Replies to P.H., D.B., B.McL., G.M., J.C

Paul Hahn wrote,

>> Of course the major mode was not one
>> of the favorite modes of the diatonic scale until triadic harmony had
been
>> in use for some time -- the tritone resolves to the tonic triad only in
the
>> major and minor modes.

>A minor quibble--this is true if the voices are led in contrary motion,
>but using oblique motion the tritones in the Phrygian and Lydian modes
>resolve to the tonic fifth.

I would contend that in a homophonic style, any note in the scale may appear
(if only in passing) with any chord, and this can create a tritone unless
the chord is the tonic chord of either the major or minor mode. Hence the
"resolution" of interest is where both voices move. The Mixolydian and
Dorian modes are common in jazz, where the tritone blends with the tonic
triad to form approximate 7-limit tetrads (often with added notes). The
behavior of the tritone here is more "static" as opposed to its "dynamic"
behavior in the major and minor modes (the harmonic minor has two dynamic
tritones). No, this has nothing to do with "dynamic just intonation."

David Beardsley wrote
>he who only calls himself "PAULE" writes:
I know I must come across as pretentious sometimes, but my name is Paul
Erlich, and my e-mail address it not a clever attempt to conceal it but an
unfortunate creation of my co-workers in the systems department.

Brian -- I tried to include all the world music tuning systems I was
acquainted with (not many) in my proposed curriculum, and would certainly
think well over 5 hours could be devoted to those and others without
changing the outline. (I conceived it as a 30-, not 15-, hour course). The
history and diversity of tuning theory is of course very interesting, but
would be outside the scope of this course; ideas such as 31-tet and JI have
arisen again and again from many highly incompatible theories and so should
be included despite their "theoretical" nature. As long as you're mentioning
55-tet (1/6-comma meantone), don't forget 43-tet (1/5-comma meantone), which
was also discussed in the 18th century, I believe. I'm not aware of 34
having been discussed in the Renaissance, but 53 was (Mercator was 16th
century?) so 34 could have been. As for non-octave scales, etc., the field
at this point has very little that can be "taught" as "this is the way it
is," but certainly the intent of putting it last is to say, "This is the
avant garde, you can throw away the presumptions of all the previous lessons
if you want to." Those interested can use the remaining free-form weeks to
explore these ideas if they wish, and go on to take an advanced course in
the subject taught by Prof. McLaren. :)

Gary wrote,

>That brings to mind a curious question: How many underlying bases (as in
>plural of basis), structurally speaking, are there for the major scale,
>historical or not? I can think of three right off the top of my head:
>1. Identical upper and lower tetrachords
>2. Circle of fifths (one down and five up)
>3. Primary triads

I'd add
4. Maximal evenness (this works only in 12-tet and 19-tet)
5. Unique interval vector (this only works in 12-tet); Balzano liked this
one.
6-infinity. Lots of other 12-tet set theory garbage; Xenakis has some
construction in the appendix of his book "Formalized Music" that looks more
complicated than just saying what all the notes are!

John Chalmers wrote,

>It seems
>therefore, that the major-minor system presupposes some sort of tuning
>with acceptable thirds that ignores the syntonic comma. Meantone, ET
>and various irregular systems do just this.
>(Redfield's just scale does have this property as it consists of three
>minor triads on 1/1, 4/3 and 3/2. Every mode of this scale contains a
>note differing by a comma from a Ptolemaic mode. The generating tetrachord
>is 10/9 x 9/8 x16/15 rather than the Greek 16/15 x 9/8 x 10/9. )

But this scale has only two major triads, so by your definition of a major
scale, it is not a satisfactory tuning for the major-minor system. Or am I
missing something?


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