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Delta-rational chord: Difference between revisions

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==== Classification ====
==== Classification ====
===== Class i =====
===== Class I =====
The simplest isoharmonic chords are built by stepping up the harmonic series by single steps (adjacent steps in the harmonic series). Take, for instance, 4:5:6:7, the harmonic seventh chord. We may call these class i isoharmonic chords. There is one class i series (the harmonic series), which looks like this:
The simplest isoharmonic chords are built by stepping up the harmonic series by single steps (adjacent steps in the harmonic series). Take, for instance, 4:5:6:7, the harmonic seventh chord. We may call these class I isoharmonic chords. There is one class i series (the harmonic series), which looks like this:


{| class="wikitable"
{| class="wikitable"
|-
|-
| | harmonic
! Harmonic
| | 1
| 1
| |  
|  
| | 2
| 2
| |  
|  
| | 3
| 3
| |  
|  
| | 4
| 4
| |  
|  
| | 5
| 5
| |  
|  
| | 6
| 6
| |  
|  
| | 7
| 7
| |  
|  
| | 8
| 8
| |  
|  
| | 9
| 9
| |  
|  
| | 10
| 10
| |  
|  
| | 11
| 11
| |  
|  
| | 12
| 12
| |  
|  
| | 13
| 13
| |  
|  
| | 14
| 14
| |  
|  
| | 15
| 15
| |  
|  
| | 16
| 16
|-
|-
| | cents diff
! Cents diff
| |  
|  
| | 1200
| 1200
| |  
|  
| | 702
| 702
| |  
|  
| | 498
| 498
| |  
|  
| | 386
| 386
| |  
|  
| | 316
| 316
| |  
|  
| | 267
| 267
| |  
|  
| | 231
| 231
| |  
|  
| | 204
| 204
| |  
|  
| | 182
| 182
| |  
|  
| | 165
| 165
| |  
|  
| | 151
| 151
| |  
|  
| | 139
| 139
| |  
|  
| | 128
| 128
| |  
|  
| | 119
| 119
| |  
|  
| | 112
| 112
| |  
|  
|}
|}


Some "scales" built this way: [[otones12-24]], [[otones20-40]]...
Some "scales" built this way: [[otones12-24]], [[otones20-40]]...


===== Class ii =====
===== Class II =====
The next simplest isoharmonic chords are built by stepping up the harmonic series by two (skipping every other harmonic). This gives us chords such as 3:5:7:9 (the primary tetrad in the [[BP|Bohlen-Pierce]] tuning system) and 9:11:13:15. Note that if you start on an even number, your chord is equivalent to a class i harmonic chord: 4:6:8:10 = 2:3:4:5. Thus, there is one class ii series (the series of all odd harmonics):
The next simplest isoharmonic chords are built by stepping up the harmonic series by two (skipping every other harmonic). This gives us chords such as 3:5:7:9 (the primary tetrad in the [[BP|Bohlen-Pierce]] tuning system) and 9:11:13:15. Note that if you start on an even number, your chord is equivalent to a class i harmonic chord: 4:6:8:10 = 2:3:4:5. Thus, there is one class ii series (the series of all odd harmonics):


{| class="wikitable"
{| class="wikitable"
|-
|-
| | harmonic
! Harmonic
| | 1
| 1
| |  
|  
| | 3
| 3
| |  
|  
| | 5
| 5
| |  
|  
| | 7
| 7
| |  
|  
| | 9
| 9
| |  
|  
| | 11
| 11
| |  
|  
| | 13
| 13
| |  
|  
| | 15
| 15
| |  
|  
| | 17
| 17
| |  
|  
| | 19
| 19
| |  
|  
| | 21
| 21
| |  
|  
| | 23
| 23
| |  
|  
| | 25
| 25
| |  
|  
| | 27
| 27
| |  
|  
| | 29
| 29
| |  
|  
| | 31
| 31
|-
! Cents diff
| | cents diff
|  
| |  
| 1902
| | 1902
|  
| |  
| 884
| | 884
|  
| |  
| 583
| | 583
|  
| |  
| 435
| | 435
|  
| |  
| 347
| | 347
|  
| |  
| 289
| | 289
|  
| |  
| 248
| | 248
|  
| |  
| 217
| | 217
|  
| |  
| 193
| | 193
|  
| |  
| 173
| | 173
|  
| |  
| 157
| | 157
|  
| |  
| 144
| | 144
|  
| |  
| 133
| | 133
|  
| |  
| 124
| | 124
|  
| |  
| 115
| | 115
|  
| |  
|}
|}


