Breaking stride (airplay, perhaps?)

As I think I've mentioned before, my weekly radio program on the local
Pacifica affiliate usually collects and sequences a bunch of music with
little regard for genre. The usual wheezy joke pictures me as enticing
impressionable youth to a lifetime of Darmstadt honkin' and tweetin'.
I think that a look at http://www.msn.fullfeed.com/~gtaylor/RTQE.html
ought to provide a more nuanced view. Well, lately I've been turning
over the thought of doing a *whole* fenced-off special of work in
other tunings; a program which might include your work.

There are a couple of things that led me to this decision. First, I've
been routinely getting queries lately whenever I run non12TET work
(most recently, it happened that some David First and Jukka Tiensuu
and Alistair Riddell wound up coming one after another completely
by accident), and I'm reminded that I've not done an all non12TET
show for years.

Second, I think that my colleague Arthur and I will be trying to procure
an extra hour+ from our late-night/red-eye pal gwillard and running
all of LaMonte Young's "The Well-Tuned Piano" at some point in the near
future. It's been six or seven years since we've done it, and the time
seems right.

Finally, I asked local wunderkind Bill Sethares if he'd be interested in
coming on RTQE and playing some music and talking about his work
before he turned world traveller. I'll be getting to it sometime this
fall or early winter.

This clustering suggests that maybe it's time to solicit your advice
and your recordings, should you be interested in contributing to
such a project. If you've got any advice on what you think would be
a nice "here's what's possible outside of 12TET" show [for an erudite
but "normal" radio audience], drop me a line. I may already have
some examples of your work [certainly Larry Polansky, and Ted
Mook and and the Artifact folks have been great, and Carter Scholz
has kept me up on the occasional Just Intonation tape releases,
for example. I know that you're probably not in the habit of doling
out promos like candy, so it's best to be prudent before you start
posting me stuff I already have]. Length will eventually be a concern;
you'd be surprised at just how quickly 2 hours of uninterrupted
airtime can fill up. I'd very much like to keep the announcer's
blathering to a minimum in order to maximize the time for
cortical damage, as well.

In a sense, I'd also be interested in what some of you view as more
or less "canonical" pieces [for example, which Wendy Carlos from
Beauty in the Beast would I run?] as well. I think it might be best
to take this to email.

I'll let you know as things progress.

_
I would go to her, lay it all out, unedited. The plot was a simple one,
paraphrasable by the most ingenuous of nets. The life we lead is our only
maybe. The tale we tell is the must that we make by living it. [Richard
Powers, "Galatea 2.2"] Gregory Taylor/Heurikon Corporation/Madison, WI



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Daniel,

I'm not familiar with Fux's fundamental bass, but I know that I would
disagree with Rameau where he does not allow the fundamental bass to
progress by second.

>Thus, only pitch complexes constructed from (lower)
>harmonic series members will be significantly affected by masking, such
>that difficulties in discerning individual tones from the entire complex
>occur. (Anyone who has ever tried to dictate ensemble musics has
>experienced the masking problem).

My point about combination tones was that the masking that occurs with
complexes constructed from subharmonic series members will be at least as
bad as that which occurs with harmonic complexes.

>as the first visual observation was of different
>strings (of different tension or thickness) moving faster and slower and
>yielding different pitches (this is immediately observable in gut or wire
>strings throughout the frequency ranges presumably used in ancient music),
>and then the compared lengths of a single string were then used to quantify
>this observation.

And this quantification would have led to subharmonic relationships forming
an arithmetic progression, the simplest progression to understand (and to
continue to infinity).

>Oscillatory periods and wave lengths are not observable
>as such without special apparati

True, but note the frequency is one over the ocsillatory preiod; therefore
if temporal measurements are basic, as you say, oscillatory periods are
logically prior to frequencies.

>This is a weak argument, but I give it anyways: why is it that
>microtonal music theorist structure their discourse almost universally in
>terms of frequency ratios and not stringlengths, wavelengths, or periods?
>Either this is just a convention (which I doubt because each individual
>theorist seems to like building from first principles), or there is indeed
>an intuitive quality to frequency ratios not shared by _alternative_
>descriptions.

Note that only very recently is this the case; as recent and as
scientifically astute a microtonal theorist as Huygens used the inverse
representation (which could have referred to stringlengths, wavelengths, or
periods).

I'm sure you knew what my answer would be: (a) frequency ratios describe the
perception of the chord by the central pitch processor, so important
characteristics such as the root of the chord can be inferred from this
representation; (b) the combination tones will only make this representation
more relevant in terms of the implied root. The importance and special
status of the harmonic series, _especially_ when used to organize sine
waves, cannot be denied. It comes into play every time we hear a musical
instrument and we think we know what note it's playing.

