Kwarttoons gelijkzwevende stemmingen:
Vergelijking verdeling oktaaf in 24 delen, of duodeciem in 38 delen, gelijkzwevend
noot= 24 24-root 2 freq= 32.7032 38-root 3 freq= 32.7032
noot= 24.5 24-root 2 freq= 33.17888 38-root 3 freq= 33.17937
noot= 25 24-root 2 freq= 33.66148 38-root 3 freq= 33.66248
noot= 25.5 24-root 2 freq= 34.1511 38-root 3 freq= 34.15262
noot= 26 24-root 2 freq= 34.64783 38-root 3 freq= 34.64989
noot= 26.5 24-root 2 freq= 35.1518 38-root 3 freq= 35.15441
noot= 27 24-root 2 freq= 35.66309 38-root 3 freq= 35.66628
noot= 27.5 24-root 2 freq= 36.18182 38-root 3 freq= 36.18559
noot= 28 24-root 2 freq= 36.7081 38-root 3 freq= 36.71247
noot= 28.5 24-root 2 freq= 37.24203 38-root 3 freq= 37.24701
noot= 29 24-root 2 freq= 37.78373 38-root 3 freq= 37.78934
noot= 29.5 24-root 2 freq= 38.33331 38-root 3 freq= 38.33957
noot= 30 24-root 2 freq= 38.89088 38-root 3 freq= 38.89782
noot= 30.5 24-root 2 freq= 39.45656 38-root 3 freq= 39.46418
noot= 31 24-root 2 freq= 40.03047 38-root 3 freq= 40.0388
noot= 31.5 24-root 2 freq= 40.61272 38-root 3 freq= 40.62178
noot= 32 24-root 2 freq= 41.20345 38-root 3 freq= 41.21325
noot= 32.5 24-root 2 freq= 41.80277 38-root 3 freq= 41.81333
noot= 33 24-root 2 freq= 42.4108 38-root 3 freq= 42.42215
noot= 33.5 24-root 2 freq= 43.02768 38-root 3 freq= 43.03983
noot= 34 24-root 2 freq= 43.65354 38-root 3 freq= 43.66651
noot= 34.5 24-root 2 freq= 44.28849 38-root 3 freq= 44.30231
noot= 35 24-root 2 freq= 44.93268 38-root 3 freq= 44.94737
noot= 35.5 24-root 2 freq= 45.58624 38-root 3 freq= 45.60182
noot= 36 24-root 2 freq= 46.24931 38-root 3 freq= 46.26581
noot= 36.5 24-root 2 freq= 46.92202 38-root 3 freq= 46.93946
noot= 37 24-root 2 freq= 47.60452 38-root 3 freq= 47.62291
noot= 37.5 24-root 2 freq= 48.29694 38-root 3 freq= 48.31632
noot= 38 24-root 2 freq= 48.99944 38-root 3 freq= 49.01983
noot= 38.5 24-root 2 freq= 49.71215 38-root 3 freq= 49.73358
noot= 39 24-root 2 freq= 50.43523 38-root 3 freq= 50.45772
noot= 39.5 24-root 2 freq= 51.16883 38-root 3 freq= 51.1924
noot= 40 24-root 2 freq= 51.9131 38-root 3 freq= 51.93779
noot= 40.5 24-root 2 freq= 52.66819 38-root 3 freq= 52.69402
noot= 41 24-root 2 freq= 53.43426 38-root 3 freq= 53.46126
noot= 41.5 24-root 2 freq= 54.21149 38-root 3 freq= 54.23968
noot= 42 24-root 2 freq= 55.00001 38-root 3 freq= 55.02944
noot= 42.5 24-root 2 freq= 55.8 38-root 3 freq= 55.83069
noot= 43 24-root 2 freq= 56.61163 38-root 3 freq= 56.64361
noot= 43.5 24-root 2 freq= 57.43507 38-root 3 freq= 57.46836
noot= 44 24-root 2 freq= 58.27048 38-root 3 freq= 58.30512
noot= 44.5 24-root 2 freq= 59.11804 38-root 3 freq= 59.15407
noot= 45 24-root 2 freq= 59.97794 38-root 3 freq= 60.01538
noot= 45.5 24-root 2 freq= 60.85033 38-root 3 freq= 60.88922
noot= 46 24-root 2 freq= 61.73542 38-root 3 freq= 61.7758
noot= 46.5 24-root 2 freq= 62.63338 38-root 3 freq= 62.67528
noot= 47 24-root 2 freq= 63.54441 38-root 3 freq= 63.58786
noot= 47.5 24-root 2 freq= 64.46868 38-root 3 freq= 64.51372
