Solid angle - Revision history

en>Fjackson: /* Solid angles in arbitrary dimensions */ removed "sr" and "rad" from section per talk page

2015-01-07T17:45:49Z

Solid angles in arbitrary dimensions: removed "sr" and "rad" from section per talk page

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	I'm a 39 years old and work at the college (Art History).<br>In my [https://www.google.com/search?hl=en&gl=us&tbm=nws&q=free+time&btnI=lucky free time] [http://simultaneousinterpretation.com/michael.html michael kors outlet] I try to teach myself Swedish. I have been twicethere and look forward to returning sometime in the future. I love to read, preferably [http://simultaneousinterpretation.com/michael.html michael kors outlet] on [+s2+ michael kors outlet] my ebook reader. I really love to watch The Big Bang Theory [http://simultaneousinterpretation.com/michael.html michael kors outlet] and The Simpsons as well as docus about nature. I enjoy Sculpting.<br><br>		Hi there! :) My name is Boyd, I'm a student studying Chinese Studies from Favarella, Italy.<br><br>Here is my homepage [http://www.inretea.ro/member/39706/blog/view/45463/ Fifa 15 coin generator]

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en>R. J. Mathar: /* References */ complemented bibcodes

2014-02-21T21:26:15Z

References: complemented bibcodes

← Older revision		Revision as of 23:26, 21 February 2014
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	~~{{Wiktionary}}~~		I'm a 39 years old and work at the college (Art History).<br>In my [https://www.google.com/search?hl=en&gl=us&tbm=nws&q=free+time&btnI=lucky free time] [http://simultaneousinterpretation.com/michael.html michael kors outlet] I try to teach myself Swedish. I have been twicethere and look forward to returning sometime in the future. I love to read, preferably [http://simultaneousinterpretation.com/michael.html michael kors outlet] on [+s2+ michael kors outlet] my ebook reader. I really love to watch The Big Bang Theory [http://simultaneousinterpretation.com/michael.html michael kors outlet] and The Simpsons as well as docus about nature. I enjoy Sculpting.<br><br>
	~~In modern [[musical notation]] and [[musical tuning\|tuning]], an~~ '~~''enharmonic equivalent''' is~~ a ~~[[note]], [[interval~~ (~~music~~)~~\|interval]], or~~ [~~[key signature]~~] ~~that is~~ [~~[Equivalence class (music)\|equivalent]~~] to ~~some other note, interval, or key signature but "spelled", or named differently~~. ~~Thus,~~ the ~~'''enharmonic spelling''' of a written note, interval, or chord is an alternative way~~ to ~~write that note~~, ~~interval, or chord~~. ~~For example, in twelve-tone [[equal temperament~~]~~] (the currently predominant system of [~~[~~musical tuning]~~] ~~in Western music), the notes C{{Music\|sharp}} and D{{Music\|flat}} are ''enharmonic'' (or ''enharmonically equivalent'') notes~~. ~~Namely, they are the same key on a~~ [~~[musical keyboard\|keyboard]~~], and ~~thus they are identical in pitch, although they have different names and different [[Diatonic function\|role]] in harmony and chord progressions~~.

	In other words, if two notes have the same [[pitch (music)\|pitch]] but are represented by different letter names and [[accidental (music)\|accidentals]], they are enharmonic.<ref name="B&S54">Benward & Saker (2003). ''Music in Theory and Practice, Vol. I'', p.7 & 360. ISBN 978-0-07-294262-0.</ref> "''Enharmonic intervals'' are intervals with the same sound that are spelled differently...[resulting], of course, from enharmonic tones."<ref>Benward & Saker (2003), p.54.</ref>		Also visit my [http://simultaneousinterpretation.com/michael.html michael kors outlet] webpage - [http://simultaneousinterpretation.com/michael.html michael kors outlet]

	Prior to this modern meaning, "enharmonic" referred to relations in which there is no exact equivalence in pitch between a sharpened note such as F{{Music\|sharp}} and a flattened note such as G{{Music\|flat}}.<ref>Louis Charles Elson (1905) Elson's Music Dictionary, p.100. O. Ditson Company. "The relation existing between two chromatics, when, by the elevation of one and depression of the other, they are united into one".</ref> as in [[enharmonic scale]].

