Reply to Graham Breed on Temperaments

You were correct that I didn't intend for my statement to be
taken as dogmatically as it sounded.

[Graham Breed:]
> ...Other reasons [for accepting equal temperament] are:
>
> 1) You like the sound of it.
>
> The only reason you need.

OK, fair enough -- I have to grant you this one.

> 2) It simplifies notation.
>
> ...

I agree here too -- when I said "easier to play on most instruments" that
also included notational aspects. (I mentioned
notation somewhere in this discussion -- can't find it now.)

> 3) It allows ambiguous chords.
>
> I remember you (Monzo, TD1403) agreeing that the chord
> below only works in meantone. Did you really mean to imply
> (above) that this isn't a use for temperament?
>
> A---E---B
> \ / \
> G---D---A

The chord referred to was originally given as an example by
Paul Erlich in TD1400:
>
> [Example] 3. 6/9 (meaning root, maj. 3rd, perf. 5th. maj. 6th,
> maj. 9th)
>
> Here, we have a chain of 5ths, but also a major 3rd and two minor
> thirds. If all these intervals are to be consonant, no just
> interpretation will really do. This chord requires a tuning where
> the 80:81 syntonic comma vanishes.

I replied in TD1403 that I agreed with this statement, and I
suppose you are at least partly correct in pigeonholing Erlich's
broader definition of "a tuning where the 80:81 syntonic comma
vanishes" as "meantone".

I represented this chord with a lattice diagram that would look
like this in your version:

B
/ \
G---D---A---E---B

with the _caveat_ that the A and E a syntonic comma lower would
probably also be implied, but I didn't have space to include them.
That diagram would look like:
A---E---B
\ / \
G---D---A---E---B

so again, your citation is basically, though not literally,
correct.

Anyway, the opening phrase of my posting (from TD1415) which
you quoted and to which you replied, was: "the only reasons I can
see for accepting any equal-temperament...". I did specify that I
was talking about _equal_ temperaments, not well or meantone.

Of course, you do have a good point though. I guess this one is
a personal decision -- I prefer chords that are _not_ ambiguous.
There's already so much numerical interplay possible in JI, who
needs notes that are "close but no cigar"? Chords can be tuned
in JI with a variety of ratios that are close enough to a "target"
that many of these ambiguities can be explored rationally.

> 4) You can't be bothered to work out the ratios.
>
> As implied in the Schoenberg paragraph, most
> composers don't want to worry about ratios. They want a
> resourceful scale to compose in. Give them a meantone, and
> they'll find good harmonies in it. They might even give
> them to musicians who'll render them in JI. Do you really
> expect the composer to sit and work out where the commas
> should be?

My dissatisfaction with working directly with ratios was exactly
the stimulus which caused me to notate ratios as prime factors with
exponents. To me, this, especially when graphed on lattice
diagrams, is by far the simplest way to accurately represent
JI harmonic concepts and pitches. I usually find it useful to
describe temperaments this way too, since often composers who
favor temperaments _do_ wish to imply JI harmonies.

> Where were you when Gregg Gibson was stomping all over
> the list?

Who's Gregg Gibson? I've been receiving the Tuning Digest since
# 1314. (Email me something -- I'd like to see it).

> Then again, perhaps Schoenberg had a piano.

I took this as an attempt at humor. As I suggested, there are far
more profound reasons why Schoenberg so enthusiastically
embraced the 12-Eq scale. He intuitively understood much more
about rational harmonic concepts than his limited mathematically
ability demonstrates; and this is important because Schoenberg's
compositions and theory have had a huge impact on a lot of
this century's music, both academic and commercial.


Joseph L. Monzo
monz@juno.com
4940 Rubicam St., Philadelphia, PA 19144-1809, USA
phone 215 849 6723

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