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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:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2011-09-14 21:02:48 UTC</tt>.<br>
| Color name = w2, wa 2nd
: The original revision id was <tt>254170168</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 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.


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]], and [[31edo]].
'''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.


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.
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.


See: [[Gallery of Just Intervals]]</pre></div>
A stack of six intervals of 9/8 exceeds the octave by the [[Pythagorean comma]].
<h4>Original HTML 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;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;9/8 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;
== History ==
&lt;br /&gt;
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]].
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;, and &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt;.&lt;br /&gt;
 
&lt;br /&gt;
== Temperaments ==
9/8 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;
In [[meantone]], 9/8 is equated with [[10/9]], so that two instances of 9/8~10/9 stack to ~[[5/4]]. [[Superpyth]] instead sharpens 9/8 to equate it with [[8/7]].
&lt;br /&gt;
 
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>
Since 9/8 is reached by stacking two instances of [[3/2]], temperaments in subgroups that include 3 cannot be generated by ~9/8. However, it can be a generator in subgroups such as [[2.9.5.7 subgroup|2.9.5.7]], where it generates [[Subgroup temperaments #Baldy|baldy]] for example.
 
== Approximation ==
9/8 is well-represented in [[6edo]] and its multiples, though only multiples of [[12edo]] (up to [[300edo]]) map 9/8 to 1\6 by [[patent val]]. [[Edo]]s which tune [[3/2]] close to just, such as [[29edo]], [[41edo]], and [[53edo]], will tune 9/8 close to just as well.
 
{{Interval edo approximation|9/8}}
 
== Notation ==
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 the notes A–B, C–D, D–E, F–G, and G–A.
 
The scale is structured with the following step pattern:
* A to B: [[9/8|whole tone]]
* B to C: [[256/243|limma]]
* C to D: [[9/8|whole tone]]
* 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]]
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