one more time: augmented and diminished

>> [Daniel Wolf 453.12]
>>
>> Free style is plain just intonation without a fixed gamut
>> of pitches, adaptive tuning is a mixture of just and
>> tempered, with most vertical sonorities in JI (some schemes
>> make augmented and diminished chords equal tempered) and
>> melodic intervals tempered from just to reduce drift.

> [Gerald Eskelin, TD 455.21]
>
> ... diminished triads, contrary to Mr. Wolf's
> description, in my experience seem NOT to conform to
> a tempered configuration. A diminished triad most
> commonly contains a functional tritone (presumably
> the same one that drove Medieval theorists nuts) that
> acoustically must make up its mind regarding its allegiance.
> It is only the tempered keyboard tritone that is immune.
> An acoustic tritone (5:7 or 7:10) appears to lean one way
> or the other and tunes accordingly. For example, a B-F
> tritone (diminished fifth) relates to a C tonic while a
> B-E# tritone (augmented fourth) relates to an F# tonic.

Gerald: I agree with your description here as it pertains
to JI dyads [= 'intervals'] and triads, at least as far as
Renaissance-to-'common practice' and perhaps much of the JI
or quasi-JI pop and folk music of every era.

But note that Daniel Wolf did not say that 'diminished triads
... conform to a tempered configuration'. He simply noted
that 'some schemes [of adaptive JI] make augmented and diminished
chords equal tempered'.

I think what Mr. Wolf really meant was 'diminished *7th*'
chords. His point was (one made in this List over and over
again) that chords which divide the 'octave' into a certain
number of identical intervals, for example, an 'augmented
triad' with 3 'major 3rds' or a 'diminished 7th' with 4
'minor 3rds', generally only 'work' when tuned in an appropriate
ET; 12-tET in the case of the two examples given.

Some have argued that composers would not have used these
two particular chords in the particular ways they did (and do),
especially around 100 years ago (say from around Wagner to
Schoenberg and Debussy), if 12-tET had not become the accepted
norm and thus influenced these composers's thinking.

I think it's significant that Schoenberg in particular found
so much to explore among the 'augmented triads', 'diminished 7th
chords', and '4th-chords', all of which pretty much require
12-tET, since it is the smallest ET which can produce them all
within a closed system.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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Joe Monzo wrote,

>His point was (one made in this List over and over
>again) that chords which divide the 'octave' into a certain
>number of identical intervals, for example, an 'augmented
>triad' with 3 'major 3rds' or a 'diminished 7th' with 4
>'minor 3rds', generally only 'work' when tuned in an appropriate
>ET; 12-tET in the case of the two examples given.

That's because, in these cases, the identical intervals in question are
clear approximations of simple-integer consonances, and trying to improve
one of them by moving it toward JI will force you to worsen another by
moving it away from JI. On the other hand, a "chord" that divides the octave
into 2 equal parts is clearly improved by making one of the parts 7:5 and
the other 10:7, rather than ET.

>I think it's significant that Schoenberg in particular found
>so much to explore among the 'augmented triads', 'diminished 7th
>chords', and '4th-chords', all of which pretty much require
>12-tET, since it is the smallest ET which can produce them all
>within a closed system.

Don't see that for '4th-chords' unless they include 12 different notes.
'4th-chords' of 3-5 notes are especially cool in 22-tET or any tuning with
~709-cent fifths, since the intevals produced approximate 7:4, 7:3, and 7:9.
'5th-chords' of 4-5 notes are especially nice in meantones, where you get
approximations to 5:3 and 5:2.

Merry Christmas to all!!!!!!!