Kai's House of Odds and Ends

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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 ◻32 ◻3 2 ◻22 ◻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 ◻22 ◻32
◻2 ◻3 ◻4
◻2