Wikispaces>Andrew_Heathwaite |
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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{interwiki |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | | en = 5L 2s |
| : This revision was by author [[User:Andrew_Heathwaite|Andrew_Heathwaite]] and made on <tt>2009-11-04 15:00:44 UTC</tt>.<br>
| | | de = 5L2s |
| : The original revision id was <tt>100207569</tt>.<br>
| | | es = |
| : The revision comment was: <tt></tt><br>
| | | ja = 5L 2s |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| | | ko = 5L2s (Korean) |
| <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">=5L 2s - "diatonic"=
| | {{Infobox MOS}} |
| | {{Wikipedia|Diatonic scale}} |
|
| |
|
| One way of distinguishing the "diatonic" scale is by considering it a [[MOSScales|moment of symmetry]] scale produced by a chain of "fifths". This will include [[12edo]]'s diatonic scale along with the Pythagorean diatonic scale, while excluding just intonation scales that use more than one size of "tone".
| | {{MOS intro}} |
|
| |
|
| This produces a generalized diatonic scale with the form:
| | The familiar pattern of 5 whole steps and 2 half steps, commonly written as WWHWWWH for the major scale, takes on a generalized form of LLsLLLs, where the large and small steps—denoted as ''L''{{'s}} and ''s''{{`s}}—represent whole number step sizes, thus producing different [[edo]]s. These [[step ratio]]s affect the sizes of the diatonic scale's intervals and correspond to different tuning systems. |
| L L s L L L s
| |
|
| |
|
| Insert 2 for L and 1 for s and you'll get the 12edo diatonic of standard practice.
| | Among the most well-known forms of this scale are the Pythagorean diatonic scale, and scales produced by meantone systems (including [[12edo]]). |
| 2 2 1 2 2 2 1
| |
|
| |
|
| When L=3, s=1, you have [[17edo]]:
| | == Name == |
| 3 3 1 3 3 3 1
| | {{TAMNAMS name}} "Mosdiatonic" may also be used for the sake of specificity. |
|
| |
|
| When L=3, s=2, you have [[19edo]]:
| | == Notation == |
| 3 3 2 3 3 3 2
| | : ''This article assumes [[TAMNAMS]] for naming step ratios.'' |
|
| |
|
| When L=4, s=1, you have [[22edo]]:
| | == Scale characteristics == |
| 4 4 1 4 4 4 1
| | {{TAMNAMS use}} |
|
| |
|
| When L=4, s=3, you have [[26edo]]:
| | === Intervals === |
| 4 4 3 4 4 4 3
| | {{MOS intervals}} |
|
| |
|
| When L=5, s=1, you have [[27edo]]:
| | === Generator chain === |
| 5 5 1 5 5 5 1
| | {{MOS genchain}} |
|
| |
|
| When L=5, s=2, you have [[29edo]]:
| | === Modes === |
| 5 5 2 5 5 5 2
| | {{MOS mode degrees}} |
|
| |
|
| When L=5, s=3, you have [[31edo]]:
| | Diatonic modes have standard names from classical music theory. |
| 5 5 3 5 5 5 3
| | {{MOS modes}} |
|
| |
|
| When L=5, s=4, you have [[33edo]]:
| | === Note names === |
| 5 5 4 5 5 5 4
| | Note names are identical to that of standard notation. Thus, the basic gamut for 5L 2s is the following: |
| | {{MOS gamut}} |
|
| |
|
| So you have scales where L and s are nearly equal, which approach [[7edo]]:
| | == Theory == |
| 1 1 1 1 1 1 1
| | === Temperament interpretations === |
| | {{Main| {{PAGENAME}}/Temperaments }} |
| | 5L 2s has several rank-2 temperament interpretations, such as: |
| | * [[Meantone]], with generators around 696.2{{c}}. This includes: |
| | ** [[Flattone]], with generators around 693.7{{c}}. |
| | * [[Schismic]], with generators around 702{{c}}. |
| | * [[Leapfrog]], with generators around 704.7{{c}}. |
| | * [[Archy]], with generators around 709.3{{c}}. This includes: |
| | ** Supra, with generators around 707.2{{c}} |
| | ** [[Superpyth]], with generators around 710.3{{c}} |
| | ** [[Ultrapyth]], with generators around 713.7{{c}}. |
|
| |
|
| And you have scales where s becomes so small it approaches zero, which would give us [[5edo]]:
| | === Generator chain === |
| 1 1 0 1 1 1 0 or 1 1 1 1 1
| | {{MOS genchain}} |
|
| |
|
| So if 3\7 (three degrees of 7edo) is at one extreme and 2\5 (two degrees of 5edo) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators. Thus, between 3\7 and 2\5 you have (3+2)\(7+5) = 5\12, five degrees of 12edo:
| | === Warped diatonic scales === |
| | Because of most listeners' familiarity with the 5L 2s diatonic scale, listeners may sometimes experience an effect like pareidolia, hearing 5L 2s even when it isn’t there. |
|
| |
|
| || 3\7 || ||
| | A larger scale can be constructed so that it contains chains of 5L 2s, but then breaks the pattern, exploiting that pareidolic effect to surprise and disorient the listener. Scales which have this effect are called [[warped diatonic]] scales. |
| || || 5\12 ||
| |
| || 2\5 || ||
| |
|
| |
|
| If we carry this freshman-summing out a little further, new, larger [[edo]]s pop up in our continuum.