===== Class iii =====
===== Class III =====
Class iii isoharmonic chords are less common and more complex sounding. They include chords such as 7:10:13:16 and 14:17:20:23. Note that if you start on a number divisible by three, you'll again get a chord reducible to class i (e.g. 9:12:15 = 3:4:5). There are two series for class iii:
Class III isoharmonic chords are less common and more complex sounding. They include chords such as 7:10:13:16 and 14:17:20:23. Note that if you start on a number divisible by three, you'll again get a chord reducible to class i (e.g. 9:12:15 = 3:4:5). There are two series for class iii:


{| class="wikitable"
{| class="wikitable"
|-
|-
| | harmonic
! Harmonic
| | 1
| 1
| |  
|  
| | 4
| 4
| |  
|  
| | 7
| 7
| |  
|  
| | 10
| 10
| |  
|  
| | 13
| 13
| |  
|  
| | 16
| 16
| |  
|  
| | 19
| 19
| |  
|  
| | 22
| 22
| |  
|  
| | 25
| 25
| |  
|  
| | 28
| 28
| |  
|  
| | 31
| 31
| |  
|  
| | 34
| 34
| |  
|  
| | 37
| 37
| |  
|  
| | 40
| 40
| |  
|  
| | 43
| 43
| |  
|  
| | 46
| 46
|-
|-
| | cents diff
! Cents diff
| |  
|  
| | 2400
| 2400
| |  
|  
| | 969
| 969
| |  
|  
| | 617
| 617
| |  
|  
| | 454
| 454
| |  
|  
| | 359
| 359
| |  
|  
| | 298
| 298
| |  
|  
| | 254
| 254
| |  
|  
| | 221
| 221
| |  
|  
| | 196
| 196
| |  
|  
| | 176
| 176
| |  
|  
| | 160
| 160
| |  
|  
| | 146
| 146
| |  
|  
| | 135
| 135
| |  
|  
| | 125
| 125
| |  
|  
| | 117
| 117
| |  
|  
|}
|}


{| class="wikitable"
{| class="wikitable"
|-
|-
| | harmonic
! Harmonic
| | 2
| 2
| |  
|  
| | 5
| 5
| |  
|  
| | 8
| 8
| |  
|  
| | 11
| 11
| |  
|  
| | 14
| 14
| |  
|  
| | 17
| 17
| |  
|  
| | 20
| 20
| |  
|  
| | 23
| 23
| |  
|  
| | 26
| 26
| |  
|  
| | 29
| 29
| |  
|  
| | 32
| 32
| |  
|  
| | 35
| 35
| |  
|  
| | 38
| 38
| |  
|  
| | 41
| 41
| |  
|  
| | 44
| 44
| |  
|  
| | 47
| 47
|-
|-
| | cents diff
! Cents diff
| |  
|  
| | 1586
| 1586
| |  
|  
| | 814
| 814
| |  
|  
| | 551
| 551
| |  
|  
| | 418
| 418
| |  
|  
| | 336
| 336
| |  
|  
| | 281
| 281
| |  
|  
| | 242
| 242
| |  
|  
| | 212
| 212
| |  
|  
| | 189
| 189
| |  
|  
| | 170
| 170
| |  
|  
| | 155
| 155
| |  
|  
| | 142
| 142
| |  
|  
| | 132
| 132
| |  
|  
| | 122
| 122
| |  
|  
| | 114
| 114
| |  
|  
|}
|}


