Harmonic seventh

harmonic seventh

Inverse	Septimal major second
Name
Other names	septimal minor seventh, subminor seventh, acute diminished just seventh, quarter comma augmented sixth
Abbreviation	m 7, ^H7, min 7, ^accdim 7, ^Aug6
Size
Semitones	~9.7
Interval class	~2.3
Just interval	7:4[1]
Cents
Just intonation	968.826

The harmonic seventh interval, also known as the septimal minor seventh,[2][3] or subminor seventh,[4][5][6] is one with an exact 7:4 ratio[7] (about 969 cents).[8] This is somewhat narrower than and is, "particularly sweet",[9] "sweeter in quality" than an "ordinary"[10] just minor seventh, which has an intonation ratio of 9:5[11] (about 1018 cents).

Harmonic seventh, septimal seventh

The harmonic seventh arises from the 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).

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 scale. As a pitch relation (968.826 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 5-limit just intonation the harmonic 7th is very near precisely an accute diminished seventh: 7^￪ .[a]
• In multiple slight variations of quarter comma meantone, the harmonic seventh is accurately rendered by the augmented sixth interval (rather than a seventh).[b]

Actual use in musical practice

Use of the seventh harmonic in the prologue to Britten's Serenade for Tenor, Horn and Strings

When played on the natural horn, the note is often adjusted to 16:9 of the root as a compromise (for C maj^7♭, the substituted note is B♭-, 996.09 cents), but some pieces call for the pure harmonic seventh, including Britten's Serenade for Tenor, Horn and Strings.[12] Composer Ben Johnston uses a small "7" as an accidental to indicate a note is lowered 49 cents (1018 − 969 = 49), or an upside-down "7" to indicate a note is raised 49 cents. Thus, in C major, "the seventh partial", or harmonic seventh, is notated as ♭ note with "7" written above the flat.[13][14]

Inverse, septimal major second on B7♭

The harmonic seventh is also expected from barbershop quartet singers, when they tune dominant seventh chords (harmonic seventh chord), and is considered an essential aspect of the barbershop style.[15][16][c][17]

Origin of large and small seconds and thirds in harmonic series.[18]

In quarter-comma meantone tuning, standard in the Baroque and earlier, the augmented sixth is 965.78 cents – only 3 cents below 7:4, well within normal tuning error and vibrato. 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 centuries the formerly harmonic G_maj^7♭ and B^♭_maj^7♭ became "lost chords" (among other chords).

The harmonic seventh differs from the just 5-limit augmented sixth of ⁠ 225 / 128 ⁠ by a septimal kleisma (⁠ 225 / 224 ⁠, 7.71 cents), or about ⁠ 1 / 3 ⁠ Pythagorean comma.[19] The harmonic seventh note is about ⁠ 1 / 3 ⁠ semitone ( ≈ 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 I^7) and functions as a "fully resolved" final chord.[20]

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

Notes

1. A just accute diminished seventh is a just seventh (⁠ 15 / 8 ⁠) flattened twice (first flat is min 7, second flat is dim 7, each just flat lowers the pitch by 70.672 ¢) sharpened by a syntonic comma ("accute") (raises pitch by about 21.506 ¢), hence:

   7^￪ ${\displaystyle ={\tfrac {15}{\ 8\ }}\times \left({\tfrac {\ 24\ }{25}}\right)^{2}\times {\tfrac {\ 81\ }{80}}={\tfrac {3^{7}}{\ 2\times 5^{4}\ }}={\frac {2187}{\ 1250\ }}~.}$

   ${\displaystyle \operatorname {cents} \!\left({\tfrac {7}{\ 4\ }}\right)\quad =968.826}$ ¢; compare this to

   ${\displaystyle \operatorname {cents} \!\left({\tfrac {2187}{\ 1250\ }}\right)=968.430}$ ¢, only 0.396 ¢ flat.

   Sadly, regardless of how accurately it reproduces the interval of a seventh harmonic, a 5-limit justly intoned accute diminished seventh is only a theoretical pitch: The pitch's position in the just 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 accute diminished seventh cannot be reached from its tonic in any feasible justly intoned octave made up of only 12 notes.
2. A small modification of meantone – the fifth 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 ${\displaystyle \ \left(56\right)^{1/10}\ ,}$ about 696.883 ¢ instead of ${\displaystyle \ \left(5\right)^{1/4}\ ,}$ or about 696.578  ¢, used for conventional quarter comma meantone (which produces pure major thirds by letting fifths fall a quarter-comma flat).
3. Hagerman & Sundberg (1980)[17] present empirical data that challenges the accuracy of the claim.

