Introductory examples in Sagittal notation

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[[toc]]

This page lists a few elementary examples that hopefully shed some light on the philosophy (and the pitfalls!) of [[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= 
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]]. Other elementary commas appearing along the overtone series are: in [[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]], 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:
[[image:SagittalOvertoneSeries.jpg]]
For a complete list of all comma symbols see [[http://sagittal.org/]] or [[file:xenharmonic/Sagittal.pdf|sagittal.pdf]].

=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:
# 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.
# 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.

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.

In [[22edo]] (third example), however, 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 must stay.

[todo]

=Equal temperaments (2): an 11edo scale= 
[todo]

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<html><head><title>Introductory examples in Sagittal notation</title></head><body><!-- ws:start:WikiTextTocRule:6:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:6 --><!-- ws:start:WikiTextTocRule:7: --><div style="margin-left: 1em;"><a href="#Just intonation: notating an overtone scale">Just intonation: notating an overtone scale</a></div>
<!-- ws:end:WikiTextTocRule:7 --><!-- ws:start:WikiTextTocRule:8: --><div style="margin-left: 1em;"><a href="#Equal temperaments (1): comparison of notation in different equal temperaments">Equal temperaments (1): comparison of notation in different equal temperaments</a></div>
<!-- ws:end:WikiTextTocRule:8 --><!-- ws:start:WikiTextTocRule:9: --><div style="margin-left: 1em;"><a href="#Equal temperaments (2): an 11edo scale">Equal temperaments (2): an 11edo scale</a></div>
<!-- ws:end:WikiTextTocRule:9 --><!-- ws:start:WikiTextTocRule:10: --></div>
<!-- ws:end:WikiTextTocRule:10 --><br />
This page lists a few elementary examples that hopefully shed some light on the philosophy (and the pitfalls!) of <a class="wiki_link" href="/Sagittal%20notation">Sagittal notation</a>. For a detailed introduction into Sagittal notation the document <a href="/file/view/Sagittal.pdf/243193787/Sagittal.pdf" onclick="ws.common.trackFileLink('/file/view/Sagittal.pdf/243193787/Sagittal.pdf');">Sagittal.pdf</a> is the reference.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Just intonation: notating an overtone scale"></a><!-- ws:end:WikiTextHeadingRule:0 -->Just intonation: notating an overtone scale</h1>
 As the introduction <a href="/file/view/Sagittal.pdf/243193787/Sagittal.pdf" onclick="ws.common.trackFileLink('/file/view/Sagittal.pdf/243193787/Sagittal.pdf');">Sagittal.pdf</a> 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 <a class="wiki_link" href="/3-limit">Pythagorean tuning</a>. The meaning of conventional sharps and flats is Pythagorean as well, i.e. they stand for raising the corresponding note by a <a class="wiki_link" href="/2187_2048">Pythagorean chromatic semitone (apotome)</a>, a &quot;large&quot; semitone 113.7 cents in size.<br />
<br />
For the notation of notes in higher <a class="wiki_link" href="/Harmonic%20Limit">limits</a>, additional symbols are introduced. The intervals these symbols stand for are mostly <a class="wiki_link" href="/Comma">commas</a> - the maybe most elementary example is the <a class="wiki_link" href="/81_80">syntonic comma</a> (ratio 81/80, 21.506 cents), the difference between a pythagorean and a just major third, necessary for just intonation in <a class="wiki_link" href="/5-limit">5-limit</a>. Other elementary commas appearing along the overtone series are: in <a class="wiki_link" href="/7-limit">7-limit</a> the <a class="wiki_link" href="/64_63">septimal comma or Architas' comma</a> (64/63, 27.264 Cents), the difference between a minor and a harmonic seventh, and, in <a class="wiki_link" href="/11-limit">11-limit</a>, the <a class="wiki_link" href="/33_32">undecimal comma or al-Farabi quarter-tone</a> (33/32, 53.2729 cents), the difference between an undecimal semi-augmented and a perfect fourth.<br />
<br />
With the Sagittal symbos for these three commas, a scale consisting of the overtones 4 to 11 can be written as follows:<br />
<!-- ws:start:WikiTextLocalImageRule:17:&lt;img src=&quot;/file/view/SagittalOvertoneSeries.jpg/557089547/SagittalOvertoneSeries.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/SagittalOvertoneSeries.jpg/557089547/SagittalOvertoneSeries.jpg" alt="SagittalOvertoneSeries.jpg" title="SagittalOvertoneSeries.jpg" /><!-- ws:end:WikiTextLocalImageRule:17 --><br />
For a complete list of all comma symbols see <a class="wiki_link_ext" href="http://sagittal.org/" rel="nofollow">http://sagittal.org/</a> or <a href="http://xenharmonic.wikispaces.com/file/view/Sagittal.pdf/243193787/Sagittal.pdf" onclick="ws.common.trackFileLink('http://xenharmonic.wikispaces.com/file/view/Sagittal.pdf/243193787/Sagittal.pdf');">sagittal.pdf</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Equal temperaments (1): comparison of notation in different equal temperaments"></a><!-- ws:end:WikiTextHeadingRule:2 -->Equal temperaments (1): comparison of notation in different equal temperaments</h1>
 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:<br />
<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><br />
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.<br />
<br />
Below is an example how the standard notation systems for some equal termperaments differ.<br />
<!-- ws:start:WikiTextLocalImageRule:18:&lt;img src=&quot;/file/view/SagittalEDOExample.jpg/557089549/SagittalEDOExample.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/SagittalEDOExample.jpg/557089549/SagittalEDOExample.jpg" alt="SagittalEDOExample.jpg" title="SagittalEDOExample.jpg" /><!-- ws:end:WikiTextLocalImageRule:18 --><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 />
<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 />
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 />
<br />
In <a class="wiki_link" href="/22edo">22edo</a> (third example), however, 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 must stay.<br />
<br />
[todo]<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Equal temperaments (2): an 11edo scale"></a><!-- ws:end:WikiTextHeadingRule:4 -->Equal temperaments (2): an 11edo scale</h1>
 [todo]</body></html>