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15,677 edits
→Delta-rational harmony: Good to demand a little more
Line 403: Line 403:


== Delta-rational harmony ==
== Delta-rational harmony ==
The tables below show chords that approximate 3-integer-limit [[delta-rational]] chords with least-squares error less than 0.0015.  
The tables below show chords that approximate 3-integer-limit [[delta-rational]] chords with least-squares error less than 0.0011.  
=== Fully delta-rational triads ===
=== Fully delta-rational triads ===
{| class="wikitable"
{| class="wikitable"
Line 429: Line 429:
|[1, 1]
|[1, 1]
|0.00082
|0.00082
|-
|(0,2,5)
|[2, 3]
|0.00129
|-
|-
|(0,3,4)
|(0,3,4)
|[3, 1]
|[3, 1]
|0.00056
|0.00056
|-
|(0,3,5)
|[3, 2]
|0.00120
|-
|-
|(0,3,11)
|(0,3,11)
Line 453: Line 445:
|[3, 2]
|[3, 2]
|0.00081
|0.00081
|-
|(0,6,16)
|[1, 2]
|0.00111
|-
|(0,7,13)
|[1, 1]
|0.00125
|-
|(0,7,16)
|[2, 3]
|0.00129
|-
|-
|(0,8,18)
|(0,8,18)
|[2, 3]
|[2, 3]
|0.00076
|0.00076
|-
|(0,8,26)
|[1, 3]
|0.00131
|-
|-
|(0,9,13)
|(0,9,13)
Line 485: Line 461:
|[1, 2]
|[1, 2]
|0.00100
|0.00100
|-
|(0,10,13)
|[3, 1]
|0.00140
|-
|(0,10,22)
|[2, 3]
|0.00114
|-
|-
|(0,11,20)
|(0,11,20)
Line 517: Line 485:
|[2, 1]
|[2, 1]
|0.00079
|0.00079
|-
|(0,18,27)
|[3, 2]
|0.00135
|-
|-
|(0,18,31)
|(0,18,31)
Line 544: Line 508:
|[1, 2]
|[1, 2]
|0.00094
|0.00094
|-
|(0,1,2,5)
|[1, 3]
|0.00125
|-
|-
|(0,1,3,4)
|(0,1,3,4)
|[1, 1]
|[1, 1]
|0.00080
|0.00080
|-
|(0,1,3,5)
|[1, 2]
|0.00132
|-
|-
|(0,1,4,5)
|(0,1,4,5)
|[1, 1]
|[1, 1]
|0.00107
|0.00107
|-
|(0,1,5,6)
|[1, 1]
|0.00136
|-
|(0,1,15,16)
|[2, 3]
|0.00133
|-
|(0,1,15,17)
|[1, 3]
|0.00147
|-
|-
|(0,1,16,17)
|(0,1,16,17)
Line 624: Line 568:
|[2, 3]
|[2, 3]
|0.00090
|0.00090
|-
|(0,1,22,24)
|[1, 3]
|0.00133
|-
|(0,1,23,24)
|[2, 3]
|0.00125
|-
|(0,1,29,30)
|[1, 2]
|0.00140
|-
|-
|(0,1,30,31)
|(0,1,30,31)
Line 652: Line 584:
|[2, 1]
|[2, 1]
|0.00109
|0.00109
|-
|(0,2,5,6)
|[2, 1]
|0.00142
|-
|-
|(0,2,6,11)
|(0,2,6,11)
Line 668: Line 596:
|[1, 3]
|[1, 3]
|0.00069
|0.00069
|-
|(0,2,11,12)
|[3, 2]
|0.00119
|-
|-
|(0,2,12,13)
|(0,2,12,13)
Line 700: Line 624:
|[3, 2]
|[3, 2]
|0.00031
|0.00031
|-
|(0,2,15,18)
|[1, 2]
|0.00136
|-
|-
|(0,2,16,17)
|(0,2,16,17)
|[3, 2]
|[3, 2]
|0.00071
|0.00071
|-
|(0,2,17,18)
|[3, 2]
|0.00111
|-
|(0,2,17,19)
|[2, 3]
|0.00147
|-
|-
|(0,2,17,21)
|(0,2,17,21)
Line 740: Line 652:
|[2, 3]