Line 690: Line 689:
{| class="wikitable"
{| class="wikitable"
|-
|-
| | harmonic
! Harmonic
| | 1
| 1
| |  
|  
| | 5
| 5
| |  
|  
| | 9
| 9
| |  
|  
| | 13
| 13
| |  
|  
| | 17
| 17
| |  
|  
| | 21
| 21
| |  
|  
| | 25
| 25
| |  
|  
| | 29
| 29
| |  
|  
| | 33
| 33
| |  
|  
| | 37
| 37
| |  
|  
| | 41
| 41
| |  
|  
| | 45
| 45
| |  
|  
| | 49
| 49
| |  
|  
| | 53
| 53
| |  
|  
| | 57
| 57
| |  
|  
| | 61
| 61
|-
|-
| | cents diff
! Cents diff
| |  
|  
| | 2786
| 2786
| |  
|  
| | 1018
| 1018
| |  
|  
| | 637
| 637
| |  
|  
| | 464
| 464
| |  
|  
| | 366
| 366
| |  
|  
| | 302
| 302
| |  
|  
| | 257
| 257
| |  
|  
| | 224
| 224
| |  
|  
| | 198
| 198
| |  
|  
| | 178
| 178
| |  
|  
| | 161
| 161
| |  
|  
| | 147
| 147
| |  
|  
| | 136
| 136
| |  
|  
| | 126
| 126
| |  
|  
| | 117
| 117
| |  
|  
|}
|}


{| class="wikitable"
{| class="wikitable"
|-
|-
| | harmonic
! Harmonic
| | 3
| 3
| |  
|  
| | 7
| 7
| |  
|  
| | 11
| 11
| |  
|  
| | 15
| 15
| |  
|  
| | 19
| 19
| |  
|  
| | 23
| 23
| |  
|  
| | 27
| 27
| |  
|  
| | 31
| 31
| |  
|  
| | 35
| 35
| |  
|  
| | 39
| 39
| |  
|  
| | 43
| 43
| |  
|  
| | 47
| 47
| |  
|  
| | 51
| 51
| |  
|  
| | 55
| 55
| |  
|  
| | 59
| 59
| |  
|  
| | 63
| 63
|-
|-
| | cents diff
! Cents diff
| |  
|  
| | 1467
| 1467
| |  
|  
| | 782
| 782
| |  
|  
| | 537
| 537
| |  
|  
| | 409
| 409
| |  
|  
| | 331
| 331
| |  
|  
| | 278
| 278
| |  
|  
| | 239
| 239
| |  
|  
| | 210
| 210
| |  
|  
| | 187
| 187
| |  
|  
| | 169
| 169
| |  
|  
| | 154
| 154
| |  
|  
| | 141
| 141
| |  
|  
| | 131
| 131
| |  
|  
| | 122
| 122
| |  
|  
| | 114
| 114
| |  
|  
|}
|}


Line 830: Line 829:
{| class="wikitable"
{| class="wikitable"
|-
|-
| | harmonic
! Harmonic
| | 1
| 1
| |  
|  
| | 6
| 6
| |  
|  
| | 11
| 11
| |  
|  
| | 16
| 16
| |  
|  
| | 21
| 21
| |  
|  
| | 26
| 26
| |  
|  
| | 31
| 31
| |  
|  
| | 36
| 36
| |  
|  
| | 41
| 41
| |  
|  
| | 46
| 46
| |  
|  
| | 51
| 51
| |  
|  
| | 56
| 56
| |  
|  
| | 61
| 61
| |  
|  
| | 66
| 66
| |  
|  
| | 71
| 71
| |  
|  
| | 76
| 76
|-
|-
| | cents diff
! Cents diff
| |  
|  
| | 3102
| 3102
| |  
|  
| | 1049
| 1049
| |  
|  
| | 649
| 649
| |  
|  
| | 471
| 471
| |  
|  
| | 370
| 370
| |  
|  
| | 306
| 306
| |  
|  
| | 259
| 259
| |  
|  
| | 225
| 225
| |  
|  
| | 199
| 199
| |  
|  
| | 179
| 179
| |  
|  
| | 162
| 162
| |  
|  
| | 148
| 148
| |  
|  
| | 136
| 136
| |  
|  
| | 126
| 126
| |  
|  
| | 118
| 118
| |  
|  
|}
|}


