the 6th chord and odd-limit theory (re-post)

As I don't seem to be the only person who missed Digest 1122,
I'm re-posting this with revisions. Presumably, it will be
accompanied by the resurrected digest just to fill up all our
mail boxes.

All the numbers we'll be meeting today will be odd, so I won't say
so every time. Numbers with more than one prime factor have a
special significance, so I'll call them "rhubarb numbers" for want
of knowing a better term.

For an n-limit chord, first choose a number m. Either n or m, or
both, must be composite. mfactor. Providing neither n nor m are rhubarb, a saturated n-limit
chord can be constructed form all the divisors of n*m except n*m
and 1. This chord will be symmetrical: it's own utonal analog.

Examples are the two 9-limit symmetric chords already found.
Also, the 11-limit s-chord 33:11:9:3, the 25-limit s-chord
75:45:25:15:9:5:3 and the 35-limit s-chord
105:63:45:35:21:15:9:7:5:3.

When mj is rhubarb, we have a problem. The intervals n*i/j and
n*j/i will be beyond the n-limit. The same problem occurs when
nj is rhubarb unless i>m and j>m, so that i*mThis problem complicates matters, but is not insoluble as I
shall demonstrate with the example n, m

[Or that's what I though when I orginally posted. Squares of
primes are not a problem because n*i/i1. However, with
mi*i, n*i*i/ii/1 will exceed the n-limit. In practice,
treat 3*3*3', 3*3*3*3� and 5*5*55 as rhubarb.]

This gives us the chord 35:21:15:7:5:3. The intervals 35/3 and
21/5 are >15-limit. Therefore [21 or 5] and [35 or 3] must be
removed. If 35 and 21 are removed, we have a 15-limit otonality.
Removing 5 and 3 gives its utonal analog. The choice is then
removing [21 and 3] or [35 and 5]. The resulting chords 15-limit
s-chords 35:15:7:5 and 21:15:7:3 are then otonal/utonal analogs.

I think otonal/utonal pairs will arise whenever the rhubarb
problem occurs. I haven't looked at any cases where n and m are
both rhubarb. The first of these is n5, m3. Doubly rhubarb
numbers can safely be ignored until someone starts writing 105-
limit music.

That should be all you need to know.

Graham

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Lucy wrote:

'' Assuming that he is correct (I have no reason to doubt him on this), and the fashion moved from meantone into well temperaments, and eventually 12tET, I wonder why meantone tunings were abandoned.''

My post clearly did not say this, but rather that the well temperaments
were an abandoned cul-de-sac within a generally meantone era, and that the
repertoire associated with the well temperaments had certain attributes
that were not taken up by the immediately following generations of
composers, who were apparently satisfied with the tonal resources of
meantone (whether or not they actually used meantone is another question!).
From my posting, I wanted to suggest that 12tet was not an historicallyconsequent development out of the well temperaments, but rather a later
development directly out of meantone, and reflecting different musical
priorities than those which gave rise to the well temperaments. Lucy also writes:

''The difficulty of quickly retuning meantones would have severely
restricted the harmonic possibilities, yet well temperaments immediately offered keyboard players both tuning and intervallic novelties, in different keys.''

Which is not quite right. Tuning meantone and Werckmeister III or
Kirnberger III involve setting _exactly_ the same tempered series, and then
tuning series of just intervals - in meantone, a series of 5/4 thirds, inthe well temperaments, a series of 3/2 fifths. Therefore there is no
question of the speed of retuning here. Without retuning, the well
temperaments offer 12 usable key signatures, but if the target is
''intervallic novelties'' then using more than then the 6 ''in tune''
signatures in meantone will certainly greater novelty of the sort. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl
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>I wonder why meantone tunings were abandoned.
>Could it have been due to the limitations imposed on composers and
>performers, when using twelve fixed pitches per octave?

Almost certainly that was the most basic reason.

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>...but rather that the well temperaments
>were an abandoned cul-de-sac within a generally meantone era,

In terms of required resources that makes sense, but not in terms of
actual usage to the best of my knowledge. Correct me if I'm wrong, but I
know of no evidence that meantone was never actually used in any
appreciable abundance in the Classical Era or later.

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