[Harp-L] new diatonic tuning for chromatic play with chords

[Roger Myerson <myerson@xxxxxxxxxxxx>]

Suppose that we want to find a way to tune a diatonic harmonica to allow 
us to get the whole chromatic scale with only draw bends (no overblows), 
and we want at least one major scale to be playable without any bends at 
all.  We also want a simple pattern that can be repeated each octave, 
with notes always getting higher as we go from left to right in the harp 
(no back and forth).   Andy Newton showed that there were essentially 
only three ways of designing such a good chromatic-in-diatonic tuning, 
and none of them seemed to provide good easily playable chords.   But he 
included one other assumption in his construction: that in each hole the 
draw note should always be higher than the blow note.

If we drop this last assumption, then some variants of the tunings that 
Andy Newton found actually can have an easily playable major chord.  In 
fact, I have found a tuning that satisfies all these good properties and 
has both a major chord and a minor chord are easily playable (in three 
consecutive holes with the same breath direction).  I believe that there 
is essentially only one way to do this, and that is to use a variant of 
Newton's less-known "twokey" pattern.   Here is the layout for a 10-hole 
twokey harp that plays the G and C major keys and also A harmonic minor 
without bending and that has a playable G major chord and playable A 
minor chord:

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

The missing chromatic notes are Bb and C#, and they are available by 
simple draw bends, without any need for overblowing.

This tuning has five-hole octaves, which are necessary for any such 
diatonic tuning that allows chromatic play without overblowing and that 
allows at least one major key to be played without bends (as is 
explained athttp://4keyharps.wordpress.com/ ).  The blow and draw notes 
in each hole here differ either by a wholetone interval with draw note 
high, or by a halftone interval with the blow note high.  The reversal 
of the halftone intervals is the change from Andy Newton's twokey 
pattern that allows us to get the G major and A minor chords.

The twokey pattern would have put the notes E and D# in hole 1, but my 
suggested layout here breaks with the pattern in hole 1 to get two full 
octaves in this 10-hole harp (without losing any other desiderata).  The 
missing D# in hole 1 is available by a draw bend.

I think that this is new.   Have you seen this tuning or anything like 
it before?
-Roger

--
Roger B. Myerson, Glen A. Lloyd Distinguished Service Professor
Department of Economics, University of Chicago
1126 East 59th Street, Chicago, IL 60637
Phone: 773-834-9071, Fax: 773-702-8490
http://home.uchicago.edu/~rmyerson/