2 pianos, 19 tones at Harvard

So I promised I'd write some more about an upcoming concert at Harvard's
Paine Hall, on March 1st at 8 pm, which we're retuning two pianos for. I
suggested the project to the Harvard Group for New Music maybe 18 months
ago, and a few other composers were keen to write for the combination.
Unfortunately in the end, after the concert was postponed once last year,
there are only two pieces to be played. I reluctantly had to withdraw mine
due to another commission that interfered with the deadline. The two
composers having pieces done are Bob Hasagewa and Chris Honnett, and as
long as the space is available before the concert I'll be giving a little
talk and demonstration at about 7:30 (I'll confirm onlist beforehand).
There will be other new music on the program, don't know if any of it is
microtonal though.

I notice there has been some discussion on the list about notating pitches
as they sound, vs notating them how they'd have to be "fingered" - in this
case the two coincide. The performers read off regular notation (so they
can practice on regularly tuned pianos if need be), and anyone looking at
the score can identify the pitches intended.

I only know of Mandelbaum's scheme for 19 tones on two pianos, and this is
slightly different, but the advantages seemed to outweigh the benefit of
having it the same. In the future I guess it would be great if a standard
emerged.

The following is a little description of the system as I envisaged it, for
composers wondering whether to write for the combination. If I'd been
writing for a tuning-savvy audience I'd have said things differently.

--[begin excerpt]--
The 19-tone scale is written as follows:

C C# Db D D# Eb E E#/Fb F F# Gb G G# Ab A A# Bb B B#/Cb (C...)

so the familiar white-note diatonic scale is preserved, but with two
pitches where each black note would usually be (old enharmonic
equivalences disappear: F# and Gb are now two distinct pitches, for
example), and one new pitch between E and F (E#, alias Fb, a new
enharmonic equivalence), and between B and C (B# or Cb). The 19 pitches
are equally spaced within the octave.

Between the two pianos 24 notes are available per octave, and you
only need 19. This means 5 can be duplicated. My idea is to let the
pianos share a pentatonic scale of C, D, F, G and A mapped onto the same
keys on each piano. Then piano 1's 12-note scale would be as follows:

C C# D D# E F F# G G# A A# B

and piano 2's would be:

C Db D Eb Fb F Gb G Ab A Bb Cb

yielding all 19 pitches in combination.

Each pianist can thus play without having to learn new
note-name/key-position associations (piano 2 has perhaps the harder job,
with less obvious spellings like Fb and Cb).

Here are the tunings in Hertz for this system, given for the octave above
middle C. This presumes leaving the note C unchanged from its usual tuning
in an A 440 system; unfortunately _all_ the others, in both pianos, have
to be moved-- some up and some down. Other octaves would of course follow
this pattern, and could be tuned from the central octave by eliminating
beats.

Table of pitches in Hz

"normally" PIANO 1 PIANO 2

C 523.2 C 523.2 C 523.2
B 494.0 B 486.4 B#/Cb 504.5
Bb/A# 466.1 A# 452.2 Bb 469.0
A 440.0 A 436.0 A 436.0
G#/Ab 415.3 G# 405.3 Ab 420.4
G 392.0 G 390.8 G 390.8
F#/Gb 370.0 F# 363.3 Gb 376.8
F 349.2 F 350.3 F 350.3
E 329.6 E 325.6 E#/Fb 337.7
Eb/D# 311.1 D# 302.7 Eb 314.0
D 293.6 D 291.9 D 291.9
C#/Db 277.2 C# 271.3 Db 281.4
C 261.6 C 261.6 C 261.6

and here is a table of cents adjustments each piano needs on each note
(the basic step of 19-tone music is 63.15 cents)

piano 1 piano 2

C 0 C 0
C# -37 Db +26
D -11 D -11
D# -47 Eb +16
E -21 Fb +42
F + 5 F + 5
F# -32 Gb +32
G - 5 G -5
G# -42 Ab +21
A -16 A -16
A# -53 Bb +11
B -26 Cb +37

These deviations could be somewhat adjusted across the board if necessary
- as it is they are somewhat skewed towards the flat side (i.e. piano 1 is
flatter than piano 2 is sharp). For example fixing the pitch G to its
usual value, instead of C, all cents deviations would be adjusted
sharpwards by 5 cents.

--[end excerpt]--

as it turns out, our tuner is actually going to be tuning a variety of
1/3-comma meantone (by ear) instead, pretty much aurally indistinguishable
from the equal-tempered version: I checked his practice tuning (he did a
two-manual harpsichord, which was fun), and couldn't tell which of E#-G#
or Db-Fb was intended as pure, they both sounded fine.

More info if anyone wants it!

Jon