noot= 48 24-root 2 freq= 65.4064 38-root 3 freq= 65.45307
noot= 48.5 24-root 2 freq= 66.35776 38-root 3 freq= 66.40609
noot= 49 24-root 2 freq= 67.32296 38-root 3 freq= 67.37299
noot= 49.5 24-root 2 freq= 68.30219 38-root 3 freq= 68.35397
noot= 50 24-root 2 freq= 69.29567 38-root 3 freq= 69.34923
noot= 50.5 24-root 2 freq= 70.3036 38-root 3 freq= 70.35898
noot= 51 24-root 2 freq= 71.32619 38-root 3 freq= 71.38344
noot= 51.5 24-root 2 freq= 72.36365 38-root 3 freq= 72.42281
noot= 52 24-root 2 freq= 73.41621 38-root 3 freq= 73.47732
noot= 52.5 24-root 2 freq= 74.48407 38-root 3 freq= 74.54718
noot= 53 24-root 2 freq= 75.56747 38-root 3 freq= 75.63261
noot= 53.5 24-root 2 freq= 76.66662 38-root 3 freq= 76.73386
noot= 54 24-root 2 freq= 77.78176 38-root 3 freq= 77.85114
noot= 54.5 24-root 2 freq= 78.91312 38-root 3 freq= 78.98468
noot= 55 24-root 2 freq= 80.06094 38-root 3 freq= 80.13473
noot= 55.5 24-root 2 freq= 81.22545 38-root 3 freq= 81.30152
noot= 56 24-root 2 freq= 82.40691 38-root 3 freq= 82.4853
noot= 56.5 24-root 2 freq= 83.60554 38-root 3 freq= 83.68632
noot= 57 24-root 2 freq= 84.82161 38-root 3 freq= 84.90483
noot= 57.5 24-root 2 freq= 86.05537 38-root 3 freq= 86.14108
noot= 58 24-root 2 freq= 87.30708 38-root 3 freq= 87.39533
noot= 58.5 24-root 2 freq= 88.57698 38-root 3 freq= 88.66785
noot= 59 24-root 2 freq= 89.86536 38-root 3 freq= 89.95888
noot= 59.5 24-root 2 freq= 91.17248 38-root 3 freq= 91.26872
noot= 60 24-root 2 freq= 92.49862 38-root 3 freq= 92.59763
noot= 60.5 24-root 2 freq= 93.84405 38-root 3 freq= 93.94589
noot= 61 24-root 2 freq= 95.20904 38-root 3 freq= 95.31378
noot= 61.5 24-root 2 freq= 96.59389 38-root 3 freq= 96.70159
noot= 62 24-root 2 freq= 97.99888 38-root 3 freq= 98.1096
noot= 62.5 24-root 2 freq= 99.4243 38-root 3 freq= 99.53812
noot= 63 24-root 2 freq= 100.8705 38-root 3 freq= 100.9874
noot= 63.5 24-root 2 freq= 102.3376 38-root 3 freq= 102.4578
noot= 64 24-root 2 freq= 103.8262 38-root 3 freq= 103.9497
noot= 64.5 24-root 2 freq= 105.3364 38-root 3 freq= 105.4632
noot= 65 24-root 2 freq= 106.8685 38-root 3 freq= 106.9988
noot= 65.5 24-root 2 freq= 108.423 38-root 3 freq= 108.5568
noot= 66 24-root 2 freq= 110 38-root 3 freq= 110.1374
noot= 66.5 24-root 2 freq= 111.6 38-root 3 freq= 111.741
noot= 67 24-root 2 freq= 113.2233 38-root 3 freq= 113.368
noot= 67.5 24-root 2 freq= 114.8701 38-root 3 freq= 115.0187
noot= 68 24-root 2 freq= 116.541 38-root 3 freq= 116.6934
noot= 68.5 24-root 2 freq= 118.2361 38-root 3 freq= 118.3925
noot= 69 24-root 2 freq= 119.9559 38-root 3 freq= 120.1164
noot= 69.5 24-root 2 freq= 121.7007 38-root 3 freq= 121.8653
noot= 70 24-root 2 freq= 123.4708 38-root 3 freq= 123.6397
noot= 70.5 24-root 2 freq= 125.2668 38-root 3 freq= 125.44
noot= 71 24-root 2 freq= 127.0888 38-root 3 freq= 127.2664
noot= 71.5 24-root 2 freq= 128.9374 38-root 3 freq= 129.1195
noot= 72 24-root 2 freq= 130.8128 38-root 3 freq= 130.9995
noot= 72.5 24-root 2 freq= 132.7155 38-root 3 freq= 132.9069
noot= 73 24-root 2 freq= 134.6459 38-root 3 freq= 134.8421
