Wikispaces>Andrew_Heathwaite |
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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{Infobox MOS}} |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | {{MOS intro|Scale Signature=7L 3s}} |
| : This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-11-29 13:15:55 UTC</tt>.<br>
| | 7L 3s represents [[temperament]]s such as [[mohajira]]/[[mohaha]]/[[mohoho]], among others, whose generators are around a neutral third. The [[Mohaha7|seven]] and [[Mohaha10|ten-note]] forms of mohaha/mohoho form a [[Chromatic pairs#Mohaha|chromatic pair]]. |
| : The original revision id was <tt>105851523</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4>
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| <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">=7+3 - "unfair mosh" - "neutral third scales"=
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| "7L 3s" refers to the structure of [[MOSScales|moment of symmetry scales]] built from a 10-tone chain of neutral thirds (assuming a period of an octave):
| | == Name == |
| | {{TAMNAMS name}} |
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| L s L L L s L L s L
| | == Scale properties == |
| | {{TAMNAMS use}} |
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| Graham Breed has a [[http://x31eq.com/7plus3.htm|page on his website]] dedicated to 7+3 scales. He proposes calling the large step "t" for "tone", lowercase because the large step is a narrow neutral tone, and the small step "q" for "quartertone", because the small step is often close to a quartertone. (Note that the small step is not a quartertone in every instance of 7+3, so do not take that "q" literally.) Thus we have:
| | === Intervals === |
| | {{MOS intervals}} |
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| t q t t t q t t q t
| | === Generator chain === |
| | {{MOS genchain}} |
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| =Interval ranges= | | === Modes === |
| The generator (g) will fall between 343 cents (2\7 - two degrees of [[7edo]] and 360 cents (3\10 - three degrees of [[10edo]]) - the perceptual zone of a "neutral third".
| | {{MOS mode degrees}} |
| 2g, then, will fall between 686 cents (4\7) and 720 cents (3\5) - the perceptual zone of a "perfect fifth".
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| The "large step" will fall between 171 cents (1\7) and 120 cents (1\10) - the perceptual zone of "neutral second".
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| The "small step" will fall between 0 cents and 120 cents, sometimes sounding like a minor second, and sometimes sounding like a quartertone or smaller microtone.
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| The most frequent interval, then is the neutral third (and its inversion, the neutral sixth), followed by the perfect fourth and fifth. Thus, 7+3 combines the familiar sound of perfect fifths and fourths with the unfamiliar sounds of neutral intervals. They are compatible with Arabic and Turkish scales, but not with traditional Western ones.
| | == Theory == |
| | === Neutral intervals === |
| | 7L 3s combines the familiar sound of perfect fifths and fourths with the unfamiliar sounds of neutral intervals, thus making it compatible with [[Arabic, Turkish, Persian|Arabic]] and [[Arabic, Turkish, Persian|Turkish]] scales, but not with traditional Western scales. Notable intervals include: |
| | * The '''perfect 3-mosstep''', the scale's dark generator, whose range is around that of a neutral third. |
| | * The '''perfect 7-mosstep''', the scale's bright generator, the inversion of the perfect 3-mosstep, whose range is around that of a neutral sixth. |
| | * The '''minor mosstep''', or '''small step''', which ranges form a [[quartertone]] to a minor second. |
| | * The '''major mosstep''', or '''large step''', which ranges from a submajor second to a [[sinaic]], or trienthird (around 128{{c}}). |
| | * The '''major 4-mosstep''', whose range coincides with that of a perfect fourth. |
| | * The '''minor 6-mosstep''', the inversion of the major 4-mosstep, whose range coincides with that of a perfect 5th. |
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| =A continuum of compatible edos= | | === Quartertone and tetrachordal analysis=== |
| The generator range reflects two extremes: one where L=s (3\10), and another where s=0 (2\7). Between these extremes, there is an infinite continuum of possible generator sizes. By taking freshman sums of the two edges (adding the numerators, then adding the denominators), we can fill in this continuum with compatible edos, increasing in number of tones as we continue filling in the in-betweens. Thus, the smallest in-between edo would be (3+2)\(10+7) = 5\17 -- five degrees of [[17edo]]:
| | Due to the presence of [[quartertone]]-like intervals, [[Graham Breed]] has proposed the terms ''tone'' (abbreviated as ''t'') and ''quartertone'' (abbreviated as ''q'') as alternatives for large and small steps. This interpretation only makes sense for step ratios in which the small step approximates a quartertone. Additionally, Breed has also proposed a larger tone size, abbreviated using a capital ''T'', to refer to the combination of ''t'' and ''q''. Through this addition of a larger step, 7-note subsets of 7L 3s can be constructed. Some of these subsets are identical to that of 3L 4s, such as {{dash|''T, t, T, t, T, t, t''}}, but Breed states that non-MOS patterns are possible, such as {{dash|''T, t, t, T, t, t, T''}}. |
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| || || || generator || | | Additionally, due to the presence of fourth and fifth-like intervals, 7L 3s can be analyzed as a [[tetrachord|tetrachordal scale]]. Since the major 4-dicostep, the fourth-like interval, is reached using 4 steps rather than 3 (3 tones and 1 quartertone), Andrew Heathwaite offers an additional step ''A'', for ''augmented second'', to refer to the combination of two tones (''t''). Thus, the possible tetrachords can be built as a combination of a (large) tone and two (regular) tones ({{dash|''T'', ''t'', ''t''}}), or an augmented step, small tone, and quartertone ({{dash|''A'', ''t'', ''q''}}). |
| || 3\10 || || 360 ||
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| || || 5\17 || 353 ||
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| || 2\7 || || 343 ||
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| The scale produced by stacks of 5\17 is the [[17edo neutral scale]].
