Isoharmonic chord: Difference between revisions

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=isoharmonic chords=
#REDIRECT [[Delta-rational chord#Isoharmonic chord]]
 
In [[JustIntonation|just intonation]], Isoharmonic chords are built by successive jumps up the [[OverToneSeries|harmonic series]] by some number of steps. Since the harmonic series is arranged such that each higher step is smaller than the one before it, all isoharmonic chords have this same shape -- with diminishing step size as one ascends. All isoharmonic chords are equal-hertz chords, meaning that the frequencies of the notes are in an arithmetic sequence with an equal difference in cycles per second between successive notes. However, not all equal-hertz chords are isoharmonic chords, since the ratios between the notes need not be integers. An isoharmonic "chord" may function more like a "scale" than a chord (depending on the composition of course), but we will use the word "chord" on this page for consistency.
 
===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:
 
{| class="wikitable"
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| | harmonic
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| | cents diff
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| | 1200
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| | 702
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| | 498
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| | 386
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| | 267
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| | 231
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| | 139
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| | 128
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|}
 
Some "scales" built this way: [[otones12-24|otones12-24]], [[otones20-40|otones20-40]]...
 
===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):
 
{| class="wikitable"
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| | harmonic
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| | 31
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| | cents diff
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| | 1902
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| | 884
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| | 583
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| | 435
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| | 347
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| | 289
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| | 248
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| | 217
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| | 193
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| | 173
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| | 157
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| | 144
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| | 133
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| | 124
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| | 115
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|}
 
===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 (eg. 9:12:15 = 3:4:5). There are two series for class iii:
 
{| class="wikitable"
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| | 46
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| | cents diff
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| | 2400
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| | 969
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| | 617
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| | 454
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| | 359
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| | 298
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| | 254
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| | 221
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| | 196
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| | 176
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| | 160
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| | 146
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| | 135
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| | 125
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| | 117
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|}
 
{| class="wikitable"
|-
| | harmonic
| | 2
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| | 5
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| | 41
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| | 44
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| | 47
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| | cents diff
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| | 1586
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| | 814
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| | 551
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| | 418
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| | 336
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| | 281
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| | 242
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| | 212
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| | 189
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| | 170
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| | 155
| |
| | 142
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| | 132
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| | 122
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| | 114
| |
|}
 
===class iv===
 
{| class="wikitable"
|-
| | harmonic
| | 1
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| | 5
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| | 9
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| | 13
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| | 17
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| | 21
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| | 25
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| | 33
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| | 37
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| | 41
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| | 45
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| | 49
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| | 53
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| | 57
| |
| | 61
|-
| | cents diff
| |
| | 2786
| |
| | 1018
| |
| | 637
| |
| | 464
| |
| | 366
| |
| | 302
| |
| | 257
| |
| | 224
| |
| | 198
| |
| | 178
| |
| | 161
| |
| | 147
| |
| | 136
| |
| | 126
| |
| | 117
| |
|}
 
{| class="wikitable"
|-
| | harmonic
| | 3
| |
| | 7
| |
| | 11
| |
| | 15
| |
| | 19
| |
| | 23
| |
| | 27
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| | 31
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| | 35
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| | 39
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| | 43
| |
| | 47
| |
| | 51
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| | 55
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| | 59
| |
| | 63
|-
| | cents diff
| |
| | 1467
| |
| | 782
| |
| | 537
| |
| | 409
| |
| | 331
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| | 278
| |
| | 239
| |
| | 210
| |
| | 187
| |
| | 169
| |
| | 154
| |
| | 141
| |
| | 131
| |
| | 122
| |
| | 114
| |
|}
 
===class v===
 
{| class="wikitable"
|-
| | harmonic
| | 1
| |
| | 6
| |
| | 11
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| | 16
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| | 21
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| | 26
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| | 31
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| | 36
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| | 46
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| | 51
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| | 56
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| | 61
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| | 66
| |
| | 71
| |
| | 76
|-
| | cents diff
| |
| | 3102
| |
| | 1049
| |
| | 649
| |
| | 471
| |
| | 370
| |
| | 306
| |
| | 259
| |
| | 225
| |
| | 199
| |
| | 179
| |
| | 162
| |
| | 148
| |
| | 136
| |
| | 126
| |
| | 118
| |
|}
 
{| class="wikitable"
|-
| | harmonic
| | 2
| |
| | 7
| |
| | 12
| |
| | 17
| |
| | 22
| |
| | 27
| |
| | 32
| |
| | 37
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| | 42
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| | 47
| |
| | 52
| |
| | 57
| |
| | 62
| |
| | 67
| |
| | 72
| |
| | 77
|-
| | cents diff
| |
| | 2169
| |
| | 933
| |
| | 603
| |
| | 446
| |
| | 355
| |
| | 294
| |
| | 251
| |
| | 219
| |
| | 195
| |
| | 175
| |
| | 159
| |
| | 146
| |
| | 134
| |
| | 125
| |
| | 116
| |
|}
 
{| class="wikitable"
|-
| | harmonic
| | 3
| |
| | 8
| |
| | 13
| |
| | 18
| |
| | 23
| |
| | 28
| |
| | 33
| |
| | 38
| |
| | 43
| |
| | 48
| |
| | 53
| |
| | 58
| |
| | 63
| |
| | 68
| |
| | 73
| |
| | 78
|-
| | cents diff
| |
| | 1698
| |
| | 841
| |
| | 563
| |
| | 424
| |
| | 341
| |
| | 284
| |
| | 244
| |
| | 214
| |
| | 190
| |
| | 172
| |
| | 156
| |
| | 143
| |
| | 132
| |
| | 123
| |
| | 115
| |
|}
 
{| class="wikitable"
|-
| | harmonic
| | 4
| |
| | 9
| |
| | 14
| |
| | 19
| |
| | 24
| |
| | 29
| |
| | 34
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| | 39
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| | 44
| |
| | 49
| |
| | 54
| |
| | 59
| |
| | 64
| |
| | 69
| |
| | 74
| |
| | 79
|-
| | cents diff
| |
| | 1404
| |
| | 765
| |
| | 529
| |
| | 404
| |
| | 328
| |
| | 275
| |
| | 238
| |
| | 209
| |
| | 186
| |
| | 168
| |
| | 153
| |
| | 141
| |
| | 130
| |
| | 121
| |
| | 113
| |
|}
[[Category:chord]]
[[Category:isoharmonic]]
[[Category:list]]
[[Category:overtone]]
[[Category:theory]]

Latest revision as of 03:46, 21 February 2024

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