Introductory examples in Sagittal notation - Revision history

BudjarnLambeth: Links

2024-12-24T06:11:41Z

Links

ArrowHead294 at 14:21, 10 October 2024

2024-10-10T14:21:41Z

Zhenlige: /* Just intonation: notating a harmonic scale */fix typo

2024-06-01T14:19:41Z

Just intonation: notating a harmonic scale: fix typo

← Older revision		Revision as of 14:19, 1 June 2024
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	The second example shows the best approximation for the same chord in [[12edo\|12edo]]. Here, both the syntonic as the septimal comma are tempered out, so none of the additional symbols are necessary. The best approximation of the otonal tetrad is the same as the best approximation for a Pythagorean dominant seventh chord and can be written the same way. We see that Sagittal notation, when used for the western standard tuning, is identical to conventional notation.		The second example shows the best approximation for the same chord in [[12edo\|12edo]]. Here, both the syntonic as the septimal comma are tempered out, so none of the additional symbols are necessary. The best approximation of the otonal tetrad is the same as the best approximation for a Pythagorean dominant seventh chord and can be written the same way. We see that Sagittal notation, when used for the western standard tuning, is identical to conventional notation.

	In [[22edo\|22edo]] (third example), the septimal comma is tempered out, but not the syntonic comma. Therefore the symbol at the Bb note can be omitted, but the symbol at the E note has to stay. The ~~diffference~~ between the approximations of Pythagorean and just major third is one 22edo step, which is the best approximation of the syntonic comma in 22edo (more than twice as large as the just syntonic comma, though).		In [[22edo\|22edo]] (third example), the septimal comma is tempered out, but not the syntonic comma. Therefore the symbol at the Bb note can be omitted, but the symbol at the E note has to stay. The difference between the approximations of Pythagorean and just major third is one 22edo step, which is the best approximation of the syntonic comma in 22edo (more than twice as large as the just syntonic comma, though).

	Another property of 22edo is that the undecimal comma is approximated by one step as well . i.e. undecimal and syntonic comma are the same in 22edo, which makes one of the symbols unnecessary again. Overall, only one additional symbol is needed for the notation of 22edo (or, more precisely, two - one up and one down), representing a modification by one 22edo step. The syntonic comma symbol has been defined as the recommended standard symbol for 22edo.		Another property of 22edo is that the undecimal comma is approximated by one step as well . i.e. undecimal and syntonic comma are the same in 22edo, which makes one of the symbols unnecessary again. Overall, only one additional symbol is needed for the notation of 22edo (or, more precisely, two - one up and one down), representing a modification by one 22edo step. The syntonic comma symbol has been defined as the recommended standard symbol for 22edo.
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	=Equal temperaments (2): 11edo scale=		=Equal temperaments (2): 11edo scale=
	As second example, an [[11edo\|11edo]] scale is shown below. The task of defining a standard notation presents another possible pitfall here: 11edo doesn't have a good approximation of the perfect fifth. Building the notation following guideline 2 on series of fifths doesn't make much sense in this case - it would lead to strange effects, even contradicting guideline 1 (the written note E would sound ~~llike~~ a D, the written note Eb like an E). This problem can be avoided using the symbols of a finer division - in the current case of 11edo, the symbols of [[22edo\|22edo]] are used. For the same reason, the recommended symbols for [[16edo\|16edo]] are those of [[48edo\|48edo]].		As second example, an [[11edo\|11edo]] scale is shown below. The task of defining a standard notation presents another possible pitfall here: 11edo doesn't have a good approximation of the perfect fifth. Building the notation following guideline 2 on series of fifths doesn't make much sense in this case - it would lead to strange effects, even contradicting guideline 1 (the written note E would sound like a D, the written note Eb like an E). This problem can be avoided using the symbols of a finer division - in the current case of 11edo, the symbols of [[22edo\|22edo]] are used. For the same reason, the recommended symbols for [[16edo\|16edo]] are those of [[48edo\|48edo]].

	{\| class="wikitable"		{\| class="wikitable"

BudjarnLambeth: Categorised this uncategorised page

2023-04-23T21:14:27Z

Categorised this uncategorised page

← Older revision		Revision as of 21:14, 23 April 2023
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	Observe that both lines show the same scale! The different appearance is caused by enharmonic equivalence effects. The upper line uses only conventional symbols, which is valid but gives the wrong impression of an up and down movement - it is in fact a simple ascending scale made of 11edo steps. The second line requires additional symbols but has a more intuitive appearance.		Observe that both lines show the same scale! The different appearance is caused by enharmonic equivalence effects. The upper line uses only conventional symbols, which is valid but gives the wrong impression of an up and down movement - it is in fact a simple ascending scale made of 11edo steps. The second line requires additional symbols but has a more intuitive appearance.
			[[Category:Sagittal notation]]

Cmloegcmluin: remove link to old version of article

2022-07-22T01:57:49Z

remove link to old version of article

← Older revision		Revision as of 01:57, 22 July 2022
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	This page lists a few elementary examples that hopefully shed some light on the philosophy (and the pitfalls!) of [[Sagittal_notation\|Sagittal notation]]. For a detailed introduction into Sagittal notation the document [[:~~File:Sagittal~~.pdf\|Sagittal.pdf]] is the ~~reference~~.		This page lists a few elementary examples that hopefully shed some light on the philosophy (and the pitfalls!) of [[Sagittal_notation\|Sagittal notation]]. For a detailed introduction into Sagittal notation, see the document [https://sagittal.org/sagittal.pdf Sagittal.pdf] on Sagittal's main website, which is the latest version of the original article which introduced Sagittal to the world, that was published in Xenharmonikôn in 2006.

	=Just intonation: notating a harmonic scale=		=Just intonation: notating a harmonic scale=
	As the introduction ~~[[:File:~~Sagittal.pdf~~\|Sagittal.pdf]]~~ says, Sagittal notation uses a conventional staff on which the natural notes are in a single series of fifths, i.e, in the case of just intonation, in [[3-limit\|Pythagorean tuning]]. The meaning of conventional sharps and flats is Pythagorean as well, i.e. they stand for raising the corresponding note by a [[2187/2048\|Pythagorean chromatic semitone (apotome)]], a "large" semitone 113.7 cents in size.		As the introduction to Sagittal.pdf says, Sagittal notation uses a conventional staff on which the natural notes are in a single series of fifths, i.e, in the case of just intonation, in [[3-limit\|Pythagorean tuning]]. The meaning of conventional sharps and flats is Pythagorean as well, i.e. they stand for raising the corresponding note by a [[2187/2048\|Pythagorean chromatic semitone (apotome)]], a "large" semitone 113.7 cents in size.

