Introductory examples in Sagittal notation: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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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.

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).

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.

=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"]]

<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 />

<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 />

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 />

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 />

<h1 id="toc2"><a name="Equal temperaments (2): 11edo scale"></a>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 />

<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><br />

<span style="background-color: #ffffff;">(Rendering Juhani Nuorvala)</span><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>

@@ 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>2015-08-23 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>
 : 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 32: / Line 32: @@
 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).
-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.
 =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"]]
 &lt;span style="background-color: #ffffff;"&gt;(Rendering Juhani Nuorvala)&lt;/span&gt;
@@ Line 73: / Line 73: @@
 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;
+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 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.&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. 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;
+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.&lt;br /&gt;
 &lt;br /&gt;
 Finally, &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; (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 &lt;a class="wiki_link" href="/Meantone%20family#Septimal%20meantone"&gt;septimal meantone&lt;/a&gt; system.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Equal temperaments (2): 11edo scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Equal temperaments (2): 11edo scale&lt;/h1&gt;
-  As second example, an &lt;a class="wiki_link" href="/11edo"&gt;11edo&lt;/a&gt; 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 &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; are used. For the same reason, the recommended symbols for &lt;a class="wiki_link" href="/16edo"&gt;16edo&lt;/a&gt; are those of &lt;a class="wiki_link" href="/48edo"&gt;48edo&lt;/a&gt;.&lt;br /&gt;
+  As second example, an &lt;a class="wiki_link" href="/11edo"&gt;11edo&lt;/a&gt; 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 &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; are used. For the same reason, the recommended symbols for &lt;a class="wiki_link" href="/16edo"&gt;16edo&lt;/a&gt; are those of &lt;a class="wiki_link" href="/48edo"&gt;48edo&lt;/a&gt;.&lt;br /&gt;
 &lt;!-- ws:start:WikiTextLocalImageRule:19:&amp;lt;img src=&amp;quot;http://xenharmonie.wikispaces.com/file/view/Sagittal11EDO.jpg/556903777/496x283/Sagittal11EDO.jpg&amp;quot; alt=&amp;quot;Sagittal11EDO.jpg&amp;quot; title=&amp;quot;Sagittal11EDO.jpg&amp;quot; style=&amp;quot;height: 283px; width: 496px;&amp;quot; /&amp;gt; --&gt;&lt;table class="captionBox"&gt;&lt;tr&gt;&lt;td class="captionedImage"&gt;&lt;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;" /&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td class="imageCaption"&gt;Sagittal11EDO.jpg&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;&lt;!-- ws:end:WikiTextLocalImageRule:19 --&gt;&lt;br /&gt;
 &lt;span style="background-color: #ffffff;"&gt;(Rendering Juhani Nuorvala)&lt;/span&gt;&lt;br /&gt;
 &lt;br /&gt;
 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.&lt;/body&gt;&lt;/html&gt;</pre></div>