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