	~~[[Image:Enharmonic F-sharp G-flat.png\|thumb\|right\|The notes F{{Music\|sharp}} and G{{Music\|flat}} are enharmonic equivalents.]]~~
	~~[[Image:Non enharmonic E-sharp F-flat.png\|thumb\|right\|E{{Music\|sharp}} and F{{Music\|flat}}, however, are not enharmonic equivalents, because E{{Music\|sharp}} is enharmonic with F{{Music\|natural}}.]]~~
	~~[[Image:Enharmonic GX Bbb.png\|thumb\|right\|G{{Music\|doublesharp}} and B{{Music\|doubleflat}} are enharmonic equivalents, both the same as A{{Music\|natural}}.]]~~
	[[Image:Enharmonic key sig B Cb.png\|thumb\|right\|Enharmonically equivalent key signatures of B{{Music\|natural}} and C{{Music\|flat}} major, each followed by its respective [[tonic (music)\|tonic]] chord]]

	Some [[key signature]]s have an enharmonic equivalent that represents a scale identical in sound but spelled differently. The number of sharps and flats of two enharmonically equivalent keys sum to twelve. For example, the key of [[B major]], with 5 sharps, is enharmonically equivalent to the key of [[C-flat major]] with 7 flats, and 5 (sharps) + 7 (flats) = 12. Keys past 7 sharps or flats exist only theoretically and not in practice. The enharmonic keys are six pairs, three major and three minor: B major/C-flat major, [[G-sharp minor]]/[[A-flat minor]], [[F-sharp major]]/[[G-flat major]], [[D-sharp minor]]/[[E-flat minor]], [[C-sharp major]]/[[D-flat major]] and [[A-sharp minor]]/[[B-flat minor]]. There are no works composed in keys that require double sharping or double flatting ''in the key signature'', except in jest. In practice, musicians learn and practice 15 major and 15 minor keys, three more than 12 due to the enharmonic spellings.

	~~[[Image:Enharmonic tritone.png\|thumb\|right\|Enharmonic tritones: A4 = d5 on C {{audio\|Tritone on C.mid\|Play}}.]]~~
	For example the intervals of a minor sixth on C, on B{{music\|#}}, and an augmented fifth on C are all enharmonic intervals {{audio\|Minor sixth on C.mid\|Play}}. The most common enharmonic intervals are the augmented fourth and diminished fifth, or [[tritone]], for example C-F{{music\|#}} = C-G{{music\|b}}.<ref name="B&S54"/>

	Enharmonic equivalence is not to be confused with [[octave]] equivalence, nor are enharmonic intervals to be confused with [[inversion (music)\|inverted]] or [[Interval (music)#Simple and compound\|compound intervals]].

	~~==Tuning enharmonics==~~
	In principle, the modern musical use of the word ''enharmonic'' to mean identical tones is correct only in [[equal temperament]], where the octave is divided into 12 equal semitones; but even in other tuning systems enharmonic associations can be perceived by listeners and exploited by composers.<ref>Rushton, Julian (2001). "Enharmonic", ''The New Grove Dictionary of Music and Musicians''. Second edition, edited by [[Stanley Sadie]] and [[John Tyrrell (professor of music)\|John Tyrrell]]. London: Macmillan Publishers. ISBN 0-19-517067-9.</ref> This is in contrast to the ancient use of the word in the context of unequal temperaments, such as [[quarter-comma meantone]] intonation, in which enharmonic notes differ slightly in pitch. It should be noted, however, that enharmonic equivalents occur in any equal temperament system, such as [[19 equal temperament]] or [[31 equal temperament]], if it can be and is used as a [[meantone temperament]]. The specific equivalences define the equal temperament. 19 equal is characterized by E{{Music\|sharp}} = F{{Music\|flat}} and 31 equal by D{{Music\|doublesharp}} = F{{Music\|doubleflat}}, for instance; in these tunings it is ''not'' true that E{{Music\|sharp}} = F{{music\|natural}}, which is characteristic only of 12 equal temperament.{{Or\|date=October 2010}}