|[2, 3]
|0.00046
|0.00046
|-
|(0,2,20,24)
|[1, 3]
|0.00142
|-
|(0,2,21,23)
|[2, 3]
|0.00113
|-
|(0,2,31,32)
|[1, 1]
|0.00127
|-
|-
|(0,3,4,5)
|(0,3,4,5)
Line 760: Line 660:
|[2, 3]
|[2, 3]
|0.00108
|0.00108
|-
|(0,3,4,12)
|[1, 3]
|0.00117
|-
|(0,3,5,6)
|[3, 1]
|0.00131
|-
|-
|(0,3,5,9)
|(0,3,5,9)
Line 784: Line 676:
|[1, 2]
|[1, 2]
|0.00037
|0.00037
|-
|(0,3,9,16)
|[1, 3]
|0.00114
|-
|-
|(0,3,10,17)
|(0,3,10,17)
|[1, 3]
|[1, 3]
|0.00009
|0.00009
|-
|(0,3,11,18)
|[1, 3]
|0.00134
|-
|(0,3,17,18)
|[2, 1]
|0.00146
|-
|-
|(0,3,17,23)
|(0,3,17,23)
Line 808: Line 688:
|[2, 1]
|[2, 1]
|0.00103
|0.00103
|-
|(0,3,18,20)
|[1, 1]
|0.00140
|-
|(0,3,18,21)
|[2, 3]
|0.00128
|-
|-
|(0,3,18,22)
|(0,3,18,22)
Line 852: Line 724:
|[2, 3]
|[2, 3]
|0.00070
|0.00070
|-
|(0,3,20,24)
|[1, 2]
|0.00136
|-
|-
|(0,3,21,22)
|(0,3,21,22)
Line 868: Line 736:
|[2, 1]
|[2, 1]
|0.00078
|0.00078
|-
|(0,3,23,24)
|[2, 1]
|0.00126
|-
|(0,3,26,31)
|[1, 3]
|0.00119
|-
|-
|(0,3,27,32)
|(0,3,27,32)
Line 896: Line 756:
|[2, 3]
|[2, 3]
|0.00023
|0.00023
|-
|(0,4,9,14)
|[2, 3]
|0.00112
|-
|-
|(0,4,11,20)
|(0,4,11,20)
|[1, 3]
|[1, 3]
|0.00023
|0.00023
|-
|(0,4,12,13)
|[3, 1]
|0.00131
|-
|-
|(0,4,12,18)
|(0,4,12,18)
|[1, 2]
|[1, 2]
|0.00101
|0.00101
|-
|(0,4,12,21)
|[1, 3]
|0.00145
|-
|-
|(0,4,13,14)
|(0,4,13,14)
|[3, 1]
|[3, 1]
|0.00091
|0.00091
|-
|(0,4,13,15)
|[3, 2]
|0.00132
|-
|(0,4,13,16)
|[1, 1]
|0.00126
|-
|-
|(0,4,13,19)
|(0,4,13,19)
Line 972: Line 812:
|[1, 3]
|[1, 3]
|0.00043
|0.00043
|-
|(0,4,18,19)
|[3, 1]
|0.00121
|-
|(0,4,18,26)
|[1, 3]
|0.00124
|-
|-
|(0,4,19,23)
|(0,4,19,23)
Line 988: Line 820:
|[2, 3]
|[2, 3]
|0.00094
|0.00094
|-
|(0,4,21,26)
|[1, 2]
|0.00129
|-
|-
|(0,4,22,27)
|(0,4,22,27)
Line 1,000: Line 828:
|[1, 3]
|[1, 3]
|0.00022
|0.00022
|-
|(0,4,25,32)
|[1, 3]
|0.00146
|-
|-
|(0,5,6,9)
|(0,5,6,9)
Line 1,032: Line 856:
|[1, 1]
|[1, 1]
|0.00018
|0.00018
|-
|(0,5,11,22)
|[1, 3]
|0.00142
|-
|-
|(0,5,12,14)
|(0,5,12,14)
|[2, 1]
|[2, 1]
|0.00011
|0.00011
|-
|(0,5,12,16)
|[1, 1]
|0.00118
|-
|-
|(0,5,12,23)
|(0,5,12,23)
Line 1,052: Line 868:
|[2, 1]
|[2, 1]
|0.00067
|0.00067
|-
|(0,5,14,16)
|[2, 1]
|0.00147
|-
|-
|(0,5,16,23)
|(0,5,16,23)
Line 1,068: Line 880:
|[2, 3]
|[2, 3]
|0.00051
|0.00051
|-
|(0,5,20,25)
|[2, 3]
|0.00119
|-
|-
|(0,5,22,31)
|(0,5,22,31)
|[1, 3]
|[1, 3]
|0.00057
|0.00057
|-
|(0,5,24,25)
|[3, 1]
|0.00122
|-
|-
|(0,5,24,30)
