formulasearchengine - User contributions [en]

TSL color space

2014-05-01T03:39:54Z

24.22.235.130: /* Conversion between RGB and TSL */ Fixed TSL->RGB equation

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Proposed redefinition of SI base units

2013-12-31T17:14:20Z

24.22.217.233: /* Ampere */

In [[quantum field theory in curved spacetime]], there is a unique quantum state over a background [[de Sitter space]] which is invariant under all the [[isometries]]. This is a [[thermal state]], which is at the [[de Sitter temperature]] of <math>T = H/2\pi</math> where ''H'' is the [[Hubble constant]]. This state is called the '''Bunch–Davies state'''.<ref>
{{cite journal
| last1 = Bunch
| first1 = Timothy Stephen
| authorlink1 = Timothy Stephen Bunch
| last2 = Davies
| first2 = Paul
| authorlink2 = Paul Davies
| date =
| year = 1978
| month =
| title = Quantum Field Theory In De Sitter Space: Renormalization By Point Splitting
| journal = Proceedings of the Royal Society of London
| volume = 360
| series = A
| pages = 117
| doi =
}}
</ref>

This explains the origin of [[cosmological perturbation]] fluctuations in inflationary models.

== See also ==

* [[Quantum field theory in curved spacetime]]
* [[Unruh effect]]
* [[Hawking radiation]]
* [[Inflation (cosmology)]]

== References ==

<references />

{{quantum gravity}}

{{DEFAULTSORT:Bunch-Davies vacuum}}
[[Category:Quantum field theory]]
[[Category:Theory of relativity]]

{{relativity-stub}}

Surface energy

2013-11-06T22:54:16Z

24.22.29.0:

__NOTOC__
{{Infobox chord|
chord_name=minor triad|
first_interval=[[root (chord)|root]]|
second_interval=[[minor third]]|
third_interval=[[perfect fifth]]|
tuning=[[just intonation|10:12:15]]<ref>Shirlaw, Matthew (). ''The Theory of Harmony'', p.81. ISBN 978-1-4510-1534-8. "20:24:30"</ref>|
forte_number=3-11|
complement=9-11
}}

[[Image:Minor chord on C.png|thumb|right|Minor chord on C {{Audio|Minor chord on C.mid|play}}.]]

In [[music theory]], a '''minor chord''' ({{audio|D minor triad.mid|play D minor chord}}) is a [[chord (music)|chord]] having a [[root (chord)|root]], a [[minor third]], and a [[perfect fifth]].<ref>Miller, Michael. [http://books.google.com/books?id=sTMbuSQdqPMC ''The Complete Idiot's Guide to Music Theory, 2nd ed''], p. 114. [Indianapolis, IN]: Alpha, 2005. ISBN 1-59257-437-8.</ref>
When a chord has these three notes alone, it is called a '''minor triad'''.
Some minor [[Triad (music)|triads]] with additional notes, such as the [[minor seventh chord]], may also be called minor chords.

[[Image:Minor and major thirds.png|center|thumb|Minor and major third in a minor chord: minor third on bottom, major third on top {{audio|Minor and major thirds.mid|Play}}.]]

A minor triad can also be described as a minor third [[Interval (music)|interval]] with a [[major third]] interval on top or as a root note, a note 3 [[semitone]]s higher than the root, and a note 7 semitones higher than the root. Hence it can be represented by the [[Pitch class#Integer notation|integer notation]] {0, 3, 7}.

A [[major chord]] ({{Audio|Major triad on C.mid|play}}) differs from a minor chord in having a major third above the root instead of a minor third.
It can also be described as a major third with a minor third on top, in contrast to a minor chord, which has a minor third with a major third on top. They both contain fifths, because a major third (4 semitones) plus a minor third (3 semitones) equals a fifth (7 semitones).

A [[Diminished triad|diminished chord]] is a minor chord with a lowered fifth. {{Audio|Diminished triad on C.mid|play}}

An example of a minor chord is the C minor chord, which consists of the notes C (root), E{{music|b}} (minor third) and G (perfect fifth):
:{{audio|Minor_triad_on_C.mid|Play C minor chord in root position}}.