	~~===Pythagorean===~~
	~~{{Main\|Pythagorean tuning}}~~

	In Pythagorean tuning, all pitches are generated from a series of [[Just intonation\|justly tuned]] [[perfect fifth]]s, each with a ratio of 3 to 2. If the first note in the series is an A{{Music\|flat}}, the thirteenth note in the series, G{{Music\|sharp}}, will be ''higher'' than the seventh octave (octave = ratio of 1 to 2, seven octaves is 1 to 2<sup>7</sup> = 128) of the A{{Music\|flat}} by a small interval called a [[~~Pythagorean comma]]. This interval is expressed mathematically as~~:

	~~:<math>\frac{\hbox{twelve fifths}}{\hbox{seven octaves}}~~
	~~=\left(\tfrac32\right)^{12} \!\!\bigg/\, 2^{7}~~
	~~= \frac{3^{12}}{2^{19}}~~
	~~= \frac{531441}{524288}~~
	~~= 1.0136432647705078125~~
	~~\!</math>~~

	~~===Meantone===~~
	~~{{Main\|Meantone temperament}}~~
	~~In 1/4 comma meantone, on the other hand, consider G{{Music\|sharp}} and A{{Music\|flat}}. Call middle C's frequency <math>x</math>. Then high C has a frequency of <math>2x<~~/~~math>. The 1~~/~~4 comma meantone has just (i.e., perfectly tuned) major thirds, which means [[major third]]s with a frequency ratio of exactly 4 to 5~~.

	~~In order to form a just major third with the C above it, A{{Music\|flat}} and high C need to be in the ratio 4 to 5, so A{{Music\|flat}} needs to have the frequency~~

	~~:<math>\frac {8x}{5} = 1.6 x. \!<~~/~~math>~~

	In order to form a just major third above E, however, G{{Music\|sharp}} needs to form the ratio 5 to 4 with E, which, in turn, needs to form the ratio 5 to 4 with C. Thus the frequency of G{{Music\|sharp}} is
	~~:<math>\left(\frac{5}{4}\right)^2x = \left(\frac{25}{16}\right)x = 1~~.~~5625 x</math>~~

	~~Thus, G{{Music\|sharp}} and A{{Music\|flat}} are not the same note; G{{Music\|sharp}} is, in fact 41 [[cent (music)\|cent~~]]s lower in pitch (41% of a semitone, not quite a quarter of a tone). The difference is the interval called the enharmonic [[diesis]], or a frequency ratio of <math>\frac{128}{125}</math>. On a piano tuned in equal temperament, both G{{Music\|sharp}} and A{{Music\|flat}} are played by striking the same key, so both have a frequency <math>2^\frac{8}{12}x = 2^\frac{2}{3} \approx 1.5874 x</math>. Such small differences in pitch can escape notice when presented as melodic intervals. However, when they are sounded as chords, the difference between meantone intonation and equal-~~tempered intonation can be quite noticeable, even to untrained ears.~~

	The reason that — despite the fact that in recent Western music, A{{Music\|flat}} is exactly the same pitch as G{{Music\|sharp}} — we label them differently is that in [[tonality\|tonal]] music notes are named for their harmonic function, and retain the names they had in the meantone tuning era.{{Citation needed\|date=October 2010}} This is called [[diatonic functionality]]. One can however label enharmonically equivalent pitches with one and only one name, sometimes called [[integer notation]], often used in [[serialism]] and [[set theory (music)\|musical set theory]] and employed by the [[MIDI]] interface.

