Introductory examples in Sagittal notation: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2015-08-22 10:03:27 UTC</tt>.

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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.

Below ~~there~~ 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.

[[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.

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.

[todo]

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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.

Below ~~there~~ 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.

<img src="/file/view/SagittalEDOExample.jpg/557089549/SagittalEDOExample.jpg" alt="SagittalEDOExample.jpg" title="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.

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 <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.

[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 09:47:37 UTC</tt>.<br>
+: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2015-08-22 10:03:27 UTC</tt>.<br>
-: The original revision id was <tt>557162621</tt>.<br>
+: The original revision id was <tt>557163045</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 26: / Line 26: @@
 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.
-Below there 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.
 [[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.
+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.
 [todo]
@@ Line 53: / Line 58: @@
 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;
 &lt;br /&gt;
-Below there is an example how the standard notation systems for some equal termperaments differ.&lt;br /&gt;
+Below is an example how the standard notation systems for some equal termperaments differ.&lt;br /&gt;
 &lt;!-- ws:start:WikiTextLocalImageRule:13:&amp;lt;img src=&amp;quot;/file/view/SagittalEDOExample.jpg/557089549/SagittalEDOExample.jpg&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/SagittalEDOExample.jpg/557089549/SagittalEDOExample.jpg" alt="SagittalEDOExample.jpg" title="SagittalEDOExample.jpg" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:13 --&gt;&lt;br /&gt;
 &lt;br /&gt;
+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.&lt;br /&gt;
+&lt;br /&gt;
+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.&lt;br /&gt;
+&lt;br /&gt;
+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;
 [todo]&lt;br /&gt;