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{| class="wikitable"
{{interwiki
|-
| de = Naturseptime
| | [[File:glyph_7_4.png|alt=glyph 7 4.png|124x112px|glyph 7 4.png]]
| en = 7/4
|-
| es =
| | JI glyph for 7/4
| ja =
|}
| ro = 7/4 (ro)
 
}}
'''7/4'''
{{Infobox Interval
|-2 0 0 1>
| Name = harmonic seventh, natural seventh, septimal minor seventh, subminor seventh
 
| Color name = z7, zo 7th
968.82591 cents
| Sound = jid_7_4_pluck_adu_dr220.mp3
 
}}
[[File:jid_7_4_pluck_adu_dr220.mp3]] [[:File:jid_7_4_pluck_adu_dr220.mp3|sound sample]]
{{Wikipedia|Harmonic seventh}}


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]]s, named '''harmonic seventh''' or '''natural seventh''', represents the interval between the 4th and 7th harmonics in the [[harmonic series]]. It is also called a '''septimal (sub)minor 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]] or [[16/9]], [[12edo]]'s 1000-cent interval, or a minor seventh found in a meantone system). It is traditionally seen as a minor seventh, though it may show up as an augmented sixth in some cases.


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 ptolemaic 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]]
|}
|}


This chord is similar to the "dominant seventh chord" in 12edo, the most significant difference being the mistuning of the harmonic seventh. In 12edo, the interval closest to the harmonic seventh is the minor seventh at 1000 cents. The difference of about 31 cents is striking, and especially noticable when the chords are presented next to one another. While the "dominant seventh chord" of 12edo is treated as a dissonance that needs to resolve (usually in the chord pattern V7 to I), the "harmonic seventh chord" has a much different flavor and is often treated by composers in Just Intonation as a consonance.
This chord is similar to the "dominant seventh chord" in 12edo, the most significant difference being the mistuning of the harmonic seventh. In 12edo, the interval closest to the harmonic seventh is the minor seventh at 1000 cents. The difference of about 31 cents is striking, and especially noticable when the chords are presented next to one another. While the "dominant seventh chord" of 12edo is treated as a dissonance that needs to resolve (usually in the chord pattern V7 to I), the "harmonic seventh chord" has a much different flavor and is often treated by composers in Just Intonation as a consonance.


Another interval found in a harmonic seventh chord is the septimal tritone of [[7/5|7:5]], which represents the interval between the major third (5) and the harmonic seventh (7). This interval, at 583 cents, sounds distinct from 12edo's half-octave tritone of 600 cents. In just intonation, 7:5 is treated as a ''consonant tritone'', and has a much mellower and sweeter sound than the 600-cent tritone we are used to hearing.
Another interval found in a harmonic seventh chord is the septimal tritone of [[7/5]], which represents the interval between the major third (5) and the harmonic seventh (7). This interval, at 583 cents, sounds distinct from 12edo's half-octave tritone of 600 cents. In just intonation, 7/5 is treated as a ''consonant tritone'', and has a much mellower and sweeter sound than the 600-cent tritone we are used to hearing.


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


[[6/5|6:5]] becomes 300 cents.
== Meantone augmented sixth ==
{{See also| Meantone }}


[[7/6|7:6]] becomes 300 cents.
In [[Meantone family #Septimal meantone|meantone systems]] – which are generated by repeatedly stacking a slightly flattened (from just) [[perfect fifth]] such that four fifths gives a near-just major third of 5/4 – 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]]) 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. The so-called [[German sixth chord]], 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. Systems on the flat end of reasonable meantone tunings, flatter than [[19edo]], have the augmented sixth closer to [[12/7]], while the diminished seventh is closer to 7/4. Mapping the harmonic seventh to A6 is known as [[septimal meantone]] and mapping it to d7 is known as [[flattone]].


=Meantone Augmented Sixth=
== Approximations by EDOs ==
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).
Following [[EDO]]s (up to 200) contain good approximations<ref>error magnitude below 7, both, absolute (in ¢) and relative (in r¢)</ref> of the interval 7/4. Errors are given by magnitude, the arrows in the table show if the EDO representation is sharp (&uarr;) or flat (&darr;).


