7/4: Difference between revisions

Line 12:

__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]] 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).

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 [[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]]" 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.

Line 54:

== 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).

In [[Meantone family|meantone systems]] -- which are generated by repeatedly stacking a slightly flatted (from just) [[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]]) 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]] (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.

:''See also [~~http~~:~~//en.wikipedia.org/wiki/~~Septimal_meantone_temperament Septimal meantone temperament - Wikipedia].''

:''See also: [[wikipedia:Septimal_meantone_temperament|Septimal meantone temperament - Wikipedia]].''

== Approximations ==

EDOs containing good approximations of the interval 7:4 are (pre-ordered by relative delta):

[[EDO|EDOs]] containing good approximations of the interval 7:4 are (pre-ordered by [[Relative error|relative delta]]):

{| class="wikitable sortable"

{| class="wikitable sortable center-1 right-2 right-3"

|-

! [[~~EDO|~~EDO]]

! [[EDO]]

! Abs Delta

! Abs Delta ([[Cent|¢]])

! Rel Delta

! Rel Delta ([[Relative cent|r¢]])

! class="unsortable" | Prominent Multiples

|-

| [[~~26edo|~~26edo]]

| [[26edo]]

| 0.40486 ~~[[cent|¢]]~~

| 0.40486

| 0.87720 ~~[[Relative_cent|r¢]]~~

| 0.87720

| [[~~78edo|~~78edo]]

| [[78edo]]

|-

| [[~~83edo|~~83edo]]

| [[83edo]]

| 0.15121 ¢

| 0.15121

| 1.0459 r¢

| 1.0459

| [[~~166edo|~~166edo]]

| [[166edo]]

|-

| [[~~57edo|~~57edo]]

| [[57edo]]

| 0.40485 ¢

| 0.40485

| 1.9231 r¢

| 1.9231

|

|-

| [[~~31edo|~~31edo]]

| [[31edo]]

| 1.0839 ¢

| 1.0839

| 2.8003 r¢

| 2.8003

|

|-

| [[5edo| <span style="display:none">0</span>5edo]]

| 8.8259 ¢

| 8.8259

| 3.6775 r¢

| 3.6775

| [[10edo]], [[15edo]], [[20edo]], [[25edo]]

|-

| [[~~21edo|~~21edo]]

| [[21edo]]

| 2.6026 ¢

| 2.6026

| 4.5547 r¢

| 4.5547

|

|-

| [[~~88edo|~~88edo]]

| [[88edo]]

| 0.6441 ¢

| 0.6441

| 4.7233 r¢

| 4.7233

|

|-

| [[~~47edo|~~47edo]]

| [[47edo]]

| 1.3868 ¢

| 1.3868

| 5.4319 r¢

| 5.4319

| [[~~94edo|~~94edo]]

| [[94edo]]

|-

| [[~~73edo|~~73edo]]

| [[73edo]]

| 1.0371 ¢

| 1.0371

| 6.3091 r¢

| 6.3091

|

|-

| [[~~36edo|~~36edo]]

| [[36edo]]

| 2.1592 ¢

| 2.1592

| 6.4777 r¢

| 6.4777

| [[~~72edo|~~72edo]]

| [[72edo]]

|-

| [[~~16edo|~~16edo]]

| [[16edo]]

| 6.1741 ¢

| 6.1741

| 8.2321 r¢

| 8.2321

|

|-

| [[~~68edo|~~68edo]]

| [[68edo]]

| 0.14686 ¢

| 0.14686

| 9.9865 r¢

| 9.9865

|

|-

| [[~~11edo|~~11edo]]

| [[11edo]]

| 12.992 ¢

| 12.992

| 11.910 r¢

| 11.910

| [[~~22edo|~~22edo]]

| [[22edo]]

|-

| [[~~89edo|~~89edo]]

| [[89edo]]

| 1.9606 ¢

| 1.9606

| 14.541 r¢

| 14.541

|

|}

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

:''See also: [[wikipedia:Harmonic_seventh|Harmonic seventh - Wikipedia]]''

[[Category:7-limit]]

[[Category:Harmonic]]

[[Category:Just interval]]

Line 150:

[[Category:Theory]]

[[Category:Seventh]]

[[Category:minor seventh]]

[[de:Naturseptime]]

