In just intonation, an isoharmonic chord is a chord built by successive jumps up the 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 isodifferential chords (or 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 isodifferential chords are isoharmonic chords, since the ratios between the notes need not be rational numbers.
An isoharmonic "chord" may function more like a "scale" than a chord (depending on the composition of course), but the word "chord" is used on this page for consistency.
Notation
Some complex isoharmonic chords can be expressed with an offset from a simpler isoharmonic chord, so it is useful to notate them in a compact and readable way. For example, 41:51:61 is very similar to 4:5:6, so it can be notated as (4:5:6)[+0.1]. Similarly, (20:22:24:27:30:33:36)[+0.339] can be expanded to 20339:22339:24339:27339:30339:33339:36339.
Irrational isodifferential chords can be expressed with the same notation by using irrational numbers within the square brackets. For example, the chord (1:2:3)[+φ] can be expanded to (1+φ):(2+φ):(3+φ), which is approximately equal to 1.618:2.618:3.618.
Classification
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:
| harmonic
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1
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2
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3
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4
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5
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6
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7
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8
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9
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10
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11
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12
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13
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14
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15
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16
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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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316
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267
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231
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204
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182
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165
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151
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139
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128
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119
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112
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Some "scales" built this way: otones12-24, 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 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):
| harmonic
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1
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3
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5
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7
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9
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11
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13
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15
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17
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19
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21
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23
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25
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27
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29
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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 (e.g. 9:12:15 = 3:4:5). There are two series for class iii:
| harmonic
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1
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4
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7
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10
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13
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16
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22
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25
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28
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31
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34
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37
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40
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43
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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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2
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5
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8
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11
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14
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17
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20
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23
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26
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29
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32
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35
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38
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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
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142
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132
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122
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114
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Class iv
| harmonic
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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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29
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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
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61
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| cents diff
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2786
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1018
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637
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464
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366
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302
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257
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224
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198
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178
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161
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147
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136
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126
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117
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| harmonic
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3
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7
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11
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15
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19
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23
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27
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31
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35
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39
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43
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47
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51
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55
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59
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63
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| cents diff
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1467
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782
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537
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409
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331
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278
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239
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210
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187
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169
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154
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141
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131
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122
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114
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Class v
| harmonic
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1
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6
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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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41
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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
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71
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76
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| cents diff
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3102
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1049
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649
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471
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370
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306
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259
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225
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199
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179
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162
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148
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136
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126
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118
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| harmonic
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2
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7
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12
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17
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22
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27
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32
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37
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42
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47
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52
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57
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62
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67
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72
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|
77
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| cents diff
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2169
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933
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603
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446
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355
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294
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251
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219
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195
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175
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159
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146
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134
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125
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116
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| harmonic
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3
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8
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13
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18
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23
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28
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33
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38
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43
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48
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53
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58
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63
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|
68
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73
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|
78
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| cents diff
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1698
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841
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563
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|
424
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341
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|
284
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244
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214
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190
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|
172
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156
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143
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132
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123
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115
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| harmonic
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4
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9
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14
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19
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24
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29
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34
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39
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44
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49
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54
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59
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64
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69
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74
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79
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| cents diff
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1404
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765
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529
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404
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328
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275
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|
238
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|
209
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|
186
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|
168
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|
153
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|
141
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|
130
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121
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|
113
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|
See also