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.
{| class="wikitable sortable right-1 center-2 right-3 right-4 center-5"
 
|-
See: [http://en.wikipedia.org/wiki/Septimal_meantone_temperament Septimal Meantone Temperament on Wikipedia].
! [[EDO]]
 
! class="unsortable" | deg\edo
=Approximations=
! Absolute <br> error ([[Cent|¢]])
EDOs containing good approximations of the interval 7:4 are (ordered by relative delta):
! Relative <br> error ([[Relative cent|r¢]])
 
! &#8597;
{| class="wikitable"
! class="unsortable" | Equally acceptable multiples <ref>Super EDOs up to 200 within the same error tolerance</ref>
|-
| [[21edo|21]]
| 17\21
| 2.6026
| 4.5547
| &uarr;
|
|-
| [[26edo|26]]
| 21\26
| 0.4049
| 0.8772
| &uarr;
| [[52edo|42\52]], [[78edo|63\78]], [[104edo|84\104]], [[130edo|105\130]], [[156edo|126\156]], [[182edo|147\182]]
|-
|-
! | [[EDO|EDO]]
| [[31edo|31]]
! | Abs Delta
| 25\31
! | Rel Delta
| 1.0839
! | Prominent Multiples
| 2.8003
| &darr;
| [[62edo|50\62]]
|-
|-
| | [[26edo|26edo]]
| [[36edo|36]]
| | 0.40486 [[cent|¢]]
| 29\36
| | 0.87720 [[Relative_cent|r¢]]
| 2.1592
| | [[78edo|78edo]]
| 6.4777
| &darr;
|
|-
|-
| | [[83edo|83edo]]
| [[47edo|47]]
| | 0.15121 ¢
| 38\47
| | 1.0459 r¢
| 1.3868
| | [[166edo|166edo]]
| 5.4319
| &uarr;
|
|-
|-
| | [[57edo|57edo]]
| [[57edo|57]]
| | 0.40485 ¢
| 46\57
| | 1.9231 r¢
| 0.4049
| |  
| 1.9231
| &darr;
| [[114edo|92\114]], [[171edo|138\171]]
|-
|-
| | [[31edo|31edo]]
| [[73edo|73]]
| | 1.0839 ¢
| 59\73
| | 2.8003 r¢
| 1.0371
| |  
| 6.3091
| &uarr;
|
|-
|-
| | [[5edo|5edo]]
| [[83edo|83]]
| | 8.8259 ¢
| 67\83
| | 3.6775 r¢
| 0.1512
| | [[10edo|10edo]], [[15edo|15edo]], [[20edo|20edo]], [[25edo|25edo]]
| 1.0459
| &darr;
| [[166edo|134\166]]
|-
|-
| | [[21edo|21edo]]
| [[88edo|88]]
| | 2.6026 ¢
| 71\88
| | 4.5547 r¢
| 0.6441
| |  
| 4.7233
| &darr;
|
|-
|-
| | [[88edo|88edo]]
| [[109edo|109]]
| | 0.6441 ¢
| 88\109
| | 4.7233 r¢
| 0.0186
| |  
| 0.1687
| &darr;
|
|-
|-
| | [[47edo|47edo]]
| [[135edo|135]]
| | 1.3868 ¢
| 109\135
| | 5.4319 r¢
| 0.0630
| | [[94edo|94edo]]
| 0.7086
| &uarr;
|
|-
|-
| | [[73edo|73edo]]
| [[140edo|140]]
| | 1.0371 ¢
| 113\140
| | 6.3091 r¢
| 0.2545
| |  
| 2.9689
| &darr;
|
|-
|-
| | [[36edo|36edo]]
| [[145edo|145]]
| | 2.1592 ¢
| 117\145
| | 6.4777 r¢
| 0.5500
| | [[72edo|72edo]]
| 6.6464
| &darr;
|
|-
|-
| | [[16edo|16edo]]
| [[161edo|161]]
| | 6.1741 ¢
| 130\161
| | 8.2321 r¢
| 0.1182
| |  
| 1.5858
| &uarr;
|  
|-
|-
| | [[68edo|68edo]]
| [[187edo|187]]
| | 0.14686 ¢
| 151\187
| | 9.9865 r¢
| 0.1581
| |  
| 2.4630
| &uarr;
|
|-
|-
| | [[11edo|11edo]]
| [[192edo|192]]
| | 12.992 ¢
| 155\192
| | 11.910 r¢
| 0.0759
| | [[22edo|22edo]]
| 1.2145
| &darr;
|
|-
|-
| | [[89edo|89edo]]
| [[197edo|197]]
| | 1.9606 ¢
| 159\197
| | 14.541 r¢
| 0.2980
| |  
| 4.8920
| &darr;
|
|}
|}


[http://en.wikipedia.org/wiki/Harmonic_seventh http://en.wikipedia.org/wiki/Harmonic_seventh]
<references/>
 
== See also ==
* [[8/7]] – its [[octave complement]]
* [[12/7]] – its [[twelfth complement]]
* [[Ed7/4]]
* [[Gallery of just intervals]]


[[Category:7-limit]]
[[Category:Seventh]]
[[Category:harmonic]]
[[Category:Subminor seventh]]
[[Category:interval]]
[[Category:just]]
[[Category:just_interval]]
[[Category:listen]]
[[Category:overtone]]
[[Category:theory]]
Retrieved from "https://en.xen.wiki/w/7/4"