	For the notation of notes in higher [[Harmonic_Limit\|limits]], additional symbols are introduced. The intervals these symbols stand for are mostly [[Comma\|commas]] - the maybe most elementary example is the [[81/80\|syntonic comma]] (ratio 81/80, 21.506 cents), the difference between a pythagorean and a just major third, necessary for just intonation in [[5-limit\|5-limit]]. Other elementary commas appearing along the harmonic series are: in [[7-limit\|7-limit]] the [[64/63\|septimal comma or Architas' comma]] (64/63, 27.264 Cents), the difference between a minor and a harmonic seventh, and, in [[11-limit\|11-limit]], the [[33/32\|undecimal comma or al-Farabi quarter-tone]] (33/32, 53.2729 cents), the difference between an undecimal semi-augmented and a perfect fourth.		For the notation of notes in higher [[Harmonic_Limit\|limits]], additional symbols are introduced. The intervals these symbols stand for are mostly [[Comma\|commas]] - the maybe most elementary example is the [[81/80\|syntonic comma]] (ratio 81/80, 21.506 cents), the difference between a pythagorean and a just major third, necessary for just intonation in [[5-limit\|5-limit]]. Other elementary commas appearing along the harmonic series are: in [[7-limit\|7-limit]] the [[64/63\|septimal comma or Architas' comma]] (64/63, 27.264 Cents), the difference between a minor and a harmonic seventh, and, in [[11-limit\|11-limit]], the [[33/32\|undecimal comma or al-Farabi quarter-tone]] (33/32, 53.2729 cents), the difference between an undecimal semi-augmented and a perfect fourth.
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	[[File:SagittalOvertoneSeries.jpg\|alt=SagittalOvertoneSeries.jpg\|SagittalOvertoneSeries.jpg]]		[[File:SagittalOvertoneSeries.jpg\|alt=SagittalOvertoneSeries.jpg\|SagittalOvertoneSeries.jpg]]

	For a complete list of all comma symbols see [http://sagittal.org/ http://sagittal.org/] or ~~[[:File:~~Sagittal.pdf~~\|sagittal.pdf]]~~.		For a complete list of all comma symbols see [http://sagittal.org/ http://sagittal.org/] or Sagittal.pdf.

	=Equal temperaments (1): comparison of notation in different equal temperaments=		=Equal temperaments (1): comparison of notation in different equal temperaments=
	A special attraction of Sagittal notation is that it has been designed to notate both rational intervals (e.g. just intonation) and all kinds equal divisions of the octave. Basic guidelines for the latter, as defined in ~~[[:File:~~Sagittal.pdf~~\|Sagittal.pdf]]~~, are as follows:		A special attraction of Sagittal notation is that it has been designed to notate both rational intervals (e.g. just intonation) and all kinds equal divisions of the octave. Basic guidelines for the latter, as defined in Sagittal.pdf, are as follows:

	<ol><li>An interval in an equal temperament is to be notated in the same way as a just ratio for which the equal interval is the best approximation.</li><li>Conventional staff notation (natural notes, sharps and flats) indicates tones in a series built on the equal division’s best approximation of a fifth.</li></ol>		<ol><li>An interval in an equal temperament is to be notated in the same way as a just ratio for which the equal interval is the best approximation.</li><li>Conventional staff notation (natural notes, sharps and flats) indicates tones in a series built on the equal division’s best approximation of a fifth.</li></ol>

Wolftune: fixed erroneous use of "overtone" vs "harmonic"

2022-01-22T20:33:48Z

fixed erroneous use of "overtone" vs "harmonic"

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	This page lists a few elementary examples that hopefully shed some light on the philosophy (and the pitfalls!) of [[Sagittal_notation\|Sagittal notation]]. For a detailed introduction into Sagittal notation the document [[:File:Sagittal.pdf\|Sagittal.pdf]] is the reference.		This page lists a few elementary examples that hopefully shed some light on the philosophy (and the pitfalls!) of [[Sagittal_notation\|Sagittal notation]]. For a detailed introduction into Sagittal notation the document [[:File:Sagittal.pdf\|Sagittal.pdf]] is the reference.

	=Just intonation: notating ~~an overtone~~ scale=		=Just intonation: notating a harmonic scale=
	As the introduction [[:File:Sagittal.pdf\|Sagittal.pdf]] says, Sagittal notation uses a conventional staff on which the natural notes are in a single series of fifths, i.e, in the case of just intonation, in [[3-limit\|Pythagorean tuning]]. The meaning of conventional sharps and flats is Pythagorean as well, i.e. they stand for raising the corresponding note by a [[2187/2048\|Pythagorean chromatic semitone (apotome)]], a "large" semitone 113.7 cents in size.		As the introduction [[:File:Sagittal.pdf\|Sagittal.pdf]] says, Sagittal notation uses a conventional staff on which the natural notes are in a single series of fifths, i.e, in the case of just intonation, in [[3-limit\|Pythagorean tuning]]. The meaning of conventional sharps and flats is Pythagorean as well, i.e. they stand for raising the corresponding note by a [[2187/2048\|Pythagorean chromatic semitone (apotome)]], a "large" semitone 113.7 cents in size.

	For the notation of notes in higher [[Harmonic_Limit\|limits]], additional symbols are introduced. The intervals these symbols stand for are mostly [[Comma\|commas]] - the maybe most elementary example is the [[81/80\|syntonic comma]] (ratio 81/80, 21.506 cents), the difference between a pythagorean and a just major third, necessary for just intonation in [[5-limit\|5-limit]]. Other elementary commas appearing along the ~~overtone~~ series are: in [[7-limit\|7-limit]] the [[64/63\|septimal comma or Architas' comma]] (64/63, 27.264 Cents), the difference between a minor and a harmonic seventh, and, in [[11-limit\|11-limit]], the [[33/32\|undecimal comma or al-Farabi quarter-tone]] (33/32, 53.2729 cents), the difference between an undecimal semi-augmented and a perfect fourth.		For the notation of notes in higher [[Harmonic_Limit\|limits]], additional symbols are introduced. The intervals these symbols stand for are mostly [[Comma\|commas]] - the maybe most elementary example is the [[81/80\|syntonic comma]] (ratio 81/80, 21.506 cents), the difference between a pythagorean and a just major third, necessary for just intonation in [[5-limit\|5-limit]]. Other elementary commas appearing along the harmonic series are: in [[7-limit\|7-limit]] the [[64/63\|septimal comma or Architas' comma]] (64/63, 27.264 Cents), the difference between a minor and a harmonic seventh, and, in [[11-limit\|11-limit]], the [[33/32\|undecimal comma or al-Farabi quarter-tone]] (33/32, 53.2729 cents), the difference between an undecimal semi-augmented and a perfect fourth.

