Major sixth: Difference between revisions

major sixth
Inverse	minor third
Name
Other names	septimal major sixth, supermajor sixth, major hexachord, greater hexachord, hexachordon maius
Abbreviation	M6
Size
Semitones	9
Interval class	3
Just interval	5:3, 12:7 (septimal), 27:16
Cents
12-Tone equal temperament	900
Just intonation	884, 933, 906

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In music from Western culture, a sixth is a musical interval encompassing six note letter names or staff positions (see Interval number for more details), and the major sixth is one of two commonly occurring sixths. It is qualified as major because it is the larger of the two. The major sixth spans nine semitones. Its smaller counterpart, the minor sixth, spans eight semitones. For example, the interval from C up to the nearest A is a major sixth. It is a sixth because it encompasses six note letter names (C, D, E, F, G, A) and six staff positions. It is a major sixth, not a minor sixth, because the note A lies nine semitones above C. Diminished and augmented sixths (such as C♯ to A♭ and C to A♯) span the same number of note letter names and staff positions, but consist of a different number of semitones (seven and ten, respectively).

The intervals from the tonic (keynote) in an upward direction to the second, to the third, to the sixth, and to the seventh scale degrees (of a major scale are called major.^[2]

A commonly cited example of a melody featuring the major sixth as its opening is "My Bonnie Lies Over the Ocean".^[3]

The major sixth is one of the consonances of common practice music, along with the unison, octave, perfect fifth, major and minor thirds, minor sixth, and (sometimes) the perfect fourth. In the common practice period, sixths were considered interesting and dynamic consonances along with their inverses the thirds. In medieval times theorists always described them as Pythagorean major sixths of 27/16 and therefore considered them dissonances unusable in a stable final sonority. How major sixths actually were sung in the Middle Ages is unknown. In just intonation, the (5/3) major sixth is classed as a consonance of the 5-limit.

A major sixth is also used in transposing music to E-flat instruments, like the alto clarinet, alto saxophone, E-flat tuba, trumpet, natural horn, and alto horn when in E-flat, as a written C sounds like E-flat on those instruments.

Assuming close-position voicings for the following examples, the major sixth occurs in a first inversion minor triad, a second inversion major triad, and either inversion of a diminished triad. It also occurs in the second and third inversions of a dominant seventh chord.

The septimal major sixth (12/7) is approximated in 53 tone equal temperament by an interval of 41 steps or 928 cents.

Major sixth (equal temperament)

The file plays middle C, followed by A (a tone 900 cents sharper than C), followed by both tones together.

Problems playing this file? See media help.

Frequency proportions

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 (play^ⓘ), approximately 884 cents.

In 12-tone equal temperament, a major sixth is equal to nine semitones, exactly 900 cents, 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,^[4] 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,^[5]^{[failed verification]} 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). 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.^[4] 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 = 32/19 = 902.49 cents.

References

^ Jan Haluska, The Mathematical Theory of Tone Systems (New York: Marcel Dekker; London: Momenta; Bratislava: Ister Science, 2004), p.xxiii. ISBN 978-0-8247-4714-5. Septimal major sixth.
^ Bruce Benward and Marilyn Nadine Saker, Music: In Theory and Practice, Vol. I, seventh edition (^{[full citation needed]} 2003): p. 52. ISBN 978-0-07-294262-0.
^ Blake Neely, Piano For Dummies, second edition (Hoboken, NJ: Wiley Publishers, 2009), p. 201. ISBN 978-0-470-49644-2.
^ ^a ^b 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.
^ Oscar Paul, 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.