See also

• augmented sixth
• quarter comma meantone
• 31 tone equal temperament

Citations

1. Haluska, Jan (2003). "Harmonic seventh". The Mathematical Theory of Tone Systems. CRC Press. p. xxiii. ISBN 0-8247-4714-3.
2. Gann, Kyle (1998). "Anatomy of an octave". kylegann.com. Just Intonation Explained.
3. Partch, Harry (1979). Genesis of a Music. p. 68. ISBN 0-306-80106-X.
4. von Helmholtz, H.L.F.; Ellis, A.J. (2007). On the Sensations of Tone. Ellis, A.J. translator of English ed., editor, and author of an extensive appendix (reprint ed.). Cosimo. p. 456. ISBN 978-1-60206-639-7.
5. Ellis, A.J. (1880). "Notes of observations on musical beats". Proceedings of the Royal Society of London. 30 (200–205): 520–533. doi:10.1098/rspl.1879.0155.
6. Ellis, A.J. (1877). "On the measurement and settlement of musical pitch". Journal of the Society of Arts. 25 (1279): 664–687. JSTOR 41335396.
7. Horner, Andrew; Ayres, Lydia (2002). Cooking with Csound: Woodwind and brass recipes. A-R Editions. p. 131. ISBN 0-89579-507-8.
8. Bosanquet, R.H.M. (1876). An Elementary Treatise on Musical Intervals and Temperament. Houten, NL: Diapason Press. pp. 41–42. ISBN 90-70907-12-7.
9. Brabner, John H.F. (1884). The National Encyclopædia. Vol. 13. London, UK. p. 135 – via Google books.
10. Breakspeare, Eustace J. (1886–1887). "On certain novel aspects of harmony". Proceedings of the Musical Association. Royal Musical Association / Oxford University Press. p. 119.
11. Perrett, Wilfrid (1931–1932). "The heritage of Greece in music". Proceedings of the Musical Association. Royal Musical Association / Oxford University Press. p. 89.
12. Fauvel, J.; Flood, R.; Wilson, R.J. (2006). Music and Mathematics. Oxford University Press. pp. 21–22. ISBN 9780199298938.
13. Keislar, Douglas; Blackwood, Easley; Eaton, John; Harrison, Lou; Johnston, Ben; Mandelbaum, Joel; Schottstaedt, William (Winter 1991). "Six American composers on nonstandard tunings". Perspectives of New Music. 1. 29 (1): 176–211 (esp. 193). doi:10.2307/833076. JSTOR 833076.
14. Fonville, J. (Summer 1991). "Ben Johnston's extended Just Intonation: A guide for interpreters". Perspectives of New Music. 29 (2): 106–137. doi:10.2307/833435. JSTOR 833435.
15. "Definition of barbershop harmony". About Us. barbershop.org.
16. Richards, Jim, Dr. "The physics of barbershop sound". shop.barbershop.org.
17. Hagerman, B.; Sundberg, J. (1980). "Fundamental frequency adjustment in barbershop singing" (PDF). STL-QPSR (Speech Transmission Laboratory. Quarterly Progress and Status Reports). 21 (1): 28–42. Retrieved 13 August 2021.
18. Harrison, Lou (1988). Miller, Leta E. (ed.). Lou Harrison: Selected keyboard and chamber music, 1937–1994. p. xliii. ISBN 978-0-89579-414-7.
19. Bosanquet, R.H.M. (1876–1877). "On some points in the harmony of perfect consonances". Proceedings of the Musical Association. Royal Musical Association / Oxford University Press. p. 153.
20. Mathieu, W.A. (1997). Harmonic Experience. Rochester, VT: Inner Traditions International. pp. 318–319. ISBN 0-89281-560-4.

Further reading

• Hewitt, Michael (2000). The Tonal Phoenix: A study of tonal progression through the prime numbers three, five, and seven. Orpheus-Verlag. ISBN 978-3922626961.