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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
42 64.70780907334513
43 65.11517705642518
44 65.51317942364757
45 65.9022371559561
46 66.28274347268416
47 66.65506622013166
48 67.01955000865388
49 67.3765181293825
50 67.7262742772967
51 68.06910410365795
52 68.40527661769312
53 68.73504545475839
54 69.05865002596163
55 69.37631656229593
56 69.68825906469125
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
76 74.97513016132304
77 75.20143848833882
78 75.42482662634698
79 75.64536897812523
80 75.86313713864834
81 76.0782000346155
82 76.29062405541701
83 76.5004731761631
84 76.70780907334512
85 76.91269123365242
86 77.11517705642518
87 77.31532195018474
88 77.51317942364759
89 77.70880117159678
90 77.9022371559561
91 78.09353568238436
92 78.28274347268416
93 78.46990573329639
94 78.65506622013166
95 78.83826729997138
96 79.01955000865388
97 79.19895410624554
98 79.3765181293825
99 79.55227944095532
100 79.72627427729671
101 79.89853779302155
102 80.06910410365793
103 80.23800632619862
104 80.4052766176931
105 80.57094621199347
106 80.73504545475839
107 80.89760383681376
108 81.05865002596163
109 81.21821189732312
110 81.37631656229593
111 81.53299039620127
112 81.68825906469124
113 81.84214754898227
114 81.9946801699769
115 82.1458806113325
116 82.29577194153086
117 82.44437663500085
118 82.5917165923421
119 82.73781315969532
120 82.88268714730222
121 83.02635884729514
122 83.16884805075463
123 83.31017406407088
124 83.4503557246425
125 83.58941141594505
126 83.727359081999
127 83.86421624126599
128 84
129 84.13472706507905
130 84.26841375634146
131 84.4010760184494
132 84.53272943230144
133 84.66338922601427
134 84.79307028549326
135 84.92178716460998
136 85.04955409500408
137 85.17638499552632
138 85.30229348133804
139 85.42729287268209
140 85.55139620333959
141 85.67461622878552
142 85.79696543405619
143 85.91845604134068
144 86.03910001730776
145 86.15890908017921
146 86.2778947065602
147 86.39606813803637
148 86.51344038754739
149 86.63002224554594
150 86.74582428595056
151 86.86085687190095
152 86.97513016132304
153 87.08865411231183
154 87.20143848833882
155 87.31349286329086
156 87.42482662634698
157 87.53544898669952
158 87.64536897812523
159 87.75459546341227
160 87.86313713864834
161 87.97100253737541
162 88.07820003461549
163 88.18473785077293
164 88.29062405541701
165 88.39586657094978
166 88.5004731761631
167 88.60445150968863
168 88.70780907334513
169 88.81055323538621
170 88.91269123365242
171 89.01423017863078
172 89.11517705642518
173 89.2155387316407
174 89.31532195018474
175 89.41453334198796
176 89.51317942364757
177 89.61126660099598
178 89.70880117159678
179 89.80578932717108
180 89.9022371559561
181 89.99815064499847
182 90.09353568238436
183 90.18839805940853
184 90.28274347268416
185 90.37657752619575
186 90.46990573329639
187 90.56273351865164
188 90.65506622013166
189 90.74690909065288
190 90.83826729997138
191 90.92914593642898
192 91.01955000865388
193 91.10948444721697
194 91.19895410624554
195 91.28796376499534
196 91.37651812938249
197 91.46462183347651
198 91.55227944095532
199 91.63949544652378
200 91.7262742772967
201 91.81262029414715
202 91.89853779302155
203 91.98403100622212
204 92.06910410365793
205 92.15376119406535
206 92.23800632619862
207 92.3218434899919
208 92.40527661769312
209 92.4883095849706
210 92.57094621199347
211 92.65319026448623
212 92.73504545475839
213 92.81651544271007
214 92.89760383681376
215 92.97831419507352
216 93.05865002596163
217 93.13861478933376
218 93.21821189732312
219 93.29744471521408
220 93.37631656229593
221 93.45483071269717
222 93.53299039620127
223 93.61079879904366
224 93.68825906469124
225 93.76537429460444
226 93.84214754898227
227 93.918581847491
228 93.9946801699769
229 94.07044545716333
230 94.1458806113325
231 94.22098849699269
232 94.29577194153086
233 94.37023373585136
234 94.44437663500085
235 94.51820335878
236 94.59171659234211
237 94.66491898677913
238 94.73781315969534
239 94.810401695769
240 94.88268714730222
241 94.95467203475954
242 95.02635884729514
243 95.09775004326937
244 95.16884805075463
245 95.23965526803086
246 95.31017406407088
247 95.38040677901614
248 95.4503557246425
249 95.52002318481696
250 95.58941141594505
251 95.65852264740926
252 95.727359081999
253 95.79592289633172
254 95.86421624126599
255 95.93224124230629
256 96
257 96.06749459032655
258 96.13472706507905
259 96.20169945223864
260 96.26841375634146
261 96.33487195883862
262 96.4010760184494
263 96.46702787150763
264 96.53272943230145
265 96.59818259340675
266 96.66338922601429