[[Image:Minor chord root and inversions.PNG|thumb|center|400px |An A minor chord (consisting of notes A, C, E) in its [[root position]], [[first inversion]], and [[second inversion]], respectively {{audio|Minor_triad_inversions_A.mid|Play A minor chord and inversions}}]]

The minor chord, along with the major chord, is one of the basic building blocks of [[tonality|tonal]] music and the [[common practice period]]. In Western music, a minor chord, in comparison, "sounds darker than a major chord"<ref name="Kamien">[[Roger Kamien|Kamien, Roger]] (2008). ''Music: An Appreciation'', 6th Brief Edition, p.46. ISBN 978-0-07-340134-8.</ref> but is still considered highly [[consonance and dissonance|consonant]], stable, or as not requiring [[resolution (music)|resolution]].

== Acoustic consonance of the minor chord ==

A unique particularity of the minor chord is that this is the only chord of three notes in which the three notes have one harmonic - hearable and with a not too high row - in common (more or less exactly, depending on the tuning system used) : This harmonic, common to the three notes, is situated 2 octaves above the high note of the chord : This is the harmonic of row 6 of the fundamental of the chord, the one of row 5 of middle note, the one of row 4 of the high note:

:In the example ''do, mi<math>\flat</math>, sol'' : a ''sol'', 2 octaves above.

Demonstration :
* Minor third = 6/5 = 12/10
* Major third = 5/4 = 15/12
* So the ratios of Minor chord : 10:12:15
* And the explication of the unique harmonic in common, between the three notes, is verified by : 10*6 = 12*5 = 15*4

== Just intonation ==


[[Image:Harmonic Series.png|thumb|center|440px|An illustration of the harmonic series as musical notation. The numbers above the harmonic indicate the number of [[Cent (music)|cents]] it deviates from [[equal temperament]]. Red notes are sharp. Blue notes are flat.]]

In [[just intonation]], a minor chord is often (but not exclusively) tuned in the frequency ratio 10:12:15 ({{Audio|Just minor triad on C.mid|play}}).<ref name="J&G">Johnston, Ben and Gilmore, Bob (2006). "A Notation System for Extended Just Intonation" (2003), ''"Maximum clarity" and Other Writings on Music'', p.78. ISBN 978-0-252-03098-7. D-, F, A (10/9-4/3-5/3).</ref> This is the first occurrence of a minor triad in the [[harmonic series (music)|harmonic series]] (if on C: E-G-B).<ref>Hauptmann, Moritz (1888). ''The Nature of Harmony and Metre'', p.15. Swan Sonnenschein</ref> This may be found on iii, vi, {{music|b}}vi, {{music|b}}iii, and vii.<ref>Wright, David (2009). ''Mathematics and Music'', p.140-41. ISBN 978-0-8218-4873-9.</ref> In 12-TET, or twelve-tone [[equal temperament]] (now the most common tuning system in the West), a minor chord has 3 [[semitone]]s between the root and third, 4 between the third and fifth, and 7 between the root and fifth. It is represented by the [[pitch class|integer notation]] 0,3,7. The 12-TET fifth (700 [[cent (music)|cents]]) is only two cents narrower than the just perfect fifth (3:2, 701.9 cents), but the 12-TET minor third (300 cents) is noticeably (about 16 cents) narrower than the just minor third (6:5, 315.6 cents). The 12-TET minor third (300 cents) more closely approximates the [[19-limit]] ([[Limit (music)]]) minor third 16:19 {{audio|19th harmonic on C.mid|Play}} (297.5 cents, the nineteenth [[harmonic]]) with only 2 cents error.<ref>Alexander J. Ellis (translating Hermann Helmholtz): ''On the Sensations of Tone as a Physiological Basis for the Theory of Music'', page 455. Dover Publications, Inc., New York, 1954.</ref> Ellis proposes that the conflict between mathematicians and physicists on one hand and practicing musicians on the other regarding the supposed inferiority of the minor chord and scale to the major may be explained due to physicists' comparison of just minor and major triads, in which case minor comes out the loser, versus the musicians' comparison of the equal tempered triads, in which case minor comes out the winner since the ET major third is 14 cents sharp from the just major third while the ET minor third closely approximates the consonant 19:16 minor third, which many find pleasing.<ref>Ellis (1954), p.298. In the 16th through 18th centuries, prior to 12-TET, the minor third in [[Quarter-comma meantone#Construction of the chromatic scale|meantone temperament]] was 310 cents {{audio|Quarter-comma meantone minor third on C.mid|Play}} and much rougher than the 300 cent ET minor third.</ref> Other just minor chord tunings include the supertonic triad in just intonation (27:32:40)<ref name="J&G"/> the '''false minor triad''',<ref>Shirlaw (), p.375.</ref> {{audio|Supertonic minor triad on C.mid|Play}}, 16:19:24<ref name="Ruland">Ruland, Heiner (1992). ''Expanding Tonal Awareness'', p.39. ISBN 978-1-85584-170-3.</ref> {{audio|19th harmonic minor triad on C.mid|Play}}, 12:14:18 (6:7:9)<ref>Hermann von Helmholtz (1885). ''On the sensations of tone as a physiological basis for the theory of music'', p.468. Longmans, Green.</ref><ref>William Smythe Babcock Mathews (1805). ''Music: A Monthly Magazine Devoted to the Art, Science, Technic and Literature of Music, Volume 7, Volume 7'', p.608. W.S.B. Mathews. "The tones re, fa, and la, as given on the accordion, are vibrationally 6:7:9. This is ''not'' a minor triad, nor anything very near it although its fifth is just the same as in the minor and the major, and the ratio 6:9 being simply 2:3."</ref> {{audio|Septimal minor triad on C.mid|Play}} ([[Septimal minor third]]), and the Pythagorean minor triad<ref name="Ruland"/> (54:64:81) {{audio|Pythagorean minor triad on C.mid|Play}}. More tunings of the minor chord are also available in various equal temperaments other than 12-TET.

