# Isoharmonic chord

(Redirected from Isodifferential chord)

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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 cents diff 1200 702 498 386 316 267 231 204 182 165 151 139 128 119 112

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 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 cents diff 1902 884 583 435 347 289 248 217 193 173 157 144 133 124 115

### 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 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 cents diff 2400 969 617 454 359 298 254 221 196 176 160 146 135 125 117
 harmonic 2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 cents diff 1586 814 551 418 336 281 242 212 189 170 155 142 132 122 114

### Class iv

 harmonic 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 cents diff 2786 1018 637 464 366 302 257 224 198 178 161 147 136 126 117
 harmonic 3 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 cents diff 1467 782 537 409 331 278 239 210 187 169 154 141 131 122 114

### Class v

 harmonic 1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 cents diff 3102 1049 649 471 370 306 259 225 199 179 162 148 136 126 118
 harmonic 2 7 12 17 22 27 32 37 42 47 52 57 62 67 72 77 cents diff 2169 933 603 446 355 294 251 219 195 175 159 146 134 125 116
 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
 harmonic 4 9 14 19 24 29 34 39 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