Two schools of ET

>>I'll grant that equal temperaments are easy to hear in melodic terms

Carl Lumma wrote,

>Whoa! Wait a minute? Where did this come from?

I don't know where it came from, but it's true. Even Mathieu admits that
a 12-tone chromatic scale is melodically smoother in 12tET than in an
unequal tuning.

>This brings up the subject of the two schools of Equal Temperament. On the
>left we have those who want to approximate Just Intonation. Erlich and
>Hahn are of this school. They want a tuning be consistent and fairly
>accurate at a given limit before considering it usable at any higher limit.

Note that in my paper, as clarified by you (Carl Lumma), I do not care
about the 5-limit accuracy of 22tET, only that its 7-limit accuracy be
at least as great as the 5-limit accuracy of 12tET. Also, I am
interested in scales that fulfill some but not all of the properties of
diatonicity as outlined in my paper, such as Blackwood's 10-note scale
in 15tET and a couple of 14-note scales in 26tET.

>On the right we have those who insist that any ET is usable and "good".
>Students of this school include Ivor Darreg and Easley Blackwood.

Actually, Easley Blackwood thinks that 12tET is much more usable and
"good" than any other tuning. See the interview in PNM. The funny thing
is that Blackwood's microtonal music is generally much better than
Darreg's.

>the best 11/9 will be off by the absolute value of the sums of
>the errors of the 11/8 and 9/8, consistency or no

If you mean the best 11/8 and the best 9/8, then that might not be true,
although consistency will guarantee that it's true. If you don't mean
the best 11/8 and the best 9/8, then what do you mean?

>On the other hand, 25
>isn't consistent past the five limit, and it has horrible fifths, but its
>strong 3's and 7's make it great for the 7-limit, as Paul Rapaport proved
>in his "Study in Fives".

Did you mean horrible thirds or strong 5s?

Topic No. 10

Hello:

There's no such thing as atonal serialism.
Reading shabby translations of Sch*nberg will not help
one understand serialism.
One must approach Babbitt and the time-point system to
have a grasp of serialism.

>Do you make a distinction between serialism and
>atonal serialism? Since then I've heard at
>least 10 different people speak on what they thought serialism was

Paul Erlich wrote:

> I think the fact that the tritone in major nearly forms a 4:5:6:7
> with the dominant, and that the tritones in minor nearly form
> an 8:10:12:14:17 with the dominant, were not inconsequential
> for the development of tonal harmony.

I don't think the fact that the tritones are close to septimal
consonances has anything to do with the development of
tonal harmony. Tonal harmony, as used during the so-called
"common practice" period 1600-1900, developed mainly out of
the fixing of pitches as absolute values (especially in notation),
the recognition of 5-limit ratios as consonances, and the intuitive
realization that because ratios have two terms, they are
related to other ratios in two different ways (Partch's
"Basic Monophonic Concept #2": "every ratio of a Monophonic
system is at least a dual identity"; "Genesis", p. 88), giving
rise to the major/ minor system.

In fact, assuming a tuning in 5-limit JI (or a meantone
which approximates 5-ratios well), the tritone could be one
of four ratios, all of which, while much less dissonant than
the Pythagorean tritones:
3^6 729/512 6.12 aug 4th
3^-6 1024/729 5.88 dim 5th
were pretty much at, if not beyond, the limit of what could
be considered consonant:
3^2 * 5^-2 36/25 6.31 dim 5th
3^-2 * 5^-1 64/45 6.10 dim 5th
3^2 * 5^1 45/32 5.90 aug 4th
3^-2 * 5^2 25/18 5.69 aug 4th

The 64/45 would normally be considered the tritone which appears
in the 5-limit Dominant 7th chord of 36:45:54:64. 4:5:6:7 is the
same as 36:45:54:63.
---------------------------------------------------------
> [Monzo:]
>> (I'd really like to see more feedback about microtonality in
>> the blues.)

> [Erlich:]
> I think this is a great topic, but I don't think ratios of large
numbers
> have much to do with it. I do think that many blues performers do
> distinguish between 6/5 and 7/6, however, as well as approaching a
> pentatonic scale in 7tET.

I'm certain that some blues vocalists distinguish between 6/5 and
7/6, and in Robert Johnson's vocals, I've heard a "sharp 9th" which
sounds to me like it starts as a 7/6 then slides upward to a 19/16 -
exactly the pitches we debated in the "Hendrix Chord", so my ears
are especially sensitive to this distinction. Johnson in particular
exhibits
such minute pitch discrimination in his vocals that I'm sure his "odd
limit"
is _well_ above 7 or 9.
======================================
Mark Nowitzky wrote:

> P.S.: Bassoons, on a sustained low note, also sometimes
> sound to me like some sort of electrical problem.

You should hear some of the sounds Johnny Reinhard
gets out of his axe! I've dubbed him the "Hendrix of
the Bassoon".

Joseph L. Monzo
monz@juno.com


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