	~~==Enharmonic genus==~~
	~~{{Main\|Enharmonic genus}}~~

	~~In [[ancient Greek music]] the enharmonic was one of the three Greek [[genus (music)\|genera]] in music in which the [[tetrachords]] are divided (descending) as a~~ [[major third\|ditone]] plus two [[microtone]]s. The ditone can be anywhere from 16/13 to 9/7 (3.55 to 4.35 [[semitone]]s) and the microtones can be anything smaller than 1 semitone.{{Citation needed\|date=October 2010}}<!--Mathiesen 2001 states that only the chromatic and diatonic genera had different shades (chroai), though to be sure he is describing only the Aristoxenian system.--> Some examples of enharmonic genera are
	~~:1. 1/1 36/35 16/15 4/3~~
	~~:2. 1/1 28/27 16/15 4/3~~
	~~:3. 1/1 64/63 28/27 4/3~~
	:~~4. 1/1 49/48 28~~/~~27 4~~/3
	:5. 1/~~1 25/24 13/12 4/3~~

	~~==Tetrachords in Byzantine music==~~
	In [[Byzantine music]], ''enharmonic'' describes a kind of [[tetrachord]] and the [[echos]] that contain them. As in the ancient Greek system, enharmonic tetrachords are distinct from [[diatonic]] and [[chromatic scale\|chromatic]]. However Byzantine enharmonic tetrachords bear no resemblance to ancient Greek enharmonic tetrachords. Their largest division is between a [[whole-tone]] and a tone-and-a-quarter in size, and their smallest is between a [[quarter-tone]] and a [[semitone]]. These are called "improper diatonic" or "hard diatonic" tetrachords in modern western usage.{{Citation needed\|date=October 2010}}<!--Whole paragraph needs citations.-->

	~~==See also==~~
	*[[Enharmonic scale]]
	*[[Music theory]]
	*[[Music notation]]
	*[[Accidental (music)\|Accidental]]
	*[[Octave]] equivalence, [[transposition (music)\|Transpositional]] equivalence, and [[inversion (music)\|inversional]] equivalence
	*[[Diatonic and chromatic]]

	~~==Sources==~~
	~~{{Reflist}}~~

	~~==Further reading==~~
	*Mathiesen, Thomas J. 2001. "Greece, §I: Ancient". ~~''The New Grove Dictionary of Music and Musicians'', second edition, edited by [[Stanley Sadie]] and [[John Tyrrell (professor of music)\|John Tyrrell]]. London: Macmillan Publishers.~~

	~~{{Pitch (music)}}~~
	~~[[Category:Intervals]]~~
	~~[[Category:Musical genera]]~~
	~~[[Category:Musical notes]~~]

en>Smsarmad: reverted test edit. Please use Sandbox for any tests. To learn how to edit, see this Tutorial

2014-01-05T12:20:29Z

reverted test edit. Please use Sandbox for any tests. To learn how to edit, see this Tutorial