|(0,5,24,30)
Line 1,112: Line 916:
|[3, 1]
|[3, 1]
|0.00035
|0.00035
|-
|(0,5,27,29)
|[3, 2]
|0.00117
|-
|-
|(0,5,28,29)
|(0,5,28,29)
|[3, 1]
|[3, 1]
|0.00090
|0.00090
|-
|(0,5,29,30)
|[3, 1]
|0.00146
|-
|-
|(0,6,7,17)
|(0,6,7,17)
Line 1,136: Line 932:
|[1, 1]
|[1, 1]
|0.00031
|0.00031
|-
|(0,6,10,15)
|[1, 1]
|0.00133
|-
|-
|(0,6,11,18)
|(0,6,11,18)
|[2, 3]
|[2, 3]
|0.00093
|0.00093
|-
|(0,6,12,19)
|[2, 3]
|0.00113
|-
|-
|(0,6,12,21)
|(0,6,12,21)
Line 1,156: Line 944:
|[1, 3]
|[1, 3]
|0.00032
|0.00032
|-
|(0,6,14,17)
|[3, 2]
|0.00145
|-
|-
|(0,6,15,18)
|(0,6,15,18)
Line 1,180: Line 964:
|[2, 3]
|[2, 3]
|0.00062
|0.00062
|-
|(0,6,20,22)
|[2, 1]
|0.00124
|-
|-
|(0,6,20,24)
|(0,6,20,24)
|[1, 1]
|[1, 1]
|0.00052
|0.00052
|-
|(0,6,20,26)
|[2, 3]
|0.00144
|-
|-
|(0,6,21,23)
|(0,6,21,23)
|[2, 1]
|[2, 1]
|0.00031
|0.00031
|-
|(0,6,21,25)
|[1, 1]
|0.00111
|-
|-
|(0,6,21,32)
|(0,6,21,32)
Line 1,212: Line 984:
|[1, 2]
|[1, 2]
|0.00034
|0.00034
|-
|(0,7,8,10)
|[3, 1]
|0.00141
|-
|(0,7,8,12)
|[3, 2]
|0.00122
|-
|-
|(0,7,8,14)
|(0,7,8,14)
Line 1,240: Line 1,004:
|[3, 2]
|[3, 2]
|0.00019
|0.00019
|-
|(0,7,9,20)
|[1, 2]
|0.00113
|-
|-
|(0,7,10,12)
|(0,7,10,12)
Line 1,264: Line 1,024:
|[1, 3]
|[1, 3]
|0.00041
|0.00041
|-
|(0,7,13,21)
|[2, 3]
|0.00138
|-
|(0,7,14,24)
|[1, 2]
|0.00128
|-
|(0,7,15,25)
|[1, 2]
|0.00144
|-
|-
|(0,7,16,30)
|(0,7,16,30)
Line 1,296: Line 1,044:
|[3, 2]
|[3, 2]
|0.00010
|0.00010
|-
|(0,7,24,27)
|[3, 2]
|0.00133
|-
|(0,7,28,30)
|[2, 1]
|0.00129
|-
|-
|(0,7,28,32)
|(0,7,28,32)
Line 1,316: Line 1,056:
|[2, 1]
|[2, 1]
|0.00091
|0.00091
|-
|(0,8,12,24)
|[1, 2]
|0.00118
|-
|-
|(0,8,12,29)
|(0,8,12,29)
Line 1,328: Line 1,064:
|[2, 3]
|[2, 3]
|0.00051
|0.00051
|-
|(0,8,15,19)
|[3, 2]
|0.00120
|-
|-
|(0,8,15,21)
|(0,8,15,21)
Line 1,364: Line 1,096:
|[2, 1]
|[2, 1]
|0.00098
|0.00098
|-
|(0,8,18,20)
|[3, 1]
|0.00112
|-
|-
|(0,8,19,27)
|(0,8,19,27)
Line 1,380: Line 1,108:
|[1, 1]
|[1, 1]
|0.00020
|0.00020
|-
|(0,9,10,15)
|[3, 2]
|0.00135
|-
|-
|(0,9,11,16)
|(0,9,11,16)
Line 1,392: Line 1,116:
|[1, 1]
|[1, 1]
|0.00002
|0.00002
|-
|(0,9,13,26)
|[1, 2]
|0.00148
|-
|-
|(0,9,14,24)
|(0,9,14,24)
Line 1,432: Line 1,152:
|[2, 1]
|[2, 1]
|0.00093
|0.00093
|-
|(0,9,24,26)
|[3, 1]
|0.00115
|-
|-
|(0,10,11,26)
|(0,10,11,26)
Line 1,448: Line 1,164:
|[1, 1]
|[1, 1]
|0.00098
|0.00098
|-
|(0,10,13,17)
|[2, 1]
|0.00113
|-
|-
|(0,10,14,18)
