harmonics.md

Harmonics

🡪 iicaras

Image from Electrical4U ¹.

Harmonics can be understood from standing sine waves. A sine wave has 3 nodes. Therefore, the 'biggest' standing wave is the one where the first and second nodes define a constraining length $L$ such that half of its wavelength fits inside $L$. This is the fundamental. The second harmonic will have its entire wavelength fit inside $L$, since the end of the wavelength is now the node which sits at $L$. For the third harmonic, the next node is halfway through the second wavelength etc. Perhaps it is best visualised with a string. As such for wavelengths $\lambda_n$ with $n$ the harmonic number,

$$L = \tfrac{1}{2} \lambda_1 = \tfrac{2}{2} \lambda_2 = \tfrac{3}{2} \lambda_3 = \tfrac{n}{2} \lambda_n,$$

etc. With instruments, these different wavelengths translate to different overtones. If $v$ is the travelling speed of the wave, then the time it takes to complete one wavelength is given by

$$T_n = \frac{\lambda_n}{v},$$

such that for frequency,

$$f_n = \frac{1}{T_n} = \frac{v}{\lambda_n} = \frac{nv}{2L} = nf_1,$$

with $f_1$ the fundamental frequency, the first harmonic. Harmonics are therefore integer multiples of the fundamental ²³.

To find the amount of semitones $x_n$ ($1 \textup{ st} = 1 \textup{ minor second}$) from the fundamental for a specific harmonic in 12-TET, note first that semitones are defined in 12-TET as

$$f_x = 2^{\tfrac{x}{12 \textup{ st}}}f \quad \implies \quad x = 12 \frac{\ln{f_x} - \ln{f}}{\ln{2}} \textup{ st},$$

with $f_x$ the frequency $x$ semitones away from $f$ (cents can also be used; $1 \textup{ st} = 100 \textup{ cents}$) ². With these relations, frequencies and semitones can be calculated relative to each other. As such, for harmonics $f_x = f_n = nf_1$ from the fundamental $f = f_1$, the semitone deviation $x = x_n$ can be calculated as

$$nf_1 = 2^{\tfrac{x_n}{12\textup{ st}}}f_1 \quad \implies \quad n = 2^{\tfrac{x_n}{12\textup{ st}}} \quad \implies \quad \ln{n} = \ln{2^{\tfrac{x_n}{12\textup{ st}}}} = \tfrac{x_n}{12\textup{ st}}\ln{2} \quad \implies \quad x_n = 12\cdot\tfrac{\ln{n}}{\ln(2)} \textup{ st}.$$

The table below shows the semitone values corresponding to harmonics from the fundamental, along with interval relationships for the first few harmonics.

from numpy import log as ln
def get_harmonics_semitones():
    return {f'{n}': ln(n) / ln(2) * 12 for n in range(1, 1024 + 1)}

Harmonic $n$	Semitones from Fundamental $x_n$ [st]	JI Interval (approx)
1	0	P1
2	12	P8
3	19.019550008653876	P8 + P5
4	24	2 P8
5	27.863137138648348	2 P8 + M3
6	31.019550008653873	2 P8 + M3 + m3
7	33.68825906469125	2 P8 + M3 + m3 + m3↓
8	36	3 P8
9	38.03910001730775	3 P8 + M2
10	39.863137138648355	3 P8 + M2 + m2
11	41.51317942364757
12	43.01955000865388
13	44.405276617693104
14	45.688259064691245
15	46.88268714730222
16	48
17	49.04955409500408
18	50.039100017307746
19	50.97513016132302
20	51.863137138648355
21	52.70780907334513
22	53.51317942364757
23	54.28274347268416
24	55.01955000865388
25	55.726274277296696
26	56.40527661769311
27	57.058650025961626
28	57.688259064691245
29	58.29577194153087
30	58.88268714730222
31	59.450355724642506
32	60
33	60.53272943230144
34	61.04955409500408
35	61.55139620333959
36	62.039100017307746
37	62.513440387547405
38	62.97513016132302
39	63.424826626346984
40	63.863137138648355
41	64.290624055417
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43	65.11517705642518
44	65.51317942364757
45	65.9022371559561
46	66.28274347268416
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50	67.7262742772967
51	68.06910410365795
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53	68.73504545475839
54	69.05865002596163
55	69.37631656229593
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57	69.9946801699769
58	70.29577194153086
59	70.5917165923421
60	70.88268714730222
61	71.16884805075463
62	71.4503557246425
63	71.727359081999
64	72
65	72.26841375634146
66	72.53272943230144
67	72.79307028549326
68	73.04955409500408
69	73.30229348133804
70	73.5513962033396
71	73.79696543405619
72	74.03910001730775
73	74.2778947065602
74	74.5134403875474
75	74.74582428595056
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128	84
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255	95.93224124230629
256	96
257	96.06749459032655
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Bibliography

Footnotes

1. Electrical4U, ‘Fundamental Frequency And Harmonics: What Are They?’ Accessed: Jan. 12, 2025. [Online]. Available: https://www.electrical4u.com/fundamental-frequency-and-harmonics/ ↩

2. J. W. S. B. Rayleigh, The Theory of Sound. in The Theory of Sound, no. v. 1. Macmillan and Company, 1877. [Online]. Available: https://books.google.nl/books?id=3ls9AAAAYAAJ ↩ ↩²

3. S. W. Smith, The scientist and engineer’s guide to digital signal processing, 2. ed. San Diego, Calif: California Technical Pub, 1999. Accessed: Jan. 12, 2025. [Online]. Available: http://www.dspguide.com/pdfbook.htm ↩