	With the Sagittal symbos for these three commas, a scale consisting of the ~~overtones~~ 4 to 11 can be written as follows:		With the Sagittal symbos for these three commas, a scale consisting of the harmonics 4 to 11 can be written as follows:

	[[File:SagittalOvertoneSeries.jpg\|alt=SagittalOvertoneSeries.jpg\|SagittalOvertoneSeries.jpg]]		[[File:SagittalOvertoneSeries.jpg\|alt=SagittalOvertoneSeries.jpg\|SagittalOvertoneSeries.jpg]]
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	[[File:SagittalEDOExample.jpg\|alt=SagittalEDOExample.jpg\|SagittalEDOExample.jpg]]		[[File:SagittalEDOExample.jpg\|alt=SagittalEDOExample.jpg\|SagittalEDOExample.jpg]]

	What is displayed is the tetrad consisting of the ~~overtones~~ 4 to 7 (respectively, in equal temperaments, of their approximations), a chord resembling a dominant seventh chord. It appears first in mixed Sagittal notation for just intonation, with the syntonic comma symbol at the E note and the septimal comma symbol at the Bb note.		What is displayed is the tetrad consisting of the harmonics 4 to 7 (respectively, in equal temperaments, of their approximations), a chord resembling a dominant seventh chord. It appears first in mixed Sagittal notation for just intonation, with the syntonic comma symbol at the E note and the septimal comma symbol at the Bb note.

	In equal temperaments, we could, in theory, write it always like that, following guideline 1. But depending on the concrete equal division there will be harmonic equivalences that suggest certain simplifications.		In equal temperaments, we could, in theory, write it always like that, following guideline 1. But depending on the concrete equal division there will be harmonic equivalences that suggest certain simplifications.

PreImportHandFixes>Battaglia01: Mike - changed unnecessary external image to local copy

2018-08-19T19:28:45Z

Mike - changed unnecessary external image to local copy

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	~~External image: files/external-0cc572fe6f9d51891e872b33d70fe9f8<br>~~
	~~: <small><b>WARNING</b>: MediaWiki doesn't have very good support for external images.</small><br>~~
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	: <small>The file name files/external-0cc572fe6f9d51891e872b33d70fe9f8 was autogenerated by Wikispaces from its cache, but is not in this import because Mediawiki requires filenames to have extensions.</small><br>
	: <small>However, you can get the original external URL for this image by clicking "View history", going to the last pre-import history revision, and looking at the original code for this image tag.</small><br>
	~~: <small>The cached image is available for reference in the original xenharmonic export, available~~ [~~https~~:~~//archive~~.~~org/details/xenwikiultimatebackup here].</small>~~
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Wikispaces>FREEZE at 00:00, 17 July 2018

2018-07-17T00:00:00Z

Show changes

Wikispaces>hstraub: Imported revision 557207307 - Original comment:

2015-08-23T12:38:11Z

**Imported revision 557207307 - Original comment: **

Wikispaces>hstraub: Imported revision 557207111 - Original comment:

2015-08-23T12:33:03Z

**Imported revision 557207111 - Original comment: **

← Older revision		Revision as of 06:11, 24 December 2024
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	This page lists a few elementary examples that hopefully shed some light on the philosophy (and the pitfalls!) of [[Sagittal_notation\|Sagittal notation]]. For a detailed introduction into Sagittal notation, see the document [https://sagittal.org/sagittal.pdf Sagittal.pdf] on Sagittal's main website, which is the latest version of the original article which introduced Sagittal to the world, that was published in Xenharmonikôn in 2006.		This page lists a few elementary examples that hopefully shed some light on the philosophy (and the pitfalls!) of [[Sagittal_notation\|Sagittal notation]]. For a detailed introduction into Sagittal notation, see the document [https://sagittal.org/sagittal.pdf Sagittal.pdf] on Sagittal's main website, which is the latest version of the original article which introduced Sagittal to the world, that was published in [[Xenharmonikôn]] in 2006.

	= Just intonation: notating a harmonic scale =		= Just intonation: notating a harmonic scale =
	As the introduction to Sagittal.pdf says, Sagittal notation uses a conventional staff on which the natural notes are in a single series of fifths, i.e, in the case of just intonation, in [[~~3-limit\|~~Pythagorean tuning]]. The meaning of conventional sharps and flats is Pythagorean as well, i.e. they stand for raising the corresponding note by a [[2187/2048\|Pythagorean chromatic semitone (apotome)]], a "large" semitone 113.7 cents in size.		As the introduction to Sagittal.pdf says, Sagittal notation uses a conventional staff on which the natural notes are in a single series of fifths, i.e, in the case of [[just intonation]], in [[Pythagorean tuning]]. The meaning of conventional sharps and flats is Pythagorean as well, i.e. they stand for raising the corresponding note by a [[2187/2048\|Pythagorean chromatic semitone (apotome)]], a "large" semitone 113.7 [[cents]] in size.

	For the notation of notes in higher [[Harmonic_Limit\|limits]], additional symbols are introduced. The intervals these symbols stand for are mostly [[Comma\|commas]]—the maybe most elementary example is the [[81/80\|syntonic comma]] (ratio 81/80, 21.506 cents), the difference between a pythagorean and a just major third, necessary for just intonation in [[5-limit\|5-limit]]. Other elementary commas appearing along the harmonic series are: in [[7-limit\|7-limit]] the [[64/63\|septimal comma or Architas' comma]] (64/63, 27.264 Cents), the difference between a minor and a harmonic seventh, and, in [[11-limit\|11-limit]], the [[33/32\|undecimal comma or al-Farabi quarter-tone]] (33/32, 53.2729 cents), the difference between an undecimal semi-augmented and a perfect fourth.		For the notation of notes in higher [[Harmonic_Limit\|limits]], additional symbols are introduced. The intervals these symbols stand for are mostly [[Comma\|commas]]—the maybe most elementary example is the [[81/80\|syntonic comma]] (ratio 81/80, 21.506 cents), the difference between a pythagorean and a just major third, necessary for just intonation in [[5-limit\|5-limit]]. Other elementary commas appearing along the [[harmonic series]] are: in [[7-limit\|7-limit]] the [[64/63\|septimal comma or Architas' comma]] (64/63, 27.264 Cents), the difference between a minor and a harmonic seventh, and, in [[11-limit\|11-limit]], the [[33/32\|undecimal comma or al-Farabi quarter-tone]] (33/32, 53.2729 cents), the difference between an undecimal semi-augmented and a perfect fourth.

