A major aspect of the history of tuning theory is mankind's effort to understand the finity of pitch-perception.
Throughout recorded history, from the Sumerians to the present day, humans have striven to understand and classify the audible frequency range in a bewildering variety of ways, mostly by using math ... and music-theorists have always been quick to take advantage of every new advance in mathematics.
With the advent of home computer technology in the 1980s and the internet in the 1990s, tuning theorists of the early 21st century have been able to discover, discuss, and classify an enormous number of tunings. It was readily perceived that certain groups of tunings share common characteristics and thus belong to a "family".
As an example, below is a table listing some of the various 5-limit families down the left column, with subsequent columns representing equal-temperaments in order of cardinality and showing the ET when it is a member of the family in that row.
3,5-monzo comma name family name EDOs: [ 4, -1 > syntonic-comma meantone .. .. 12 .. .. .. .. .. .. 19 .. .. .. .. 24 .. 26 .. .. .. .. 31 .. .. .. .. 36 .. .. .. .. .. .. 43 .. .. .. .. .. .. 50 .. .. .. .. 55 [12, 0 > Pythagorean-comma aristoxenean .. .. 12 .. .. .. .. .. .. .. .. .. .. .. 24 .. .. .. .. .. .. .. .. .. .. .. 36 .. .. .. .. .. .. .. .. .. .. .. 48 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 72 [ 7, 6 > ampersand-comma miracle 10 11 .. .. .. .. .. .. .. .. 20 21 .. .. .. .. .. .. .. .. .. 31 .. .. .. .. .. .. .. .. .. 41 .. .. .. .. .. .. .. .. .. 51 52 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 72 [ 8, 1 > skhisma schismic .. .. 12 .. .. .. .. 17 .. .. .. .. .. .. 24 .. .. .. .. 29 .. .. .. .. .. .. 36 .. .. .. .. 41 .. .. .. .. .. .. .. .. .. .. .. 53 .. .. .. .. .. .. .. .. .. .. .. 65 .. .. .. .. .. .. .. . . . 118 [ 0, -3 > diesis augmented .. .. 12 .. .. 15 .. .. 18 .. .. 21 .. .. 24 .. .. 27 .. .. .. .. .. .. .. .. 36 [ 4, -4 > major diesis diminished .. .. 12 .. .. .. 16 .. .. .. .. .. .. .. 24 .. .. .. 28 .. .. .. .. .. .. .. 36 .. .. .. 40 .. .. .. .. .. .. .. .. .. .. .. 52 .. .. .. .. .. .. .. .. .. .. .. 64 [-4, -2 > diaschisma diaschismic 10 .. 12 .. 14 .. .. .. .. .. 20 .. 22 .. 24 .. .. .. .. .. .. .. .. .. 34 .. 36 .. .. .. .. .. .. .. .. .. 46 .. .. .. .. .. .. .. 54 [-1, 5 > magic-comma .. .. .. .. .. .. 16 .. .. 19 .. .. 22 .. .. 25 .. .. .. .. .. .. .. .. .. 35 .. .. .. .. .. 41 [-5, 6 > kleisma kleismic .. .. .. .. .. 15 .. .. .. 19 .. .. .. 23 .. .. .. .. .. .. 30 .. .. .. 34 .. .. .. 38 .. .. .. .. .. .. .. .. .. .. .. .. .. .. 53 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 72
Below is a table made by Dave Keenan of 5-limit family names current in July 2004:
For instance, the comma sequence for the 11-limit version of meantone called huygens is [81/80, 126/125, 99/98] whereas the sequence for meanpop is [81/80, 126/125, 385/384]. These are therefore sisters, with their mother standard septimal meantone, with sequence [81/80, 126/125] and grandmother 5-limit meantone with sequence [81/80]. The temperament with sequence [81/80, 525/512] is Aunt Flattone, and the one with sequence [81/80, 2401/2400] is the scapegrace Uncle Squares, the illegitimate son of 5-limit Grandma Meantone. Uncle Mothra, with sequence [81/80, 1029/1024] is another illegitimate son of Grandma Meantone, and his legitimate daughter, Cousin Cynder, is therefore a relation also, of course, with sequence [81/80, 1029/1024, 99/98].
Gene's examples of the relationships in the meantone family are shown below on a family-tree, where the top generation is 5-limit and each succeeding generation is the next higher prime-limit:
(grandmother)
5-limit meantone
[81/80]
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(aunt) (mother) (illegitimate uncle) (illegitimate uncle)
Flattone standard septimal meantone mothra squares
[81/80, 525/512] [81/80, 126/125] [81/80, 1029/1024] [81/80, 2401/2400]
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(daughter) (daughter) (female cousin)
meantone-huygens meantone-meanpop cynder
[81/80, 126/125, 99/98] [81/80, 126/125, 385/384] [81/80, 1029/1024, 99/98]
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