Editing Major sixth (section)

== Frequency proportions ==
<!--[[19th subharmonic]] and [[nineteenth subharmonic]] redirect here.-->
Many intervals in a various tuning systems qualify to be called "major sixth", sometimes with additional qualifying words in the names. The following examples are sorted by increasing width.

In [[just intonation]], the most common major sixth is the pitch ratio of 5:3 ({{audio|Just major sixth on C.mid|play}}), approximately 884 cents.

In 12-tone [[equal temperament]], a major sixth is equal to nine [[semitone]]s, exactly 900 [[cent (music)|cent]]s, with a frequency ratio of the (9/12) root of 2 over 1.

Another major sixth is the '''Pythagorean major sixth''' with a ratio of 27:16, approximately 906 cents,<ref name="Helmholtz-Ellis"/> called "Pythagorean" because it can be constructed from three just perfect fifths (C-A = C-G-D-A = 702+702+702-1200=906). It is the inversion of the [[Pythagorean minor third]], and corresponds to the interval between the 27th and the 16th harmonics. The 27:16 Pythagorean major sixth arises in the C Pythagorean [[major scale]] between F and D,<ref>Oscar Paul, ''[https://books.google.com/books?id=4WEJAQAAMAAJ&q=musical+interval+%22pythagorean+major+third%22 A Manual of Harmony for Use in Music-Schools and Seminaries and for Self-Instruction]'', trans. Theodore Baker (New York: G. Schirmer, 1885), p. 165.</ref>{{Failed verification|date=June 2017|reason=The Pythagorean major 6th is mentioned on p. 164, not 165. Oscar Paul describes it as the inversion of the Pythagorean minor third D-F, which is not exactly what is claimed here.}} as well as between C and A, G and E, and D and B. In the [[5-limit]] [[justly tuned major scale]], it occurs between the 4th and 2nd degrees (in C major, between F and D).
{{audio|Pythagorean major sixth in scale.mid|Play}}

Another major sixth is the 12:7 '''septimal major sixth''' or '''[[supermajor sixth]]''', the inversion of the [[septimal minor third]], of approximately 933 cents.<ref name="Helmholtz-Ellis">Alexander J. Ellis, Additions by the translator to Hermann L. F. Von Helmholtz (2007). ''On the Sensations of Tone'', p.456. {{ISBN|978-1-60206-639-7}}.</ref> The septimal major sixth (12/7) is approximated in 53-tone equal temperament by an interval of 41 steps, giving an actual frequency ratio of the (41/53) root of 2 over 1, approximately 928 cents.

The '''nineteenth subharmonic''' is a major sixth, A{{music|U19}} = 32/19 = 902.49 cents.