
>
> If they were not harmonizing but singing in unison, I seriously
> doubt that they gravitated towards a scale with unequal whole
> steps, though Helmholtz was clearly biased in favor of believing
> that they did!

I dare say that your own bias is clearly showing here, and
that Helomholtz was probably more observant than you and I
combined. I suppose one could rerun the experiment now
to get another opinion.

> As for quantifying dissonance...

Thanks for the math references.

Matt Nathan

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From: PAULE

>I will hopefully live tomorrow as well, and 7, 11, 13, 17,
>and 19 are important to me too, so my music will reflect
>this. My ear doesn't seem to want to do anything with 23 or

>above right now, in fact, it kind of ignores 13 and 17 and
>jumps right from 11 to 19. This may be because I keep
>hearing 12tET minor thirds which are within 3 cents of 19/16.

Yes, this bears repeating because I really believe it is true:

Pieces in minor tended to end with a Picardy (major tonic) chord in the days
when the minor triad was tuned 10:12:15. When the tuning became much closer
to 16:19:24, it became ok to end on a minor triad. I claim, because I hear
this (maybe it's just difference tones, maybe not) that this is due to that
fact the the fundamental of 16:19:24 is octave-equivalent to the root,
establishing a more stable sonority. The distinction between the two chords
is enough to give them a different nature, both useful, while the
distinction between a just and an equal-tempered major triad is merely that
the former is smoother than the latter. So JI composers might have little
use for two versions of the major triad (4:5:6 vs. 1/24:1/19:1/16), while
relishing the distinctions between 10:12:15, 16:19:24, and also 6:7:9.

>> The paradigm of similar tetrachords seems to be important
>> for melodic comprehensibility.

>I don't know where you get the data for statements as
>general as this. All I can say is that I can find examples
>of comprehensible melodies without simliar tetrachords.

let me add "of melodies that exceed a perfect fourth in range" to the above.
Also, I said important, I didn't mean necessary, I meant helpful.

>> The diatonic scale simply takes this concept to its extreme,
>> as tetrachordality manifests itself in every octave species.

>Neither ascending melodic minor nor harmonic minor have
>similar tetrachords. Each of these has modes which have
>also been used. Maybe you wouldn't considered these diatonic?

In the West, these modes (which I love using and talking about; don't forget
the harmonic major scale) arose in the era of triadic harmony. The melody
has deferred to the harmony. I guess that just proves your point, doesn't
it? Well, I still think there is room for an "unaccompanied melody"
aesthetic in harmonic music. More on this later.

Modes like these and even more complex ones arose in the Middle East and
India, though. However, the modes which do exhibit some tetrachordal
similarity are considered more basic in these cultures. They use altered
modes as a means of achieving the musical depth and variety for which the
West relies on harmony. Sometimes I really prefer the purely melodic
approach, and find classical modulatory contrivances boring. Other times
there is nothing more breathtaking or subtlely beautiful that a
well-conceived modulation.

>[snipped more interesting stuff which I don't really have a response to]


>> ...Deriving the
>> diatonic scale from the tonic, subdominant, and dominant triads is
utterly
>> inaccurate, in my opinion. There are too many pieces in major that focus
on
>> I-ii-I progressions, and pieces in minor (aeolian, to be more academic)
that
>> focus on i-VII-i progressions, for this to be an accurate explanation,
for
>> not even the tuning of the chords will be correctly specified by such an
>> explanation.

>Here would be cases, like I-ii-I, which I would say are
>implying more than 7 pitches,

A severe violation of Occam's razor!

>or at least a different
>7 pitches than say I-IV-I,

Well, I don't see a single pitch that would have to be different, but if you
meant I-V-I, yes, you're correct.

>which is what I meant when
>I said that much diatonic music implies more than 7
>pitch classes and will be played that way when the
>ear is given sway.

So you're saying that even if the melody really stands up on its own,
different harmonizations may produce different pitch inflections in the
melody. That may be so in slow tempi, but it must be recognized that if the
melody is the basis of the composition, these inflections do not represent
different pitch classes but just acoustically favorable dispositions of some
pitch classes. The melody should have a description, in terms of slighlty
mutable pitch classes, that reflects the fact that it is the same melody
regardless of harmonization. If you treat the different JI dispositions of a
given pitch class as different pitch classes, your analysis of thematic
development in common-practice works would end up reaching ridiculous
conclusions, since a downward drift is so common in JI realizations of the
repertoire. The home key is the home key, not a key seven commas flat.

