Definitions of tuning terms

© 1998 by Joseph L. Monzo

All definitions by Joe Monzo unless otherwise cited

major 3rd, major third, M3


    An interval approximating 400 cents. Major thirds include the 5/4 (386¢) of Just Intonation, the Pythagorean Ditone of 81/64 (408¢), and more complex intervals such as 19/15 (409¢) and the Skhismic Diminished Fourth, 8192/6561 (384¢) of extended Pythagorean tuning.

    [from John Chalmers, Divisions of the Tetrachord]

    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

    Successively closer small-integer approximations of the 12-EDO "major 3rd" are: 

    
 ratio    ~cents

 14/11  417.5079641
  5/4   386.3137139
 19/15  409.2443014 (good approximation of Pythagorean 3rd)
 24/19  404.4419847
 29/23  401.3028469
 34/27  399.0904069
 63/50  400.1084805 (only ~1/9 cent wider than 2(4/12))

    The intervals given in Chalmers's definition are all rational ones. Other 'major 3rds' can be defined mathematically by irrational numbers, generally in the context of some form of temperament.

    Examples of 'major 3rds' in some common meantone tunings:

    Meantone adjustment of 81/64
    [Pythagorean 'major 3rd']    		 prime-factoringapprox. cents
    1/3-comma ((3/2)4) / (22) / ((81/80)(4*1/3)) = 2(2/3)*3-(4/3)*5(4/3) = ~379
    2/7-comma ((3/2)4) / (22) / ((81/80)(4*2/7)) = 2-(10/7)*3-(4/7)*5(8/7) = ~383
    7/26-comma ((3/2)4) / (22) / ((81/80)(4*7/26)) = 2(-22/13)*3(-4/13)*5(14/13) = ~385
    1/4-comma ((3/2)4) / (22) / ((81/80)(4*1/4)) = 2-2*51 = ~386 (the 5/4 ratio)
    1/5-comma ((3/2)4) / (22) / ((81/80)(4*1/5)) = 2-(14/5)*3(4/5)*5(4/5) = ~391

    Note the frequent occurence of '4' in the numerators of the fractional powers: this is a result of the place of the Pythagorean 'major 3rd' as the fourth member in the 'cycle of 5ths', and the implied Pythagorean ratio (2-6*)34 for the fourth member in the meantone cycle.

    The golden meantone "major-3rd" has the ratio 2[ (10 - 4F) / 11 ], where F = (1 + 51/2) / 2, and is ~384.8578958 cents.


    Examples of 'major 3rds' in some common equal temperaments:

    EDO '3rd' cents
    12 24/12 400
    13 24/13 ~369
    15 25/15 400
    16 25/16 375
    17 26/17 ~424 (note that 17-EDO has nothing that resembles the JI 'major 3rd' of 5/4 ratio; also note the 'neutral 3rd' of 2(5/17) = ~353 cents)
    19 26/19 ~379
    22 27/22 ~382
    24 28/24 400 (also note the 'neutral 3rd' of 2(7/24) = 350 cents)
    31 210/31 ~387
    41 213/41 ~380 (also note the pseudo-Pythagorean '3rd' of 2(14/41) = ~410 cents)
    43 214/43 ~391
    50 216/50 384 (also note the pseudo-Pythagorean '3rd' of 2(17/50) = 408 cents)
    53 217/53 ~385 (also note the pseudo-Pythagorean '3rd' of 2(18/53) = ~407.5 cents)
    55 218/55 ~393
    72 223/72 383.&1/3

    (Note that the cents values of some ET pitches are exact.)

updated:

2003.06.09 -- reformatted table of meantone "major-3rds"
2002.09.12 -- added 53-EDO to the table of EDOs
2001.11.6
2000.6.24

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