Fourth Based Keyboard Mapping

The conventional 7+5 keyboard mapping works very well for meantone scales with a 12 note octave. The problem is, when you add more notes or use a different kind of scale, the layout becomes illogical. It does still work for a 29 note schismic scale, though. That's what I'll explain here.

The advantage of using this mapping is that you have much more percise control over intonation than usual. It should also be possible to get it to work with a dynamic tuning algorithm to produce just intonation. The problem doing this with meantones is that there's an ambiguity in diatonic scales because of the vanishing of syntonic commas.

If you aren't familiar with schismic temperaments, well, a quick explanation is in order. For meantones, a third is associated with four fifths, octave reduced. In 12-equal, it is also 8 octave reduced fourths, because 4+8=12. This second association is used for schismic scales. 53-equal is a schismic scale, and is a very good approximation to 5-limit JI.

The fifth can be very close to just in schismic scales. So, there are good Pythagorean scales tucked away in there. That means, with schismic temperament, you can choose between the wide thirds and narrow semitones of Pythagorean tuning and the smoother harmonies of the 5-limit approximations. Schismic notation can also be extended to non-schismic Pythagorean music. And vice-versa.

The intervals can all be defined in terms of an octave and a fifth. I favour a just octave and a fifth of exactly 702 cents. This scale is good for 7-limit harmony (maximum error 5.5 cents for 7/5).

You could define the scale with any basis of 2 notes, though. Here's an example, see (my explanation of tuning matrices for the formalism):

```(ρ) = (  8 -5)H'
(π)   (-19 12)
```

For the tuning above, ρ=90 cents and π=24 cents.

Now, you fit the scale to the keyboard as follows:

 Pitch C C# D Eb E Keyboard note name C D E F G A B C Intervals π π ρ-π π ρ-π π ρ-π π π ρ-π π ρ-π

The letters A to G here in the "Keyboard note name" row, and for the rest of this document, refer to keys and not pitches. The scale repeats over this fourth. The "Pitch" row gives you an idea of how the notes are tuned. The pattern carries on with a chromatic scale on the black keys. The keys are tuned as follows in cents:

 C 0000 0498 0996 1494 C# 0024 0522 1020 1518 D 0048 0546 1044 1542 Eb 0114 0612 1110 1608 E 0138 0636 1134 1632 F 0204 0702 1200 1698 F# 0228 0726 1224 1722 G 0294 0792 1290 1788 GA 0318 0816 1314 1812 A 0342 0840 1338 0836 Bb 0408 0906 1404 1902 B 0432 0930 1428 1926

That gives you a full 4 octave keyboard, which is enough for most of us. A lot of synths will allow you to tune up and down in octaves so that you can choose the right register.

Now, there are 12 black notes to an octave in his mapping. I think of these as being the base scale, whith the white notes as alterations. If you worry about absolute pitch, set one of the Bb keys to 440 Hz. You could retune to a 17 note scale on the white notes, but I prefer to hit black notes with these extended mappings.

This has all the same diatonic logic as a 12 note meantone octave. I find it really liberating to be able to try out comma shifts so easily.

A fifth is 17, a major third 9 and a minor third 8 notes. A triad formed from these intervals will be in tune provided no more than one note is white.

7/6 covers 6 notes. For this to be in tune, the lower note must be either the C#, D, E, F#, A or B key. You can work out 7-limit chords from that.

Seriously folks, if you've got full keyboard tuning, try this out. It's the next best thing to a generalized keyboard.