Rather than directly from the [[harmonic series (music)|harmonic series]], [[Georg Andreas Sorge|Sorge]] derived the minor chord from joining two major triads; for example the A minor triad being the confluence of the F and C major triads.<ref name="Lester">Lester, Joel (1994). ''Compositional theory in the eighteenth century'', p.194. ISBN 978-0-674-15523-7.</ref> A-C-E= f-A-C-E-g. Given justly tuned major triads this produces a justly tuned minor triad: 10:12:15 on 8/5.

==Minor chord table==
{| class="wikitable"
!bgcolor=#dddddd|Chord
!bgcolor=#dddddd|Root
!bgcolor=#dddddd|Minor Third
!bgcolor=#dddddd|Perfect Fifth
|-
!Cm
|C
|E{{music|b}}
|G
|-
!C{{music|#}}m
|C{{music|#}}
|E
|G{{music|#}}
|-
!D{{music|b}}m
|D{{music|b}}
|F{{music|b}} (E)
|A{{music|b}}
|-
!Dm
|D
|F
|A
|-
!D{{music|#}}m
|D{{music|#}}
|F{{music|#}}
|A{{music|#}}
|-
!E{{music|b}}m
|E{{music|b}}
|G{{music|b}}
|B{{music|b}}
|-
!Em
|E
|G
|B
|-
!E{{music|#}}m
|E{{music|#}}
|G{{music|#}}
|B{{music|#}}
|-
!Fm
|F
|A{{music|b}}
|C
|-
!F{{music|#}}m
|F{{music|#}}
|A
|C{{music|#}}
|-
!G{{music|b}}m
|G{{music|b}}
|B{{music|bb}} (A)
|D{{music|b}}
|-
!Gm
|G
|B{{music|b}}
|D
|-
!G{{music|#}}m
|G{{music|#}}
|B
|D{{music|#}}
|-
!A{{music|b}}m
|A{{music|b}}
|C{{music|b}} (B)
|E{{music|b}}
|-
!Am
|A
|C
|E
|-
!A{{music|#}}m
|A{{music|#}}
|C{{music|#}}
|E{{music|#}} (F)
|-
!B{{music|b}}m
|B{{music|b}}
|D{{music|b}}
|F
|-
!Bm
|B
|D
|F{{music|#}}
|-
|}

==See also==
* [[Major and minor]]
* [[Musical tuning]]
* [[Major chord]]
* [[Otonality and Utonality]]

==References==
{{Reflist}}

{{Chords}}

{{DEFAULTSORT:Minor Chord}}
[[Category:Chords]]

Pareto distribution

2012-09-01T00:35:18Z

24.22.62.222: /* Relation to the exponential distribution */

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Multiphase particle-in-cell method

2012-08-10T02:08:04Z

24.22.93.255:

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