| | ==Scale tree== |
| | {{MOS tuning spectrum |
| | | 6/5 = [[Restles]] ↑ |
| | | 7/5 = [[Beatles]] |
| | | 3/2 = [[Suhajira]] / ringo |
| | | 12/5 = [[Hemif]] / [[hemififths]] |
| | | 5/2 = [[Mohaha]] / [[neutrominant]] / [[mohamaq]] |
| | | 13/5 = [[Hemif]] / [[salsa]] / [[karadeniz]] |
| | | 4/1 = [[Mohaha]] / [[migration]] / [[mohajira]] |
| | | 6/1 = [[Mohaha]] / [[ptolemy]] |
| | | 13/8 = Golden [[suhajira]] |
| | }} |
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| Other compatible edos include: [[37edo]], [[27edo]], [[44edo]], [[41edo]], [[24edo]], [[31edo]].
| | == External links== |
| | * [http://x31eq.com/7plus3.htm Graham Breed's page on 7L 3s] (which covers 3L 7s to an extent) |
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| You can also build this scale by stacking neutral thirds that are not members of edos -- for instance, frequency ratios 11:9, 49:40, 27:22, 16:13 -- or the square root of 3:2 (a bisected just perfect fifth).
| | [[Category:10-tone scales]] |
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| =7-note subsets=
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| If you stop the chain at 7 tones, you have a heptatonic scale of the form [[3L 4s]]:
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| L s s L s L s
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| The large steps here consist of t+s of the 10-tone system, and the small step is the same as t. Graham proposes calling the large step here T for "tone," uppercase because it is a wider tone than t. Thus, we have:
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| T t t T t T t
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| This scale (and its rotations) is not the only possible heptatonic scale. Graham also gives us:
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| T t t T t t T
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| which is not a complete moment of symmetry scale in itself, but a subset of one.
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| =Tetrachordal structure=
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| Due to the frequency of perfect fourths and fifths in this scale, it can also be analyzed as a [[tetrachord|tetrachordal scale]]. The perfect fourth can be traversed by 3 t's and a q, or 2 t's and a T.
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| I ([[user:Andrew_Heathwaite]]) offer "a" to refer to a step of 2t (for "augmented second")
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| Thus, the possible tetrachords are:
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| T t t
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| t T t
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| t t T
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| a q t
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| a t q
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| t a q
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| t q a
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| q a t
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| q t a
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| T, which usually functions as a "whole tone", can be used to</pre></div>
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| <h4>Original HTML content:</h4>
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| <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>7L 3s</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x7+3 - &quot;unfair mosh&quot; - &quot;neutral third scales&quot;"></a><!-- ws:end:WikiTextHeadingRule:0 -->7+3 - &quot;unfair mosh&quot; - &quot;neutral third scales&quot;</h1>
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| <br />
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| &quot;7L 3s&quot; refers to the structure of <a class="wiki_link" href="/MOSScales">moment of symmetry scales</a> built from a 10-tone chain of neutral thirds (assuming a period of an octave):<br />
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| <br />
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| L s L L L s L L s L<br />
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| <br />
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| Graham Breed has a <a class="wiki_link_ext" href="http://x31eq.com/7plus3.htm" rel="nofollow">page on his website</a> dedicated to 7+3 scales. He proposes calling the large step &quot;t&quot; for &quot;tone&quot;, lowercase because the large step is a narrow neutral tone, and the small step &quot;q&quot; for &quot;quartertone&quot;, because the small step is often close to a quartertone. (Note that the small step is not a quartertone in every instance of 7+3, so do not take that &quot;q&quot; literally.) Thus we have:<br />
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| <br />
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| t q t t t q t t q t<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Interval ranges"></a><!-- ws:end:WikiTextHeadingRule:2 -->Interval ranges</h1>
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| The generator (g) will fall between 343 cents (2\7 - two degrees of <a class="wiki_link" href="/7edo">7edo</a> and 360 cents (3\10 - three degrees of <a class="wiki_link" href="/10edo">10edo</a>) - the perceptual zone of a &quot;neutral third&quot;.<br />
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| 2g, then, will fall between 686 cents (4\7) and 720 cents (3\5) - the perceptual zone of a &quot;perfect fifth&quot;.<br />
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| The &quot;large step&quot; will fall between 171 cents (1\7) and 120 cents (1\10) - the perceptual zone of &quot;neutral second&quot;.<br />
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| The &quot;small step&quot; will fall between 0 cents and 120 cents, sometimes sounding like a minor second, and sometimes sounding like a quartertone or smaller microtone.<br />
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| <br />
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| The most frequent interval, then is the neutral third (and its inversion, the neutral sixth), followed by the perfect fourth and fifth. Thus, 7+3 combines the familiar sound of perfect fifths and fourths with the unfamiliar sounds of neutral intervals. They are compatible with Arabic and Turkish scales, but not with traditional Western ones.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="A continuum of compatible edos"></a><!-- ws:end:WikiTextHeadingRule:4 -->A continuum of compatible edos</h1>
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| The generator range reflects two extremes: one where L=s (3\10), and another where s=0 (2\7). Between these extremes, there is an infinite continuum of possible generator sizes. By taking freshman sums of the two edges (adding the numerators, then adding the denominators), we can fill in this continuum with compatible edos, increasing in number of tones as we continue filling in the in-betweens. Thus, the smallest in-between edo would be (3+2)\(10+7) = 5\17 -- five degrees of <a class="wiki_link" href="/17edo">17edo</a>:<br />
