Introductory examples in Sagittal notation: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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# Conventional staff notation (natural notes, sharps and flats) indicates tones in a series built on the equal division’s best approximation of a fifth.

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 ~~equal systems~~, as the syntonic comma in [[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]] 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.

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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.

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.

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.

[todo]

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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 <a href="/file/view/Sagittal.pdf/243193787/Sagittal.pdf" onclick="ws.common.trackFileLink('/file/view/Sagittal.pdf/243193787/Sagittal.pdf');">Sagittal.pdf</a>, 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>

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 <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Enharmonic" rel="nofollow">enharmonic equivalences</a>. 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 ~~equal systems~~, as the syntonic comma in <a class="wiki_link" href="/meantone">meantone</a> 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 <a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Enharmonic" rel="nofollow">enharmonic equivalences</a>. 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 <a class="wiki_link" href="/meantone">meantone</a> 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.

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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.

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.

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).

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.

[todo]

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2015-08-22 10:40:23 UTC</tt>.<br>
+: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2015-08-22 13:14:35 UTC</tt>.<br>
-: The original revision id was <tt>557164235</tt>.<br>
+: The original revision id was <tt>557169903</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>
@@ Line 23: / Line 23: @@
 # Conventional staff notation (natural notes, sharps and flats) indicates tones in a series built on the equal division’s best approximation of a fifth.
-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 equal systems, as the syntonic comma in [[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]] 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.
@@ Line 34: / Line 34: @@
 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.
-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.
+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.
 [todo]
@@ Line 59: / Line 60: @@
   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 &lt;a href="/file/view/Sagittal.pdf/243193787/Sagittal.pdf" onclick="ws.common.trackFileLink('/file/view/Sagittal.pdf/243193787/Sagittal.pdf');"&gt;Sagittal.pdf&lt;/a&gt;, are as follows:&lt;br /&gt;
 &lt;ol&gt;&lt;li&gt;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.&lt;/li&gt;&lt;li&gt;Conventional staff notation (natural notes, sharps and flats) indicates tones in a series built on the equal division’s best approximation of a fifth.&lt;/li&gt;&lt;/ol&gt;&lt;br /&gt;
-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 &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Enharmonic" rel="nofollow"&gt;enharmonic equivalences&lt;/a&gt;. 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 equal systems, as the syntonic comma in &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; 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.&lt;br /&gt;
+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 &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Enharmonic" rel="nofollow"&gt;enharmonic equivalences&lt;/a&gt;. 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 &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; 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.&lt;br /&gt;
 &lt;br /&gt;
 Below is an example how the standard notation systems for some equal termperaments differ.&lt;br /&gt;
@@ Line 70: / Line 71: @@
 The second example shows the best approximation for the same chord in &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;. 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.&lt;br /&gt;
 &lt;br /&gt;
-In &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; (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.&lt;br /&gt;
+In &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; (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).&lt;br /&gt;
+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.&lt;br /&gt;
 &lt;br /&gt;
 [todo]&lt;br /&gt;