View source for Harmonic seventh

{{Infobox interval
| main_interval_name = harmonic seventh
| inverse = [[Septimal major second]]
| complement = [[complement (music)]]
| other_names = septimal minor seventh, subminor seventh, acute diminished just seventh, quarter comma augmented sixth
| abbreviation = m 7, [[harmonic|{{sup|H }}]]7, {{sub|min }}7, [[wikt:acute|{{sup|acc}}]][[diminished 7th|{{sub|dim }}7]], [[augmented sixth|{{sup|Aug }}]]6
| semitones = ~9.7
| interval_class = ~2.3
| just_interval = 7:4<ref>
{{cite book
 |last = Haluska |first = Jan
 |year=2003
 |section = Harmonic seventh
 |title = The Mathematical Theory of Tone Systems
 |page = {{mvar|xxiii}}
 |publisher = CRC Press
 |isbn = 0-8247-4714-3
}}
</ref>
| cents_equal_temperament =
| cents_24T_equal_temperament =
| cents_just_intonation = 968.826
}}
The '''harmonic seventh interval''', also known as the '''septimal minor seventh''',<ref>
{{cite web
 |last = Gann |first = Kyle |author-link = Kyle Gann
 |year = 1998
 |title = Anatomy of an octave
 |series = Just Intonation Explained
 |website = kylegann.com
 |url=http://www.kylegann.com/Octave.html
}}
</ref><ref>
{{cite book
 |last=Partch |first=Harry  |author-link=Harry Partch
 |year=1979
 |title=Genesis of a Music
 |title-link=Genesis of a Music
 |page=68
 |isbn=0-306-80106-X
}}
</ref>
or '''subminor seventh''',<ref>
{{cite book
 |last1=von Helmholtz |first1=H.L.F. |author1-link=Hermann von Helmholtz
 |last2=Ellis         |first2=A.J.   |author2-link=Alexander John Ellis
 |others=[[Alexander John Ellis|Ellis, A.J.]] translator of English ed., editor, and author of an extensive appendix
 |year=2007
 |title=On the Sensations of Tone  |edition=reprint  |lang=en-US
 |title-link=Sensations of Tone
 |page=456
 |publisher=Cosimo |isbn=978-1-60206-639-7
}}
</ref><ref>
{{cite journal
 |last=Ellis |first=A.J. |author-link=Alexander John Ellis
 |year=1880
 |title=Notes of observations on musical beats
 |journal=[[Proceedings of the Royal Society of London]]
 |volume=30  |issue=200–205  |pages=520–533
 |doi=10.1098/rspl.1879.0155 |doi-access=
}}
</ref><ref>
{{cite journal
 |last=Ellis  |first=A.J.  |author-link=Alexander John Ellis
 |year=1877
 |title=On the measurement and settlement of musical pitch
 |journal=[[Journal of the Society of Arts]]
 |volume=25  |issue=1279  |pages=664–687
 |jstor=41335396
}}
</ref>
is one with an exact 7:4&nbsp;ratio<ref>
{{cite book
 |first1=Andrew |last1=Horner
 |first2=Lydia  |last2=Ayres
 |year=2002
 |title=Cooking with Csound: Woodwind and brass recipes
 |page=131
 |publisher=A-R Editions
 |isbn=0-89579-507-8
}}
</ref>
(about 969&nbsp;[[cent (music)|cents]]).<ref>
{{cite book
 |last=Bosanquet |first=R.H.M.   |author-link=Robert Holford Macdowall Bosanquet
 |year=1876
 |title=An Elementary Treatise on Musical Intervals and Temperament
 |pages=41–42
 |publisher=Diapason Press
 |place=Houten, NL
 |isbn=90-70907-12-7
}}
</ref>
This is about 32 cents narrower, with a more stable and consonant sound, than a [[minor seventh]] in [[equal temperament]], and is up to 49 cents narrower than and is, "particularly sweet",<ref>
{{cite book
 |last=Brabner |first=John H.F.
 |year=1884
 |title=The National Encyclopædia
 |volume=13 |page=135
 |place=London, UK
 |url=https://books.google.com/books?id=cy2P8q3RRI0C&pg=PA134
 |via=Google books
}}
</ref>
"sweeter in quality" than an "ordinary"<ref>
{{cite conference
 |first = Eustace J. |last = Breakspeare
 |year = 1886–1887
 |title = On certain novel aspects of harmony
 |page = 119
 |book-title = [[Proceedings of the Musical Association]]
 |section = 13th&nbsp;session, pp.&nbsp;113–131
 |publisher = [[Royal Musical Association]] / [[Oxford University Press]]
}}
</ref>
[[minor seventh|just minor seventh]], which has an intonation ratio of 9:5<ref>
{{cite conference
 |first = Wilfrid |last = Perrett
 |year = 1931–1932
 |title = The heritage of Greece in music
 |page = 89 
 |book-title = [[Proceedings of the Musical Association]]
 |section = 58th&nbsp;session pp.&nbsp;85–103
 |publisher = [[Royal Musical Association]] / [[Oxford University Press]]
}}
</ref>
(about 1018&nbsp;cents).

