Perhaps it might be useful to offer here a short summary of how Andy Newton came up with the idea of fourkey tuning, for people who might wonder what’s so good about it.
Most other ideas about tunings had been driven by a desire to get some good playable chords, but Andy Newton’s innovative approach concentrated on melodic properties. He systematically listed several desirable properties that we might hope to find in a diatonic harmonica tuning.
Let me list here a slightly modified version of these desirable properties, as follows:
1. All twelve notes in the chromatic scale should be playable as straight blows or draws or as draw bends (no need for overblowing).
2. The notes should always get higher as we move from left to right and from blow to draw (no back and forth).
3. The blow/draw note pattern should repeat in the same way for every octave.
4. At least one major key should be playable in the harmonica without bending.
5. The preceding properties should be satisfied with the fewest possible holes per octave.
6. The preceding properties should be satisfied with as many major keys as possible being playable without bends.
These six properties uniquely characterize Newton fourkey tuning.
Properties 1 and 2 directly imply that the blow note in each hole must be one semitone higher than the draw note to its left. If not, then a missing note in the chromatic scale would require overblowing.
Diminished tuning satisfies properties 1 to 3, with four holes per octave, but unfortunately you cannot play any major scale without bends in diminished tuning.
In fact, there is no way to satisfy properties 1 to 4 in a tuning with four holes per octave. Let me explain why. There are seven notes in a major scale, which we may think of as the white keys of the piano for the C major scale. In four holes, we have eight reeds to play eight straight (blow or draw) notes. So to include all the white notes in four holes, we have only one reed left over to play one black note. But for each note, our tuning must also include at least one note a semitone higher or lower (depending on whether the note is a blow or draw). So to satisfy property 4 in the C major scale, the D note would require us to include either C# or D#, and the G note would require us to include either F# or G#. So we must include at least two black notes without bending, which is more than we can include with seven white notes in four holes.
Thus, we need at least five holes per octave to satisfy properties 1 through 4. In five holes, we have ten straight notes, and so two of the twelve chromatic notes must be available only as bends in a five-hole-octave tuning.
Let me now show why we cannot play more than four major keys in such a five-hole-octave tuning when two notes per octave are unavailable.
A major scale is a sequence of seven consecutive notes in the circle of fifths. When one note is dropped from the chromatic scale, we can play only five major keys in the eleven remaining notes, because among these eleven notes there are only five ways of picking seven notes consecutively in the circle of fifths. When we drop a second note from the chromatic scale, we must lose at least one more major key. But we can hope to still have four major keys playable in the ten remaining notes, if the two omitted notes are separated by a fifth, so that the ten remaining notes are a consecutive sequence in the circle of fifths.
Thus, to satisfy properties 1 to 4, we must have at least five holes per octave, and we cannot have more than four major keys available as straight notes in these five-hole octaves. In fact, Newton fourkey tuning achieves these bounds of four major keys in five holes, by having a pentatonic scale in the blow notes and another pentatonic scale one semitone lower in the draw notes.
A pentatonic scale is a sequence of five notes that are consecutive in the circle of fifths. Lowering notes by a semitone corresponds to rotating clockwise five steps in the circle of fifths. So when we have omitted two consecutive notes in the circle of fifths (say Bb and Eb), we can divide the remaining ten notes into two pentatonic scales (say F-G-A-C-D and E-F#-G#-B-C#) that differ by a semitone. We put the higher pentatonic scale in the blow notes and the lower pentatonic scale in the draw notes, offsetting the draw notes one hole to the left. Thus we get the basic [blow/draw] pattern for a fourkey harmonica in which the major keys of C, G, D, and A are playable without bends:
[F/F#] [G/G#] [A/B] [C/C#] [D/E]
The two omitted chromatic notes Bb and Eb are available here as simple draw bends.
References:
http://en.wikipedia.org/wiki/Circle_of_fifths
http://archive.harmonicasessions.com/aug07/Newton.html
http://archive.harmonicasessions.com/oct07/Newton.html
A SHORTER DERIVATION
Let me suggest one other way to think about these tunings.
Suppose we have a diatonic harmonica tuning that has these good properties: all twelve notes in the chromatic scale are playable as straight blows or draws or as draw bends (without overblowing), the notes always get higher as we move from left to right and from blow to draw, and at least one major key can be played without any bends.
Transposing if necessary, let us assume that the C major scale can be played in our harmonica without bends. So any bend notes must be black notes of the piano.