noot= 73.5 24-root 2 freq= 136.6044 38-root 3 freq= 136.8055
noot= 74 24-root 2 freq= 138.5913 38-root 3 freq= 138.7974
noot= 74.5 24-root 2 freq= 140.6072 38-root 3 freq= 140.8184
noot= 75 24-root 2 freq= 142.6524 38-root 3 freq= 142.8687
noot= 75.5 24-root 2 freq= 144.7273 38-root 3 freq= 144.949
noot= 76 24-root 2 freq= 146.8324 38-root 3 freq= 147.0595
noot= 76.5 24-root 2 freq= 148.9681 38-root 3 freq= 149.2007
noot= 77 24-root 2 freq= 151.1349 38-root 3 freq= 151.3732
noot= 77.5 24-root 2 freq= 153.3332 38-root 3 freq= 153.5772
noot= 78 24-root 2 freq= 155.5635 38-root 3 freq= 155.8134
noot= 78.5 24-root 2 freq= 157.8262 38-root 3 freq= 158.0821
noot= 79 24-root 2 freq= 160.1219 38-root 3 freq= 160.3838
noot= 79.5 24-root 2 freq= 162.4509 38-root 3 freq= 162.719
noot= 80 24-root 2 freq= 164.8138 38-root 3 freq= 165.0883
noot= 80.5 24-root 2 freq= 167.2111 38-root 3 freq= 167.4921
noot= 81 24-root 2 freq= 169.6432 38-root 3 freq= 169.9308
noot= 81.5 24-root 2 freq= 172.1107 38-root 3 freq= 172.4051
noot= 82 24-root 2 freq= 174.6142 38-root 3 freq= 174.9154
noot= 82.5 24-root 2 freq= 177.154 38-root 3 freq= 177.4622
noot= 83 24-root 2 freq= 179.7307 38-root 3 freq= 180.0461
noot= 83.5 24-root 2 freq= 182.345 38-root 3 freq= 182.6677
noot= 84 24-root 2 freq= 184.9972 38-root 3 freq= 185.3274
noot= 84.5 24-root 2 freq= 187.6881 38-root 3 freq= 188.0258
noot= 85 24-root 2 freq= 190.4181 38-root 3 freq= 190.7636
noot= 85.5 24-root 2 freq= 193.1878 38-root 3 freq= 193.5412
noot= 86 24-root 2 freq= 195.9978 38-root 3 freq= 196.3592
noot= 86.5 24-root 2 freq= 198.8486 38-root 3 freq= 199.2183
noot= 87 24-root 2 freq= 201.7409 38-root 3 freq= 202.119
noot= 87.5 24-root 2 freq= 204.6753 38-root 3 freq= 205.0619
noot= 88 24-root 2 freq= 207.6524 38-root 3 freq= 208.0477
noot= 88.5 24-root 2 freq= 210.6728 38-root 3 freq= 211.077
noot= 89 24-root 2 freq= 213.7371 38-root 3 freq= 214.1503
noot= 89.5 24-root 2 freq= 216.8459 38-root 3 freq= 217.2684
noot= 90 24-root 2 freq= 220 38-root 3 freq= 220.432
noot= 90.5 24-root 2 freq= 223.2 38-root 3 freq= 223.6415
noot= 91 24-root 2 freq= 226.4465 38-root 3 freq= 226.8978
noot= 91.5 24-root 2 freq= 229.7403 38-root 3 freq= 230.2016
noot= 92 24-root 2 freq= 233.0819 38-root 3 freq= 233.5534
noot= 92.5 24-root 2 freq= 236.4722 38-root 3 freq= 236.954
noot= 93 24-root 2 freq= 239.9117 38-root 3 freq= 240.4042
noot= 93.5 24-root 2 freq= 243.4013 38-root 3 freq= 243.9046
noot= 94 24-root 2 freq= 246.9417 38-root 3 freq= 247.4559
noot= 94.5 24-root 2 freq= 250.5335 38-root 3 freq= 251.059
noot= 95 24-root 2 freq= 254.1776 38-root 3 freq= 254.7145
noot= 95.5 24-root 2 freq= 257.8747 38-root 3 freq= 258.4232
noot= 96 24-root 2 freq= 261.6256 38-root 3 freq= 262.186
noot= 96.5 24-root 2 freq= 265.431 38-root 3 freq= 266.0035
noot= 97 24-root 2 freq= 269.2918 38-root 3 freq= 269.8766
noot= 97.5 24-root 2 freq= 273.2088 38-root 3 freq= 273.8062
noot= 98 24-root 2 freq= 277.1827 38-root 3 freq= 277.7929
