Re: Ives's tuning: still more on Ives's stretched 'larger scale'

Regarding p 108-110 of 'Scrapbook', from Ives's writings:

Of course, it's important to keep in mind that this fragment only
once specifies 48-tET, and there's no way to be sure how exact
Ives's tuning of that was on his father's 'glasses'. He never
really was more specific, at least not in these fragments, than
to say that they were 'tuned in different intervals larger and
less than quarter tones'.

Ives certainly did not manipulate the mathematics of these tunings
as carefully as I have, but there's no reason to dismiss the
possibility that he could have drawn geometrical diagrams of
his stretched-'octaves' in various subdivisions and tuned the
'glasses' accordingly. Thus, it is certainly possibly that he
tuned them to the 1/3-tones I analyzed as being in the second
scale in the musical example, and tho less probable, still
possible that he tuned them something like the other scales.

----------------

Here's yet another idea for the tuning of Ives's stretched-'octave'
'larger scale'. Let's assume Johnny Reinhard is correct that
Pythagorean was Ives's 'own personal tuning', i.e., that the
Pythagorean pitches and intervals were more-or-less generally
the ones Ives thought of when he composed his music.

It will help, in visualizing the mathematical procedure carried
out here, to once again use Ives's diagram. We add the Pythagorean
ratios of the notated pitches on the left, and for the 'quarter-'
and '1/8-tones', divide either the 'limma' [= (2^8)/(3^5) = 256/243
= ~0.90 Semitone] or the 'apotome' [= (3^7)/(2^11) = 2187/2048
= ~1.14 Semitones] semitone, whichever is appropriate at that
particular spot in the gamut:

Notes in Divisions of Pythagorean
intervals Semitones
old scale

(2^6)/(3^3) Eb 31 -+---- 8 Doh (2^6)/(3^3) ~14.94
/ |
limma 30 -+
\ |
(3^2)/(2^2) D 29 -+
/ |---- 7 Te
(3^2)/(2^2)/(((2^8)/(3^5))^(1/4)) ~13.81
limma 28 -+
\ |
(3^7)/(2^10) C# 27 -+
/ |
apotome 26 -+
\ |
n^0 8 C 25 -+
/ |
limma 24 -+
\ |---- 6 Lah
((3^5)/(2^7))*(((2^8)/(3^5))^(1/4)) ~11.32
(3^5)/(2^7) B 23 -+
/ |
limma 22 -+
\ |
(3^10)/(2^15) A# 21 -+
/ |
apotome 20 -+
\ |
(3^3)/(2^4) 6 A 19 -+
/ |---- 5 Soh
((3^3)/(2^4))/(((2^8)/(3^5))^(1/4)) ~ 8.83
limma 18 -+
\ |
(3^8)/(2^12) G# 17 -+
/ |
apotome 16 -+
\ |
(3^1)/(2^1) 5 G 15 -+
/ |
limma 14 -+
\ |---- 4 Fah
((3^6)/(2^9))*(((2^8)/(3^5))^(1/4)) ~ 6.34
(3^6)/(2^9) F# 13 -+
/ |
apotome 12 -+
\ |
(2^2)/(3^1) 4 F 11 -+---- 3 Me (2^2)/(3^1) ~ 4.98
/ |
limma 10 -+
\ |
(3^4)/(2^6) 3 E 9 -+
/ |-- Re
limma 8 -+
\ |
(3^9)/(2^14) D# 7 -+
/ |
apotome 6 -+---- 2 Ray
((3^2)/(2^3))*(((3^7)/(2^11))^(1/2)) ~ 2.61
\ |
(3^2)/(2^3) 2 D 5 -+
/ |
limma 4 -+
\ |-- De
(3^7)/(2^11) C# 3 -+
/ |
apotome 2 -+
\ |
n^0 1 C 1 -+---- 1 Doh n^0 0.00

(I hope that diagram didn't get mangled with extra carriage
returns.)

The MIDI-file of this scale is at:
http://www.ixpres.com/interval/monzo/ives/lg-pythg.mid

-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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