Kai's House of Odds and Ends
Hai, I'm Kai! This is a place for musings, messings, and novelties of mine that I'm not sure deserve their own page, or that I'm having trouble fitting into an existing page, but that I think will be of interest to my fellow xen-heads anyways!
La Monte Young's tuning from The Well-Tuned Piano, in Rational Comma Notation (and brand spankin' new KAIGI SIGNATURES!)
With Concert Pitch (1/1) being C4@256Hz, Eb4[7] = 7/6 = (298+2/3)Hz serves nicely as the tonic in Rational Comma Notation (RCN). Originally, Young tuned his "A", 189/128, to 440Hz, giving Eb of (297+187/189)Hz. These two pitches are less than 0.7Hz apart, a difference that can be essentially ignored, unless you're playing along to a recording with the same reference pitch, or feel like giving an officially sanctioned performance of the WTP (yeah right.)
The octave above the Eb[7] reference pitch is as follows:
"Eb" | "E" | "F" | "F#" | "G" | "G#" | "A" | "Bb" | "B" | "C" | "C#" | "D" | "Eb" | |
RATIO | 1/1 | 567/512 | 9/8 | 147/128 | 21/16 | 1323/1024 | 189/128 | 3/2 | 49/32 | 7/4 | 441/256 | 63/32 | 2/1 |
CENTS | 0.000000 | 176.645910 | 203.910002 | 239.606814 | 470.780907 | 443.516816 | 674.690909 | 701.955001 | 737.651813 | 968.825906 | 941.561815 | 1172.735908 | 1200.000000 |
KAIGI SIGNATURE | – | ◻4▲ | ◻2 | ◻▲2 | ◻▲ | ◻3▲2 | ◻3▲ | ◻ | ▲2 | ▲ | ◻2▲2 | ◻2▲ | – |
ABS. PITCH (RCN) | Eb4[7] | F4[49] | F4[7] | Gb4[343] | Ab4[49] | Ab4[343] | Bb4[49] | Bb4[7] | Cb4[343] | Db4[49] | Db4[343] | Eb5[49] | Eb5[7] |
FREQUENCY (Hz) | 298+2/3 | 330+3/4 | 336 | 343 | 392 | 385+7/8 | 441 | 448 | 457+1/3 | 522+2/3 | 514+1/2 | 588 | 597+1/3 |
The pitch classes of this tuning as a lattice in RCN:
Cb[343] | Gb[343] | Db[343] | Ab[343] | |
Db[49] | Ab[49] | Eb[49] | Bb[49] | F[49] |
Eb[7] | Bb[7] | F[7] |
The same lattice in tonic-agnostic Kaigi Signatures, which makes the logic of the system clear:
▲2 | ◻▲2 | ◻2▲2 | ◻3▲2 | |
▲ | ◻▲ | ◻2▲ | ◻3▲ | ◻4▲ |
– | ◻ | ◻2 |