the stack is four notes over a 110 hz drone and i have used it on six pieces. here it is as ratios, because writing it out as note names throws away the only information that matters.
| ratio | cents | 12-tet | difference | hz over 110 |
|---|---|---|---|---|
| 1/1 | 0.000 | 0 | none | 110.00 |
| 9/8 | 203.910 | 200 | 3.910 sharp | 123.75 |
| 5/4 | 386.314 | 400 | 13.686 flat | 137.50 |
| 3/2 | 701.955 | 700 | 1.955 sharp | 165.00 |
| 15/8 | 1088.269 | 1100 | 11.731 flat | 206.25 |
the conversion is one line:
so , and the tempered major third at 400 sits 13.686 cents above it. that is not a rounding error. it is about a seventh of a semitone, and in a slow piece it is the loudest decision in the chord.
take the root at 110 hz. its fifth harmonic is 550 hz exactly. take the just third, 137.5 hz, and its fourth harmonic is also 550 hz exactly. same number. nothing beats.
the tempered third is 138.5913 hz. four times that is 554.365 hz. against the root's 550 you get 4.365 beats a second. i write chords that sit for forty seconds, so that is about 175 beats of roughness inside one chord, and you do hear all of them, not as tuning but as texture. the note sounds like it has a small motor in it.
the fifth is nearly free by comparison. three times the root is 330 hz. twice the tempered fifth is 329.628 hz. 0.372 beats a second, one beat every 2.7 seconds, slow enough that you will read it as vibrato and probably like it.
so a tempered fifth costs almost nothing and a tempered third costs everything. that is why a piano sounds settled playing fifths and never quite settles on a held major chord.
stack four just fifths, drop two octaves, and you land on 81/64, which is 407.820 cents. the just third is 386.314. the gap is 21.506 cents and it has a name, the syntonic comma. every awkward thing in western tuning is that number refusing to be zero. pure fifths or pure thirds. not both, not with twelve notes.
equal temperament pays the whole comma to the thirds and keeps the fifths within two cents of just. @bindery/justified-text-on-the-web-is-a-lie-we-agreed-to is about the same shape of trade, and it is a good trade if you need every key. i need one chord for four minutes. i am not paying for the other eleven keys.
i type frequencies in. two oscillators and a slowed tape for the drone, and the numbers go in as numbers. tuning by ear against a chromatic tuner does not work here, because the tuner is answering a different question: it reads 137.5 hz as a c sharp that is 14 cents flat, which is true and useless. £18 for a device whose job is to tell you your correct note is wrong.
what i did not expect was how much slower everything had to get. a 5/4 third with no beating in it has nothing moving, so the movement has to come from somewhere else, and tape drift turned out to be the answer. the cheap machine wanders about a quarter of a semitone across a side, roughly 1.5% in speed. a chord tuned to the cent that then drifts as a whole block is a different sound entirely from a chord that is out of tune with itself.
none of this is mystical. two numbers are the same number or they are not.
my piano is 14 cents flat at the top and has been since upstairs flooded. i do not tune the piano to any of this. i tune the tape to the piano.