	With the Sagittal symbos for these three commas, a scale consisting of the harmonics 4 to 11 can be written as follows:		With the Sagittal symbos for these three commas, a scale consisting of the harmonics 4 to 11 can be written as follows:
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	<ol><li>An interval in an equal temperament is to be notated in the same way as a just ratio for which the equal interval is the best approximation.</li><li>Conventional staff notation (natural notes, sharps and flats) indicates tones in a series built on the equal division’s best approximation of a fifth.</li></ol>		<ol><li>An interval in an equal temperament is to be notated in the same way as a just ratio for which the equal interval is the best approximation.</li><li>Conventional staff notation (natural notes, sharps and flats) indicates tones in a series built on the equal division’s best approximation of a fifth.</li></ol>

	There are a number of details to be observed here. First and most important point is that a notation defined this way is highly ambiguous. Every note of an equal-tempered system is best approximation for a whole range of just ratios—even an unlimited number of them, in fact. There are, in other words, extremely many [https://en.wikipedia.org/wiki/Enharmonic enharmonic equivalences]. This is not necessarily a problem—enharmonic equivalences exist anyway, in conventional non-microtonal notation, too. Yet there are certain simplifications it make sense to define - certain commas, for example, vanish completely in some EDOs, as the syntonic comma in [[Meantone\|meantone]] systems. The corresponding symbol is obviously superfluous in this case. Other cases of enharmonic equivalence are less obvious. The developers for Sagittal notation have defined.a standard selection of symbols to be used for each equal system; these definitions have the character of recommendations.		There are a number of details to be observed here. First and most important point is that a notation defined this way is highly ambiguous. Every note of an equal-tempered system is best approximation for a whole range of just ratios—even an unlimited number of them, in fact. There are, in other words, extremely many [https://en.wikipedia.org/wiki/Enharmonic enharmonic equivalences]. This is not necessarily a problem—enharmonic equivalences exist anyway, in conventional non-microtonal notation, too. Yet there are certain simplifications it make sense to define - certain commas, for example, vanish completely in some EDOs, as the [[syntonic comma]] in [[Meantone\|meantone]] systems. The corresponding symbol is obviously superfluous in this case. Other cases of enharmonic equivalence are less obvious. The developers for Sagittal notation have defined. A standard selection of symbols to be used for each equal system; these definitions have the character of recommendations.

	Below is an example how the standard notation systems for some equal termperaments differ.		Below is an example how the standard notation systems for some equal termperaments differ.
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	[[File:SagittalEDOExample.jpg\|alt=SagittalEDOExample.jpg\|SagittalEDOExample.jpg]]		[[File:SagittalEDOExample.jpg\|alt=SagittalEDOExample.jpg\|SagittalEDOExample.jpg]]

	What is displayed is the tetrad consisting of the harmonics 4 to 7 (respectively, in equal temperaments, of their approximations), a chord resembling a dominant seventh chord. It appears first in mixed Sagittal notation for just intonation, with the syntonic comma symbol at the E note and the septimal comma symbol at the Bb note.		What is displayed is the [[tetrad]] consisting of the harmonics 4 to 7 (respectively, in equal temperaments, of their approximations), a chord resembling a dominant seventh chord. It appears first in mixed Sagittal notation for just intonation, with the syntonic comma symbol at the E note and the septimal comma symbol at the Bb note.

	In equal temperaments, we could, in theory, write it always like that, following guideline 1. But depending on the concrete equal division there will be harmonic equivalences that suggest certain simplifications.		In equal temperaments, we could, in theory, write it always like that, following guideline 1. But depending on the concrete equal division there will be harmonic equivalences that suggest certain simplifications.

	The second example shows the best approximation for the same chord in [[12edo\|12edo]]. Here, both the syntonic as the septimal comma are tempered out, so none of the additional symbols are necessary. The best approximation of the otonal tetrad is the same as the best approximation for a Pythagorean dominant seventh chord and can be written the same way. We see that Sagittal notation, when used for the western standard tuning, is identical to conventional notation.		The second example shows the best approximation for the same chord in [[12edo\|12edo]]. Here, both the syntonic as the septimal comma are tempered out, so none of the additional symbols are necessary. The best approximation of the [[otonal]] tetrad is the same as the best approximation for a Pythagorean dominant seventh chord and can be written the same way. We see that Sagittal notation, when used for the western standard tuning, is identical to conventional notation.

	In [[22edo\|22edo]] (third example), the septimal comma is tempered out, but not the syntonic comma. Therefore the symbol at the Bb note can be omitted, but the symbol at the E note has to stay. The difference between the approximations of Pythagorean and just major third is one 22edo step, which is the best approximation of the syntonic comma in 22edo (more than twice as large as the just syntonic comma, though).		In [[22edo\|22edo]] (third example), the septimal comma is tempered out, but not the syntonic comma. Therefore the symbol at the Bb note can be omitted, but the symbol at the E note has to stay. The difference between the approximations of Pythagorean and just major third is one 22edo step, which is the best approximation of the syntonic comma in 22edo (more than twice as large as the just syntonic comma, though).
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	Another property of 22edo is that the undecimal comma is approximated by one step as well . i.e. undecimal and syntonic comma are the same in 22edo, which makes one of the symbols unnecessary again. Overall, only one additional symbol is needed for the notation of 22edo (or, more precisely, two—one up and one down), representing a modification by one 22edo step. The syntonic comma symbol has been defined as the recommended standard symbol for 22edo.		Another property of 22edo is that the undecimal comma is approximated by one step as well . i.e. undecimal and syntonic comma are the same in 22edo, which makes one of the symbols unnecessary again. Overall, only one additional symbol is needed for the notation of 22edo (or, more precisely, two—one up and one down), representing a modification by one 22edo step. The syntonic comma symbol has been defined as the recommended standard symbol for 22edo.

	Finally, [[31edo\|31edo]] (last example) is, like 12edo, a meantone system and thus tempers out the syntonic comma—but not the septimal comma, exactly opposite to 22edo in this aspect. The septimal comma, in turn coincides with the undecimal comma here—both are approximated by one 31edo step. The recommended standard symbol for this interval in 31edo is the symbol for the undecimal comma (quartertone), so the tetrad in question could be notated with a quartertone symbol at the B♭. In our example, the enharmonic equivalent A♯ is used instead, which is possible because 31edo is a [[Meantone_family#Septimal meantone\|septimal meantone]] system.		Finally, [[31edo\|31edo]] (last example) is, like 12edo, a meantone system and thus tempers out the syntonic comma—but not the septimal comma, exactly opposite to 22edo in this aspect. The septimal comma, in turn coincides with the undecimal comma here—both are approximated by one 31edo step. The recommended standard symbol for this interval in 31edo is the symbol for the undecimal comma ([[quartertone]]), so the tetrad in question could be notated with a quartertone symbol at the B♭. In our example, the enharmonic equivalent A♯ is used instead, which is possible because 31edo is a [[Meantone_family#Septimal meantone\|septimal meantone]] system.

	= Equal temperaments (2): 11edo scale =		= Equal temperaments (2): 11edo scale =

← Older revision		Revision as of 14:21, 10 October 2024
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	This page lists a few elementary examples that hopefully shed some light on the philosophy (and the pitfalls!) of [[Sagittal_notation\|Sagittal notation]]. For a detailed introduction into Sagittal notation, see the document [https://sagittal.org/sagittal.pdf Sagittal.pdf] on Sagittal's main website, which is the latest version of the original article which introduced Sagittal to the world, that was published in Xenharmonikôn in 2006.		This page lists a few elementary examples that hopefully shed some light on the philosophy (and the pitfalls!) of [[Sagittal_notation\|Sagittal notation]]. For a detailed introduction into Sagittal notation, see the document [https://sagittal.org/sagittal.pdf Sagittal.pdf] on Sagittal's main website, which is the latest version of the original article which introduced Sagittal to the world, that was published in Xenharmonikôn in 2006.