This is why I love meantone tuning. It is almost as pure as JI, and you
don't have to worry about any of this! Unfortunately, dominant sevenths are
rather ugly in meantone, so it's better for Monteverdi than for barbershop.
Augmented sixth chords in meantone are great barbershop sevenths, though!

>> >(I read, I think in Helmoholtz,
>> >about the comparative difficulty of solfege students to
>> >stay in tune as a group when accompanied by the (ET)
>> >church organ and ease when unaccompanied and allowed
>> >to sing in JI.)

>> If they were not harmonizing but singing in unison, I seriously
>> doubt that they gravitated towards a scale with unequal whole
>> steps, though Helmholtz was clearly biased in favor of believing
>> that they did!

>I dare say that your own bias is clearly showing here, and
>that Helomholtz was probably more observant than you and I
>combined. I suppose one could rerun the experiment now
>to get another opinion.

Recent data I have seen on barbershop singing points to major thirds of
around 397 cents (string players average around 405 cents), but plenty of
flattened minor sevenths. It appears musicians are unwilling to lessen the
degree to which tendency tones "lean" to their target from the
equal-tempered standard, but are quite willing to increase this leaning when
the results are acoustically favorable. In any case, understanding the fact
that the

Helmholtz, as Ellis attests, was more interested in proving his theories
than making objective measurements. He was a genius of natural physics, but
music is not a natural phenomenon.

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> From: PAULE
>
> > > ...Deriving the
> > > diatonic scale from the tonic, subdominant, and dominant
> > > triads is utterly inaccurate, in my opinion. There are too
> > > many pieces in major that focus on I-ii-I progressions,
> > > and pieces in minor (aeolian, to be more academic)
> > > that focus on i-VII-i progressions, for this to be an
> > > accurate explanation, for not even the tuning of the chords
> > > will be correctly specified by such an explanation.
>
> > Here would be cases, like I-ii-I, which I would say are
> > implying more than 7 pitches,
>
> A severe violation of Occam's razor!

Haha, yes it involves only 6 pitches! This is amusing but
a misunderstanding. The diatonic symbol IV pretty
assuredly reflects the rational structure 4/n 5/n 6/n
where n is 3. The diatonic symbol ii is a case of
confounded rational structures however. There are at
least two distinct rational structures which suffer
lossy compression in a diatonic assumption and get
lumped under "ii". You can sometimes tell by context which
of these is being implied. The ii chord can function
in context as either the relative minor of IV, which comes
from the rational structure n/4 n/5 n/6 where n is 5/3,
or as the secondary v (minor) of V, which comes from the
rational structure n/4 n/5 n/6 where n is 27/16.

A diatonic piece which uses both I-(v of V)-I and
I-IV-I (no I didn't mean I-V-I) implies at least
the 8-pitch set 1/1 9/8 5/4 4/3 27/20 3/2 5/3 27/16.

In vector notation, this is (using fixed-width font):

cell:

5
|
1 - 3

pitch set (C tonic):

A,- E
| |
F - C - G - D - A
|
F'

Matt Nathan

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From: PAULE

>The diatonic symbol ii is a case of
>confounded rational structures however.

No, rational structures are a case of confounded diatonic theory.

>There are at
>least two distinct rational structures which suffer
>lossy compression in a diatonic assumption and get
>lumped under "ii". You can sometimes tell by context which
>of these is being implied. The ii chord can function
>in context as either the relative minor of IV, which comes
>from the rational structure n/4 n/5 n/6 where n is 5/3,
>or as the secondary v (minor) of V, which comes from the
>rational structure n/4 n/5 n/6 where n is 27/16.

The ii chord must often be both or neither. In a I-ii-I progression (common
in popular music), it is definitely neither. Rational structures fail in
these cases.

Here, Matt, I wrote a piece where the chord progression is
I-IV-ii-V-I-IV-ii-V-(ad infinitum). Analyze this piece in Just Intonation.
How many pitches are implied here? When does the piece return to the home
key? Are these results going to make any sense for musicians and composers
working in a meantone temperament (including 12-tET)? Do your
interpretations have any psychoacoustical manifestations? If not, I maintain
it's nothing more than playing with numbers. Composing in Just Intonation,
on the other hand, is a valid enterprise and I have no objection to trying
to produce diatonic structures in JI. Doing so requires extra care, however.

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