Parapyth on a Single Keyboard

Parapyth is a temperament class (or at least a tuning scheme that can be understood as a temperament class) that Margo Schulter uses on two standard keyboards. It has good approximations of prime numbers other than five. I use a mapping similar to the schismatic fourth "cassava" mapping to get it to work on a single, standard (Halberstadt) keyboard.

The Mapping

Instead of using two keyboards for two chains of sharper-than-Pythagorean fifths, I put one chain on the black notes and the other on the white notes. Both chains form MoS scales, and repeat every perfect fourth of pitch. On the black notes, these are equivalent to a 12 note octave. The pattern of large and small intervals matches the visual appearance of the keyboard: a two-step semitone (e.g. between the C# and E♭ keys) is smaller than a three-step semitone (e.g. between the E♭ and F# keys). The same pattern is duplicated on the white keys with two additional pitches per fourth (the C and F keys) rounding up to 17 pitches per octave.

If you know a chord taken from Mistress Harmonic Series, you play it with pitches involving primes lower than 5 on the black keys and primes higher than 5 on the white keys. Ratios involving the prime 5 itself don't work very well. Ratios that don't involve primes higher than 5 will work on either the black or the white keys, but not a mixture of the two. Same for ratios that only involve primes higher than 5.

The Tuning

Tune your keyboard using the TE Scala file or the pure octaves variant. These start on the C key. You can choose the absolute pitch as you like. For charts, I assume the lowest E♭ key on your keyboard is tuned to G.

The TE tuning tunes each prime correctly to within a cent. But realistic musical intervals use multiple prime factors and can be a few cents out. Note that prime 11 in the pure octaves (POTE) tuning is more than 3 cents out. This is still quite good.

The Keyboard

This chart shows how pitch names correspond to the names of keys on the keyboard.

Pitch F♯ G G♯ A B♭
Pitch B C C♯ D E♭
Pitch E F F♯ G A♭
Pitch A B♭ B C D♭
Pitch D E♭ E F G♭
Keyboard key name C D E F G A B C

The white keys don't have conventional note names, unless you consider half-sharps and half-flats to be conventional. In which case, hurrah for you. You'll need to extrapolate to cover the C and F keys.

For tuning purposes, here are the pitches in cents for nearly 5 octaves.

Keyboard key name C D E F G A B
Pitch 0.0 22.4 80.7 103.0 161.4 207.6 230.0 288.3 310.6 369.0 391.3 449.6
Pitch 495.9 518.3 576.6 598.9 657.2 703.5 725.9 784.2 806.5 864.8 887.2 945.5
Pitch 991.8 1014.1 1072.4 1094.8 1153.1 1199.4 1221.7 1280.0 1302.4 1360.7 1383.1 1441.4
Pitch 1487.7 1510.0 1568.3 1590.7 1649.0 1695.3 1717.6 1775.9 1798.3 1856.6 1879.0 1937.3
Pitch 1983.5 2005.9 2064.2 2086.6 2144.9 2191.1 2213.5 2271.8 2294.2 2352.5 2374.9 2433.2

Some Chords

Here are some charts showing chords taken from Mistress Harmonic Series. First, here's a full 4:6:7:9:11:13 chord.

The root pitch The keys you press
[period][period][period] [period][period]
B♭B♭E♭ AG♯E B
AG♯C♯ GF♯D A
GE♭G♯DC♯ AE

I added the [period] markers to show that there's quite a gap between the 4 and the 6. This is quite difficult to play if you only have two hands. Sustain pedals are useful, but you can also squash the chord to get 8:9:11:12:13:14. You can play chords like that like this

The root pitch The keys you press
GG♯C♯ AC♯ E G
DG♯C♯ AC♯ E G
AG♯C♯ AC♯ E G
FE♭ G♯E G♯ BD
CE♭ G♯E G♯ BD
GE♭ G♯E G♯ BD
AC♯ F♯D F♯ A C
AC♯ F♯D F♯ AC
BC♯ F♯D F♯ AC
F♯C♯ F♯D F♯ AC

Now for 6:7:9:11 subminor seventh chords. That is, seventh chords with a subminor third. The seventh itself is neutral but you might get away with it as a jazz chord.

The root pitch The keys you press
EC♯ GF♯D
BC♯ GF♯D
F♯C♯ GF♯D
FE♭ AG♯E
CE♭ AG♯E
GE♭ AG♯E
GG♯DC♯ A
DG♯DC♯ A
AG♯DC♯ A
A♭B♭EE♭ B
E♭B♭EE♭ B
B♭B♭EE♭ B
Tuning documents