← Older revision		Revision as of 14:20, 5 January 2014
Line 1:		Line 1:
	~~Hello~~, ~~dear friend! My~~ name is ~~Stewart~~. ~~I am pleased~~ that ~~I could unify~~ to the ~~entire world~~. ~~I live~~ in ~~Australia~~, in the ~~NSW region~~. ~~I dream~~ to ~~head~~ to the ~~various nations~~, to ~~obtain familiarized~~ with ~~interesting individuals~~.<br><br>~~My webpage - FIFA 15 coin hack~~ ([~~http~~://~~www~~.~~mittelohr~~-~~workshop~~.de/~~content~~/~~fifa~~-~~coin~~-~~generator~~-1 ~~please click~~ the ~~next webpage~~])		{{Wiktionary}}
			In modern [[musical notation]] and [[musical tuning\|tuning]], an '''enharmonic equivalent''' is a [[note]], [[interval (music)\|interval]], or [[key signature]] that is [[Equivalence class (music)\|equivalent]] to some other note, interval, or key signature but "spelled", or named differently. Thus, the '''enharmonic spelling''' of a written note, interval, or chord is an alternative way to write that note, interval, or chord. For example, in twelve-tone [[equal temperament]] (the currently predominant system of [[musical tuning]] in Western music), the notes C{{Music\|sharp}} and D{{Music\|flat}} are ''enharmonic'' (or ''enharmonically equivalent'') notes. Namely, they are the same key on a [[musical keyboard\|keyboard]], and thus they are identical in pitch, although they have different names and different [[Diatonic function\|role]] in harmony and chord progressions.

			In other words, if two notes have the same [[pitch (music)\|pitch]] but are represented by different letter names and [[accidental (music)\|accidentals]], they are enharmonic.<ref name="B&S54">Benward & Saker (2003). ''Music in Theory and Practice, Vol. I'', p.7 & 360. ISBN 978-0-07-294262-0.</ref> "''Enharmonic intervals'' are intervals with the same sound that are spelled differently...[resulting], of course, from enharmonic tones."<ref>Benward & Saker (2003), p.54.</ref>

			Prior to this modern meaning, "enharmonic" referred to relations in which there is no exact equivalence in pitch between a sharpened note such as F{{Music\|sharp}} and a flattened note such as G{{Music\|flat}}.<ref>Louis Charles Elson (1905) Elson's Music Dictionary, p.100. O. Ditson Company. "The relation existing between two chromatics, when, by the elevation of one and depression of the other, they are united into one".</ref> as in [[enharmonic scale]].

			[[Image:Enharmonic F-sharp G-flat.png\|thumb\|right\|The notes F{{Music\|sharp}} and G{{Music\|flat}} are enharmonic equivalents.]]
			[[Image:Non enharmonic E-sharp F-flat.png\|thumb\|right\|E{{Music\|sharp}} and F{{Music\|flat}}, however, are not enharmonic equivalents, because E{{Music\|sharp}} is enharmonic with F{{Music\|natural}}.]]
			[[Image:Enharmonic GX Bbb.png\|thumb\|right\|G{{Music\|doublesharp}} and B{{Music\|doubleflat}} are enharmonic equivalents, both the same as A{{Music\|natural}}.]]
			[[Image:Enharmonic key sig B Cb.png\|thumb\|right\|Enharmonically equivalent key signatures of B{{Music\|natural}} and C{{Music\|flat}} major, each followed by its respective [[tonic (music)\|tonic]] chord]]

			Some [[key signature]]s have an enharmonic equivalent that represents a scale identical in sound but spelled differently. The number of sharps and flats of two enharmonically equivalent keys sum to twelve. For example, the key of [[B major]], with 5 sharps, is enharmonically equivalent to the key of [[C-flat major]] with 7 flats, and 5 (sharps) + 7 (flats) = 12. Keys past 7 sharps or flats exist only theoretically and not in practice. The enharmonic keys are six pairs, three major and three minor: B major/C-flat major, [[G-sharp minor]]/[[A-flat minor]], [[F-sharp major]]/[[G-flat major]], [[D-sharp minor]]/[[E-flat minor]], [[C-sharp major]]/[[D-flat major]] and [[A-sharp minor]]/[[B-flat minor]]. There are no works composed in keys that require double sharping or double flatting ''in the key signature'', except in jest. In practice, musicians learn and practice 15 major and 15 minor keys, three more than 12 due to the enharmonic spellings.

