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{| class="wikitable"
{| class="wikitable"
|-
|-
| | [[File:glyph_7_4.png|alt=glyph 7 4.png|124x112px|glyph 7 4.png]]
| [[File:glyph_7_4.png|alt=glyph 7 4.png|124x112px|glyph 7 4.png]]
|-
|-
| | JI glyph for 7/4
| JI glyph for 7/4
|}
|}


'''7/4'''
'''7/4''' <br/>
|-2 0 0 1&gt;
{{Monzo| -2 0 0 1 }} <br/>
968.82591 cents


968.82591 cents
[[File:jid_7_4_pluck_adu_dr220.mp3]] [[:File:jid_7_4_pluck_adu_dr220.mp3|sound info]]


[[File:jid_7_4_pluck_adu_dr220.mp3]] [[:File:jid_7_4_pluck_adu_dr220.mp3|sound sample]]
__TOC__


Frequency ratio '''7:4''', measuring approximately 968.8 [[cent|cents]], has been given the name '''"harmonic seventh."''' It represents the interval between the 4th and 7th harmonics in the [[OverToneSeries|overtone series]]. It is also called a "septimal subminor seventh" -- the word "septimal" referring to the presence of a 7 as the highest [[prime|prime]] in the ratio, and the word "subminor" referring to the harmonic seventh's narrowness compared with a traditional minor seventh (such as [[9/5|9:5]] or [[16/9|16:9]], [[12edo|12edo]]'s 1000-cent interval, or a minor seventh found in a meantone system).
Frequency ratio '''7:4''', measuring approximately 968.8 [[cent|cents]], has been given the name '''"harmonic seventh."''' It represents the interval between the 4th and 7th harmonics in the [[OverToneSeries|overtone series]]. It is also called a "septimal subminor seventh" -- the word "septimal" referring to the presence of a 7 as the highest [[prime]] in the ratio, and the word "subminor" referring to the harmonic seventh's narrowness compared with a traditional minor seventh (such as [[9/5|9:5]] or [[16/9|16:9]], [[12edo]]'s 1000-cent interval, or a minor seventh found in a meantone system).


7:4 has seen use in blues music, barbershop quartet music, and some musical traditions of the world, but has mostly not been recognized as a "[[consonance|consonance]]" in Western music theory. In most [[Just_intonation|Just Intonation]] systems, the harmonic seventh is treated as a fundamental consonance in its own right, with its own distinct quality.
7:4 has seen use in blues music, barbershop quartet music, and some musical traditions of the world, but has mostly not been recognized as a "[[consonance]]" in Western music theory. In most [[Just Intonation]] systems, the harmonic seventh is treated as a fundamental consonance in its own right, with its own distinct quality.


=Harmonic Seventh Chord=
== Harmonic Seventh Chord ==
7:4 appears in an otonal tetrad that forms the basis of much JI music, commonly called a "harmonic seventh chord." It consists of a major triad (4:5:6) plus a harmonic seventh: 4:5:6:7(:8). This tetrad, a hallmark of blues and barbershop harmony, not to mention modern Just Intonation practice, represents a sequence of overtones from the fourth to the seventh. (8, being a doubling of 4, represents an octave above the root.) The intervals between adjacent members of the chord decrease in size:
7:4 appears in an otonal tetrad that forms the basis of much JI music, commonly called a "harmonic seventh chord." It consists of a major triad (4:5:6) plus a harmonic seventh: 4:5:6:7(:8). This tetrad, a hallmark of blues and barbershop harmony, not to mention modern Just Intonation practice, represents a sequence of overtones from the fourth to the seventh. (8, being a doubling of 4, represents an octave above the root.) The intervals between adjacent members of the chord decrease in size:


{| class="wikitable"
{| class="wikitable"
|-
|-
| | [[5/4|5:4]]
| [[5/4|5:4]]
| | approx. 386 cents
| approx. 386 cents
| | major third
| major third
| | [[File:jid_5_4_pluck_adu_dr220.mp3]]
| [[File:jid_5_4_pluck_adu_dr220.mp3]]
|-
|-
| | [[6/5|6:5]]
| [[6/5|6:5]]
| | approx. 316 cents
| approx. 316 cents
| | minor third
| minor third
| | [[File:jid_6_5_pluck_adu_dr220.mp3]]
| [[File:jid_6_5_pluck_adu_dr220.mp3]]
|-
|-
| | [[7/6|7:6]]
| [[7/6|7:6]]
| | approx. 267 cents
| approx. 267 cents
| | septimal subminor third
| septimal subminor third
| | [[File:jid_7_6_pluck_adu_dr220.mp3]]
| [[File:jid_7_6_pluck_adu_dr220.mp3]]
|-
|-
| | [[8/7|8:7]]
| [[8/7|8:7]]
| | approx. 231 cents
| approx. 231 cents
| | septimal supermajor second
| septimal supermajor second
| | [[File:jid_8_7_pluck_adu_dr220.mp3]]
| [[File:jid_8_7_pluck_adu_dr220.mp3]]
|}
|}


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Since 12edo does not distinguish between a minor and subminor third or a major and supermajor second, the intervals between adjacent members of the chord do not have the pattern of decreasing step size which characterizes the harmonic seventh chord:
Since 12edo does not distinguish between a minor and subminor third or a major and supermajor second, the intervals between adjacent members of the chord do not have the pattern of decreasing step size which characterizes the harmonic seventh chord:


[[5/4|5:4]] becomes 400 cents.
* [[5/4|5:4]] becomes 400 cents.
 
* [[6/5|6:5]] becomes 300 cents.
[[6/5|6:5]] becomes 300 cents.
* [[7/6|7:6]] becomes 300 cents.
* [[8/7|8:7]] becomes 200 cents.


[[7/6|7:6]] becomes 300 cents.
== Meantone Augmented Sixth ==
In [[Meantone family|meantone systems]] -- which are generated by repeatedly stacking a slightly flatted (from just) [[perfect_fifth|perfect fifth]] such that four fifths gives a near-just [[major third]] -- there is sometimes a good approximation of the harmonic seventh in the form of an "augmented sixth". [[Quarter-comma meantone]] (aurally identical, for most intents and purposes, to [[31edo|31edo]]) is one such system. In quarter-comma meantone, the interval of C to A# approximates a harmonic seventh, and is a distinct interval from C to Bb, a meantone minor seventh (falling somewhere between 16:9 and 9:5). The augmented sixth appears in tonal harmony in the "augmented sixth chord," and is treated as a rare and special dissonance. The so-called "German Sixth," in quarter-comma meantone, would approximate the harmonic seventh chord of 4:5:6:7(:8).


[[8/7|8:7]] becomes 200 cents.
Note that a good approximation of the harmonic seventh is not available in every meantone system. In [[19edo]] (aurally identical, more or less, to 1/3-comma meantone), the "augmented sixth" is an interval of 947 cents -- about 22 cents flat of 7:4, and so less effective as a consonance.


=Meantone Augmented Sixth=
:''See also [http://en.wikipedia.org/wiki/Septimal_meantone_temperament Septimal meantone temperament - Wikipedia].''
In [[Meantone_family|meantone systems]] -- which are generated by repeatedly stacking a slightly flatted (from just) [[perfect_fifth|perfect fifth]] such that four fifths gives a near-just [[major_third|major third]] -- there is sometimes a good approximation of the harmonic seventh in the form of an "augmented sixth". [[Quarter-comma_meantone|Quarter-comma meantone]] (aurally identical, for most intents and purposes, to [[31edo|31edo]]) is one such system. In quarter-comma meantone, the interval of C to A# approximates a harmonic seventh, and is a distinct interval from C to Bb, a meantone minor seventh (falling somewhere between 16:9 and 9:5). The augmented sixth appears in tonal harmony in the "augmented sixth chord," and is treated as a rare and special dissonance. The so-called "German Sixth," in quarter-comma meantone, would approximate the harmonic seventh chord of 4:5:6:7(:8).


