31edo: Difference between revisions

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||= [[13_11|13/11]], [[22_13|22/13]] ||= 18.242 ||

A single step of 31-edo is about 38.71¢. Intervals around this size are called //dieses// (singular '//diesis//'). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G#, to B#, but B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). In the [[11-limit]], the diesis stands in for just ratios 56:55 (31.19); 55:54 (31.77¢); 49:48 (39.70¢); 45:44 (38.91¢); 36:35 (48.77¢); 33:32 (53.27¢) and others. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. Demonstrated in [[SpiralProgressions]].

The difference between a major and minor third. The more 'expressive' of the 'half steps,' and the larger of 31's two "microtones". In meantone, it is the //chromatic semitone//, the interval that distinguishes major and minor intervals of the same generic interval class (eg. thirds). 2\31 stands in for just ratios 28:27 (62.96¢); 25:24 (70.67¢); 22:21 (80.54¢); 21:20 (84.45¢) and others. Generates [[Starling temperaments#Valentine%20temperament|valentine temperament]] - aka [[Armodue theory#Semi-equalized%20Armodue|semi-equalized Armodue]].

====MOS Scales generated by 2\31:====  

|| 16-tone || [[15L 1s]] || 2 ||  || 2 ||  || 2 ||  || 2 ||  || 2 ||  || 2 ||  || 2 ||  || 2 ||  || 2 ||  || 2 ||  || 2 ||  || 2 ||  || 2 ||  || 2 ||  || 2 ||  || 1 ||

The larger and clunkier of the 31edo semitones. In meantone, it is the //diatonic semitone// which appears in the diatonic scale between, for instance, the major third and perfect fourth, and the major seventh and octave. 3\31 stands in for just ratios 16:15 (111.73¢); 15:14 (119.44¢) and others. It is notable that two of these make an 8/7; this implies that the 3\31 is a //secor// and generates [[Gamelismic clan|miracle temperament]].

====MOS Scales generated by 3\31:====  

|| 21-tone (Blackjack) || [[11L 10s]] || 2 ||  || 1 || 2 ||  || 1 || 2 ||  || 1 || 2 ||  || 1 || 2 ||  || 1 || 2 ||  || 1 || 2 ||  || 1 || 2 ||  || 1 || 2 ||  || 1 || 2 ||  || 1 || 1 ||

Exactly one half of the minor third and twice the minor semitone. 4\31 stands in for 12:11 (150.64¢); 35:32 (155.14¢); 11:10 (165.00¢) and others. Although neutral seconds are typically associated with the 11-limit, 4\31 approximates the [[7-limit]] interval 35/32 quite well, as the 5th harmonic of the 7th harmonic or vice versa, both of which are closely approximated in 31edo. And although 31 is not extremely accurate in the 11-limit, it is notable that since 11 and 3 are both flat, the interval that distinguishes them (12/11) is only about 4.5¢ off. Generates [[Starling temperaments|nusecond temperament]].

====MOS Scales generated by 4\31:====  

|| 23-tone || [[8L 15s]] || 1 || 1 || 2 ||  || 1 || 1 || 2 ||  || 1 || 1 || 2 ||  || 1 || 1 || 2 ||  || 1 || 1 || 2 ||  || 1 || 1 || 2 ||  || 1 || 1 || 2 ||  || 1 || 2 ||  ||

A rather smallish whole tone. Sometimes called melodically dull. As it falls between (and functions as) just whole tones 9:8 and 10:9, 5\31 is considered a "meantone". Two meantones make a near-just major third. Perhaps it is worth noting that its relative narrowness (to JI 9/8) makes it easier to distinguish from the 8/7 approximation. And although it is over 10¢ flat of 9/8, 5\31 can function as a somewhat "active" (as opposed to perfectly stable) harmonic ninth, and it can be effective in combination with the also-narrow 11th harmonic. Indeed, the 11/9 approximation is excellent. Try, for instance 31's version of a 4:6:9:11 chord (steps 0-18-36-45). Generates [[Gamelismic clan|hemithirds temperament]] and [[Wuerschmidt family|hermiwuerschmidt temperament]].

====MOS Scales generated by 5\31:====  

|| 25-tone || [[6L 19s]] || 2 ||  || 1 || 1 || 1 || 2 ||  || 1 || 1 || 1 || 2 ||  || 1 || 1 || 1 || 2 ||  || 1 || 1 || 1 || 2 ||  || 1 || 1 || 1 || 2 ||  || 1 || 1 || 1 || 1 ||

Exactly one half of a narrow fourth, twice a major semitone, or thrice a minor semitone. In 7-limit tonal music, 6\31 closely represents 8:7 (231.17¢). In meantone, it is a diminished third, eg. C to Ebb. Generates [[Meantone family|mothra temperament]].

