9/8: Difference between revisions

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~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

{{Infobox Interval

~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~

| Name = Pythagorean whole tone, Pythagorean major second

~~: This revision was by author [[User:spt3125~~|~~spt3125]] and made on <tt>2014-06-07 12:30:03 UTC</tt>.<br>~~

| Color name = w2, wa 2nd

~~: The original revision id was <tt>513184734</tt>.<br>~~

| Sound = jid_9_8_pluck_adu_dr220.mp3

~~: The revision comment was: <tt></tt><br>~~

| Comma = yes

~~The revision contents are below~~, ~~presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>~~

}}

~~<h4>Original Wikitext content:</h4>~~

~~<div style~~="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">**9/8**

|~~-3 2>~~

~~203.91000 cents~~

~~[[media type="file" key~~="jid_9_8_pluck_adu_dr220.mp3~~"]]~~

9/8 is the Pythagorean whole tone, measuring approximately 203.9¢. It can be arrived at by stacking two just perfect fifths ([[~~3_2|~~3/2]]) and reducing the result by one octave. However, it is also a relatively low overtone in its own right, octave-reduced. It can be treated as a dissonance or a consonance, depending on compositional context.

'''9/8''', the '''Pythagorean whole tone''' or '''major second''', is an interval measuring approximately 203.9¢. It can be arrived at by stacking two just perfect fifths ([[3/2]]) and reducing the result by one octave. However, it is also a relatively low overtone in its own right, octave-reduced. It can be treated as a dissonance or a consonance, depending on compositional context, though because of its relatively close proximity to the [[unison]], it is the largest [[superparticular]] interval known to cause crowding, which lends more to it being considered a type of dissonance- at least in historical Western Classical traditions and in the xenharmonic traditions derived from them.

Two 9/8's stacked produce [[~~81_64|~~81/64]], the Pythagorean major third, a rather bright major third of approximately 407.8¢. However, a 9/8 plus the minor whole tone [[~~10_9|~~10/9]] yields [[~~5_4|~~5/4]]. This distinction, between a major whole tone and minor whole tone, has been completely obliterated in [[12edo]], and so we are unaccustomed to thinking of more than one size of whole tone comprising a major third. Other systems ~~which~~ temper out this difference (which is [[~~81_80|~~81/80]], the syntonic comma of about 21.5¢) ~~include~~ [[19edo]], [[26edo]], [[31edo]], ~~and all~~ [[meantone]] temperaments.

Two 9/8's stacked produce [[81/64]], the Pythagorean major third, a rather bright major third of approximately 407.8¢. However, a 9/8 plus the minor whole tone [[10/9]] yields [[5/4]]. This distinction, between a major whole tone and minor whole tone, has been completely obliterated in [[12edo]], and so we are unaccustomed to thinking of more than one size of whole tone comprising a major third. Other systems that temper out this difference (which is [[81/80]], the syntonic comma of about 21.5¢), such as [[19edo]], [[26edo]], and [[31edo]], are called [[meantone]] temperaments.

9/8 is well-represented in [[6edo]] and its multiples. [[Edo]]s which tune [[~~3_2~~]] close to just ([[29edo]], [[41edo]], [[53edo]], to name three) will tune 9/8 close as well.

9/8 is well-represented in [[6edo]] and its multiples. [[Edo]]s which tune [[3/2]] close to just ([[29edo]], [[41edo]], [[53edo]], to name three) will tune 9/8 close to just as well. The difference between six instances of 9/8 and the octave is the [[Pythagorean comma]].

~~See: [[Gallery of Just Intervals]]</pre></div>~~

== History ==

~~<h4>Original HTML content:</h4>~~

The (whole) tone as an interval measure was already known in Ancient Greece. {{w|Aristoxenus}} (fl. 335 BC) defined the tone as the difference between the [[3/2|just fifth (3/2)]] and the [[4/3|just fourth (4/3)]]. From this base size, he derived the size of other intervals as multiples or fractions of the tone, so for instance the just fourth was 2½ tones in size, which implies [[12edo]].

~~<div style~~=~~"width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style~~=~~"margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class~~=~~"old-revision-html"><html><head><title>9_8</title></head><body><strong>9/8</strong><br />~~

~~|-3 2&gt;<br />~~

== Temperaments ==

~~203~~.~~91000 cents<br />~~

When this ratio is taken as a comma to be [[tempering out|tempered out]], it produces [[Very low accuracy temperaments #Antitonic|antitonic]] temperament. Edos that temper it out include [[2edo]] and [[4edo]]. If it is instead used as a generator, it produces, among others, [[Subgroup temperaments #Baldy|baldy]].