Any child has noticed that the five black notes of the piano are grouped in two clusters: a cluster of two notes {C#,Eb}, and a cluster of three notes {F#,G#,Bb}. Take either one of these clusters, and consider any pair of black notes within the cluster. If you chose the pair F# and Bb, then your pair has three chromatic notes in between them. Any other pair of black notes within a cluster must be adjacent black notes that have one note between them. So any pair of black notes within a cluster must always have an odd number of chromatic notes in between them.
Now pick a chromatic note that is playable only as a bend in our tuning, and then go up the chromatic scale to the next higher chromatic note that is also playable only as a bend. How many other chromatic notes will there be in between these two bends? If these two bends were in the same hole, then the answer would be 0. If the two bends are in adjacent holes, then the answer will be 2 notes: the draw note in the hole with the lower bend, and the blow note in the hole with the higher bend. If the harmonica has other holes in between those where our two bends occur, then each of these intervening holes will add a matched pair of blow and draw notes. But in any case, the number of chromatic notes in between our two bend notes must always be an even number.
This shows that we cannot have two bend notes together in one cluster. Thus, our tuning can have at most one bend note in {C#,Eb}, and at most one bend note in {F#,G#,Bb}. With only two bend notes, an octave needs 5 holes in our tuning, to include a reed for each of the other ten chromatic notes.
There are six ways of picking one bend note from each cluster. Of these, four ways yield a pair of bend notes that are separated by a fifth, and these correspond to the fourkey tunings that can play in C major without bending. The other two ways yield a pair of bend notes that are separated by a minor third, and these correspond to the “twokey” tunings that Andy Newton also discovered (but which, I believe, nobody has ever tried to play).
RETHINKING THE BASIC DIATONIC HARMONICA: PENTABENDER TUNING
To better understand fourkey tuning, it might be helpful to consider another closely related tuning which has substantial interest in its own right.
Let me list here a slightly modified version of our desirable properties, as follows:
1. The notes should always get higher as we move from blow to draw within one hole and the notes should never get lower as we move from left to right (so no back and forth).
2. In each octave, the blow and draw notes should include all seven notes of a major scale but no other notes (so you can easily play tunes that stay “in key” and you don’t have to worry about accidentally playing something out of the key).
3. The other five notes of the chromatic scale should all be playable as draw bends (so no need for overblowing).
People who are buying their first diatonic harmonica might often believe that it has these three properties: that a “C” diatonic harmonic would have, within its range, reeds to play all the white notes of the piano and only the white notes, and that all the black notes in this range could be played by the technique called bending. Of course, the standard diatonic harmonica (Richter tuning) does not actually have these properties, but we can ask what kinds of harmonicas do.
The answer is that only one tuning satisfies all three of these properties, and it is called “pentabender” tuning. For a ten-hole pentabender harmonica in the key of C, the layout would be:
BLOW C D F G A C D F G A
DRAW D E G A B D E G A B
For a harmonica in the key of C, our three desirable properties can be satisfied only if each black note can be played as a draw bend in a hole where the blow note is the next lower white note and the draw note is the next higher white note. A piano has five black notes per octave, and so our harmonica must have five holes per octave, one for each black note. Then the blow and draw notes in each hole are a wholetone apart, respectively a semitone below and above the draw-bend black note in this hole.
The black notes together form a pentatonic scale (F#, G#, A#, C#, D#), and so the blow notes and the draw notes in this tuning also form a pentatonic scales, the blow notes being a pentatonic scale one semitone below the bend notes, and the draw notes being a pentatonic scale one semitone above the bend notes. In our C pentabender harmonica, the blow notes form an F-major (or D-minor) pentatonic scale, and the draw notes form a G-major (or E-minor) pentatonic scale. Essentially we have decomposed the seven-note major scale into two overlapping pentatonic scales, whose paired notes enclose the other five chromatic notes.
The F and G pentatonic scales both include the notes D, G, and A, and so these three notes are available as both blow and draw notes. Such enharmonic alternatives may seem redundant but they can be helpful for reducing the number of changes of breath direction when playing a tune.
If we eliminated pentabender’s blow-draw enharmonic pairs by lowering each redundant draw note one semitone (so that, instead of repeating the next blow note to the right, the draw reed would yield the black note that the pentabender had as a bend in this hole), then we would get a harmonica in fourkey tuning.
Although the properties that characterize pentabender tuning are very natural, this tuning does not seem to have been noticed before 2014:
http://www.slidemeister.com/forums/index.php?topic=1205.msg121897#msg121897
http://harp-l.org/pipermail/harp-l/2014-June/msg00398.html
(Posting revised November 2016)
Fourkey harps 