{| class="wikitable"
{| class="wikitable"
|-
|-
| | harmonic
! Harmonic
| | 2
| 2
| |  
|  
| | 7
| 7
| |  
|  
| | 12
| 12
| |  
|  
| | 17
| 17
| |  
|  
| | 22
| 22
| |  
|  
| | 27
| 27
| |  
|  
| | 32
| 32
| |  
|  
| | 37
| 37
| |  
|  
| | 42
| 42
| |  
|  
| | 47
| 47
| |  
|  
| | 52
| 52
| |  
|  
| | 57
| 57
| |  
|  
| | 62
| 62
| |  
|  
| | 67
| 67
| |  
|  
| | 72
| 72
| |  
|  
| | 77
| 77
|-
|-
| | cents diff
! Cents diff
| |  
|  
| | 2169
| 2169
| |  
|  
| | 933
| 933
| |  
|  
| | 603
| 603
| |  
|  
| | 446
| 446
| |  
|  
| | 355
| 355
| |  
|  
| | 294
| 294
| |  
|  
| | 251
| 251
| |  
|  
| | 219
| 219
| |  
|  
| | 195
| 195
| |  
|  
| | 175
| 175
| |  
|  
| | 159
| 159
| |  
|  
| | 146
| 146
| |  
|  
| | 134
| 134
| |  
|  
| | 125
| 125
| |  
|  
| | 116
| 116
| |  
|  
|}
|}


{| class="wikitable"
{| class="wikitable"
|-
|-
| | harmonic
! Harmonic
| | 3
| 3
| |  
|  
| | 8
| 8
| |  
|  
| | 13
| 13
| |  
|  
| | 18
| 18
| |  
|  
| | 23
| 23
| |  
|  
| | 28
| 28
| |  
|  
| | 33
| 33
| |  
|  
| | 38
| 38
| |  
|  
| | 43
| 43
| |  
|  
| | 48
| 48
| |  
|  
| | 53
| 53
| |  
|  
| | 58
| 58
| |  
|  
| | 63
| 63
| |  
|  
| | 68
| 68
| |  
|  
| | 73
| 73
| |  
|  
| | 78
| 78
|-
|-
| | cents diff
! Cents diff
| |  
|  
| | 1698
| 1698
| |  
|  
| | 841
| 841
| |  
|  
| | 563
| 563
| |  
|  
| | 424
| 424
| |  
|  
| | 341
| 341
| |  
|  
| | 284
| 284
| |  
|  
| | 244
| 244
| |  
|  
| | 214
| 214
| |  
|  
| | 190
| 190
| |  
|  
| | 172
| 172
| |  
|  
| | 156
| 156
| |  
|  
| | 143
| 143
| |  
|  
| | 132
| 132
| |  
|  
| | 123
| 123
| |  
|  
| | 115
| 115
| |  
|  
|}
|}


{| class="wikitable"
{| class="wikitable"
|-
|-
| | harmonic
! Harmonic
| | 4
| 4
| |  
|  
| | 9
| 9
| |  
|  
| | 14
| 14
| |  
|  
| | 19
| 19
| |  
|  
| | 24
| 24
| |  
|  
| | 29
| 29
| |  
|  
| | 34
| 34
| |  
|  
| | 39
| 39
| |  
|  
| | 44
| 44
| |  
|  
| | 49
| 49
| |  
|  
| | 54
| 54
| |  
|  
| | 59
| 59
| |  
|  
| | 64
| 64
| |  
|  
| | 69
| 69
| |  
|  
| | 74
| 74
| |  
|  
| | 79
| 79
|-
|-
| | cents diff
! Cents diff
| |  
|  
| | 1404
| 1404
| |  
|  
| | 765
| 765
| |  
|  
| | 529
| 529
| |  
|  
| | 404
| 404
| |  
|  
| | 328
| 328
| |  
|  
| | 275
| 275
| |  
|  
| | 238
| 238
| |  
|  
| | 209
| 209
| |  
|  
| | 186
| 186
| |  
|  
| | 168
| 168
| |  
|  
| | 153
| 153
| |  
|  
| | 141
| 141
| |  
|  
| | 130
| 130
| |  
|  
| | 121
| 121
| |  
|  
| | 113
| 113
| |  
|  
|}
|}


Retrieved from "https://en.xen.wiki/w/Delta-rational_chord"