| | === Interval categories === |
| | ''See [[5L 2s/Interval categories]]''. |
|
| |
|
| || 3\7 || || || || || ||
| | == Tuning ranges == |
| || || || || || || ||
| | {{Todo|Verify|inline=1|text=Populate/verify tables}} |
| || || || || || 14\33 || ||
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| || || || || 11\26 || || ||
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| || || || || || 19\35 || ||
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| || || || 8\19 || || || ||
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| || || || || || 21\50 || ||
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| || || || || 13\31 || || ||
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| || || || || || 18\43 || ||
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| || || 5\12 || || || || ||
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| || || || || || 17\41 || ||
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| || || || || 12\29 || || ||
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| || || || || || 19\46 || ||
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| || || || 7\17 || || || ||
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| || || || || || 16\39 || ||
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| || || || || 9\22 || || ||
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| || || || || || 11\27 || ||
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| || 2\5 || || || || || ||</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>5L 2s</title></head><body><!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="x5L 2s - &quot;diatonic&quot;"></a><!-- ws:end:WikiTextHeadingRule:0 -->5L 2s - &quot;diatonic&quot;</h1>
| |
| <br />
| |
| One way of distinguishing the &quot;diatonic&quot; scale is by considering it a <a class="wiki_link" href="/MOSScales">moment of symmetry</a> scale produced by a chain of &quot;fifths&quot;. This will include <a class="wiki_link" href="/12edo">12edo</a>'s diatonic scale along with the Pythagorean diatonic scale, while excluding just intonation scales that use more than one size of &quot;tone&quot;.<br />
| |
| <br />
| |
| This produces a generalized diatonic scale with the form:<br />
| |
| L L s L L L s<br />
| |
| <br />
| |
| Insert 2 for L and 1 for s and you'll get the 12edo diatonic of standard practice.<br />
| |
| 2 2 1 2 2 2 1<br />
| |
| <br />
| |
| When L=3, s=1, you have <a class="wiki_link" href="/17edo">17edo</a>:<br />
| |
| 3 3 1 3 3 3 1<br />
| |
| <br />
| |
| When L=3, s=2, you have <a class="wiki_link" href="/19edo">19edo</a>:<br />
| |
| 3 3 2 3 3 3 2<br />
| |
| <br />
| |
| When L=4, s=1, you have <a class="wiki_link" href="/22edo">22edo</a>:<br />
| |
| 4 4 1 4 4 4 1<br />
| |
| <br />
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| When L=4, s=3, you have <a class="wiki_link" href="/26edo">26edo</a>:<br />
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| 4 4 3 4 4 4 3<br />
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| <br />
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| When L=5, s=1, you have <a class="wiki_link" href="/27edo">27edo</a>:<br />
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| 5 5 1 5 5 5 1<br />
| |
| <br />
| |
| When L=5, s=2, you have <a class="wiki_link" href="/29edo">29edo</a>:<br />
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| 5 5 2 5 5 5 2<br />
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| <br />
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| When L=5, s=3, you have <a class="wiki_link" href="/31edo">31edo</a>:<br />
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| 5 5 3 5 5 5 3<br />
| |
| <br />
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| When L=5, s=4, you have <a class="wiki_link" href="/33edo">33edo</a>:<br />