~~<embed src="~~/s/~~mediaplayer.swf" pluginspage="http:~~/~~/www~~.~~macromedia~~.~~com/go/getflashplayer" type="application/x-shockwave-flash" quality~~=~~"high" width~~=~~"240" height~~=~~"20" wmode~~=~~"transparent" flashvars="file=http%253A%252F%252Fxenharmonic~~.~~wikispaces~~.~~com%252Ffile%252Fview%252Fjid_9_8_pluck_adu_dr220~~.~~mp3?file_extension~~=~~mp3&autostart~~=~~false&repeat~~=~~false&showdigits~~=~~true&showfsbutton=false&width=240&height=20"></embed><!~~-- ~~ws:end:WikiTextMediaRule:0~~ --~~><br />~~

~~<br />~~

== Notation ==

~~9/8 is~~ the ~~Pythagorean~~ whole tone, ~~measuring approximately 203.9¢. It can be arrived at by stacking two just perfect fifths (<a class="wiki_link" href="/3_2">3/2</a>)~~ and ~~reducing the result by one octave. However~~, ~~it is also a relatively low overtone in its own right~~, ~~octave-reduced. It can be treated~~ as ~~a dissonance or a consonance, depending on compositional context~~.~~<br~~ /~~>~~

In musical notations that employ the [[5L 2s|diatonic]] [[chain-of-fifths notation|chain-of-fifths]], such as the [[ups and downs notation]], the whole tone is represented by the distances between A and B, between C and D, between D and E, between F and G, as well as between G and A.

~~<br~~ /~~>~~

~~Two~~ 9/8~~'s stacked produce <a class="wiki_link" href="/81_64">81/64</a>, the Pythagorean major third, a rather bright major third of approximately 407.8¢. However, a~~ 9/8 ~~plus the minor~~ whole tone ~~<a class="wiki_link" href="/10_9">10~~/9~~<~~/~~a> yields <a class="wiki_link" href="/5_4">5/4</a>. This distinction, between a major~~ whole tone ~~and minor~~ whole tone~~, has been completely obliterated in <a class="wiki_link" href="/12edo">12edo</a>, and so we are unaccustomed to thinking of more than one size~~ of whole tone ~~comprising a major third. Other systems which temper out this difference (which is <a class="wiki_link" href="~~/~~81_80">81/80</a>,~~ the ~~syntonic comma of about 21~~.~~5¢) include <a class~~=~~"wiki_link" href~~=~~"/19edo">19edo</a>, <a class~~=~~"wiki_link" href~~="/~~26edo">26edo</a>, <a class="wiki_link" href="/31edo">31edo</a>, and all <a class="wiki_link" href="/meantone">meantone</a> temperaments.<br />~~

The scale is structured with the following step pattern:

~~<br~~ /~~>~~

* A to B: [[9/8|whole tone]]

9/~~8 is well-represented in <a class="wiki_link" href="/6edo">6edo</a> and~~ its ~~multiples. <a class="wiki_link" href="/Edo">Edo</a>s which tune <a class="wiki_link" href="/3_2">3_2</a> close to~~ just ~~(<a class~~=~~"wiki_link" href~~=~~"/29edo">29edo</a>, <a class~~=~~"wiki_link" href~~="/~~41edo">41edo<~~/~~a>, <a class="wiki_link" href="~~/~~53edo">53edo<~~/~~a>, to name three) will tune 9~~/~~8 close as well~~.~~<br />~~

* B to C: [[256/243|limma]]

~~<br />~~

* C to D: [[9/8|whole tone]]

~~See~~: ~~<a class="wiki_link" href="/Gallery%20of%20Just%20Intervals">Gallery of Just Intervals</a></body></html></pre></div>~~

* D to E: [[9/8|whole tone]]

* E to F: [[256/243|limma]]

* F to G: [[9/8|whole tone]]

* G to A: [[9/8|whole tone]]

This pattern highlights the placement of the whole tone intervals between the note pairs above, distinguishing them from the [[256/243|limma]] that occurs between the other note pairs.

