/musical-notes

How do you convert musical notes / notation to sounds? We know that every sound consists of sine waves, however, how to calculate the frequencies from musical notations?

Tuning

Frequencies for equal-tempered scale, A4 = 440 Hz

Speed of Sound = 345 m/s = 1130 ft/s = 770 miles/hr

NoteFrequency (Hz)Wavelength (cm)
C0 16.35 2109.89
 C#0/Db0  17.32 1991.47
D0 18.35 1879.69
 D#0/Eb0  19.45 1774.20
E0 20.60 1674.62
F0 21.83 1580.63
 F#0/Gb0  23.12 1491.91
G0 24.50 1408.18
 G#0/Ab0  25.96 1329.14
A0 27.50 1254.55
 A#0/Bb0  29.14 1184.13
B0 30.87 1117.67
C1 32.70 1054.94
 C#1/Db1  34.65 995.73
D1 36.71 939.85
 D#1/Eb1  38.89 887.10
E1 41.20 837.31
F1 43.65 790.31
 F#1/Gb1  46.25 745.96
G1 49.00 704.09
 G#1/Ab1  51.91 664.57
A1 55.00 627.27
 A#1/Bb1  58.27 592.07
B1 61.74 558.84
C2 65.41 527.47
 C#2/Db2  69.30 497.87
D2 73.42 469.92
 D#2/Eb2  77.78 443.55
E2 82.41 418.65
F2 87.31 395.16
 F#2/Gb2  92.50 372.98
G2 98.00 352.04
 G#2/Ab2  103.83 332.29
A2 110.00 313.64
 A#2/Bb2  116.54 296.03
B2 123.47 279.42
C3 130.81 263.74
 C#3/Db3  138.59 248.93
D3 146.83 234.96
 D#3/Eb3  155.56 221.77
E3 164.81 209.33
F3 174.61 197.58
 F#3/Gb3  185.00 186.49
G3 196.00 176.02
 G#3/Ab3  207.65 166.14
A3 220.00 156.82
 A#3/Bb3  233.08 148.02
B3 246.94 139.71
C4 261.63 131.87
 C#4/Db4  277.18 124.47
D4 293.66 117.48
 D#4/Eb4  311.13 110.89
E4 329.63 104.66
F4 349.23 98.79
 F#4/Gb4  369.99 93.24
G4 392.00 88.01
 G#4/Ab4  415.30 83.07
A4 440.00 78.41
 A#4/Bb4  466.16 74.01
B4 493.88 69.85
C5 523.25 65.93
 C#5/Db5  554.37 62.23
D5 587.33 58.74
 D#5/Eb5  622.25 55.44
E5 659.25 52.33
F5 698.46 49.39
 F#5/Gb5  739.99 46.62
G5 783.99 44.01
 G#5/Ab5  830.61 41.54
A5 880.00 39.20
 A#5/Bb5  932.33 37.00
B5 987.77 34.93
C6 1046.50 32.97
 C#6/Db6  1108.73 31.12
D6 1174.66 29.37
 D#6/Eb6  1244.51 27.72
E6 1318.51 26.17
F6 1396.91 24.70
 F#6/Gb6  1479.98 23.31
G6 1567.98 22.00
 G#6/Ab6  1661.22 20.77
A6 1760.00 19.60
 A#6/Bb6  1864.66 18.50
B6 1975.53 17.46
C7 2093.00 16.48
 C#7/Db7  2217.46 15.56
D7 2349.32 14.69
 D#7/Eb7  2489.02 13.86
E7 2637.02 13.08
F7 2793.83 12.35
 F#7/Gb7  2959.96 11.66
G7 3135.96 11.00
 G#7/Ab7  3322.44 10.38
A7 3520.00 9.80
 A#7/Bb7  3729.31 9.25
B7 3951.07 8.73
C8 4186.01 8.24
 C#8/Db8  4434.92 7.78
D8 4698.63 7.34
 D#8/Eb8  4978.03 6.93
E8 5274.04 6.54
F8 5587.65 6.17
 F#8/Gb8  5919.91 5.83
G8 6271.93 5.50
 G#8/Ab8  6644.88 5.19
A8 7040.00 4.90
 A#8/Bb8  7458.62 4.63
B8 7902.13 4.37

(To convert lengths in cm to inches, divide by 2.54)

Equations for the Frequency Table

The basic formula for the frequencies of the notes of the equal tempered scale is given by

fn = f0 * (a)n

where

f0 = the frequency of one fixed note which must be defined. A common choice is setting the A above middle C (A4) at f0 = 440 Hz.

n = the number of half steps away from the fixed note you are. If you are at a higher note, n is positive. If you are on a lower note, n is negative.

fn = the frequency of the note n half steps away.

a = (2)1/12 = the twelth root of 2 = the number which when multiplied by itself 12 times equals 2 = 1.059463094359...

The wavelength of the sound for the notes is found from

Wn = c/fn

where W is the wavelength and c is the speed of sound. The speed of sound depends on temperature, but is approximately 345 m/s at "room temperature".

Examples using A4 = 440 Hz:


C5 = the C an octave above middle C. This is 3 half steps above A4 and so the frequency is
f3 = 440 * (1.059463..)3 = 523.3 Hz
If your calculator does not have the ability to raise to powers, then use the fact that
(1.059463..)3 = (1.059463..)*(1.059463..)*(1.059463..)
That is, you multiply it by itself 3 times.

Middle C is 9 half steps below A4 and the frequency is:
f -9 = 440 * (1.059463..)-9 = 261.6 Hz
If you don't have powers on your calculator, remember that the negative sign on the power means you divide instead of multiply. For this example, you divide by (1.059463..) 9 times.

Shamelessly "borrowed" from this source.