Muddy waters

Hi Paul E,

>Let those of us who don't understand Partch's work not accuse
>him of muddy thinking!!!

How about muddy writing?

>As for priority, I seriously doubt the claim that the
>prime-limit concept predated Partch's work; in fact, it is often
>merely the result of misunderstanding Partch.

Gosh, I thought Pythagorus started the whole thing (at least in
the west) with his famous 3 limit. I certainly was working with
prime limits before I ever heard of Partch.

>There are those who subscribe to a rectangular
>matrix/ROHS/LCM/Tenney harmonic distance idea. According to any
>of these philosophies, 15:1 is "the same" as 5:3,...

They produce patterns of the same length, but the patterns are
quite different. The picture frames are the same size, but the
pictures they contain are not the same. But you can use LCMs to
both determine the size of the picture frame and the contents of
the picture.

Actually, the 15:1 ratio, assuming it is truly phase-locked,
would be heard as a single tone, since it can be assumed that a
substantial amount of the acoustical energy is in the
fundamental. However, if we decompose the sum of 5:3, into its
component sine waves from the viewpoint of the 15 unit length of
the repeated pattern, we find that it has no energy in the
fundamental, and therefore would be heard as two distinct,
harmonious tones.

>i.e., a major seventh is just as dissonant/complex as a major
>sixth.

I must be missing something here. I thought a major seventh was
15:8, and a major sixth was 5:3. The last time I checked the LCM
of 5:3 was 15 and the LCM of 15:8 was 120.

In your discussion, you talk of the consonance of intervals,
which I presume means you are discussing only the consonance of
diads. How does your theory handle triads?

As a programmer, I find it more comfortable to deal with these
questions in terms of the sorting and searching of lists. All
these geometrical approaches seem to me to be only introducing
unnecessary complications. The geometry relevant to music is the
geometry of the sound waves produced, and perhaps the geometry of
the ear. JI music theory is a very quantum kind of thing, and
analyzing it in terms of lines and spaces with their infinite
numbers of points seems to me to drag in lots of irrelevant
information which just has to be filtered out later.

Using octave equivalence, there would seem to be 1320 possible
chords in a 12-tone scale. Making a list of these chords and
sorting them according to their LCMs is simple and easy on a
computer, and gives a good starting point for a discussion of
relative consonance. Certainly, the list could be reordered in
many ways, but I believe that most of these orderings would have
a high correlation with an LCM ordering.

If we want to discard octave equivalence and use instead just the
list of all notes in a multi-octave scale, or if we want to use
more notes per octave, the problem is still quite manageable.

I don't see that picking certain diads out of a scale and
analyzing them geometrically gives as full a picture as analyzing
every possible triad the scale can produce.

I believe it is good that there are many points of view, but if I
am obliged to understand Partch, does that also mean you are
obliged to understand me?

Marion

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Thu, 3 Jul 1997 21:15 +0200
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA04283; Thu, 3 Jul 1997 21:16:00 +0200
Date: Thu, 3 Jul 1997 21:16:00 +0200
Received: from ella.mills.edu by ns (smtpxd); id XA04277
Received: (qmail 20563 invoked from network); 3 Jul 1997 17:11:29 -0000
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 3 Jul 1997 17:11:29 -0000
Message-Id: <33BBA5C5.7BB6@erols.com>
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu

>1. Tempering initially referred to how one deviated about the system of
>Pythagorean tuning
>Nowadays, however, temperament and tuning seem to be synonymous.

At least among most tuning buffs, "temperament" is pretty much
synonymous with "deviation from just intonation". And that is to say JI of
whatever sort, although the Ptolemaic tuning of 1:1 9:8 5:4 4:3 3:2 5:3
15:8 2:1 is probably generally regarded as the most basic reference point
for a major scale anyway.

That is NOT to suggest, however, that Pythagorean tuning (1:1 9:8 81:64
4:3 3:2 27:16 243:128 2:1) is regarded as a temperament of that Ptolemaic
tuning. Pythagorean is another JI point of reference.




>3. Mean Temperament has two general meanings. First it means a system of
>tuning by fifths such that four ascending fifths and two descending
>octaves make up a perfect third. It also means any one of many slight
>variations about this basic system.

Meantone temperament is a tuning system based upon a consistent size of
perfect fifth, that fifth being tempered downward to make other intervals
Just, or at least a lot closer to just. Now in 99% of the cases they
involved a "wolf fifth" at the point where the circle was broken,
frequently between G# and Eb. That does not violate the
single-size-of-fifth rule, because G# to Eb is clearly a diminished sixth
instead of a P5.

The system you described above is called "quarter-comma meantone",
because its fifth is tempered flat by a quarter of a syntonic comma (81:80)
so that four of them stacked atop on another, adjusting downward by octaves
as needed, arrives at an exact, just 5:4. (Zikes! Please don't say
"perfect third"; fifths can be perfect, but not thirds; that's what the
word "just" is for.)

Notice that I said in 99% of the cases it involved a wolf fifth. There
are historically rare, but well-documented, cases of two meantone systems
being completed by carrying them beyond 12 (or occasionally 13-14) tones
per octave. The two systems that this works for a third-comma meantone, in
which the fifth is flatted by a third of an 81:80 syntonic comma to produce
a just 5:3 major sixth, and in quarter-comma meantone.

When 19 third-comma flat fifths are stacked atop one another, you arrive
at a pitch that is astoundingly close to exactly 11 octaves above where you
started - only about 1 cent off! That means that "complete" third-comma
meantone is so close to 19-tone-per-octave equal-temperament (called
"19TET") that there's no real-world point in distinguishing between the
two. And the analogous situation occurs with quarter-comma meantone and
31TET. Christian Huygens and others noticed this during the reign of
quarter-comma meantone.




>4. Bach�s WTC refers to a mean tempered clavichord; in particular,
>well-tempered means mean tempered.

I am personally aware of no evidence of that. All evidence I've heard
suggests that he wrote it for several of the Werkmeister well temperaments,
which were used by the organs in the area.

To the best of my knowledge "klavier" conceptually refers to any
keyboard instrument, although it's astoundingly unlikely that it would be
applied to the organ.

Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
with SMTP-OpenVMS via TCP/IP; Fri, 4 Jul 1997 08:38 +0200
Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA04784; Fri, 4 Jul 1997 08:38:37 +0200
Date: Fri, 4 Jul 1997 08:38:37 +0200
Received: from ella.mills.edu by ns (smtpxd); id XA04758
Received: (qmail 21446 invoked from network); 4 Jul 1997 06:37:25 -0000
Received: from localhost (HELO ella.mills.edu) (127.0.0.1)
by localhost with SMTP; 4 Jul 1997 06:37:25 -0000
Message-Id:
Errors-To: madole@mills.edu
Reply-To: tuning@eartha.mills.edu
Originator: tuning@eartha.mills.edu
Sender: tuning@eartha.mills.edu