Re: pythagorean tuning

>
> Hello, there, and here's a revision of your table showing the Pythagorean
> intervals in relation to 12-tone equal temperament (12-tet):
>
> > Interval JI5 pyth JI7
> > --------------------------------------------------------------
> maj3 5/4 81/64 5/4
> vs. 12-tet -13.7c 7.8c
>
> min3 6/5 32/27 6/5
> 15.6c -5.9c
>
> min7 9/5 16/9 7/4
> 17.6c 4.1c -31.2c

Just having started playing with 12-out-of-N scales, one non-N scale
I plugged in was Pythagorean tuning Gb Db Ab Eb Bb F C G D A E B.

Most of it sounds so much like 12tet that you can't notice (but
I play jazz standards, which are usually in flat keys). I was
surprised but probably shouldn't have been. These tones are much closer
to 12tet than some other tunings we discuss, my sound card may not
be particularly accurate at the small number of cents, my ears are
not terribly refined yet, and still ringing from my rock'n'roll days.

Bob Valentine

Robert C Valentine wrote,

>I plugged in was Pythagorean tuning Gb Db Ab Eb Bb F C G D A E B.

This scale was used quite a bit around 1450. Try the key of A major. Five of
the triads are 2 cents off 5-limit JI; only the B-minor has a wolf. Now
contrast these with the regular Pythagorean triads in the flat keys, which
are close to 12-tET but worse. Hear the difference?

This arrangement of using Pythagorean diminshed fourths to get near-just
5:4s is called schismatic tuning. A 12-note keyboard does not allow this
feature to be exploited fully in even one key without encountering wolves
(e.g., the B-F# in D major). It was superseded by meantone temperament,
where a 12-note keyboard allows 6 keys with 6 triads only 6 cents off
5-limit JI (I might as well say 6-limit).

It seems that Charles Ives envisioned an extended Pythagorean tuning for his
music. Ives wanted sharps higher than flats. He states in his Memos on the
Concord Sonata that B# is an 1/8th tone higher than C. That smells like
Pythagorean to me.

The AFMM is now planning to record Joshua Pierce playing the Concord Sonata
on 3 Discklaviers (Yamaha) to get the full range of intonational
implications. Incidentally, the AFMM performance of the Ives Universe
Symphony was performed in this tuning.

Johnny Reinhard
AFMM Director

>
> Robert C Valentine wrote,
>
> >I plugged in was Pythagorean tuning Gb Db Ab Eb Bb F C G D A E B.
>

Paul answered :

> This scale was used quite a bit around 1450. Try the key of A major. Five of
> the triads are 2 cents off 5-limit JI; only the B-minor has a wolf. Now
> contrast these with the regular Pythagorean triads in the flat keys, which
> are close to 12-tET but worse. Hear the difference?
>
> This arrangement of using Pythagorean diminshed fourths to get near-just
> 5:4s is called schismatic tuning.

Thanks for typing in what must be very old information to you. I hadn't
been looking for ANY 5-limit implications in this tuning, so I wasn't
listenning for this. Now I'm spared the embaressment of shouting
'Eureka' when I discover it!

Nonetheless, other than being slightly more 'jingly'
the flat keys were so much like their 12tet counterparts that the tunes
I was playing were not TOO noticeably different. Which is not
surprising since 12tet is a very good 3-limit tuning.

Bob Valentine

Small comment on Pythagorean tuning -- i just built a keyed monochord,
presumed early precurser to the clavichord, with twelve keys on it,
naturals from b to e'' with a b-flat near the top. One string that all the
keys operate plus an extra string to be plucked as a drone. Assuming it
existed historically it would have been circa 1400, and Pythagorean tuning
is the received wisdom for that. But in addition, i discovered a wonderful
note of simplicity: using Pythagirean tuning, all the tangent placements
could be determined using only a pair of dividers!

Judith Conrad, Clavichord Player (jconrad@tiac.net)
Music Minister, Calvary Baptist Church, Providence, RI
Director of Fall River Fipple Fluters
Piano and Harpsichord Tuner-Technician

[Johnny Reinhard:]
> The AFMM is now planning to record Joshua Pierce playing the Concord
Sonata on 3 Discklaviers (Yamaha) to get the full range of
intonational implications. Incidentally, the AFMM performance of the
Ives Universe Symphony was performed in this tuning.

Wow - two massively ambitious (and as far as I'm concerned, very
exciting) undertakings. I'm really interested to know the (exact)
process (or general philosophy) your using to assign specific
intervals to "the full range of intonational implications" in a piece
like the Concord... I would think (with the bit Ives wrote in the
Emerson prologue about the 'muddiness' of Brahms orchestration ringing
in my head) that any sort of a direct, one-to-one,
note-spelling-to-circle-of-fifths approach would be (though possibly
interesting) much too methodical or temperate here... but by "this
tuning" I don't imagine that you simply mean an extended Pythagorean
tuning, but rather some sort of Pythagorean plus (1/4, 1/8, etc.)
division of the (12e) tone (or other) hybrid?

Thanks (for any additional information along these lines you could
share),

Dan