## Monday, July 27, 2015

### [hqoeierf] Planck frequency pitch standard

We present a fanciful alternative to the musical pitch standard A440 by having some piano key note, not necessarily A4, have a frequency that is an integer perfect power multiple of the Planck time interval.

Let Pf = Planck frequency = 1/plancktime = 1/(5.3910604e-44 s) = 1.8549226e+43 Hz.

We first consider some possibilities of modifying A = 440 Hz as little as possible.  Sharpness or flatness is given in cents, where 100 cents = 1 semitone.

F3 = Pf / 725926^7 = 174.6141 Hz, or A = 440.0000 Hz, offset = -0.00003 cents
G3 = Pf / 714044^7 = 195.9977 Hz, or A = 440.0000 Hz, offset = -0.00013 cents
E3 = Pf / 135337^8 = 164.8137 Hz, or A = 439.9999 Hz, offset = -0.00030 cents
G3 = Pf / 132437^8 = 195.9978 Hz, or A = 440.0001 Hz, offset = 0.00045 cents
D#5 = Pf / 31416^9 = 622.2542 Hz, or A = 440.0001 Hz, offset = 0.00053 cents
A#3 = Pf / 12305^10 = 233.0825 Hz, or A = 440.0011 Hz, offset = 0.00442 cents
C#5 = Pf / 1310^13 = 554.3690 Hz, or A = 440.0030 Hz, offset = 0.01176 cents
A#3 = Pf / 360^16 = 233.0697 Hz, or A = 439.9770 Hz, offset = -0.09058 cents
A#1 = Pf / 77^22 = 58.2814 Hz, or A = 440.0824 Hz, offset = 0.32419 cents
D#4 = Pf / 50^24 = 311.2044 Hz, or A = 440.1095 Hz, offset = 0.43060 cents
E1 = Pf / 40^26 = 41.1876 Hz, or A = 439.8303 Hz, offset = -0.66769 cents
B5 = Pf / 22^30 = 990.0232 Hz, or A = 441.0052 Hz, offset = 3.95060 cents
F#3 = Pf / 10^41 = 185.4923 Hz, or A = 441.1774 Hz, offset = 4.62660 cents
A7 = Pf / 7^47 = 3537.6749 Hz, or A = 442.2094 Hz, offset = 8.67126 cents
G6 = Pf / 3^84 = 1549.3174 Hz, or A = 434.7625 Hz, offset = -20.73121 cents
G#7 = Pf / 2^132 = 3406.9548 Hz, or A = 451.1929 Hz, offset = 43.48887 cents

Next some modifications of other pitch standards, used by continental European orchestras.

Modifications of A = 441 Hz:

C#6 = Pf / 106614^8 = 1111.2503 Hz, or A = 441.0000 Hz, offset = -0.00007 cents
F2 = Pf / 39067^9 = 87.5055 Hz, or A = 441.0000 Hz, offset = -0.00011 cents
G#2 = Pf / 38322^9 = 104.0620 Hz, or A = 440.9995 Hz, offset = -0.00184 cents
G1 = Pf / 6022^11 = 49.1109 Hz, or A = 441.0006 Hz, offset = 0.00240 cents
B5 = Pf / 22^30 = 990.0232 Hz, or A = 441.0052 Hz, offset = 0.02044 cents
F#3 = Pf / 10^41 = 185.4923 Hz, or A = 441.1774 Hz, offset = 0.69644 cents
A7 = Pf / 7^47 = 3537.6749 Hz, or A = 442.2094 Hz, offset = 4.74110 cents
E7 = Pf / 5^57 = 2673.2253 Hz, or A = 446.0410 Hz, offset = 19.67702 cents
G6 = Pf / 3^84 = 1549.3174 Hz, or A = 434.7625 Hz, offset = -24.66137 cents
G#7 = Pf / 2^132 = 3406.9548 Hz, or A = 451.1929 Hz, offset = 39.55871 cents

Modifications of A = 442 Hz:

D#6 = Pf / 547981^7 = 1250.1649 Hz, or A = 442.0000 Hz, offset = 0.00014 cents
G6 = Pf / 530189^7 = 1575.1097 Hz, or A = 442.0002 Hz, offset = 0.00086 cents
G#6 = Pf / 525832^7 = 1668.7709 Hz, or A = 442.0003 Hz, offset = 0.00116 cents
F#4 = Pf / 122256^8 = 371.6759 Hz, or A = 441.9996 Hz, offset = -0.00170 cents
A5 = Pf / 30214^9 = 883.9990 Hz, or A = 441.9995 Hz, offset = -0.00194 cents
F#4 = Pf / 11744^10 = 371.6767 Hz, or A = 442.0006 Hz, offset = 0.00242 cents
A7 = Pf / 217^17 = 3535.9843 Hz, or A = 441.9980 Hz, offset = -0.00769 cents
D2 = Pf / 151^19 = 73.7503 Hz, or A = 442.0024 Hz, offset = 0.00939 cents
A2 = Pf / 62^23 = 110.4885 Hz, or A = 441.9539 Hz, offset = -0.18072 cents
D#3 = Pf / 38^26 = 156.2976 Hz, or A = 442.0764 Hz, offset = 0.29903 cents
D#4 = Pf / 37^26 = 312.6662 Hz, or A = 442.1768 Hz, offset = 0.69244 cents
A7 = Pf / 7^47 = 3537.6749 Hz, or A = 442.2094 Hz, offset = 0.81985 cents
E7 = Pf / 5^57 = 2673.2253 Hz, or A = 446.0410 Hz, offset = 15.75576 cents
G6 = Pf / 3^84 = 1549.3174 Hz, or A = 434.7625 Hz, offset = -28.58262 cents
G#7 = Pf / 2^132 = 3406.9548 Hz, or A = 451.1929 Hz, offset = 35.63745 cents

Modifications of A = 443 Hz:

F#5 = Pf / 590036^7 = 745.0342 Hz, or A = 443.0000 Hz, offset = 0.00003 cents
C7 = Pf / 508595^7 = 2107.2749 Hz, or A = 443.0000 Hz, offset = -0.00007 cents
F7 = Pf / 488038^7 = 2812.8743 Hz, or A = 442.9999 Hz, offset = -0.00020 cents
B2 = Pf / 140193^8 = 124.3126 Hz, or A = 442.9998 Hz, offset = -0.00093 cents
A5 = Pf / 109676^8 = 885.9985 Hz, or A = 442.9992 Hz, offset = -0.00296 cents
B7 = Pf / 25564^9 = 3978.0160 Hz, or A = 443.0012 Hz, offset = 0.00456 cents
G#1 = Pf / 5988^11 = 52.2668 Hz, or A = 442.9982 Hz, offset = -0.00722 cents
B1 = Pf / 391^16 = 62.1581 Hz, or A = 443.0125 Hz, offset = 0.04895 cents
A6 = Pf / 226^17 = 1772.0760 Hz, or A = 443.0190 Hz, offset = 0.07422 cents
F7 = Pf / 163^18 = 2811.5701 Hz, or A = 442.7946 Hz, offset = -0.80308 cents
A#3 = Pf / 60^23 = 234.8805 Hz, or A = 443.3954 Hz, offset = 1.54462 cents
E6 = Pf / 35^26 = 1326.0401 Hz, or A = 442.5128 Hz, offset = -1.90507 cents
E2 = Pf / 34^27 = 82.8696 Hz, or A = 442.4704 Hz, offset = -2.07100 cents
C#2 = Pf / 18^33 = 69.8768 Hz, or A = 443.6902 Hz, offset = 2.69500 cents
A7 = Pf / 7^47 = 3537.6749 Hz, or A = 442.2094 Hz, offset = -3.09255 cents
E7 = Pf / 5^57 = 2673.2253 Hz, or A = 446.0410 Hz, offset = 11.84337 cents
G#7 = Pf / 2^132 = 3406.9548 Hz, or A = 451.1929 Hz, offset = 31.72506 cents

Planck time is not known to high precision due to uncertainty of the gravitational constant G.  Fortunately coincidentally, musical instruments are not tuned to greater than 7 significant digits of precision, either.

Source code in Haskell. The algorithm is not clever; it simply brute forces every perfect integer power multiple of Planck time, with base less than 1 million, and within the range of an 88-key piano.  The code also can base the fundamental frequency off the hydrogen 21 cm line or off the frequency of cesium used for atomic clocks.

Inspired by Scientific pitch, which set C4 = 2^8 Hz = 256 Hz, or A = 430.538964609902 Hz, offset = -37.631656229590796 cents.