|(0,10,14,18)
Line 1,468: Line 1,180:
|[3, 2]
|[3, 2]
|0.00055
|0.00055
|-
|(0,10,18,25)
|[1, 1]
|0.00148
|-
|(0,10,19,26)
|[1, 1]
|0.00135
|-
|-
|(0,10,19,29)
|(0,10,19,29)
Line 1,488: Line 1,192:
|[1, 1]
|[1, 1]
|0.00106
|0.00106
|-
|(0,10,27,30)
|[2, 1]
|0.00146
|-
|-
|(0,10,27,31)
|(0,10,27,31)
Line 1,504: Line 1,204:
|[2, 1]
|[2, 1]
|0.00016
|0.00016
|-
|(0,10,28,32)
|[3, 2]
|0.00128
|-
|-
|(0,10,29,31)
|(0,10,29,31)
Line 1,524: Line 1,220:
|[3, 1]
|[3, 1]
|0.00044
|0.00044
|-
|(0,11,15,27)
|[2, 3]
|0.00143
|-
|-
|(0,11,16,31)
|(0,11,16,31)
Line 1,536: Line 1,228:
|[1, 1]
|[1, 1]
|0.00091
|0.00091
|-
|(0,11,18,22)
|[2, 1]
|0.00136
|-
|-
|(0,11,19,23)
|(0,11,19,23)
|[2, 1]
|[2, 1]
|0.00045
|0.00045
|-
|(0,11,19,30)
|[2, 3]
|0.00120
|-
|-
|(0,11,21,26)
|(0,11,21,26)
|[3, 2]
|[3, 2]
|0.00074
|0.00074
|-
|(0,11,24,31)
|[1, 1]
|0.00132
|-
|(0,12,13,18)
|[2, 1]
|0.00145
|-
|-
|(0,12,14,31)
|(0,12,14,31)
Line 1,584: Line 1,260:
|[3, 1]
|[3, 1]
|0.00095
|0.00095
|-
|(0,12,19,31)
|[2, 3]
|0.00133
|-
|(0,12,21,29)
|[1, 1]
|0.00129
|-
|-
|(0,12,23,27)
|(0,12,23,27)
|[2, 1]
|[2, 1]
|0.00083
|0.00083
|-
|(0,12,24,28)
|[2, 1]
|0.00118
|-
|-
|(0,12,26,31)
|(0,12,26,31)
Line 1,608: Line 1,272:
|[1, 1]
|[1, 1]
|0.00019
|0.00019
|-
|(0,13,15,29)
|[2, 3]
|0.00121
|-
|(0,13,16,21)
|[2, 1]
|0.00135
|-
|-
|(0,13,17,22)
|(0,13,17,22)
|[2, 1]
|[2, 1]
|0.00085
|0.00085
|-
|(0,13,19,32)
|[2, 3]
|0.00145
|-
|-
|(0,13,21,27)
|(0,13,21,27)
Line 1,656: Line 1,308:
|[3, 2]
|[3, 2]
|0.00054
|0.00054
|-
|(0,14,26,29)
|[3, 1]
|0.00114
|-
|-
|(0,14,27,30)
|(0,14,27,30)
Line 1,668: Line 1,316:
|[3, 1]
|[3, 1]
|0.00055
|0.00055
|-
|(0,15,17,21)
|[3, 1]
|0.00125
|-
|-
|(0,15,17,28)
|(0,15,17,28)
Line 1,684: Line 1,328:
|[1, 1]
|[1, 1]
|0.00039
|0.00039
|-
|(0,15,24,29)
|[2, 1]
|0.00123
|-
|(0,15,25,30)
|[2, 1]
|0.00136
|-
|-
|(0,16,18,26)
|(0,16,18,26)
Line 1,724: Line 1,360:
|[3, 1]
|[3, 1]
|0.00066
|0.00066
|-
|(0,17,24,28)
|[3, 1]
|0.00141
|-
|(0,18,26,30)
|[3, 1]
|0.00126
|-
|(0,18,26,32)
|[2, 1]
|0.00144
|-
|-
|(0,18,27,31)
|(0,18,27,31)
Line 1,744: Line 1,368:
|[2, 1]
|[2, 1]
|0.00065
|0.00065
|-
|(0,20,21,26)
|[3, 1]
|0.00131
|-
|(0,20,22,27)
|[3, 1]
|0.00120
|-
|-
|(0,20,24,31)
|(0,20,24,31)
Line 1,768: Line 1,384:
|[3, 1]
|[3, 1]
|0.00038
|0.00038
|-
|(0,25,26,32)
|[3, 1]
|0.00116
|}
|}


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