267 96.72835118025066
268 96.79307028549326
269 96.85754835067951
270 96.92178716460998
271 96.98578849624647
272 97.04955409500408
273 97.11308569103824
274 97.17638499552632
275 97.23945370094428
276 97.30229348133801
277 97.36490599259027
278 97.42729287268207
279 97.48945574195025
280 97.55139620333958
281 97.61311584265113
282 97.67461622878552
283 97.7358989139826
284 97.7969654340562
285 97.85781730862526
286 97.91845604134068
287 97.97888312010826
288 98.03910001730775
289 98.09910819000817
290 98.15890908017921
291 98.21850411489942
292 98.2778947065602
293 98.33708225306698
294 98.39606813803636
295 98.45485373099045
296 98.51344038754739
297 98.5718294496092
298 98.63002224554594
299 98.68802009037728
300 98.74582428595058
301 98.80343612111642
302 98.86085687190095
303 98.9180878016754
304 98.97513016132302
305 99.03198518940297
306 99.08865411231181
307 99.14513814444214
308 99.2014384883388
309 99.2575563348525
310 99.31349286329086
311 99.36924924156725
312 99.42482662634698
313 99.48022616319142
314 99.53544898669952
315 99.59049622064735
316 99.64536897812525
317 99.70006836167289
318 99.75459546341227
319 99.80895136517843
320 99.86313713864834
321 99.91715384546762
322 99.97100253737541
323 100.02468425632709
324 100.07820003461549
325 100.1315508949898
326 100.18473785077293
327 100.237761905977
328 100.290624055417
329 100.3433252848229
330 100.39586657094979
331 100.44824888168661
332 100.50047317616308
333 100.55254040485516
334 100.60445150968864
335 100.65620742414163
336 100.70780907334512
337 100.75925737418231
338 100.81055323538622
339 100.86169755763612
340 100.91269123365242
341 100.96353514829008
342 101.01423017863077
343 101.06477719407374
344 101.11517705642518
345 101.16543061998638
346 101.2155387316407
347 101.26550223093912
348 101.31532195018474
349 101.36499871451583
350 101.41453334198795
351 101.46392664365473
352 101.51317942364756
353 101.56229247925417
354 101.61126660099596
355 101.66010257270455
356 101.70880117159678
357 101.75736316834919
358 101.80578932717108
359 101.85408040587662
360 101.9022371559561
361 101.95026032264605
362 101.99815064499847
363 102.04590885594901
364 102.09353568238436
365 102.14103184520854
366 102.18839805940851
367 102.23563503411856
368 102.28274347268416
369 102.32972407272476
370 102.37657752619575
371 102.42330451944963
372 102.46990573329637
373 102.51638184306287
374 102.56273351865164
375 102.60896142459892
376 102.65506622013166
377 102.70104855922398
378 102.74690909065288
379 102.79264845805311
380 102.83826729997139
381 102.88376624991989
382 102.929145936429
383 102.97440698309948
384 103.01955000865388
385 103.06457562698719
386 103.10948444721697
387 103.15427707373291
388 103.19895410624554
389 103.2435161398343
390 103.28796376499533
391 103.33229756768824
392 103.37651812938249
393 103.42062602710328
394 103.46462183347653
395 103.50850611677359
396 103.55227944095532
397 103.5959423657155
398 103.6394954465238
399 103.68293923466817
400 103.72627427729671
401 103.76950111745906
402 103.81262029414714
403 103.85563234233561
404 103.89853779302155
405 103.94133717326383
406 103.98403100622212
407 104.02661981119496
408 104.06910410365793
409 104.11148439530099
410 104.15376119406537
411 104.1959350041802
412 104.23800632619862
413 104.27997565703336
414 104.32184348999192
415 104.36361031481144
416 104.40527661769312
417 104.44684288133598
418 104.48830958497061
419 104.52967720439221
420 104.57094621199349
421 104.61211707679675
422 104.65319026448623
423 104.69416623743939
424 104.73504545475839
425 104.77582837230077
426 104.81651544271007
427 104.85710711544588
428 104.89760383681376
429 104.93800604999454
430 104.97831419507352
431 105.01852870906916
432 105.05865002596164
433 105.09867857672069
434 105.13861478933376
435 105.17845908883308
436 105.21821189732313
437 105.25787363400718
438 105.29744471521408
439 105.3369255544243
440 105.37631656229593
441 105.41561814669026
442 105.45483071269717
443 105.49395466266022
444 105.53299039620126
445 105.57193831024512
446 105.61079879904366
447 105.64957225419982
448 105.68825906469124
449 105.72685961689378
450 105.76537429460444
451 105.80380347906458
452 105.84214754898225
453 105.88040688055483
454 105.918581847491
455 105.9566728210327
456 105.99468016997692
457 106.03260426069684
458 106.07044545716334
459 106.10820412096571
460 106.1458806113325
461 106.18347528515207
462 106.2209884969927
463 106.25842059912284
464 106.29577194153087
465 106.33304287194471
466 106.37023373585136
467 106.40734487651596
468 106.44437663500085
469 106.48132935018451