>_Mathematically_, however, the arithmetic divisions (subhamonic)
>were recognized as closed segments of the harmonic division; each
>individual arithmetic division yielded a finite number of pitches and was
>exhausted,

What on earth do you mean?

>while the harmonic division was continuable indefinitely.

This seems utterly nonsensical. Clearly the harmonic divisions _had_ to be
expressed in terms of the arithmetic divisions, not vice versa, in order to
instruct makers of instruments where to put the frets. Your "set theoretic"
distinction of one set as superior or more general than the other is
likewise nonsense.

>(5) I havent a degree in physics but I do recognize that the region
>between the smallest particle (10^-33 or so) and the vacuum (null) is not
>precisely mirrored by its inversion.

10^-33 what? 10^-33 meters? Are meters a unit of frequency? No, they are a
unit of length. Therefore, it is at small lengths, of strings or of waves,
where you would expect a "boundary problem"; i.e., high freqencies. Your
argument falls flat on its face.

-Paul


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I think the disagreement/agreement between the fundamental bass and the
actual bass is a very important source of tension/resolution in music by any
of the composers you mentioned.

>2) In that only pairs of tones will share harmonic series in a subharmonic
>chord, the masking will be at a lower amplitude than for a harmonic chord
>where all tones share a single series.

Then you must be talking about a type of masking different from that which
is only a function of pitch proximity. I thought you weren't, which is why I
pointed out that subharmonic series will, ceteris paribus, lead to more
masking, due to the ubiquitous combination tones. However, you must be
talking about a phenomenon related to the virtual pitch phenomenon, wherein
partials in a harmonic relationship are subsumed into a single perception,
making the components difficult to hear. This phenomenon is quite easy to

demonstrate, it is what distinguishes harmonic series harmonies of sine
waves, and what makes harmonic-series chords easier for the brain to
understand than subharmonic series chords.

>3) A division of a string into aliquot parts (subharmonic) is finite.
>Successive harmonic divisions are not. Example: the division of the string
>into four equal parts yields four pitches of the subharmonic sequence 1/1,
>4/3, 2/1, 4/1 and no more.

This is because you have defined the string as the lowest possible pitch.
Clearly a series that proceeds downwards will reach a limit when you have
this constraint in place, while a series proceeding upwards will not.

But you can always get a longer string to see what 5/4 or 55/4 of the
original string length sounds like. However, you will eventually reach
impossibly microscopic string lengths with a harmonic division. This
argument is going nowhere.

>4) The period (for example, one second) is a function of our notation.

And what isn't?


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>There are some musical contexts (and some temperaments as well -
>the tuning of the TX81Z comes to mind)

What's this?

>where 9 _has_ a harmonic function
>distinct from 3^2.

I couldn't agree more: that's why 9-limit harmony is different from 7-limit
harmony. But to graph the same pitch at two different points on a lattice
seems too confusing. That's just the way my brain works, though. More:

>Wilson has [...] worked with [...] 3(1,3,5,7,9,11),

I don't dispute the musical validity of this construct. In fact it supports
everything I've been saying about odd numbers and about including a factor
of 1 in the CPS. It's just that in graphing this along with 2(1,3,5,7,9,11),
you'd have no idea that 1*9 is the same pitch as 1*3*3, if you have a
separate axis for 9.


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From: PAULE

>>'' Pieces in minor tended to end with a Picardy (major tonic) chord in the
>>days
>>when the minor triad was tuned 10:12:15.''

>Can you identify ''those days'' with any exactitude?

No, there is no single set of dates. I would say Renaissance through Middle
Baroque. Well-temperaments brought the 16:19:24 tuning into currency in the
Late Baroque period. Please don't make this into another black-and-white
issue, I'm trying to explain tendencies, not impose rules. Having played in
these tunings during the 20th century, I can't give experimental proof that
my own tendencies in this regard are similar to historical ones indepentenly
of having grown up bombarded with classical music on the radio (which
included some authentically-tuned performances of Renaissance music). But I
feel them strongly.


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>}I doubt, however, that
>}special preference would be given to movement by like intervals,
>}particularly in the _contrary_ motion resolution of an augmented sixth as=
>
>}this would seem to be extremely difficult for the ear to judge accurately=
>
>I think you got my meaning wrong. What I meant was that preference is given
>to resolution by like intervals *in the style as a whole*, not in a given
>chord. When the listener of tonal music hears an augmented sixth resolve for
>the first time, he/she hears both voices doing something very familiar. It is
>only the harmonic context which is unfamiliar.