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| <table class="wiki_table">
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| <tr>
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| <td><br />
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| </td>
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| <td><br />
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| </td>
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| <td>generator<br />
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| </td>
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| </tr>
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| <tr>
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| <td>3\10<br />
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| </td>
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| <td><br />
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| </td>
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| <td>360<br />
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| </td>
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| </tr>
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| <tr>
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| <td><br />
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| </td>
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| <td>5\17<br />
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| </td>
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| <td>353<br />
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| </td>
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| </tr>
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| <tr>
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| <td>2\7<br />
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| </td>
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| <td><br />
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| </td>
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| <td>343<br />
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| </td>
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| </tr>
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| </table>
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| <br />
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| The scale produced by stacks of 5\17 is the <a class="wiki_link" href="/17edo%20neutral%20scale">17edo neutral scale</a>.<br />
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| <br />
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| Other compatible edos include: <a class="wiki_link" href="/37edo">37edo</a>, <a class="wiki_link" href="/27edo">27edo</a>, <a class="wiki_link" href="/44edo">44edo</a>, <a class="wiki_link" href="/41edo">41edo</a>, <a class="wiki_link" href="/24edo">24edo</a>, <a class="wiki_link" href="/31edo">31edo</a>.<br />
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| <br />
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| You can also build this scale by stacking neutral thirds that are not members of edos -- for instance, frequency ratios 11:9, 49:40, 27:22, 16:13 -- or the square root of 3:2 (a bisected just perfect fifth).<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:6:&lt;h1&gt; --><h1 id="toc3"><a name="x7-note subsets"></a><!-- ws:end:WikiTextHeadingRule:6 -->7-note subsets</h1>
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| If you stop the chain at 7 tones, you have a heptatonic scale of the form <a class="wiki_link" href="/3L%204s">3L 4s</a>:<br />
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| <br />
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| L s s L s L s<br />
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| <br />
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| The large steps here consist of t+s of the 10-tone system, and the small step is the same as t. Graham proposes calling the large step here T for &quot;tone,&quot; uppercase because it is a wider tone than t. Thus, we have:<br />
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| <br />
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| T t t T t T t<br />
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| <br />
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| This scale (and its rotations) is not the only possible heptatonic scale. Graham also gives us:<br />
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| <br />
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| T t t T t t T<br />
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| <br />
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| which is not a complete moment of symmetry scale in itself, but a subset of one.<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:8:&lt;h1&gt; --><h1 id="toc4"><a name="Tetrachordal structure"></a><!-- ws:end:WikiTextHeadingRule:8 -->Tetrachordal structure</h1>
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| Due to the frequency of perfect fourths and fifths in this scale, it can also be analyzed as a <a class="wiki_link" href="/tetrachord">tetrachordal scale</a>. The perfect fourth can be traversed by 3 t's and a q, or 2 t's and a T.<br />
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| <br />
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| I (<!-- ws:start:WikiTextUserlinkRule:00:[[user:Andrew_Heathwaite]] --><span class="membersnap">- <a class="userLink" href="http://www.wikispaces.com/user/view/Andrew_Heathwaite" style="outline: none;"><img src="http://www.wikispaces.com/user/pic/Andrew_Heathwaite-lg.jpg" width="16" height="16" alt="Andrew_Heathwaite" class="userPicture" /></a> <a class="userLink" href="http://www.wikispaces.com/user/view/Andrew_Heathwaite" style="outline: none;">Andrew_Heathwaite</a></span><!-- ws:end:WikiTextUserlinkRule:00 -->) offer &quot;a&quot; to refer to a step of 2t (for &quot;augmented second&quot;)<br />
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| <br />
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| Thus, the possible tetrachords are:<br />
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| <br />
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| T t t<br />
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| t T t<br />
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| t t T<br />
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| a q t<br />
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| a t q<br />
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| t a q<br />
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| t q a<br />
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| q a t<br />
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| q t a<br />
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| <br />
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| T, which usually functions as a &quot;whole tone&quot;, can be used to</body></html></pre></div>
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