== See also ==

* [[16/9]] – its [[octave complement]]

* [[4/3]] – its [[fifth complement]]

* [[32/27]] – its [[fourth complement]]

* [[Gallery of just intervals]]

* [[List of superparticular intervals]]

== External links ==

* [http://www.tonalsoft.com/monzo/aristoxenus/aristoxenus.aspx The measurement of Aristoxenus's Divisions of the Tetrachord] on [[Tonalsoft Encyclopedia]]

[[Category:Second]]

[[Category:Whole tone]]

[[Category:Ancient Greek music]]

[[Category:Commas named after their interval size]]

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Infobox Interval
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+| Name = Pythagorean whole tone, Pythagorean major second
-: This revision was by author [[User:spt3125|spt3125]] and made on <tt>2014-06-07 12:30:03 UTC</tt>.<br>
+| Color name = w2, wa 2nd
-: The original revision id was <tt>513184734</tt>.<br>
+| Sound = jid_9_8_pluck_adu_dr220.mp3
-: The revision comment was: <tt></tt><br>
+| Comma = yes
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
+}}
-<h4>Original Wikitext content:</h4>
+{{Wikipedia|Major second}}
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">**9/8**
-|-3 2&gt;
-.91000 cents
-[[media type="file" key="jid_9_8_pluck_adu_dr220.mp3"]]
-/8 is the Pythagorean whole tone, measuring approximately 203.9¢. It can be arrived at by stacking two just perfect fifths ([[3_2|3/2]]) and reducing the result by one octave. However, it is also a relatively low overtone in its own right, octave-reduced. It can be treated as a dissonance or a consonance, depending on compositional context.
+'''9/8''', the '''Pythagorean whole tone''' or '''major second''', is an interval measuring approximately 203.9¢. It can be arrived at by stacking two just perfect fifths ([[3/2]]) and reducing the result by one octave. However, it is also a relatively low overtone in its own right, octave-reduced. It can be treated as a dissonance or a consonance, depending on compositional context, though because of its relatively close proximity to the [[unison]], it is the largest [[superparticular]] interval known to cause crowding, which lends more to it being considered a type of dissonance- at least in historical Western Classical traditions and in the xenharmonic traditions derived from them.
-Two 9/8's stacked produce [[81_64|81/64]], the Pythagorean major third, a rather bright major third of approximately 407.8¢. However, a 9/8 plus the minor whole tone [[10_9|10/9]] yields [[5_4|5/4]]. This distinction, between a major whole tone and minor whole tone, has been completely obliterated in [[12edo]], and so we are unaccustomed to thinking of more than one size of whole tone comprising a major third. Other systems which temper out this difference (which is [[81_80|81/80]], the syntonic comma of about 21.5¢) include [[19edo]], [[26edo]], [[31edo]], and all [[meantone]] temperaments.
+Two 9/8's stacked produce [[81/64]], the Pythagorean major third, a rather bright major third of approximately 407.8¢. However, a 9/8 plus the minor whole tone [[10/9]] yields [[5/4]]. This distinction, between a major whole tone and minor whole tone, has been completely obliterated in [[12edo]], and so we are unaccustomed to thinking of more than one size of whole tone comprising a major third. Other systems that temper out this difference (which is [[81/80]], the syntonic comma of about 21.5¢), such as [[19edo]], [[26edo]], and [[31edo]], are called [[meantone]] temperaments.
-/8 is well-represented in [[6edo]] and its multiples. [[Edo]]s which tune [[3_2]] close to just ([[29edo]], [[41edo]], [[53edo]], to name three) will tune 9/8 close as well.
+/8 is well-represented in [[6edo]] and its multiples. [[Edo]]s which tune [[3/2]] close to just ([[29edo]], [[41edo]], [[53edo]], to name three) will tune 9/8 close to just as well. The difference between six instances of 9/8 and the octave is the [[Pythagorean comma]].
-See: [[Gallery of Just Intervals]]</pre></div>
+== History ==
-<h4>Original HTML content:</h4>
+The (whole) tone as an interval measure was already known in Ancient Greece. {{w|Aristoxenus}} (fl. 335 BC) defined the tone as the difference between the [[3/2|just fifth (3/2)]] and the [[4/3|just fourth (4/3)]]. From this base size, he derived the size of other intervals as multiples or fractions of the tone, so for instance the just fourth was 2½ tones in size, which implies [[12edo]].
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;9_8&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;strong&gt;9/8&lt;/strong&gt;&lt;br /&gt;
-|-3 2&amp;gt;&lt;br /&gt;
+== Temperaments ==
-.91000 cents&lt;br /&gt;