>Paul Erlich was describing the mean of two intervals while Graham Breed
w=
>as
>describing the freshman sum.

No, I only said "about halfway" and only in the case of 9:8 and 10:9.
Anyway, the difference between the two means is 0.6 cents, which is too
little to quibble over.

>This does, however, raise an interesting set of problems. Given two
ratio=
>s
>(a) will the freshman sum of the ratios (notated in specific octaves)
>always yield the peak consonance? and (b) how would one determine the
pea=
>k
>dissonance? =


>I suspect that (a) is true as long as the two intervals are within a
>certain magnitude (thus the freshman sums of 1/1 and 2/1, 3/2, or 2/2
and=

>3/2, which is 5/4, are clearly peaks, but that between 81/80 and 2/1,
>83/81, is not) while (b) is rather more complicated. Depending on
registe=
>r,
>timbre, amount of mistuning and musical context I might find a slightly
>mistuned perfect fifth to be more dissonant than a 12tet augmented
fourth=

>or minor second.

Well, a VERY slightly mistuned interval will be more consonant that the
freshman sum of the just interval with an equally simple one.


>I don't have any elegant way of distinguishing these two=

>kinds of dissonance. Any ideas?

You've talked specifically about one kind of consonance and one kind of
dissonance. What are the two kinds of dissonance you are thinking of? (I
think I may be able to offer some answers once I'm sure what you mean).

>< But the simplest interval between the 9/8 and
>>
>Could you give us the condensed version of why 19/17 should be simpler
th=
>an
>11/9?

It shouldn't! Read again what I wrote.

>I find an 11/9 much more easy to tune than a 19/17 whether with
>oscillators or strings.

Me too!

>The two types of dissonance I refer to are (1) dissonances as classes
of
>intervals independent of the exact intonation and (2) dissonances
perceiv=
>ed
>on the basis of the exact intonation.

Let's consider the definitions of dissonance formulated by Helmholtz and
Plomp & Levelt and used by Kameoka & Kuriyagawa and Sethares. For two
pure tones in a typical register, there is no dissonance at the unison,
dissonance rises dramatically at a difference of about 1/12 tone,
maximal dissonance occurs at a difference of about 1/8 tone, and falls
off gradually to zero at a difference of about a minor third. Total
dissonance is defined as the sum over all pure tone pairs of the
pairwise dissonances. Therefore, for typical harmonic timbres, the most
dissonant interval of all is the unison detuned by 1/8 tone, since every
partial of one note is 1/8 tone away from an equally loud partial of the
other note. The second most dissonant interval is the octave detuned by
1/8 tone, since every partial of the higher note is 1/8 tone away from a
somewhat quieter partial of the lower note. The third or fourth most
dissonant interval is the perfect fifth detuned by 1/8 tone, since every
other partial of the higher note is 1/8 tone from an almost as loud
partial of the lower note. With inharmonic timbres such as found in the
gamelan, this last interval is not necessarily dissonant at all, since
there is nothing resembling a third harmonic partial in the tones (the
octave is produced, to a small extent, by the ear and/or brain --
witness phenomena such as second-order beating). If one looks at the
graphs of dissonance versus interval size for complex harmonic tones in
any of these references, one will see the highest peaks lying a very
short distance from the lowest troughs (the latter are at the simplest
ratios). These peaks are very narrow because all the pairs of partials
involved rise in dissonance as one approaches the peak, and they all
fall off in dissonance as one retreats from it. So these are examples of
your type (2) dissonance.

Now consider an interval such as 11/9. The 6th partial of the lower tone
is about 1/6 tone off the 5th partial of the higher tone, and the 5th
partial of the lower tone is about 1/6 tone off the 4th partial of the
higher tone. As one enlarges the interval, the former pair will fall off
in dissonance, while the latter pair will intensify in dissonance until
one is about 1/8 tone flat of a 5/4. Similarly there is both an increase
and a decrease in dissonance as one decreases the interval to about 1/8
tone sharp of a 6/5. Since the total dissonance is the sum of the
dissonances of the pairs of partials, there is a bit of a plateau of
total dissonance around 11/9. So this is an example of your type (1)
dissonance. This plateau is not as high as the peaks that flank the
simplest ratios. If the partials through the 11th are audible (as on a
piano), one can tune the 11/9 exactly by eliminating the beating.
However, the beatings between the 6th and 5th partials of the lower note
and the 5th and 4th partials of the upper nore will be more powerful and
therefore the 11/9 will represent a very minor local minimum of
dissonance at best.

One can also consider the virtual pitch/harmonic entropy component of
dissonance, which leads to similar conclusions. I will discuss this if
you wish.