	=Just intonation: notating a harmonic scale=		= Just intonation: notating a harmonic scale =
	As the introduction to Sagittal.pdf says, Sagittal notation uses a conventional staff on which the natural notes are in a single series of fifths, i.e, in the case of just intonation, in [[3-limit\|Pythagorean tuning]]. The meaning of conventional sharps and flats is Pythagorean as well, i.e. they stand for raising the corresponding note by a [[2187/2048\|Pythagorean chromatic semitone (apotome)]], a "large" semitone 113.7 cents in size.		As the introduction to Sagittal.pdf says, Sagittal notation uses a conventional staff on which the natural notes are in a single series of fifths, i.e, in the case of just intonation, in [[3-limit\|Pythagorean tuning]]. The meaning of conventional sharps and flats is Pythagorean as well, i.e. they stand for raising the corresponding note by a [[2187/2048\|Pythagorean chromatic semitone (apotome)]], a "large" semitone 113.7 cents in size.

	For the notation of notes in higher [[Harmonic_Limit\|limits]], additional symbols are introduced. The intervals these symbols stand for are mostly [[Comma\|commas]] - the maybe most elementary example is the [[81/80\|syntonic comma]] (ratio 81/80, 21.506 cents), the difference between a pythagorean and a just major third, necessary for just intonation in [[5-limit\|5-limit]]. Other elementary commas appearing along the harmonic series are: in [[7-limit\|7-limit]] the [[64/63\|septimal comma or Architas' comma]] (64/63, 27.264 Cents), the difference between a minor and a harmonic seventh, and, in [[11-limit\|11-limit]], the [[33/32\|undecimal comma or al-Farabi quarter-tone]] (33/32, 53.2729 cents), the difference between an undecimal semi-augmented and a perfect fourth.		For the notation of notes in higher [[Harmonic_Limit\|limits]], additional symbols are introduced. The intervals these symbols stand for are mostly [[Comma\|commas]]—the maybe most elementary example is the [[81/80\|syntonic comma]] (ratio 81/80, 21.506 cents), the difference between a pythagorean and a just major third, necessary for just intonation in [[5-limit\|5-limit]]. Other elementary commas appearing along the harmonic series are: in [[7-limit\|7-limit]] the [[64/63\|septimal comma or Architas' comma]] (64/63, 27.264 Cents), the difference between a minor and a harmonic seventh, and, in [[11-limit\|11-limit]], the [[33/32\|undecimal comma or al-Farabi quarter-tone]] (33/32, 53.2729 cents), the difference between an undecimal semi-augmented and a perfect fourth.

	With the Sagittal symbos for these three commas, a scale consisting of the harmonics 4 to 11 can be written as follows:		With the Sagittal symbos for these three commas, a scale consisting of the harmonics 4 to 11 can be written as follows:
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	For a complete list of all comma symbols see [http://sagittal.org/ http://sagittal.org/] or Sagittal.pdf.		For a complete list of all comma symbols see [http://sagittal.org/ http://sagittal.org/] or Sagittal.pdf.

	=Equal temperaments (1): comparison of notation in different equal temperaments=		= Equal temperaments (1): comparison of notation in different equal temperaments =
	A special attraction of Sagittal notation is that it has been designed to notate both rational intervals (e.g. just intonation) and all kinds equal divisions of the octave. Basic guidelines for the latter, as defined in Sagittal.pdf, are as follows:		A special attraction of Sagittal notation is that it has been designed to notate both rational intervals (e.g. just intonation) and all kinds equal divisions of the octave. Basic guidelines for the latter, as defined in Sagittal.pdf, are as follows:

	<ol><li>An interval in an equal temperament is to be notated in the same way as a just ratio for which the equal interval is the best approximation.</li><li>Conventional staff notation (natural notes, sharps and flats) indicates tones in a series built on the equal division’s best approximation of a fifth.</li></ol>		<ol><li>An interval in an equal temperament is to be notated in the same way as a just ratio for which the equal interval is the best approximation.</li><li>Conventional staff notation (natural notes, sharps and flats) indicates tones in a series built on the equal division’s best approximation of a fifth.</li></ol>

	There are a number of details to be observed here. First and most important point is that a notation defined this way is highly ambiguous. Every note of an equal-tempered system is best approximation for a whole range of just ratios - even an unlimited number of them, in fact. There are, in other words, extremely many [https://en.wikipedia.org/wiki/Enharmonic enharmonic equivalences]. This is not necessarily a problem - enharmonic equivalences exist anyway, in conventional non-microtonal notation, too. Yet there are certain simplifications it make sense to define - certain commas, for example, vanish completely in some EDOs, as the syntonic comma in [[Meantone\|meantone]] systems. The corresponding symbol is obviously superfluous in this case. Other cases of enharmonic equivalence are less obvious. The developers for Sagittal notation have defined.a standard selection of symbols to be used for each equal system; these definitions have the character of recommendations.		There are a number of details to be observed here. First and most important point is that a notation defined this way is highly ambiguous. Every note of an equal-tempered system is best approximation for a whole range of just ratios—even an unlimited number of them, in fact. There are, in other words, extremely many [https://en.wikipedia.org/wiki/Enharmonic enharmonic equivalences]. This is not necessarily a problem—enharmonic equivalences exist anyway, in conventional non-microtonal notation, too. Yet there are certain simplifications it make sense to define - certain commas, for example, vanish completely in some EDOs, as the syntonic comma in [[Meantone\|meantone]] systems. The corresponding symbol is obviously superfluous in this case. Other cases of enharmonic equivalence are less obvious. The developers for Sagittal notation have defined.a standard selection of symbols to be used for each equal system; these definitions have the character of recommendations.

	Below is an example how the standard notation systems for some equal termperaments differ.		Below is an example how the standard notation systems for some equal termperaments differ.
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	In [[22edo\|22edo]] (third example), the septimal comma is tempered out, but not the syntonic comma. Therefore the symbol at the Bb note can be omitted, but the symbol at the E note has to stay. The difference between the approximations of Pythagorean and just major third is one 22edo step, which is the best approximation of the syntonic comma in 22edo (more than twice as large as the just syntonic comma, though).		In [[22edo\|22edo]] (third example), the septimal comma is tempered out, but not the syntonic comma. Therefore the symbol at the Bb note can be omitted, but the symbol at the E note has to stay. The difference between the approximations of Pythagorean and just major third is one 22edo step, which is the best approximation of the syntonic comma in 22edo (more than twice as large as the just syntonic comma, though).