+When this ratio is taken as a comma to be [[tempering out|tempered out]], it produces [[Very low accuracy temperaments #Antitonic|antitonic]] temperament. Edos that temper it out include [[2edo]] and [[4edo]]. If it is instead used as a generator, it produces, among others, [[Subgroup temperaments #Baldy|baldy]].
-&lt;!-- ws:start:WikiTextMediaRule:0:&amp;lt;img src=&amp;quot;http://www.wikispaces.com/site/embedthumbnail/file-audio/jid_9_8_pluck_adu_dr220.mp3?h=20&amp;amp;w=240&amp;quot; class=&amp;quot;WikiMedia WikiMediaFile&amp;quot; id=&amp;quot;wikitext@@media@@type=&amp;amp;quot;file&amp;amp;quot; key=&amp;amp;quot;jid_9_8_pluck_adu_dr220.mp3&amp;amp;quot;&amp;quot; title=&amp;quot;Local Media File&amp;quot;height=&amp;quot;20&amp;quot; width=&amp;quot;240&amp;quot;/&amp;gt; --&gt;&lt;embed src="/s/mediaplayer.swf" pluginspage="http://www.macromedia.com/go/getflashplayer" type="application/x-shockwave-flash" quality="high" width="240" height="20" wmode="transparent" flashvars="file=http%253A%252F%252Fxenharmonic.wikispaces.com%252Ffile%252Fview%252Fjid_9_8_pluck_adu_dr220.mp3?file_extension=mp3&amp;autostart=false&amp;repeat=false&amp;showdigits=true&amp;showfsbutton=false&amp;width=240&amp;height=20"&gt;&lt;/embed&gt;&lt;!-- ws:end:WikiTextMediaRule:0 --&gt;&lt;br /&gt;
-&lt;br /&gt;
+== Notation ==
-/8 is the Pythagorean whole tone, measuring approximately 203.9¢. It can be arrived at by stacking two just perfect fifths (&lt;a class="wiki_link" href="/3_2"&gt;3/2&lt;/a&gt;) and reducing the result by one octave. However, it is also a relatively low overtone in its own right, octave-reduced. It can be treated as a dissonance or a consonance, depending on compositional context.&lt;br /&gt;
+In musical notations that employ the [[5L 2s|diatonic]] [[chain-of-fifths notation|chain-of-fifths]], such as the [[ups and downs notation]], the whole tone is represented by the distances between A and B, between C and D, between D and E, between F and G, as well as between G and A.
-&lt;br /&gt;
-Two 9/8's stacked produce &lt;a class="wiki_link" href="/81_64"&gt;81/64&lt;/a&gt;, the Pythagorean major third, a rather bright major third of approximately 407.8¢. However, a 9/8 plus the minor whole tone &lt;a class="wiki_link" href="/10_9"&gt;10/9&lt;/a&gt; yields &lt;a class="wiki_link" href="/5_4"&gt;5/4&lt;/a&gt;. This distinction, between a major whole tone and minor whole tone, has been completely obliterated in &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, and so we are unaccustomed to thinking of more than one size of whole tone comprising a major third. Other systems which temper out this difference (which is &lt;a class="wiki_link" href="/81_80"&gt;81/80&lt;/a&gt;, the syntonic comma of about 21.5¢) include &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, &lt;a class="wiki_link" href="/26edo"&gt;26edo&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;, and all &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; temperaments.&lt;br /&gt;
+The scale is structured with the following step pattern:
-&lt;br /&gt;
+* A to B: [[9/8|whole tone]]
-/8 is well-represented in &lt;a class="wiki_link" href="/6edo"&gt;6edo&lt;/a&gt; and its multiples. &lt;a class="wiki_link" href="/Edo"&gt;Edo&lt;/a&gt;s which tune &lt;a class="wiki_link" href="/3_2"&gt;3_2&lt;/a&gt; close to just (&lt;a class="wiki_link" href="/29edo"&gt;29edo&lt;/a&gt;, &lt;a class="wiki_link" href="/41edo"&gt;41edo&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt;, to name three) will tune 9/8 close as well.&lt;br /&gt;
+* B to C: [[256/243|limma]]
-&lt;br /&gt;
+* C to D: [[9/8|whole tone]]
-See: &lt;a class="wiki_link" href="/Gallery%20of%20Just%20Intervals"&gt;Gallery of Just Intervals&lt;/a&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+* D to E: [[9/8|whole tone]]
+* E to F: [[256/243|limma]]
+* F to G: [[9/8|whole tone]]
+* G to A: [[9/8|whole tone]]
+This pattern highlights the placement of the whole tone intervals between the note pairs above, distinguishing them from the [[256/243|limma]] that occurs between the other note pairs.
+== See also ==
+* [[16/9]] – its [[octave complement]]
+* [[4/3]] – its [[fifth complement]]
+* [[32/27]] – its [[fourth complement]]
+* [[Gallery of just intervals]]
+* [[List of superparticular intervals]]
+== External links ==
+* [http://www.tonalsoft.com/monzo/aristoxenus/aristoxenus.aspx The measurement of Aristoxenus's Divisions of the Tetrachord] on [[Tonalsoft Encyclopedia]]
+[[Category:Second]]
+[[Category:Whole tone]]
+[[Category:Ancient Greek music]]
+[[Category:Commas named after their interval size]]