			[[Image:Enharmonic tritone.png\|thumb\|right\|Enharmonic tritones: A4 = d5 on C {{audio\|Tritone on C.mid\|Play}}.]]
			For example the intervals of a minor sixth on C, on B{{music\|#}}, and an augmented fifth on C are all enharmonic intervals {{audio\|Minor sixth on C.mid\|Play}}. The most common enharmonic intervals are the augmented fourth and diminished fifth, or [[tritone]], for example C-F{{music\|#}} = C-G{{music\|b}}.<ref name="B&S54"/>

			Enharmonic equivalence is not to be confused with [[octave]] equivalence, nor are enharmonic intervals to be confused with [[inversion (music)\|inverted]] or [[Interval (music)#Simple and compound\|compound intervals]].

			==Tuning enharmonics==
			In principle, the modern musical use of the word ''enharmonic'' to mean identical tones is correct only in [[equal temperament]], where the octave is divided into 12 equal semitones; but even in other tuning systems enharmonic associations can be perceived by listeners and exploited by composers.<ref>Rushton, Julian (2001). "Enharmonic", ''The New Grove Dictionary of Music and Musicians''. Second edition, edited by [[Stanley Sadie]] and [[John Tyrrell (professor of music)\|John Tyrrell]]. London: Macmillan Publishers. ISBN 0-19-517067-9.</ref> This is in contrast to the ancient use of the word in the context of unequal temperaments, such as [[quarter-comma meantone]] intonation, in which enharmonic notes differ slightly in pitch. It should be noted, however, that enharmonic equivalents occur in any equal temperament system, such as [[19 equal temperament]] or [[31 equal temperament]], if it can be and is used as a [[meantone temperament]]. The specific equivalences define the equal temperament. 19 equal is characterized by E{{Music\|sharp}} = F{{Music\|flat}} and 31 equal by D{{Music\|doublesharp}} = F{{Music\|doubleflat}}, for instance; in these tunings it is ''not'' true that E{{Music\|sharp}} = F{{music\|natural}}, which is characteristic only of 12 equal temperament.{{Or\|date=October 2010}}

			===Pythagorean===
			{{Main\|Pythagorean tuning}}

			In Pythagorean tuning, all pitches are generated from a series of [[Just intonation\|justly tuned]] [[perfect fifth]]s, each with a ratio of 3 to 2. If the first note in the series is an A{{Music\|flat}}, the thirteenth note in the series, G{{Music\|sharp}}, will be ''higher'' than the seventh octave (octave = ratio of 1 to 2, seven octaves is 1 to 2<sup>7</sup> = 128) of the A{{Music\|flat}} by a small interval called a [[Pythagorean comma]]. This interval is expressed mathematically as:

			:<math>\frac{\hbox{twelve fifths}}{\hbox{seven octaves}}
			=\left(\tfrac32\right)^{12} \!\!\bigg/\, 2^{7}
			= \frac{3^{12}}{2^{19}}
			= \frac{531441}{524288}
			= 1.0136432647705078125
			\!</math>

			===Meantone===
			{{Main\|Meantone temperament}}
			In 1/4 comma meantone, on the other hand, consider G{{Music\|sharp}} and A{{Music\|flat}}. Call middle C's frequency <math>x</math>. Then high C has a frequency of <math>2x</math>. The 1/4 comma meantone has just (i.e., perfectly tuned) major thirds, which means [[major third]]s with a frequency ratio of exactly 4 to 5.

			In order to form a just major third with the C above it, A{{Music\|flat}} and high C need to be in the ratio 4 to 5, so A{{Music\|flat}} needs to have the frequency

			:<math>\frac {8x}{5} = 1.6 x. \!</math>

			In order to form a just major third above E, however, G{{Music\|sharp}} needs to form the ratio 5 to 4 with E, which, in turn, needs to form the ratio 5 to 4 with C. Thus the frequency of G{{Music\|sharp}} is
			:<math>\left(\frac{5}{4}\right)^2x = \left(\frac{25}{16}\right)x = 1.5625 x</math>