@@ Line 7: / Line 7: @@
        semitones = 9 |
        interval_class = 3 |
+       just_interval = 5:3, 12:7 (septimal), 27:16<ref>Jan Haluska, ''The Mathematical Theory of Tone Systems'' (New York: Marcel Dekker; London: Momenta; Bratislava: Ister Science, 2004), p.xxiii. {{ISBN|978-0-8247-4714-5}}. Septimal major sixth.</ref> |
-       just_interval = 5:3 |
        cents_equal_temperament = 900|
-       cents_24T_equal_temperament = 900|
+       cents_24T_equal_temperament = |
-       cents_just_intonation = 884
+       cents_just_intonation = 884, 933, 906
 }}
 [[Image:Major sixth on C.png|thumb|right|Major sixth {{audio|Major sixth on C.mid|Play}}]]
-[[Image:Pythagorean major sixth on C.png|thumb|right|Pythagorean major sixth {{audio|Pythagorean major sixth on C.mid|Play}}, 3 Pythagorean perfect fifths on C.]]
+[[Image:Pythagorean major sixth on C.png|thumb|right|Pythagorean major sixth {{audio|Pythagorean major sixth on C.mid|Play}}, 3 Pythagorean perfect fifths on C]]
-In music from [[Western culture]], a '''sixth''' is a [[interval (music)|musical interval]] encompassing six note letter names or [[staff position]]s (see [[Interval (music)#Number|Interval number]] for more details), and the '''major sixth''' is one of two commonly occurring sixths. It is qualified as ''major'' because it is the larger of the two. The major sixth spans nine [[semitones]]. Its smaller counterpart, the [[minor sixth]], spans eight semitones. For example, the interval from C up to the nearest A is a major sixth. It is a sixth because it encompasses six note letter names (C, D, E, F, G, A) and six staff positions. It is a major sixth, not a minor sixth, because the note A lies nine semitones above C. [[Diminished sixth|Diminished]] and [[augmented sixth]]s (such as C{{Music|sharp}} to A{{Music|flat}} and C to A{{Music|sharp}}) span the same number of note letter names and staff positions, but consist of a different number of semitones (seven and ten).
+In music from [[Western culture]], a '''sixth''' is a [[interval (music)|musical interval]] encompassing six note letter names or [[staff position]]s (see [[Interval (music)#Number|Interval number]] for more details), and the '''major sixth''' is one of two commonly occurring sixths. It is qualified as ''major'' because it is the larger of the two. The major sixth spans nine [[semitones]]. Its smaller counterpart, the [[minor sixth]], spans eight semitones. For example, the interval from C up to the nearest A is a major sixth. It is a sixth because it encompasses six note letter names (C, D, E, F, G, A) and six staff positions. It is a major sixth, not a minor sixth, because the note A lies nine semitones above C. [[Diminished sixth|Diminished]] and [[augmented sixth]]s (such as C{{music|sharp}} to A{{music|flat}} and C to A{{music|sharp}}) span the same number of note letter names and staff positions, but consist of a different number of semitones (seven and ten, respectively).
+{{quote|The intervals from the tonic (keynote) in an upward direction to the second, to the third, to the sixth, and to the seventh scale degrees (of a major scale are called major.<ref>Bruce Benward and Marilyn Nadine Saker, ''Music: In Theory and Practice, Vol. I'', seventh edition ({{Full citation needed|date=June 2020|reason=location and publisher needed}} 2003): p. 52. {{ISBN|978-0-07-294262-0}}.</ref>}}
-A commonly cited example of a melody featuring the major sixth as its opening is "[[My Bonnie Lies Over the Ocean]]".<ref name="Neely">Neely, Blake (2009). ''Piano For Dummies'', p.201. ISBN 0-470-49644-4.</ref>
+A commonly cited example of a melody featuring the major sixth as its opening is "[[My Bonnie Lies Over the Ocean]]".<ref name="Neely">Blake Neely, ''Piano For Dummies'', second edition (Hoboken, NJ: Wiley Publishers, 2009), p. 201. {{ISBN|978-0-470-49644-2}}.</ref>
-The major sixth is one of the consonances of [[common practice]] music, along with the [[unison]], [[octave]], [[perfect fifth]], major and minor thirds, [[minor sixth]] and (sometimes) the [[perfect fourth]]. In the common practice period, sixths were considered interesting and dynamic consonances along with their inverses the thirds, but in [[medieval music|medieval times]] they were considered dissonances unusable in a stable final sonority; however in that period{{Vague|date=January 2013}}<!--"Medieval times" could cover everything from the 6th to the 15th centuries; surely Pythagorean tuning cannot have been used exclusively throughout such a span of time.--> they were tuned to the [[Pythagorean tuning|Pythagorean]] major sixth of 27/16. In [[just intonation]], the (5/3) major sixth is classed as a consonance of the [[limit (music)|5-limit]].
+The major sixth is one of the consonances of [[common practice]] music, along with the [[unison]], [[octave]], [[perfect fifth]], major and minor thirds, [[minor sixth]], and (sometimes) the [[perfect fourth]]. In the common practice period, sixths were considered interesting and dynamic consonances along with their [[Inversion (interval)|inverses]] the thirds. In [[Medieval music|medieval times]] theorists always described them as [[Pythagorean tuning|Pythagorean]] major sixths of 27/16 and therefore considered them dissonances unusable in a stable final sonority. How major sixths actually were sung in the Middle Ages is unknown. In [[just intonation]], the (5/3) major sixth is classed as a consonance of the [[limit (music)|5-limit]].