[[File:Harmonic seventh on C.png|thumb|left|Harmonic seventh, [[7-limit|septimal]] seventh[[File:Harmonic seventh on C.mid]]]]
The harmonic seventh arises from the [[Harmonic series (music)|harmonic series]] as the interval between the fourth harmonic (second octave of the fundamental) and the '''seventh harmonic'''; in that octave, harmonics 4, 5, 6, and 7 constitute the four notes (in order) of a purely consonant major chord (root position) with an added minor seventh (or [[augmented sixth]], depending on the tuning system used).

{{clear left}}
==Fixed pitch: Not a scale note==
Although the word "seventh" in the name suggests the seventh note in a scale, and although the seventh pitch up from the tonic is indeed used to form a harmonic seventh in a few tuning systems, the '''harmonic seventh''' is a pitch relation to the tonic, not an ordinal note position in a [[musical scale|scale]]. As a pitch relation (968.826&nbsp;[[musical cents|cents]] up from the reference or tonic note) rather than a scale-position note, a harmonic seventh is produced by different notes in different tuning systems:
* In [[equal temperament]], the harmonic seventh is about 32 cents smaller than the equal tempered [[minor seventh]].
* In 5-limit [[just intonation]] the harmonic&nbsp;7th is very near ''precisely'' an acute diminished seventh: {{nobr| 7{{sup|{{music|bb}}{{math|￪}}}} .}}{{efn|
A ''just acute diminished seventh'' is a [[just intonation|just]] seventh {{big|(}}{{small|{{math|{{sfrac| 15 | 8 }}}}}}{{big|)}} flattened twice (first flat is [[minor seventh|min 7]], second flat is [[diminished seventh|dim 7]], each just flat lowers the pitch by {{math|70.672 [[musical cents|¢]]}}) sharpened by a [[syntonic comma]] ("acute") (raises pitch by about {{math|21.506 ¢}}), hence:
: {{nobr| 7{{sup|{{music|bb}}{{math|￪}}}} }}<math>= \tfrac{ 15 }{\ 8\ } \times \left( \tfrac{\ 24\ }{ 25 }\right)^2 \times \tfrac{\ 81\ }{ 80 } = \tfrac{ 3^7 }{\ 2\times5^4\ } = \frac{ 2187 }{\ 1250\ } ~.</math>
: <math> \operatorname{cents}\!\left( \tfrac{ 7 }{\ 4\ } \right) \quad = 968.826</math>{{math| ¢}}; compare this to
: <math> \operatorname{cents}\!\left( \tfrac{ 2187 }{\ 1250\ } \right) = 968.430</math>{{math| ¢}}, only {{math|0.396 ¢}} flat.
Regardless of how accurately it reproduces the interval of a seventh harmonic, a 5-limit justly intoned ''acute diminished seventh'' is only a theoretical pitch:
The pitch's position in the just [[tonnetz|tone net]] is too far separated from its tonic for both to be played together in the same chord without many more notes in the tone network. It is a correctly specified note that does exist among the extended network of [[just intonation]] pitches, but the theoretical note cannot be put to practical use: An acute diminished seventh cannot be reached from its tonic in any feasible justly intoned octave made up of only 12&nbsp;notes.
}}
* In multiple slight variations of [[quarter comma meantone]], the harmonic seventh is accurately rendered by the [[augmented sixth]] interval (rather than a seventh).{{efn|
A small modification of meantone – the fifth about one seventh of a comma flat, slightly sharper than exactly one quarter of a comma flat – adjusts the tuning to exactly reproduce the seventh harmonic as an augmented sixth: The adjusted quarter comma uses a fifth that is <math>\ \left( 56 \right)^{1/10}\ ,</math> about  696.<u>883</u>[[musical cents|{{math| ¢}}]] instead of <math>\ \left( 5 \right)^{1/4}\ ,</math> or about 696.<u>578</u>&nbsp;{{math| ¢}}, used for conventional [[quarter comma meantone]] (which produces pure major thirds by letting fifths fall a quarter-comma flat).
}}
* In [[31 tone equal temperament]], the harmonic seventh is quite accurately rendered as 25 steps out of 31 that make up the [[octave]],{{efn|<math>\ 2^{25/31} \approx 1.749 \approx \tfrac{ 7 }{\ 4\ }  </math> which is 967.742&nbsp;cents so only {{math|1.084 ¢}} flat}} while several other [[just intonation|just intervals]] are as relatively well approximated as they are in [[quarter comma meantone]].