@@ Line 12: / Line 12: @@
 __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]] 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).
+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 [[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).
 :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.
@@ Line 54: / Line 54: @@
 == 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).
+In [[Meantone family|meantone systems]] -- which are generated by repeatedly stacking a slightly flatted (from just) [[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]]) 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]] (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.
-:''See also [http://en.wikipedia.org/wiki/Septimal_meantone_temperament Septimal meantone temperament - Wikipedia].''
+:''See also: [[wikipedia:Septimal_meantone_temperament|Septimal meantone temperament - Wikipedia]].''
 == Approximations ==
-EDOs containing good approximations of the interval 7:4 are (pre-ordered by relative delta):
+[[EDO|EDOs]] containing good approximations of the interval 7:4 are (pre-ordered by [[Relative error|relative delta]]):
-{| class="wikitable sortable"
+{| class="wikitable sortable center-1 right-2 right-3"
 |-
-! [[EDO|EDO]]
+! [[EDO]]
-! Abs Delta
+! Abs Delta ([[Cent|¢]])
-! Rel Delta
+! Rel Delta ([[Relative cent|r¢]])
 ! class="unsortable" | Prominent Multiples
 |-
-| [[26edo|26edo]]
+| [[26edo]]
-| 0.40486 [[cent|¢]]
+| 0.40486
-| 0.87720 [[Relative_cent|r¢]]
+| 0.87720
-| [[78edo|78edo]]
+| [[78edo]]
 |-
-| [[83edo|83edo]]
+| [[83edo]]
-| 0.15121 ¢
+| 0.15121
-| 1.0459 r¢
+| 1.0459
-| [[166edo|166edo]]
+| [[166edo]]
 |-
-| [[57edo|57edo]]
+| [[57edo]]
-| 0.40485 ¢
+| 0.40485
-| 1.9231 r¢
+| 1.9231
 |
 |-
-| [[31edo|31edo]]
+| [[31edo]]
-| 1.0839 ¢
+| 1.0839
-| 2.8003 r¢
+| 2.8003
 |
 |-
 | [[5edo|&nbsp;<span style="display:none">0</span>5edo]]
-| 8.8259 ¢
+| 8.8259
-| 3.6775 r¢
+| 3.6775
-| [[10edo|10edo]], [[15edo|15edo]], [[20edo|20edo]], [[25edo|25edo]]
+| [[10edo]], [[15edo]], [[20edo]], [[25edo]]
 |-
-| [[21edo|21edo]]
+| [[21edo]]
-| 2.6026 ¢
+| 2.6026
-| 4.5547 r¢
+| 4.5547
 |
 |-
-| [[88edo|88edo]]
+| [[88edo]]
-| 0.6441 ¢
+| 0.6441
-| 4.7233 r¢
+| 4.7233
 |
 |-
-| [[47edo|47edo]]
+| [[47edo]]
-| 1.3868 ¢
+| 1.3868
-| 5.4319 r¢
+| 5.4319
-| [[94edo|94edo]]
+| [[94edo]]
 |-
-| [[73edo|73edo]]
+| [[73edo]]
-| 1.0371 ¢
+| 1.0371
-| 6.3091 r¢
+| 6.3091
 |
 |-
-| [[36edo|36edo]]
+| [[36edo]]
-| 2.1592 ¢
+| 2.1592
-| 6.4777 r¢
+| 6.4777
-| [[72edo|72edo]]
+| [[72edo]]
 |-
-| [[16edo|16edo]]
+| [[16edo]]
-| 6.1741 ¢
+| 6.1741
-| 8.2321 r¢
+| 8.2321
 |
 |-
-| [[68edo|68edo]]
+| [[68edo]]
-| 0.14686 ¢
+| 0.14686
-| 9.9865 r¢
+| 9.9865
 |
 |-
-| [[11edo|11edo]]
+| [[11edo]]
-| 12.992 ¢
+| 12.992
-| 11.910 r¢
+| 11.910
-| [[22edo|22edo]]
+| [[22edo]]
 |-
-| [[89edo|89edo]]
+| [[89edo]]
-| 1.9606 ¢
+| 1.9606
-| 14.541 r¢
+| 14.541
 |
 |}
-:''See also [http://en.wikipedia.org/wiki/Harmonic_seventh Harmonic seventh - Wikipedia]''
+:''See also: [[wikipedia:Harmonic_seventh|Harmonic seventh - Wikipedia]]''
 [[Category:7-limit]]
 [[Category:Harmonic]]
 [[Category:Just interval]]
@@ Line 150: / Line 150: @@
 [[Category:Theory]]
 [[Category:Seventh]]
+[[Category:minor seventh]]
 <!-- interwiki -->
 [[de:Naturseptime]]

7/4: Difference between revisions

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