-A major sixth is also used in transposing music to [[List of E-flat instruments|E-flat]] instruments, like the [[alto clarinet]], [[alto saxophone]], E-flat [[tuba]], trumpet, [[natural horn]], and [[alto horn]] when in E-flat as a written C sounds like E-flat on those instruments.
+A major sixth is also used in transposing music to [[List of E-flat instruments|E-flat]] instruments, like the [[alto clarinet]], [[alto saxophone]], E-flat [[tuba]], trumpet, [[natural horn]], and [[alto horn]] when in E-flat, as a written C sounds like E-flat on those instruments.
-Assuming close-position [[Voicing_(music)|voicings]] for the following examples, the major sixth occurs in a first inversion minor [[Triad_(music)|triad]], a second inversion major triad, and either inversion of a diminished triad. It also occurs in the second and third inversions of a dominant seventh chord.
+Assuming close-position [[Voicing (music)|voicings]] for the following examples, the major sixth occurs in a first inversion minor [[Triad (music)|triad]], a second inversion major triad, and either inversion of a diminished triad. It also occurs in the second and third inversions of a dominant seventh chord.
 The [[7-limit|septimal]] major sixth (12/7) is approximated in [[53 tone equal temperament]] by an interval of 41 steps or 928 [[Cent (music)|cents]].
-{{Listen|filename=Sixth_ET.ogg|title=Major sixth (equal temperament)|description=The file plays [[middle C]], followed by A (a tone 900 cents sharper than C), followed by both tones together.}}
+{{listen|filename=Sixth_ET.ogg|title=Major sixth (equal temperament)|description=The file plays [[middle C]], followed by A (a tone 900 cents sharper than C), followed by both tones together.}}
 == 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.
+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 [[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'''<ref>[[John Fonville]]. "Ben Johnston's Extended Just Intonation- A Guide for Interpreters", p.124, ''Perspectives of New Music'', Vol. 29, No. 2 (Summer, 1991), pp. 106-137.</ref> with a ratio of 27:16, approximately 906 cents,<ref name="Helmholtz"/> constructed from three just perfect fifths (C-A = C-G-D-A = 702+702+702-1200=906) or by playing 27th harmonic and the next lowest octave of the fundamental frequency (16) together. The 27:16 Pythagorean major sixth arises in the C Pythagorean [[major scale]] between F and D,<ref>Paul, Oscar (1885). ''[https://books.google.com/books?id=4WEJAQAAMAAJ&dq=musical+interval+%22pythagorean+major+third%22&source=gbs_navlinks_s A manual of harmony for use in music-schools and seminaries and for self-instruction]'', p.165. Theodore Baker, trans. G. Schirmer.</ref> as well as between C and A, G and E, and D and B.{{audio|Pythagorean major sixth in scale.mid|Play}}
+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}}
-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.
-Another major sixth is the 12:7 '''septimal major sixth''' or '''[[supermajor sixth]]''' of approximately 933 cents.<ref name="Helmholtz">Hermann L. F Von Helmholtz (2007). ''On the Sensations of Tone'', p.456. ISBN 978-1-60206-639-7.</ref>
+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.
 ==See also==
-* [[musical tuning]]
+* [[Musical tuning]]
-* [[list of meantone intervals]]
+* [[List of meantone intervals]]
-* [[sixth chord]]
+* [[Sixth chord]]
-==Sources==
+==References==
 {{reflist}}
 ==Further reading==
-*Duckworth, William (1996). [untitled chapter]{{Verify source|date=November 2013}}<!--OCLC does not indicate any of the constituent articles this far into the collection were written by Duckworth, and they all seem to have titles.--> In ''Sound and Light: La Monte Young, Marian Zazeela'', edited by William Duckworth and Richard Fleming, p.&nbsp;167. Bucknell Review 40, no. 1. Lewisburg [Pa.]: Bucknell University Press; London and Cranbury, NJ: Associated University Presses. ISBN 9780838753460. Paperback reprint 2006, ISBN 0-8387-5738-3. [septimal]{{Clarify|date=November 2013}}<!--What the heck is this here for?-->
+*Duckworth, William (1996). [untitled chapter]{{Verify source|date=November 2013}}<!--OCLC does not indicate any of the constituent articles this far into the collection were written by Duckworth, and they all seem to have titles.--> In ''Sound and Light: La Monte Young, Marian Zazeela'', edited by William Duckworth and Richard Fleming, p.&nbsp;167. Bucknell Review 40, no. 1. Lewisburg [Pa.]: Bucknell University Press; Cranbury, NJ / London: Associated University Presses. {{ISBN|9780838753460}}. Paperback reprint 2006, {{ISBN|0-8387-5738-3}}. [septimal]{{Clarify|date=November 2013}}<!--What the heck is this here for?-->
 {{Intervals}}