			Thus, G{{Music\|sharp}} and A{{Music\|flat}} are not the same note; G{{Music\|sharp}} is, in fact 41 [[cent (music)\|cent]]s lower in pitch (41% of a semitone, not quite a quarter of a tone). The difference is the interval called the enharmonic [[diesis]], or a frequency ratio of <math>\frac{128}{125}</math>. On a piano tuned in equal temperament, both G{{Music\|sharp}} and A{{Music\|flat}} are played by striking the same key, so both have a frequency <math>2^\frac{8}{12}x = 2^\frac{2}{3} \approx 1.5874 x</math>. Such small differences in pitch can escape notice when presented as melodic intervals. However, when they are sounded as chords, the difference between meantone intonation and equal-tempered intonation can be quite noticeable, even to untrained ears.

			The reason that — despite the fact that in recent Western music, A{{Music\|flat}} is exactly the same pitch as G{{Music\|sharp}} — we label them differently is that in [[tonality\|tonal]] music notes are named for their harmonic function, and retain the names they had in the meantone tuning era.{{Citation needed\|date=October 2010}} This is called [[diatonic functionality]]. One can however label enharmonically equivalent pitches with one and only one name, sometimes called [[integer notation]], often used in [[serialism]] and [[set theory (music)\|musical set theory]] and employed by the [[MIDI]] interface.

			==Enharmonic genus==
			{{Main\|Enharmonic genus}}

			In [[ancient Greek music]] the enharmonic was one of the three Greek [[genus (music)\|genera]] in music in which the [[tetrachords]] are divided (descending) as a [[major third\|ditone]] plus two [[microtone]]s. The ditone can be anywhere from 16/13 to 9/7 (3.55 to 4.35 [[semitone]]s) and the microtones can be anything smaller than 1 semitone.{{Citation needed\|date=October 2010}}<!--Mathiesen 2001 states that only the chromatic and diatonic genera had different shades (chroai), though to be sure he is describing only the Aristoxenian system.--> Some examples of enharmonic genera are
			:1. 1/1 36/35 16/15 4/3
			:2. 1/1 28/27 16/15 4/3
			:3. 1/1 64/63 28/27 4/3
			:4. 1/1 49/48 28/27 4/3
			:5. 1/1 25/24 13/12 4/3

			==Tetrachords in Byzantine music==
			In [[Byzantine music]], ''enharmonic'' describes a kind of [[tetrachord]] and the [[echos]] that contain them. As in the ancient Greek system, enharmonic tetrachords are distinct from [[diatonic]] and [[chromatic scale\|chromatic]]. However Byzantine enharmonic tetrachords bear no resemblance to ancient Greek enharmonic tetrachords. Their largest division is between a [[whole-tone]] and a tone-and-a-quarter in size, and their smallest is between a [[quarter-tone]] and a [[semitone]]. These are called "improper diatonic" or "hard diatonic" tetrachords in modern western usage.{{Citation needed\|date=October 2010}}<!--Whole paragraph needs citations.-->

			==See also==
			*[[Enharmonic scale]]
			*[[Music theory]]
			*[[Music notation]]
			*[[Accidental (music)\|Accidental]]
			*[[Octave]] equivalence, [[transposition (music)\|Transpositional]] equivalence, and [[inversion (music)\|inversional]] equivalence
			*[[Diatonic and chromatic]]

			==Sources==
			{{Reflist}}

			==Further reading==
			*Mathiesen, Thomas J. 2001. "Greece, §I: Ancient". ''The New Grove Dictionary of Music and Musicians'', second edition, edited by [[Stanley Sadie]] and [[John Tyrrell (professor of music)\|John Tyrrell]]. London: Macmillan Publishers.

			{{Pitch (music)}}
			[[Category:Intervals]]
			[[Category:Musical genera]]
			[[Category:Musical notes]]

en>Netheril96 at 04:49, 2 September 2012

2012-09-02T04:49:12Z

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