Note that a good approximation of the harmonic seventh is not available in every meantone system. In [[19edo|19edo]] (aurally identical, more or less, to 1/3-comma meantone), the "augmented sixth" is an interval of 947 cents -- about 22 cents flat of 7:4, and so less effective as a consonance.
== Approximations ==
EDOs containing good approximations of the interval 7:4 are (pre-ordered by relative delta):


See: [http://en.wikipedia.org/wiki/Septimal_meantone_temperament Septimal Meantone Temperament on Wikipedia].
{| class="wikitable sortable"
 
=Approximations=
EDOs containing good approximations of the interval 7:4 are (ordered by relative delta):
 
{| class="wikitable"
|-
|-
! | [[EDO|EDO]]
! [[EDO|EDO]]
! | Abs Delta
! Abs Delta
! | Rel Delta
! Rel Delta
! | Prominent Multiples
! class="unsortable" | Prominent Multiples
|-
|-
| | [[26edo|26edo]]
| [[26edo|26edo]]
| | 0.40486 [[cent|¢]]
| 0.40486 [[cent|¢]]
| | 0.87720 [[Relative_cent|r¢]]
| 0.87720 [[Relative_cent|r¢]]
| | [[78edo|78edo]]
| [[78edo|78edo]]
|-
|-
| | [[83edo|83edo]]
| [[83edo|83edo]]
| | 0.15121 ¢
| 0.15121 ¢
| | 1.0459 r¢
| 1.0459 r¢
| | [[166edo|166edo]]
| [[166edo|166edo]]
|-
|-
| | [[57edo|57edo]]
| [[57edo|57edo]]
| | 0.40485 ¢
| 0.40485 ¢
| | 1.9231 r¢
| 1.9231 r¢
| |  
|  
|-
|-
| | [[31edo|31edo]]
| [[31edo|31edo]]
| | 1.0839 ¢
| 1.0839 ¢
| | 2.8003 r¢
| 2.8003 r¢
| |
|
|-
|-
| | [[5edo|5edo]]
| [[5edo|5edo]]
| | 8.8259 ¢
| 8.8259 ¢
| | 3.6775 r¢
| 3.6775 r¢
| | [[10edo|10edo]], [[15edo|15edo]], [[20edo|20edo]], [[25edo|25edo]]
| [[10edo|10edo]], [[15edo|15edo]], [[20edo|20edo]], [[25edo|25edo]]
|-
|-
| | [[21edo|21edo]]
| [[21edo|21edo]]
| | 2.6026 ¢
| 2.6026 ¢
| | 4.5547 r¢
| 4.5547 r¢
| |
|
|-
|-
| | [[88edo|88edo]]
| [[88edo|88edo]]
| | 0.6441 ¢
| 0.6441 ¢
| | 4.7233 r¢
| 4.7233 r¢
| |  
|  
|-
|-
| | [[47edo|47edo]]
| [[47edo|47edo]]
| | 1.3868 ¢
| 1.3868 ¢
| | 5.4319 r¢
| 5.4319 r¢
| | [[94edo|94edo]]
| [[94edo|94edo]]
|-
|-
| | [[73edo|73edo]]
| [[73edo|73edo]]
| | 1.0371 ¢
| 1.0371 ¢
| | 6.3091 r¢
| 6.3091 r¢
| |
|
|-
|-
| | [[36edo|36edo]]
| [[36edo|36edo]]
| | 2.1592 ¢
| 2.1592 ¢
| | 6.4777 r¢
| 6.4777 r¢
| | [[72edo|72edo]]
| [[72edo|72edo]]
|-
|-
| | [[16edo|16edo]]
| [[16edo|16edo]]
| | 6.1741 ¢
| 6.1741 ¢
| | 8.2321 r¢
| 8.2321 r¢
| |
|
|-
|-
| | [[68edo|68edo]]
| [[68edo|68edo]]
| | 0.14686 ¢
| 0.14686 ¢
| | 9.9865 r¢
| 9.9865 r¢
| |  
|  
|-
|-
| | [[11edo|11edo]]
| [[11edo|11edo]]
| | 12.992 ¢
| 12.992 ¢
| | 11.910 r¢
| 11.910 r¢
| | [[22edo|22edo]]
| [[22edo|22edo]]
|-
|-
| | [[89edo|89edo]]
| [[89edo|89edo]]
| | 1.9606 ¢
| 1.9606 ¢
| | 14.541 r¢
| 14.541 r¢
| |  
|  
|}
|}


[http://en.wikipedia.org/wiki/Harmonic_seventh http://en.wikipedia.org/wiki/Harmonic_seventh]
:''See also [http://en.wikipedia.org/wiki/Harmonic_seventh Harmonic seventh - Wikipedia]''


[[Category:7-limit]]
[[Category:7-limit]]
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[[Category:overtone]]
[[Category:overtone]]
[[Category:theory]]
[[Category:theory]]
<!-- interwiki -->
[[de:Naturseptime]]
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