# Isoharmonic chords

# isoharmonic chords

In just intonation, Isoharmonic chords are 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 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:

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 (eg. 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 |