==Actual use in musical practice==
[[File:Britten - Serenade prologue.png|center|thumb|upright=2.7|Use of the seventh harmonic in the prologue to Britten's ''[[Serenade for Tenor, Horn and Strings]]''[[File:Britten - Serenade prologue.mid]]]]
When played on the [[natural horn]], the note is often adjusted to 16:9 of the root as a compromise (for C&nbsp;maj<sup>7{{music|b}}</sup>, the substituted note is B{{music|b}}{{music|minus}}, 996.09&nbsp;cents), but some pieces call for the pure harmonic seventh, including [[Benjamin Britten|Britten]]'s ''[[Serenade for Tenor, Horn and Strings]]''.<ref>
{{cite book
 |last1=Fauvel  |first1=J.   |author1-link=John Fauvel
 |last2=Flood   |first2=R.   |author2-link=Raymond Flood (mathematician)
 |last3=Wilson  |first3=R.J. |author3-link=Robin Wilson (mathematician)
 |year=2006
 |title=Music and Mathematics
 |pages=21–22
 |publisher=Oxford University Press |isbn=9780199298938
}}
</ref> Composer [[Ben Johnston (composer)|Ben Johnston]] uses a small&nbsp;"7" as an accidental to indicate a note is lowered [[septimal quarter tone|49&nbsp;cents]] (1018&nbsp;&minus;&nbsp;969&nbsp;=&nbsp;49) (or 32&nbsp;cents lowered compared to [[equal temperament]], or an upside-down&nbsp;"7" to indicate a note is raised 49&nbsp;cents (or raised 32&nbsp;cents compared to [[equal temperament]]. Thus, in C&nbsp;major, "the seventh partial", or harmonic seventh, is notated as {{music|flat}} note with "7" written above the flat.<ref>
{{cite journal
 |first1=Douglas |last1=Keislar
 |first2=Easley  |last2=Blackwood    |author2-link=Easley Blackwood Jr.
 |first3=John    |last3=Eaton        |author3-link=John Eaton (composer) 
 |first4=Lou     |last4=Harrison     |author4-link=Lou Harrison 
 |first5=Ben     |last5=Johnston     |author5-link=Ben Johnston (composer) 
 |first6=Joel    |last6=Mandelbaum   |author6-link=Joel Mandelbaum
 |first7=William |last7=Schottstaedt
 |date=Winter 1991
 |title=Six American composers on nonstandard tunings
 |journal=[[Perspectives of New Music]]
 |series=1  |volume=29  |issue=1  |pages=176–211 (esp. 193)
 |doi=10.2307/833076 |jstor=833076
}}
</ref><ref>
{{cite journal
 |last=Fonville |first=J. |author-link=John Fonville
 |date=Summer 1991
 |title=Ben Johnston's extended Just Intonation: A guide for interpreters
 |journal=[[Perspectives of New Music]]
 |volume=29  |issue=2  |pages=106–137
 |doi=10.2307/833435 |jstor=833435
}}
</ref>

[[File:Septimal major second on B7b.png|thumb|left|Inverse, septimal major second on B{{sup|7}}{{music|b}}[[File:Septimal major second on B7b.mid]]]]
The harmonic seventh is also expected from [[Barbershop music|barbershop quartet]] singers, when they tune [[dominant seventh chord]]s ([[harmonic seventh chord]]), and is considered an essential aspect of the barbershop style.<ref>
{{cite web
 |title = Definition of barbershop harmony
 |department = About Us
 |website=barbershop.org
 |url=http://www.barbershop.org/about-us/definition-of-barbershop-harmony/
}}
</ref><ref>
{{cite web
 |first = Jim, Dr. |last = Richards
 |title = The physics of barbershop sound
 |website = shop.barbershop.org
 |url = http://shop.barbershop.org/books/manuals/physics-of-barbershop-sound/
}}
</ref><ref name=Hagerman-Sundberg-1980>
{{cite journal
 |last1 = Hagerman |first1 = B.
 |last2 = Sundberg |first2 = J.
 |year = 1980
 |title = Fundamental frequency adjustment in barbershop singing
 |journal = STL-QPSR (Speech Transmission Laboratory. Quarterly Progress and Status Reports)
 |volume = 21  |issue = 1  |pages = 28–42
 |url = https://www.speech.kth.se/prod/publications/files/qpsr/1980/1980_21_1_028-042.pdf
 |access-date = 13 August 2021
}}
</ref>{{efn|
Hagerman & Sundberg (1980)<ref name=Hagerman-Sundberg-1980/> present empirical data that challenges the accuracy of the claim.
}}