	Another property of 22edo is that the undecimal comma is approximated by one step as well . i.e. undecimal and syntonic comma are the same in 22edo, which makes one of the symbols unnecessary again. Overall, only one additional symbol is needed for the notation of 22edo (or, more precisely, two - one up and one down), representing a modification by one 22edo step. The syntonic comma symbol has been defined as the recommended standard symbol for 22edo.		Another property of 22edo is that the undecimal comma is approximated by one step as well . i.e. undecimal and syntonic comma are the same in 22edo, which makes one of the symbols unnecessary again. Overall, only one additional symbol is needed for the notation of 22edo (or, more precisely, two—one up and one down), representing a modification by one 22edo step. The syntonic comma symbol has been defined as the recommended standard symbol for 22edo.

	Finally, [[31edo\|31edo]] (last example) is like 12edo a meantone system and thus tempers out the syntonic comma - but not the septimal comma, exactly opposite to 22edo in this aspect. The septimal comma, in turn coincides with the undecimal comma here - both are approximated by one 31edo step. The recommended standard symbol for this interval in 31edo is the symbol for the undecimal comma (quartertone), so the tetrad in question could be notated with a quartertone symbol at the Bb. In our example, the enharmonic equivalent A# is used instead, which is possible because 31edo is a [[Meantone_family#Septimal meantone\|septimal meantone]] system.		Finally, [[31edo\|31edo]] (last example) is, like 12edo, a meantone system and thus tempers out the syntonic comma—but not the septimal comma, exactly opposite to 22edo in this aspect. The septimal comma, in turn coincides with the undecimal comma here—both are approximated by one 31edo step. The recommended standard symbol for this interval in 31edo is the symbol for the undecimal comma (quartertone), so the tetrad in question could be notated with a quartertone symbol at the B♭. In our example, the enharmonic equivalent A♯ is used instead, which is possible because 31edo is a [[Meantone_family#Septimal meantone\|septimal meantone]] system.

	=Equal temperaments (2): 11edo scale=		= Equal temperaments (2): 11edo scale =
	As second example, an [[11edo\|11edo]] scale is shown below. The task of defining a standard notation presents another possible pitfall here: 11edo doesn't have a good approximation of the perfect fifth. Building the notation following guideline 2 on series of fifths doesn't make much sense in this case - it would lead to strange effects, even contradicting guideline 1 (the written note E would sound like a D, the written note Eb like an E). This problem can be avoided using the symbols of a finer division - in the current case of 11edo, the symbols of [[22edo\|22edo]] are used. For the same reason, the recommended symbols for [[16edo\|16edo]] are those of [[48edo\|48edo]].		As second example, an [[11edo\|11edo]] scale is shown below. The task of defining a standard notation presents another possible pitfall here: 11edo doesn't have a good approximation of the perfect fifth. Building the notation following guideline 2 on series of fifths doesn't make much sense in this case—it would lead to strange effects, even contradicting guideline 1 (the written note E would sound like a D, the written note E♭ like an E). This problem can be avoided using the symbols of a finer division—in the current case of 11edo, the symbols of [[22edo\|22edo]] are used. For the same reason, the recommended symbols for [[16edo\|16edo]] are those of [[48edo\|48edo]].

	{\| class="wikitable"		{\| class="wikitable"
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	<span style="background-color: #ffffff;">(Rendering Juhani Nuorvala)</span>		<span style="background-color: #ffffff;">(Rendering Juhani Nuorvala)</span>

	Observe that both lines show the same scale! The different appearance is caused by enharmonic equivalence effects. The upper line uses only conventional symbols, which is valid but gives the wrong impression of an up and down movement - it is in fact a simple ascending scale made of 11edo steps. The second line requires additional symbols but has a more intuitive appearance.		Observe that both lines show the same scale! The different appearance is caused by enharmonic equivalence effects. The upper line uses only conventional symbols, which is valid but gives the wrong impression of an up and down movement—it is in fact a simple ascending scale made of 11edo steps. The second line requires additional symbols but has a more intuitive appearance.
	[[Category:Sagittal notation]]		[[Category:Sagittal notation]]

← Older revision		Revision as of 12:38, 23 August 2015
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:hstraub\|hstraub]] and made on <tt>2015-08-23 12:33:03 UTC</tt>.<br>		: This revision was by author [[User:hstraub\|hstraub]] and made on <tt>2015-08-23 12:38:11 UTC</tt>.<br>
	: The original revision id was <tt>~~557207111~~</tt>.<br>		: The original revision id was <tt>557207307</tt>.<br>
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	In equal temperaments, we could, in theory, write it always like that, following guideline 1. But depending on the concrete equal division there will be harmonic equivalences that suggest certain simplifications.		In equal temperaments, we could, in theory, write it always like that, following guideline 1. But depending on the concrete equal division there will be harmonic equivalences that suggest certain simplifications.

	The second example shows the best approximation for the same chord in [[12edo]]. Here, both the syntonic as the septimal comma are tempered out, so none of the additional symbols is necessary. The best approximation of the otonal tetrad is the same as the best approximation for a Pythagorean dominant seventh chord and can be written the same way. We see that Sagittal notation, when used for the western standard tuning, is identical to ~~conventioal~~ notation.		The second example shows the best approximation for the same chord in [[12edo]]. Here, both the syntonic as the septimal comma are tempered out, so none of the additional symbols are necessary. The best approximation of the otonal tetrad is the same as the best approximation for a Pythagorean dominant seventh chord and can be written the same way. We see that Sagittal notation, when used for the western standard tuning, is identical to conventional notation.

	In [[22edo]] (third example), the septimal comma is tempered out, but not the syntonic comma. Therefore the symbol at the Bb note can be omitted, but the symbol at the E note has to stay. The diffference between the approximations of Pythagorean and just major third is one 22edo step, which is the best approximation of the syntonic comma in 22edo (more than twice as large as the just syntonic comma, though).		In [[22edo]] (third example), the septimal comma is tempered out, but not the syntonic comma. Therefore the symbol at the Bb note can be omitted, but the symbol at the E note has to stay. The diffference between the approximations of Pythagorean and just major third is one 22edo step, which is the best approximation of the syntonic comma in 22edo (more than twice as large as the just syntonic comma, though).
	Another property of 22edo is that the undecimal comma is approximated by one step as well . i.e. undecimal and syntonic comma are the same in 22edo, which makes one of the symbols unnecessary again. Overall, only an additional symbol is ~~used~~ for the notation of 22edo ~~needed~~ (or, more precisely, two - one up and one down), ~~which represents~~ a modification by one 22edo step. The syntonic comma symbol has been defined as the recommended standard symbol for 22edo.		Another property of 22edo is that the undecimal comma is approximated by one step as well . i.e. undecimal and syntonic comma are the same in 22edo, which makes one of the symbols unnecessary again. Overall, only one additional symbol is needed for the notation of 22edo (or, more precisely, two - one up and one down), representing a modification by one 22edo step. The syntonic comma symbol has been defined as the recommended standard symbol for 22edo.