470 106.51820335878
471 106.5549989953534
472 106.5917165923421
473 106.62835648007274
474 106.66491898677913
475 106.70140443861973
476 106.73781315969534
477 106.77414547206615
478 106.810401695769
479 106.84658214883416
480 106.88268714730224
481 106.9187170052405
482 106.95467203475954
483 106.99055254602926
484 107.02635884729514
485 107.06209124489388
486 107.09775004326937
487 107.1333355449881
488 107.16884805075463
489 107.20428785942681
490 107.23965526803084
491 107.27495057177632
492 107.31017406407088
493 107.34532603653494
494 107.38040677901614
495 107.41541657960366
496 107.45035572464252
497 107.48522449874744
498 107.52002318481698
499 107.55475206404712
500 107.58941141594505
501 107.6240015183425
502 107.65852264740926
503 107.69297507766635
504 107.72735908199901
505 107.76167493166989
506 107.79592289633172
507 107.83010324404009
508 107.86421624126598
509 107.89826215290834
510 107.9322412423063
511 107.96615377125146
512 108
513 108.03378018728463
514 108.06749459032655
515 108.10114346484698
516 108.13472706507905
517 108.16824564377922
518 108.20169945223864
519 108.23508874029457
520 108.26841375634146
521 108.3016747473421
522 108.33487195883862
523 108.36800563496331
524 108.4010760184494
525 108.43408335064183
526 108.46702787150764
527 108.49990981964659
528 108.53272943230145
529 108.56548694536832
530 108.59818259340675
531 108.63081660964986
532 108.66338922601429
533 108.69590067311012
534 108.72835118025066
535 108.76074097546211
536 108.79307028549326
537 108.82533933582495
538 108.85754835067951
539 108.88969755303006
540 108.92178716460998
541 108.9538174059218
542 108.98578849624647
543 109.01770065365234
544 109.04955409500407
545 109.08134903597147
546 109.11308569103824
547 109.14476427351069
548 109.17638499552632
549 109.2079480680624
550 109.23945370094427
551 109.2709021028539
552 109.30229348133801
553 109.33362804281649
554 109.36490599259027
555 109.39612753484963
556 109.42729287268207
557 109.45840220808037
558 109.48945574195025
559 109.52045367411829
560 109.5513962033396
561 109.58228352730552
562 109.61311584265113
563 109.64389334496275
564 109.67461622878552
565 109.70528468763061
566 109.7358989139826
567 109.76645909930673
568 109.7969654340562
569 109.82741810767898
570 109.85781730862526
571 109.88816322435412
572 109.91845604134068
573 109.94869594508287
574 109.97888312010826
575 110.00901774998084
576 110.03910001730776
577 110.0691301037458
578 110.09910819000817
579 110.12903445587085
580 110.15890908017921
581 110.18873224085435
582 110.21850411489942
583 110.24822487840596
584 110.2778947065602
585 110.30751377364922
586 110.33708225306697
587 110.36660031732063
588 110.39606813803638
589 110.42548588596551
590 110.45485373099045
591 110.4841718421304
592 110.51344038754739
593 110.54265953455204
594 110.5718294496092
595 110.60095029834368
596 110.63002224554594
597 110.65904545517768
598 110.68802009037725
599 110.71694631346543
600 110.74582428595058
601 110.77465416853423
602 110.80343612111642
603 110.83217030280103
604 110.86085687190095
605 110.8894959859435
606 110.91808780167543
607 110.9466324750682
608 110.97513016132305
609 111.00358101487599
610 111.031985189403
611 111.06034283782476
612 111.08865411231181
613 111.11691916428938
614 111.14513814444214
615 111.17331120271922
616 111.20143848833882
617 111.22952014979303
618 111.2575563348525
619 111.28554719057118
620 111.31349286329086
621 111.34139349864579
622 111.36924924156725
623 111.39706023628803
624 111.42482662634698
625 111.45254855459339
626 111.48022616319142
627 111.50785959362447
628 111.53544898669954
629 111.56299448255149
630 111.59049622064735
631 111.6179543397906
632 111.64536897812525
633 111.6727402731401
634 111.70006836167289
635 111.72735337991435
636 111.75459546341227
637 111.7817947470756
638 111.80895136517843
639 111.83606545136394
640 111.86313713864834
641 111.8901665594249
642 111.91715384546762
643 111.94409912793537
644 111.97100253737541
645 111.9978642037274
646 112.02468425632709
647 112.0514628239101
648 112.07820003461549
649 112.10489601598967
650 112.13155089498981
651 112.15816479798761
652 112.18473785077295
653 112.21127017855719
654 112.237761905977
655 112.26421315709774
656 112.290624055417
657 112.31699472386796
658 112.3433252848229
659 112.36961586009662
660 112.39586657094979
661 112.42207753809228
662 112.44824888168661
663 112.47438072135105
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1024 120

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

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