Ives's tuning: more on my 'Ives larger scale' webpage

After looking again at the analysis of what I speculate
may have been the second scale Ives put in the musical example,
I realized that this tuning could be further reduced and
described as 9-ET:

Semitones
9-tET Ives larger scale decimal fractional

Doh 2^(12/9) 2^(4/3) 16.00 16
Te 2^(11/9) (2^(4/3))^(11/12) ~14.67 14.&2/3
Lah 2^( 9/9) (2^(4/3))^( 9/12) 12.00 12
Soh 2^( 7/9) (2^(4/3))^( 7/12) ~ 9.33 9.&1/3
Fah 2^( 5/9) (2^(4/3))^( 5/12) ~ 6.67 6.&2/3
Me 2^( 4/9) (2^(4/3))^( 4/12) ~ 5.33 5.&1/3
Ray 2^( 2/9) (2^(4/3))^( 2/12) ~ 2.67 2.&2/3
Doh 2^( 0/9) (2^(4/3))^( 0/12) 0.00

The MIDI-file is at:
http://www.ixpres.com/interval/monzo/ives/18tet.mid

=============================
And there's more stuff I've added to the webpage:
http://www.ixpres.com/interval/monzo/ives/48tet.htm

- a couple of speculations on what tuning the 'lost' larger
scale, which 'ended on a quarter-tone', would have had.
Here they are:

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

I have a hunch as to what the tuning of that 'other scale'
might have been, based on my premise that the above scale
does represent the scale illustrated on the right in the
musical example.

Most likely, the quarter-tone upon which it ended would have
been the one which falls right between the 'minor 10th' and
'major 10th' upon which the two scales in the musical example
are based. This would be the 'neutral 10th', and it would
make sense that Ives would choose this particular interval
as his 'octave' after experimenting with both of the other
two and being dissatisfied at their match with 12-tET.

So this stretched 'octave' would be 2^(31/24) = 15.50 Semitones.

As 31 is prime, there is no simpler way to describe this tuning
as a power of 2, than as a subset of 288-tET:

Semitones
288-tET Ives larger scale decimal fractional

Doh 2^(372/288) 2^(31/24) 15.50 15.&1/2
Te 2^(341/288) (2^(31/24))^(11/12) ~14.21 14.&5/24
Lah 2^(279/288) (2^(31/24))^( 9/12) ~11.63 11.&5/8
Soh 2^(217/288) (2^(31/24))^( 7/12) ~ 9.04 9.&1/24
Fah 2^(155/288) (2^(31/24))^( 5/12) ~ 6.46 6.&11/24
Me 2^(124/288) (2^(31/24))^( 4/12) ~ 5.17 5.&1/6
Ray 2^( 62/288) (2^(31/24))^( 2/12) ~ 2.58 2.&7/12
Doh 2^( 0/288) (2^(31/24))^( 0/12) 0.00 0

The MIDI-file is at:
http://www.ixpres.com/interval/monzo/ives/lg-31qt.mid

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

I wrote, two postings ago:

> Given all the talk in this forum about how Ives considered
> sharps to be higher in pitch than flats, I find it very
> interesting that his notation here of 'notes in the old scale'
> uses sharps for all of the chromatic notes except for the flat
> which marks the highest note, the 'octave' of the 'larger scale'.
> I don't know if there's any significance to that, but it
> seems noteworthy.

Alternatively, considering the oft-quoted (here) statement that
Ives considered sharps to be higher in pitch than flats, perhaps
that 'Eb' instead of 'D#' at the top of the diagram indicates
that he 'heard' (in his mind) the stretched 'octave' as being
slightly lower than the 'minor 10th'. Then perhaps the 'other
scale' was based on a stretched 'octave' of 2^(29/24) = 14.50
Semitones.

As in the above scale, 29 is also prime, so the smallest fractional
power of 2 to describe this scale is again 288-tET:

Semitones
288-tET Ives larger scale decimal fractional

Doh 2^(348/288) 2^(29/24) 14.50 14.&1/2
Te 2^(319/288) (2^(29/24))^(11/12) ~13.29 13.&7/24
Lah 2^(261/288) (2^(29/24))^( 9/12) ~10.88 10.&7/8
Soh 2^(203/288) (2^(29/24))^( 7/12) ~ 8.46 8.&11/24
Fah 2^(145/288) (2^(29/24))^( 5/12) ~ 6.04 6.&1/24
Me 2^(116/288) (2^(29/24))^( 4/12) ~ 4.83 4.&5/6
Ray 2^( 58/288) (2^(29/24))^( 2/12) ~ 2.42 2.&5/12
Doh 2^( 0/288) (2^(29/24))^( 0/12) 0.00 0

That little discrepancy between the Eb and the D# in the diagram
seems to me to indicate that this is probably the 'other' larger
scale Ives had in mind.

The MIDI-file is at:
http://www.ixpres.com/interval/monzo/ives/lg-29qt.mid

That should give performers of Ives considering alternative tunings
plenty to try out. Perhaps a good research project would be to
search thru his music looking for melodic patterns that come close
to fitting these intervallic structures. He may have had these
tunings in mind for some of that music, and just wrote it down
in the usual (Pythagorean/meantone-based) notation because of lack
of instruments so tuned.

Altho my interpretations of these latter three scales may have
been the tuning Ives had in his mind, it is more likely that
he constructed them from a practical viewpoint out of various
combinations of 1/4- and 1/8-tones. Even if Johnny Reinhard is
correct that Ives was implying Pythagorean tuning with some of
his prose comments, I have never seen *specific* reference from
Ives to any tuning outside of 12-tET except for these two scales
(24-tET and 48-tET).

-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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