	Finally, [[31edo]] (last example) is like 12edo a meantone system and thus tempers out the syntonic comma - but not the septimal comma, exactly opposite to 22edo in this aspect. The septimal comma, in turn coincides with the undecimal comma here - both are approximated by one 31edo step. The recommended standard symbol for this interval in 31edo is the symbol for the undecimal comma (quartertone), so the tetrad in question could be notated with a quartertone symbol at the Bb. In our example, the enharmonic equivalent A# is used instead, which is possible because 31edo is a [[Meantone family#Septimal%20meantone\|septimal meantone]] system.		Finally, [[31edo]] (last example) is like 12edo a meantone system and thus tempers out the syntonic comma - but not the septimal comma, exactly opposite to 22edo in this aspect. The septimal comma, in turn coincides with the undecimal comma here - both are approximated by one 31edo step. The recommended standard symbol for this interval in 31edo is the symbol for the undecimal comma (quartertone), so the tetrad in question could be notated with a quartertone symbol at the Bb. In our example, the enharmonic equivalent A# is used instead, which is possible because 31edo is a [[Meantone family#Septimal%20meantone\|septimal meantone]] system.

	=Equal temperaments (2): 11edo scale=		=Equal temperaments (2): 11edo scale=
	As second example, an [[11edo]] scale is shown below. The task of defining a standard notation presents another possible pitfall here: 11edo doesn't have a good approximation of the perfect fifth. Building the notation following guideline 2 on series of fifths doesn't make much sense in this case - it would lead to strange effects, even contradicting guideline 1 (the written note E would sound llike a D, the written note Eb like an E). This problem can be avoided using the symbols of a finer division - in the current case of 11edo, the symbols ~~for~~ [[22edo]] are used. For the same reason, the recommended symbols for [[16edo]] are those of [[48edo]].		As second example, an [[11edo]] scale is shown below. The task of defining a standard notation presents another possible pitfall here: 11edo doesn't have a good approximation of the perfect fifth. Building the notation following guideline 2 on series of fifths doesn't make much sense in this case - it would lead to strange effects, even contradicting guideline 1 (the written note E would sound llike a D, the written note Eb like an E). This problem can be avoided using the symbols of a finer division - in the current case of 11edo, the symbols of [[22edo]] are used. For the same reason, the recommended symbols for [[16edo]] are those of [[48edo]].
	[[image:xenharmonie/Sagittal11EDO.jpg width="496" height="283" caption="Sagittal11EDO.jpg"]]		[[image:xenharmonie/Sagittal11EDO.jpg width="496" height="283" caption="Sagittal11EDO.jpg"]]
	<span style="background-color: #ffffff;">(Rendering Juhani Nuorvala)</span>		<span style="background-color: #ffffff;">(Rendering Juhani Nuorvala)</span>
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	In equal temperaments, we could, in theory, write it always like that, following guideline 1. But depending on the concrete equal division there will be harmonic equivalences that suggest certain simplifications.<br />		In equal temperaments, we could, in theory, write it always like that, following guideline 1. But depending on the concrete equal division there will be harmonic equivalences that suggest certain simplifications.<br />
	<br />		<br />
	The second example shows the best approximation for the same chord in <a class="wiki_link" href="/12edo">12edo</a>. Here, both the syntonic as the septimal comma are tempered out, so none of the additional symbols is necessary. The best approximation of the otonal tetrad is the same as the best approximation for a Pythagorean dominant seventh chord and can be written the same way. We see that Sagittal notation, when used for the western standard tuning, is identical to ~~conventioal~~ notation.<br />		The second example shows the best approximation for the same chord in <a class="wiki_link" href="/12edo">12edo</a>. Here, both the syntonic as the septimal comma are tempered out, so none of the additional symbols are necessary. The best approximation of the otonal tetrad is the same as the best approximation for a Pythagorean dominant seventh chord and can be written the same way. We see that Sagittal notation, when used for the western standard tuning, is identical to conventional notation.<br />
	<br />		<br />
	In <a class="wiki_link" href="/22edo">22edo</a> (third example), the septimal comma is tempered out, but not the syntonic comma. Therefore the symbol at the Bb note can be omitted, but the symbol at the E note has to stay. The diffference between the approximations of Pythagorean and just major third is one 22edo step, which is the best approximation of the syntonic comma in 22edo (more than twice as large as the just syntonic comma, though).<br />		In <a class="wiki_link" href="/22edo">22edo</a> (third example), the septimal comma is tempered out, but not the syntonic comma. Therefore the symbol at the Bb note can be omitted, but the symbol at the E note has to stay. The diffference between the approximations of Pythagorean and just major third is one 22edo step, which is the best approximation of the syntonic comma in 22edo (more than twice as large as the just syntonic comma, though).<br />
	Another property of 22edo is that the undecimal comma is approximated by one step as well . i.e. undecimal and syntonic comma are the same in 22edo, which makes one of the symbols unnecessary again. Overall, only an additional symbol is ~~used~~ for the notation of 22edo ~~needed~~ (or, more precisely, two - one up and one down), ~~which represents~~ a modification by one 22edo step. The syntonic comma symbol has been defined as the recommended standard symbol for 22edo.<br />		Another property of 22edo is that the undecimal comma is approximated by one step as well . i.e. undecimal and syntonic comma are the same in 22edo, which makes one of the symbols unnecessary again. Overall, only one additional symbol is needed for the notation of 22edo (or, more precisely, two - one up and one down), representing a modification by one 22edo step. The syntonic comma symbol has been defined as the recommended standard symbol for 22edo.<br />
	<br />		<br />
	Finally, <a class="wiki_link" href="/31edo">31edo</a> (last example) is like 12edo a meantone system and thus tempers out the syntonic comma - but not the septimal comma, exactly opposite to 22edo in this aspect. The septimal comma, in turn coincides with the undecimal comma here - both are approximated by one 31edo step. The recommended standard symbol for this interval in 31edo is the symbol for the undecimal comma (quartertone), so the tetrad in question could be notated with a quartertone symbol at the Bb. In our example, the enharmonic equivalent A# is used instead, which is possible because 31edo is a <a class="wiki_link" href="/Meantone%20family#Septimal%20meantone">septimal meantone</a> system.<br />		Finally, <a class="wiki_link" href="/31edo">31edo</a> (last example) is like 12edo a meantone system and thus tempers out the syntonic comma - but not the septimal comma, exactly opposite to 22edo in this aspect. The septimal comma, in turn coincides with the undecimal comma here - both are approximated by one 31edo step. The recommended standard symbol for this interval in 31edo is the symbol for the undecimal comma (quartertone), so the tetrad in question could be notated with a quartertone symbol at the Bb. In our example, the enharmonic equivalent A# is used instead, which is possible because 31edo is a <a class="wiki_link" href="/Meantone%20family#Septimal%20meantone">septimal meantone</a> system.<br />
	<br />		<br />
	<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Equal temperaments (2): 11edo scale"></a><!-- ws:end:WikiTextHeadingRule:4 -->Equal temperaments (2): 11edo scale</h1>		<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Equal temperaments (2): 11edo scale"></a><!-- ws:end:WikiTextHeadingRule:4 -->Equal temperaments (2): 11edo scale</h1>
	As second example, an <a class="wiki_link" href="/11edo">11edo</a> scale is shown below. The task of defining a standard notation presents another possible pitfall here: 11edo doesn't have a good approximation of the perfect fifth. Building the notation following guideline 2 on series of fifths doesn't make much sense in this case - it would lead to strange effects, even contradicting guideline 1 (the written note E would sound llike a D, the written note Eb like an E). This problem can be avoided using the symbols of a finer division - in the current case of 11edo, the symbols ~~for~~ <a class="wiki_link" href="/22edo">22edo</a> are used. For the same reason, the recommended symbols for <a class="wiki_link" href="/16edo">16edo</a> are those of <a class="wiki_link" href="/48edo">48edo</a>.<br />		As second example, an <a class="wiki_link" href="/11edo">11edo</a> scale is shown below. The task of defining a standard notation presents another possible pitfall here: 11edo doesn't have a good approximation of the perfect fifth. Building the notation following guideline 2 on series of fifths doesn't make much sense in this case - it would lead to strange effects, even contradicting guideline 1 (the written note E would sound llike a D, the written note Eb like an E). This problem can be avoided using the symbols of a finer division - in the current case of 11edo, the symbols of <a class="wiki_link" href="/22edo">22edo</a> are used. For the same reason, the recommended symbols for <a class="wiki_link" href="/16edo">16edo</a> are those of <a class="wiki_link" href="/48edo">48edo</a>.<br />
	<!-- ws:start:WikiTextLocalImageRule:19:&lt;img src=&quot;http://xenharmonie.wikispaces.com/file/view/Sagittal11EDO.jpg/556903777/496x283/Sagittal11EDO.jpg&quot; alt=&quot;Sagittal11EDO.jpg&quot; title=&quot;Sagittal11EDO.jpg&quot; style=&quot;height: 283px; width: 496px;&quot; /&gt; --><table class="captionBox"><tr><td class="captionedImage"><img src="http://xenharmonie.wikispaces.com/file/view/Sagittal11EDO.jpg/556903777/496x283/Sagittal11EDO.jpg" alt="Sagittal11EDO.jpg" title="Sagittal11EDO.jpg" style="height: 283px; width: 496px;" /></td></tr><tr><td class="imageCaption">Sagittal11EDO.jpg</td></tr></table><!-- ws:end:WikiTextLocalImageRule:19 --><br />		<!-- ws:start:WikiTextLocalImageRule:19:&lt;img src=&quot;http://xenharmonie.wikispaces.com/file/view/Sagittal11EDO.jpg/556903777/496x283/Sagittal11EDO.jpg&quot; alt=&quot;Sagittal11EDO.jpg&quot; title=&quot;Sagittal11EDO.jpg&quot; style=&quot;height: 283px; width: 496px;&quot; /&gt; --><table class="captionBox"><tr><td class="captionedImage"><img src="http://xenharmonie.wikispaces.com/file/view/Sagittal11EDO.jpg/556903777/496x283/Sagittal11EDO.jpg" alt="Sagittal11EDO.jpg" title="Sagittal11EDO.jpg" style="height: 283px; width: 496px;" /></td></tr><tr><td class="imageCaption">Sagittal11EDO.jpg</td></tr></table><!-- ws:end:WikiTextLocalImageRule:19 --><br />
	<span style="background-color: #ffffff;">(Rendering Juhani Nuorvala)</span><br />		<span style="background-color: #ffffff;">(Rendering Juhani Nuorvala)</span><br />
	<br />		<br />
	Observe that both lines show the same scale! The different appearance is caused by enharmonic equivalence effects. The upper line uses only conventional symbols, which is valid but gives the wrong impression of an up and down movement - it is in fact a simple ascending scale made of 11edo steps. The second line requires additional symbols but has a more intuitive appearance.</body></html></pre></div>		Observe that both lines show the same scale! The different appearance is caused by enharmonic equivalence effects. The upper line uses only conventional symbols, which is valid but gives the wrong impression of an up and down movement - it is in fact a simple ascending scale made of 11edo steps. The second line requires additional symbols but has a more intuitive appearance.</body></html></pre></div>