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| 5 5 4 5 5 5 4<br />
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| <br />
| |
| So you have scales where L and s are nearly equal, which approach <a class="wiki_link" href="/7edo">7edo</a>:<br />
| |
| 1 1 1 1 1 1 1<br />
| |
| <br />
| |
| And you have scales where s becomes so small it approaches zero, which would give us <a class="wiki_link" href="/5edo">5edo</a>:<br />
| |
| 1 1 0 1 1 1 0 or 1 1 1 1 1<br />
| |
| <br />
| |
| So if 3\7 (three degrees of 7edo) is at one extreme and 2\5 (two degrees of 5edo) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking &quot;freshman sums&quot; of the two edges - adding together the numerators, then adding together the denominators. Thus, between 3\7 and 2\5 you have (3+2)\(7+5) = 5\12, five degrees of 12edo:<br />
| |
| <br />
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|
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|
| | === Simple tunings === |
| | [[17edo]] and [[19edo]] are the smallest edos that offer a greater variety of pitches than 12edo. Note that any enharmonic equivalences that 12edo has no longer hold for either 17edo or 19edo, as shown in the table below. |
| | {{MOS tunings|JI Ratios=Int Limit: 30; Complements Only: 1|Tolerance=20}} |
|
| |
|
| <table class="wiki_table">
| | === Ultrasoft tunings === |
| <tr>
| | {{See also| Superflat }} |
| <td>3\7<br />
| | In this range, the major third is so flat that it can best be approximated by [[16/13]], tempering out [[1053/1024]]. |
| </td>
| | {{MOS tunings|Step Ratios=Ultrasoft|JI Ratios=NONE}} |
| <td><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\12<br />
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| </td>
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| </tr>
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| <td>2\5<br />
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| </td>
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| <td><br />
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| </td>
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| </tr>
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| </table>
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| <br />
| | === Parasoft tunings === |
| If we carry this freshman-summing out a little further, new, larger <a class="wiki_link" href="/edo">edo</a>s pop up in our continuum.<br />
| | {{See also| Flattone }} |
| <br />
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| | Parasoft diatonic tunings (4:3 to 3:2) correspond to flattone temperaments, characterized by flattened perfect 5ths ([[3/2]], flat of 702{{c}}) to produce major 3rds that are flatter than [[5/4]] (386{{c}}). |
|
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|
| <table class="wiki_table">
| | Edos include [[19edo]], [[26edo]], [[45edo]], and [[64edo]]. |
| <tr>
| | {{MOS tunings|Step Ratios=4/3; 7/5; 10/7; 3/2|JI Ratios=Subgroup: 2.3.5.7.13; Int Limit: 27; Complements Only: 1; Tenney Height: 10|Tolerance=20}} |
| <td>3\7<br />
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| <td>13\31<br />
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| <td>18\43<br />
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| <td>5\12<br />
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| </body></html></pre></div> | | === Hyposoft tunings === |
| | {{See also| Meantone }} |
| | |
| | Hyposoft diatonic tunings (3:2 to 2:1) correspond to meantone temperaments, characterized by flattened perfect 5ths (flat of 702{{c}}) to produce diatonic major 3rds that approximate 5/4 (386{{c}}). |
| | |
| | Edos include [[19edo]], [[31edo]], [[43edo]], and [[50edo]]. |
