Major fourth and minor fifth

In music, major fourth and minor fifth are intervals from the quarter-tone scale, named by Ivan Wyschnegradsky to describe the tones surrounding the tritone (F♯/G♭) found in the more familiar twelve-tone scale,[1] as shown in the table below:

perfect fourth major fourth tritone minor fifth perfect fifth
in C: F ≊ Fhalf sharp F♯/G♭ ≊ Ghalf flat G
in cents: 500 550 600 650 700
Major fourth
InverseMinor fifth
Name
Other namesEleventh harmonic
AbbreviationM4
Size
Semitones~5½
Interval class~5½
Just interval11:8
Cents
24 equal temperament550
Just intonation551.32
Minor fifth
InverseMajor fourth
Name
Other namesEleventh subharmonic
Abbreviationm5
Size
Semitones~6½
Interval class~5½
Just interval16:11
Cents
24 equal temperament650
Just intonation648.68
The eleventh harmonic Playⓘ – shown using the Ben Johnston notation – can be approximated by the major fourth.
Just augmented fourth on C Playⓘ and its inverse, the just tritone on C Playⓘ

Major fourth

A major fourth (Playⓘ) is the interval that lies midway between the perfect fourth (500 cents) and the augmented fourth (600 cents) and is thus 550 cents (Fhalf sharp). It inverts to a minor fifth. Wyschnegradsky considered it a good approximation of the eleventh harmonic[1] (11:8 or 551.32 cents).[2] A narrower undecimal major fourth is found at 537 cents (the ratio 15:11). 31 equal temperament has an interval of 542 cents, which lies in between the two types of undecimal major fourth.

The term may also be applied to the "comma-deficient major fourth" (or "chromatic major fourth"[3]), which is the ratio 25:18, or 568.72 cents (F♯).[4]

Minor fifth

A minor fifth (Playⓘ) is the interval midway between the diminished fifth (600 cents) and the perfect fifth (700 cents) and thus 650 cents (Ghalf flat). It inverts to a major fourth. It approximates the eleventh subharmonic (G↓), 16:11 (648.68 cents).

The term may also be applied to the ratio 64:45 (G♭-) or 609.77 cents (Playⓘ), formed from the perfect fourth (4/3 = 498.04) and the major semitone (16/15 = 111.73),[3] which is sharp of the G♭ tritone. The "comma-redundant minor fifth" has the ratio 36:25 (G♭), or 631.28 cents, and is formed from two minor thirds.[4] The tridecimal minor fifth (13:9), or tridecimal tritone, is slightly larger at 636.6 cents.

Other

The term major fourth may also be applied to the follow, as minor fifth may be applied to their inversions (in the sense of augmented and diminished):

  • The "comma-deficient major fourth" (or "chromatic major fourth"[3]) is the ratio 25:18, or 568.72 cents (F♯).[4]
  • 45:32 (F♯+) or 590.22 cents (Playⓘ), formed from the major third (5/4 = 386.31) and the major tone (9/8 = 203.91) or two major tones (9:8) and one minor tone (10:9)[3]
  • 729:512 (F♯++) or 611.73 cents (Playⓘ), formed from the perfect fourth and the apotome.[3]

See also

References
  1. Skinner, Miles Leigh (2007). Toward a Quarter-tone Syntax: Analyses of Selected Works by Blackwood, Haba, Ives, and Wyschnegradsky, p.25. ProQuest. ISBN 9780542998478.
  2. Benson, Dave (2007-01-01). Music: A Mathematical Offering. Cambridge University Press. p. 370. ISBN 9780521853873.
  3. Richard Mackenzie Bacon (1821). "Manuscript Work of Francesco Bianchl", The Quarterly Musical Magazine and Review, Volume 3, p.56.
  4. (1832). The Edinburgh Encyclopaedia, Volume 9, p.249. Joseph Parker. [ISBN unspecified]


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