[[File:Origin of seconds and thirds in harmonic series.png|thumb|upright=1.4|Origin of large and small seconds and thirds in harmonic series.<ref>
{{cite book
 |first=Lou |last=Harrison |author-link=Lou Harrison
 |editor-last=Miller |editor-first=Leta E.
 |year=1988
 |title=Lou Harrison: Selected keyboard and chamber music, 1937–1994
 |page={{mvar|xliii}}
 |isbn=978-0-89579-414-7
}}
</ref>]]
In [[quarter-comma meantone]] tuning, standard in the [[Baroque]] and earlier, the augmented sixth is 965.78&nbsp;cents – only 3&nbsp;cents below 7:4, well within normal tuning error and [[vibrato]].
[[Organ (music)|Pipe organs]] were the last fixed-tuning instrument to adopt [[equal temperament]]. With the transition of organ tuning from meantone to equal-temperament in the late 19th and early 20th&nbsp;centuries the formerly harmonic G<sub>maj</sub><sup>7{{music|b}}</sup> and B<sup>{{music|b}}</sup><sub>maj</sub><sup>7{{music|b}}</sup> became "[[The Lost Chord (disambiguation)|lost chord]]s" (among other chords).

The harmonic seventh differs from the just 5-limit [[augmented sixth]] of {{small|{{sfrac| 225 | 128 }}}} by a [[septimal kleisma]] ({{small|{{sfrac| 225 | 224 }}}}, 7.71&nbsp;cents), or about {{nobr|{{small|{{sfrac| 1 | 3 }}}} [[Pythagorean comma]]}}.<ref>
{{cite conference
 |first = R.H.M.  |last = Bosanquet  |author-link=Robert Holford Macdowall Bosanquet
 |year=1876–1877
 |title = On some points in the harmony of perfect consonances
 |page = 153
 |book-title=[[Proceedings of the Musical Association]]
 |section = 3rd&nbsp;Session, pp.&nbsp;145–153
 |publisher = [[Royal Musical Association]] / [[Oxford University Press]] 
}}
</ref>
The harmonic seventh note is about {{nobr|{{small|{{sfrac| 1 | 3 }}}} semitone}} {{nobr|( ≈ 31 cents )}} flatter than an equal-tempered minor seventh. When this flatter seventh is used, the dominant seventh chord's "need to resolve" down a fifth is weak or non-existent. This chord is often used on the tonic (written as {{font|'''I<sup>7</sup>'''|font=Consolas}}) and functions as a "fully resolved" final chord.<ref>
{{cite book
 |last = Mathieu |first = W.A.  |author-link = W. A. Mathieu
 |year = 1997
 |title = Harmonic Experience
 |pages = 318–319
 |place = Rochester, VT
 |publisher = Inner Traditions International
 |isbn = 0-89281-560-4
}}
</ref>

The [[twenty-first harmonic]] (470.78&nbsp;cents) is the harmonic seventh of the dominant, and would then arise in chains of [[secondary dominant]]s (known as the [[Ragtime progression]]) in styles using harmonic sevenths, such as barbershop music.

==See also==
* [[augmented sixth]]
* [[quarter comma meantone]]
* [[31 tone equal temperament]]

==Notes==
{{notelist}}

==Citations==
{{reflist|25em}}

==Further reading==
* {{cite book
 |last = Hewitt  |first = Michael
 |year = 2000
 |title = The Tonal Phoenix: A study of tonal progression through the prime numbers three, five, and seven
 |publisher = Orpheus-Verlag
 |isbn = 978-3922626961
}}

{{Intervals}}

[[Category:7-limit tuning and intervals]]
[[Category:Harmonic series (music)]]
[[Category:Sevenths (music)]]