| | {{MOS tunings|Step Ratios=3/2; 5/3; 8/5; 7/4; 2/1|JI Ratios=Subgroup:2.3.5; Int Limit: 40; Tenney Height: 10|Tolerance=15}} |
| | |
| | === Hypohard tunings === |
| | : ''See also: [[Pythagorean tuning]] and [[Schismatic family #Schismatic aka helmholtz|schismatic temperament]]'' |
| | |
| | The range of hypohard tunings can be divided into a minihard range (2:1 to 5:2) and quasihard range (5:2 to 3:1). |
| | {{MOS tunings|Step Ratios=Hypohard|JI Ratios=NONE}} |
| | |
| | ==== Minihard tunings ==== |
| | Minihard diatonic tunings correspond to Pythagorean tuning and schismatic temperament, characterized by having a perfect 5th that is as close to just (701.96{{c}}) as possible, resulting in a major 3rd of [[81/64]] (407{{c}}). |
| | |
| | Edos include [[41edo]] and [[53edo]]. |
| | {{MOS tunings|Step Ratios=2/1; 7/3; 5/2; 9/4|JI Ratios=Prime Limit:3; Int Limit: 1024|Tolerance=10}} |
| | |
| | ==== Quasihard tunings ==== |
| | Quasihard diatonic tunings correspond to "neogothic" or "parapyth" systems whose perfect 5th is slightly sharper than just, resulting in major 3rds that are sharper than 81/64 and minor 3rds that are slightly flat of [[32/27]] (294{{c}}). |
| | |
| | Edos include [[17edo]], [[29edo]], and [[46edo]]. 17edo is considered to be on the sharper end of the neogothic spectrum, with a major 3rd that is more discordant than flatter neogothic tunings. |
| | {{MOS tunings|Step Ratios=Quasihard|JI Ratios=Subgroup: 2.3.7.11.13; Int Limit: 30; Complements Only: 1|Tolerance=15}} |
| | |
| | === Parahard and ultrahard tunings === |
| | {{See also| Archy }} |
| | |
| | Parahard (3:1 to 4:1) and ultrahard (4:1 to 1:0) diatonic tunings correspond to archy systems, with perfect 5ths that are significantly sharper than than 702{{c}}. |
| | |
| | Edos include [[17edo]], [[22edo]], [[27edo]], and [[32edo]], among others. |
| | {{MOS tunings|Step Ratios=3/1; 4/1; 5/1; 6/1|JI Ratios=Subgroup: 2.3.7 ; Int Limit: 80; Complements Only: 1|Tolerance=15}} |
| | |
| | == Scales == |
| | === Subset and superset scales === |
| | 5L 2s has a parent scale of [[2L 3s]], a pentatonic scale, meaning 2L 3s is a subset. 5L 2s also has two child scales, which are supersets of 5L 2s: |
| | * [[7L 5s]], a chromatic scale produced using soft-of-basic step ratios. |
| | * [[5L 7s]], a chromatic scale produced using hard-of-basic step ratios. |
| | 12edo, the equalized form of both 7L 5s and 5L 7s, is also a superset of 5L 2s. |
| | |
| | === MODMOS scales and muddles === |
| | {{Main|5L 2s/MODMOSes|5L 2s/Muddles}} |
| | |
| | === Scala files === |
| | * [[Meantone7]] – 19edo and 31edo tunings |
| | * [[Nestoria7]] – 171edo tuning |
| | * [[Pythagorean7]] – Pythagorean tuning |
| | * [[Garibaldi7]] – 94edo tuning |
| | * [[Cotoneum7]] – 217edo tuning |
| | * [[Edson7]] – 29edo tuning |
| | * [[Pepperoni7]] – 271edo tuning |
| | * [[Supra7]] – 56edo tuning |
| | * [[Archy7]] – 49edo tuning |
| | |
| | == Scale tree == |
| | {{MOS tuning spectrum |
| | | Depth = 6 |
| | | 7/5 = [[Flattone]] region |
| | | 21/13 = [[Golden meantone]] (696.214{{c}}) |
| | | 5/3 = [[Meantone]] region |
| | | 9/4 = [[Pythagorean tuning]] (701.955{{c}}) |
| | | 16/7 = [[Garibaldi]] / [[cassandra]] |
| | | 5/2 = [[Dominant (temperament)|Dominant]] region |
| | | 21/8 = Golden neogothic (704.096{{c}}) |
| | | 8/3 = [[Neogothic]] region |
| | | 7/2 = [[Quasisuper]] region |
| | | 9/2 = [[Superpyth]] region |
| | | 11/2 = [[Quasiultra]] region |
| | | 7/1 = [[Ultrapyth]] region |
| | }} |
| | |
| | === Step ratio diagram === |
| | [[File:5L2s.jpg|alt=5L2s.jpg|5L2s.jpg]] |
| | |
| | == See also == |
| | * [[Diatonic functional harmony]] |
| | * [[Diatonic]] (disambiguation page) |
| | |
| | [[Category:Diatonic| ]] <!-- Main article --> |
| | [[Category:7-tone scales]] |