noot= 98.5 24-root 2 freq= 281.2144 38-root 3 freq= 281.8377
noot= 99 24-root 2 freq= 285.3047 38-root 3 freq= 285.9413
noot= 99.5 24-root 2 freq= 289.4546 38-root 3 freq= 290.1048
noot= 100 24-root 2 freq= 293.6648 38-root 3 freq= 294.3288
noot= 100.5 24-root 2 freq= 297.9363 38-root 3 freq= 298.6143
noot= 101 24-root 2 freq= 302.2699 38-root 3 freq= 302.9623
noot= 101.5 24-root 2 freq= 306.6665 38-root 3 freq= 307.3736
noot= 102 24-root 2 freq= 311.127 38-root 3 freq= 311.8491
noot= 102.5 24-root 2 freq= 315.6525 38-root 3 freq= 316.3897
noot= 103 24-root 2 freq= 320.2437 38-root 3 freq= 320.9964
noot= 103.5 24-root 2 freq= 324.9018 38-root 3 freq= 325.6703
noot= 104 24-root 2 freq= 329.6276 38-root 3 freq= 330.4122
noot= 104.5 24-root 2 freq= 334.4221 38-root 3 freq= 335.2231
noot= 105 24-root 2 freq= 339.2864 38-root 3 freq= 340.1041
noot= 105.5 24-root 2 freq= 344.2215 38-root 3 freq= 345.0562
noot= 106 24-root 2 freq= 349.2283 38-root 3 freq= 350.0803
noot= 106.5 24-root 2 freq= 354.3079 38-root 3 freq= 355.1776
noot= 107 24-root 2 freq= 359.4614 38-root 3 freq= 360.3492
noot= 107.5 24-root 2 freq= 364.6899 38-root 3 freq= 365.596
noot= 108 24-root 2 freq= 369.9945 38-root 3 freq= 370.9192
noot= 108.5 24-root 2 freq= 375.3762 38-root 3 freq= 376.32
noot= 109 24-root 2 freq= 380.8362 38-root 3 freq= 381.7993
noot= 109.5 24-root 2 freq= 386.3755 38-root 3 freq= 387.3585
noot= 110 24-root 2 freq= 391.9955 38-root 3 freq= 392.9986
noot= 110.5 24-root 2 freq= 397.6972 38-root 3 freq= 398.7208
noot= 111 24-root 2 freq= 403.4818 38-root 3 freq= 404.5264
noot= 111.5 24-root 2 freq= 409.3506 38-root 3 freq= 410.4164
noot= 112 24-root 2 freq= 415.3048 38-root 3 freq= 416.3923
noot= 112.5 24-root 2 freq= 421.3455 38-root 3 freq= 422.4551
noot= 113 24-root 2 freq= 427.4741 38-root 3 freq= 428.6062
noot= 113.5 24-root 2 freq= 433.6919 38-root 3 freq= 434.8469
noot= 114 24-root 2 freq= 440.0001 38-root 3 freq= 441.1784
noot= 114.5 24-root 2 freq= 446.4 38-root 3 freq= 447.6022
noot= 115 24-root 2 freq= 452.8931 38-root 3 freq= 454.1195
noot= 115.5 24-root 2 freq= 459.4805 38-root 3 freq= 460.7316
noot= 116 24-root 2 freq= 466.1638 38-root 3 freq= 467.4401
noot= 116.5 24-root 2 freq= 472.9443 38-root 3 freq= 474.2462
noot= 117 24-root 2 freq= 479.8235 38-root 3 freq= 481.1514
noot= 117.5 24-root 2 freq= 486.8027 38-root 3 freq= 488.1572
noot= 118 24-root 2 freq= 493.8834 38-root 3 freq= 495.2649
noot= 118.5 24-root 2 freq= 501.0671 38-root 3 freq= 502.4762
noot= 119 24-root 2 freq= 508.3553 38-root 3 freq= 509.7924
noot= 119.5 24-root 2 freq= 515.7494 38-root 3 freq= 517.2153
noot= 120 24-root 2 freq= 523.2512 38-root 3 freq= 524.7461
Kwarttoonskwintencirkels
in kwinten van 13 kwarttonen (te kleine kwinten):
13 60.00
2 66.50
15 61.00
4 67.50
17 62.00
6 68.50
19 63.00
8 69.50
21 64.00
10 70.50
23 65.00
12 71.50
1 66.00
14 60.50
3 67.00
16 61.50
5 68.00
18 62.50
7 69.00
20 63.50
9 70.00
22 64.50
11 71.00
0 65.50
Hiermee kunnen we 24 kwarttonen bereiken en de gehele cirkel doorlopen.
