Dr.Godfried-Willem RAES

Kursus Akoestiek & Klankonderzoek

Hogeschool Gent : School of Arts


<Terug naar inhoudstafel kursus> Noot: de inhoud van deze paragraaf behoort tevens tot de leerstof voor het vak akoestiek.

1084-subparagraaf:

KWARTTOONSSYSTEMEN EN STEMMINGEN

Alle hier uitgevoerde berekeningen gaan uit van diapason La = 440.0Hz
Noten worden uitgedrukt als fraktionele midi getallen, waarbij noot 60
overeenkomt met de centrale Do (C) op het pianoklavier.




  
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.

Filedate: 2001-02-15 /2013-03-11

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