FREQUENCY AND PERCEPTION

Frequency and musical perception are two different things. Frequency is the number of cycles per second that a waveform completes. It determines the pitch of a note, therefore its tuning.

If a wave repeats more than about 20 times per second (20 Hz), it enters the range where the human ear begins to perceive it as sound¹. For example, a middle C corresponds to a wave that completes about 261 cycles per second.

If we look at a piano keyboard, we see that octaves are evenly spaced and relatively close to each other: C1, C2, C3, etc. One might think that there is the same number of Hz between one octave and the next, but this is not the case. Each octave is double the frequency of the previous one, or half.

If we want to find all octaves starting from a given frequency, we use a multiplication (or division) and a power with base two:

• f × 2ⁿ → higher octaves

• f ÷ 2ⁿ → lower octaves

  
  

To go up an octave, we need an increasingly higher number of cycles per second. To go down, we need fewer and fewer cycles.

	0.5	1 (16.35x2^1)	2 (16.35x2^2)	3 (16.35x2^3)	4 (16.35x2^4)	5 (16.35x2^5)	6 (16.35x2^6)	7 (16.35x2^7)	8 (16.35x2^8)
C	16,35	32,70	65,41	130,81	261,63	523,25	1046,50	2093,00	4186,01
C# / Db	17,32	34,65	69,30	138,59	277,18	554,37	1108,73	2217,46	4434,92
D	18,35	36,71	73,42	146,83	293,66	587,33	1174,66	2349,32	4698,63
D# / Eb	19,45	38,89	77,78	155,56	311,13	622,25	1244,51	2489,02	4978,03
E	20,60	41,20	82,41	164,81	329,63	659,25	1318,51	2637,02	5274,04
F	21,83	43,65	87,31	174,61	349,23	698,46	1396,91	2793,83	5587,65
F# / Gb	23,12	46,25	92,50	185,00	369,99	739,99	1479,98	2959,96	5919,91
G	24,50	49,00	98,00	196,00	392,00	783,99	1567,98	3135,96	6271,93
G# / Ab	25,96	51,91	103,83	207,65	415,30	830,61	1661,22	3322,44	6644,88
A	27,50	55,00	110,00	220,00	440,00	880,00	1760,00	3520,00	7040,00
A# / Bb	29,14	58,27	116,54	233,08	466,16	932,33	1864,66	3729,31	7458,62
B	30,87	61,74	123,47	246,94	493,88	987,77	1975,53	3951,07	7902,13

In the next image, the octaves (marked by red points) appear equally spaced, while below we can see the frequencies organized in linear blocks, 500 Hz each. The higher the note, the more cycles are required to reach the next octave. The lower the starting note, the fewer cycles are needed to reach the octave below.

CONCLUSION

From this, several reflections emerge. Many physical phenomena are not perceived in a purely linear way, but instead follow more complex patterns, often related to ratios and scaling behaviors that can be described as exponential or logarithmic (as in the case of the musical staff). The frequency of notes and musical perception exist on two different planes, yet converge into the same experience through forms of periodicity that are sometimes unexpected.

Notes

¹ Below ~20 Hz we can still perceive something, but more as vibration than as a defined pitch.