← Older revision		Revision as of 12:33, 23 August 2015
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	<h2>IMPORTED REVISION FROM WIKISPACES</h2>		<h2>IMPORTED REVISION FROM WIKISPACES</h2>
	This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>		This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
	: This revision was by author [[User:hstraub\|hstraub]] and made on <tt>2015-08-23 12:01:49 UTC</tt>.<br>		: This revision was by author [[User:hstraub\|hstraub]] and made on <tt>2015-08-23 12:33:03 UTC</tt>.<br>
	: The original revision id was <tt>~~557205817~~</tt>.<br>		: The original revision id was <tt>557207111</tt>.<br>
	: The revision comment was: <tt></tt><br>		: The revision comment was: <tt></tt><br>
	The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>		The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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	[[image:SagittalEDOExample.jpg]]		[[image:SagittalEDOExample.jpg]]

	What is displayed is the tetrad consisting of the overtones 4 to 7 (respectively, in equal temperaments, of their approximations), a chord resembling a dominant seventh chord. ~~First~~ appears in mixed Sagittal notation for just intonation, with the syntonic comma symbol at the E note and the septimal ~~vomma~~ symbol at the Bb note.		What is displayed is the tetrad consisting of the overtones 4 to 7 (respectively, in equal temperaments, of their approximations), a chord resembling a dominant seventh chord. It appears first in mixed Sagittal notation for just intonation, with the syntonic comma symbol at the E note and the septimal comma symbol at the Bb note.

	In equal temperaments, we could, in theory, write it always like that, following guideline 1. But depending on the concrete equal division there will be harmonic equivalences that suggest certain simplifications.		In equal temperaments, we could, in theory, write it always like that, following guideline 1. But depending on the concrete equal division there will be harmonic equivalences that suggest certain simplifications.
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	<br />		<br />
	What is displayed is the tetrad consisting of the overtones 4 to 7 (respectively, in equal temperaments, of their approximations), a chord resembling a dominant seventh chord. ~~First~~ appears in mixed Sagittal notation for just intonation, with the syntonic comma symbol at the E note and the septimal ~~vomma~~ symbol at the Bb note.<br />		What is displayed is the tetrad consisting of the overtones 4 to 7 (respectively, in equal temperaments, of their approximations), a chord resembling a dominant seventh chord. It appears first in mixed Sagittal notation for just intonation, with the syntonic comma symbol at the E note and the septimal comma symbol at the Bb note.<br />
	<br />		<br />
	In equal temperaments, we could, in theory, write it always like that, following guideline 1. But depending on the concrete equal division there will be harmonic equivalences that suggest certain simplifications.<br />		In equal temperaments, we could, in theory, write it always like that, following guideline 1. But depending on the concrete equal division there will be harmonic equivalences that suggest certain simplifications.<br />