APPENDIX F: Calculating Note Frequencies |

While working on this FAQ, jester had this conversation on absm:

Dj HeaVY C <craiga@cs.rmit.edu.au> wrote:

Hopefully there'll be a soul out there who will be able to give us a hand...

I remember reading somewhere about a formula that allowed you to change a note to another note (e.g. change a D to a C) but I can't remember where it was or anything about it. I know that doubling or halving the frequency of a sound moves it up or down an octave, but I remember seeing a formula somewhere to change the note...

If you can, please help me out. Just think about all the good karma it'll give you...

I'd worked it out four years ago and written a program to give me every audible note's frequency, but the computer that's on is at home -- so let's work it out here, shall we?

OK, you've already said yourself that doubling the frequency raises the note by an octave. This means that if we plot the frequency over the notes and octaves, we're going to get something looking like a square function, i.e. something like f(x) = a x^2 + b. If we use a logarithmic base 2 scale on the Y-axis (that's f(x)), the graph will look linear.

Now let's see.. A is our reference point, at 440Hz, the next one being at 880Hz. log2(440) = log(440) / log(2) = 8.78 ; log2(880) = 9.78 Hah! Cool, exactly one apart (btw, if you want the rest of the decimal places, they're: x.781359713).

Hope you haven't given up, I'll agree that I've been mumbling a bit of stuff w/o making it clear where I'm going. So here it is: Frequencies and notes have a logarithmic (base two) relationship to another. As you can see above, the base 2 logarithm of 880 is exactly that of 440 plus one. 1760 is up by another one, try it. Now, there are twelve notes in an octave, let's count them:

1 2 3 4 5 6 7 8 9 10 11 12 C C# D D# E F F# G G# A A# B

OK, those are half-notes, but they are the ones that are equally spaced frequency-wise (logarithmically, of course).

We can divide the range between two A's linearly into twelve equal segments. They are one apart, so each half-note is one twelfth away from the ones next to it. Then we take two to the power of the number we have and we get our note frequency.

Allow me to demonstrate. We're looking for the frequency of the C under the A at 440Hz (that's our reference point). It's nine half-notes further down, so we subtract nine twelfths (three quarters) from 8.781359713, leaving us with 8.031359713. Take two to the power of that, gives us: 261.6255652Hz, which is, btw, correct (one of the frequencies I know by heart from fiddling around with ST3 C4Spds two years ago a lot).

ok, now you can go figure out the rest. Or wait for the FAQ, which will contain this information too.

jester then sent me the following info:

C-0 16.35159783128741 C#0 17.3239144360545 D-0 18.35404799483797 D#0 19.44543648263006 E-0 20.60172230705437 F-0 21.82676446456274 F#0 23.12465141947715 G-0 24.49971474885933 G#0 25.95654359874657 A-0 27.5 A#0 29.13523509488062 B-0 30.86770632850776 C-1 32.70319566257483 C#1 34.64782887210901 D-1 36.70809598967595 D#1 38.89087296526012 E-1 41.20344461410874 F-1 43.65352892912549 F#1 46.2493028389543 G-1 48.99942949771866 G#1 51.91308719749314 A-1 55 A#1 58.27047018976124 B-1 61.73541265701552 C-2 65.40639132514966 C#2 69.29565774421802 D-2 73.41619197935189 D#2 77.78174593052023 E-2 82.40688922821748 F-2 87.30705785825097 F#2 92.4986056779086 G-2 97.99885899543732 G#2 103.8261743949863 A-2 110 A#2 116.5409403795225 B-2 123.470825314031 C-3 130.8127826502993 C#3 138.591315488436 D-3 146.8323839587038 D#3 155.5634918610405 E-3 164.813778456435 F-3 174.6141157165019 F#3 184.9972113558172 G-3 195.9977179908746 G#3 207.6523487899726 A-3 220 A#3 233.081880759045 B-3 246.9416506280621 C-4 261.6255653005986 C#4 277.1826309768721 D-4 293.6647679174076 D#4 311.1269837220809 E-4 329.6275569128699 F-4 349.2282314330039 F#4 369.9944227116344 G-4 391.9954359817493 G#4 415.3046975799451 A-4 440 A#4 466.1637615180899 B-4 493.8833012561241 C-5 523.2511306011972 C#5 554.3652619537442 D-5 587.3295358348151 D#5 622.2539674441618 E-5 659.2551138257398 F-5 698.4564628660078 F#5 739.9888454232688 G-5 783.9908719634985 G#5 830.6093951598903 A-5 880 A#5 932.3275230361799 B-5 987.7666025122483 C-6 1046.502261202394 C#6 1108.730523907488 D-6 1174.65907166963 D#6 1244.507934888324 E-6 1318.51022765148 F-6 1396.912925732016 F#6 1479.977690846538 G-6 1567.981743926997 G#6 1661.218790319781 A-6 1760 A#6 1864.65504607236 B-6 1975.533205024497 C-7 2093.004522404789 C#7 2217.461047814977 D-7 2349.318143339261 D#7 2489.015869776647 E-7 2637.020455302959 F-7 2793.825851464031 F#7 2959.955381693075 G-7 3135.963487853994 G#7 3322.437580639561 A-7 3520 A#7 3729.310092144719 B-7 3951.066410048993 C-8 4186.009044809578 C#8 4434.922095629953 D-8 4698.636286678521 D#8 4978.031739553295 E-8 5274.040910605919 F-8 5587.651702928062 F#8 5919.91076338615 G-8 6271.926975707988 G#8 6644.875161279122 A-8 7040 A#8 7458.620184289439 B-8 7902.132820097986 C-9 8372.018089619156 C#9 8869.844191259906 D-9 9397.272573357042 D#9 9956.06347910659 E-9 10548.08182121184 F-9 11175.30340585612 F#9 11839.8215267723 G-9 12543.85395141598 G#9 13289.75032255824 A-9 14080 A#9 14917.24036857888 B-9 15804.26564019597 C-A 16744.03617923831 C#A 17739.68838251981 D-A 18794.54514671408 D#A 19912.12695821318 E-A 21096.16364242367 F-A 22350.60681171225 F#A 23679.6430535446 G-A 25087.70790283195 G#A 26579.50064511649 A-A 28160 A#A 29834.48073715776 B-A 31608.53128039194

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