Alleen in een harmonie die deze kwinten gebruikt kan doorheen het hele toonsysteem gemoduleerd worden.
In kwinten van 15 kwarttonen (te grote kwinten):
15 60.00
6 67.50
21 63.00
12 70.50
3 66.00
18 61.50
9 69.00
0 64.50
Hiermee kunnen we 8 kwarttonen bereiken.
Met zulke kwinten bestaan er binnen het kwarttoonssysteem 3 verschillende kwintencirkels. Binnen elk kwintencirkelsysteem kunnen 8 andere intervallen
bereikt worden, maar het is niet mogelijk van de ene cirkel naar een van de andere te moduleren met de gebruikelijke muzikale technieken.
in juiste reine kwinten:
0 60.00 60.00
1 67.02 67.00
2 62.04 62.00
3 69.06 69.00
4 64.08 64.00
5 71.10 71.00
6 66.12 66.00
7 61.14 61.00
8 68.16 68.00
9 63.18 63.00
10 70.20 70.00
11 65.22 65.00
12 60.23 60.00
13 67.25 67.50
14 62.27 62.50
15 69.29 69.50
16 64.31 64.50
17 71.33 71.50
18 66.35 66.50
19 61.37 61.50
20 68.39 68.50
21 63.41 63.50
22 70.43 70.50
23 65.45 65.50
24 60.47 60.50
25 67.49 67.50
26 62.51 62.50
27 69.53 69.50
28 64.55 64.50
29 71.57 71.50
30 66.59 66.50
31 61.61 61.50
32 68.63 68.50
33 63.65 63.50
34 70.66 70.50
35 65.68 65.50
36 60.70 60.50
37 67.72 67.50
38 62.74 62.50
39 69.76 70.00
40 64.78 65.00
41 71.80 72.00
42 66.82 67.00
43 61.84 62.00
44 68.86 69.00
45 63.88 64.00
46 70.90 71.00
47 65.92 66.00
48 60.94 61.00
49 67.96 68.00
50 62.98 63.00
51 70.00 70.00
52 65.02 65.00
53 60.04 60.00
aantal noten: 53
In het gewone kwarttoonssysteem bestaan naast elkaar twee onafhankelijke kwintencirkels.
Het is niet mogelijk van het ene systeem naar het andere te moduleren via kwinten.
Alleen mits een 53-toons systeem, sluit de kwintencirkel zich doorheen alle 24 kwarttonen.
Let wel: sluiten betekent hier 'voor onze oren', want wiskundig gezien sluit het natuurlijk nooit.
Alle systemen die halve tonen verder onderverdelen, zijn verenigbaar met de traditionele
westerse muziek.
Zesde-toonssysteem: halve toon bestaat uit 3 stappen, dus 36 onderverdelingen per oktaaf: Juistheid van de 6:5 kleine terts in cent: 15.64 2 ^ ( 9/ 36). Interval: 60 63 Juistheid van de 5:4 grote terts in cent: -13.69 2 ^ ( 12/ 36). Interval: 60 64 Juistheid van de 4:3 kwart in cent:-1.95 2 ^ ( 15/ 36). Interval: 60 65 Juistheid van de 3:2 kwint in cent: 1.96 2 ^ ( 21/ 36). Interval: 60 67 Achtste toonsysteem: 48 onderverdelingen per oktaaf:
Juistheid van de 6:5 kleine terts in cent: -9.36 2 ^ ( 13/ 48). Interval: 60 63.25 Juistheid van de 5:4 grote terts in cent: 11.31 2 ^ ( 15/ 48). Interval: 60 63.75 Juistheid van de 4:3 kwart in cent:-1.95 2 ^ ( 20/ 48). Interval: 60 65 Juistheid van de 3:2 kwint in cent: 1.96 2 ^ ( 28/ 48). Interval: 60 67 We merken op dat de kwinten er bij verdere redelijke verdelingen niet beter op worden. Alleen bij de tertsen is er een merkbare verbetering naarmate de verdeling groter wordt.