====MOS Scales generated by 6\31:====  

|| 26-tone || [[5L 21s]] || 2 ||  || 1 || 1 || 1 || 1 || 2 ||  || 1 || 1 || 1 || 1 || 2 ||  || 1 || 1 || 1 || 1 || 2 ||  || 1 || 1 || 1 || 1 || 2 ||  || 1 || 1 || 1 || 1 || 1 ||

Exactly one half of a superfourth (11:8 approximation). In 7-limit tonal music, 7\31 stands in for 7:6 (266.87¢). In meantone temperament, it is an augmented 2nd, eg. C to D#. Generates [[Semicomma family|orwell temperament]].

====MOS Scales generated by 7\31:====  

A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for [[mohajira|mohajira temperament]]), and also thrice a major semitone. It is closer in quality to a minor third than a major third, but indeed, it is distinct. It is 11¢ shy of 16/13 (359.47¢), suggesting a [[13-limit]] interpretation for 31edo. However, its close proximity to 11/9 makes it hard to hear it as 16/13, which in JI has a different quality (and, as a neutral third, is more "major-like" than "minor-like"). Also, its inversion, 22\31 (851.61¢) is wide of the 13th harmonic by about 11¢, while all the lower harmonics are either near-just or narrow. This means the errors can accumulate, for instance, with 13/9 (636.62¢) represented by 17\31 (658.06¢), a good 21.4¢ sharp.

====MOS Scales generated by 9\31:====  

|| 28-tone || [[3L 25s]] || 2 ||  || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 2 ||  || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 2 ||  || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 || 1 ||

11\31 functions as 14:11 (417.51¢), 23:18 (424.36¢), 32:25 (427.37¢), 9:7 (435.08¢) and others. In meantone temperament, it is a diminished fourth, eg. C to Fb. It is notable as closely approximating an interval of the [[23-limit]], suggesting the possibility of treating 16\31 (619.35¢) as a flat version of 23/16 (628.27¢). It is perhaps also notable for being close to 6\17, the bright major third of the ever-popular [[17edo]]. Generates [[Meantone family|squares temperament]].

====MOS Scales generated by 11\31:====  

Exactly twice a supermajor second, thrice a neutral second, or four times a minor second. In the 7-limit, 12\31 functions as 21:16 (470.78¢). It is also quite close to the [[17-limit]] interval 17/13 (464.43¢), although 31edo does not offer up reasonable approximations of the 17th or 13th harmonics to help make this identity clear. This interval and its inversion 19\31 (735.48¢, a superfifth) are notable for being the only intervals in the 31edo octave larger than the 3\31 diatonic semitone (and smaller than its inversion, 28\31) that are not 11-limit consonances. Generates [[semisept]] temperament.

====MOS Scales generated by 12\31:====  

Exactly twice a subminor third. Functions as both the 11:8 (551.32¢) and 15:11 (536.95¢) undecimal superfourths (121/120 is tempered out). Thus it makes possible a symmetrical tempered version of an 8:11:15 triad. As 11/8, 14\31 is about 9¢ flat; however, it fits nicely with the also-flat 9/8, allowing a near-just 11/9. Nonetheless, most 11-limit chords in 31edo have a somewhat unstable quality which distinguishes them from their just counterparts. Generates [[Starling temperaments|casablanca temperament]].

====MOS Scales generated by 14\31:====  

In 7-limit tonal music, functions quite well as 7:5 (582.51¢). Exactly thrice a whole tone. Generates [[tritonic]] temperament.

====MOS Scales generated by 15\31:====  

===16\31 octave===  

The large tritone. Etc.

=Harmonic Scale=  

&lt;br /&gt;

  A single step of 31-edo is about 38.71¢. Intervals around this size are called &lt;em&gt;dieses&lt;/em&gt; (singular '&lt;em&gt;diesis&lt;/em&gt;'). In 31 it is equivalent to the difference between one octave and three stacked major thirds (C to E, to G#, to B#, but B# ≠ C), or four minor thirds (C to Eb to Gb to Bbb to Dbb ≠ C). In the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt;, the diesis stands in for just ratios 56:55 (31.19); 55:54 (31.77¢); 49:48 (39.70¢); 45:44 (38.91¢); 36:35 (48.77¢); 33:32 (53.27¢) and others. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. Demonstrated in &lt;a class="wiki_link" href="/SpiralProgressions"&gt;SpiralProgressions&lt;/a&gt;.&lt;br /&gt;

&lt;br /&gt;

  The difference between a major and minor third. The more 'expressive' of the 'half steps,' and the larger of 31's two &amp;quot;microtones&amp;quot;. In meantone, it is the &lt;em&gt;chromatic semitone&lt;/em&gt;, the interval that distinguishes major and minor intervals of the same generic interval class (eg. thirds). 2\31 stands in for just ratios 28:27 (62.96¢); 25:24 (70.67¢); 22:21 (80.54¢); 21:20 (84.45¢) and others. Generates &lt;a class="wiki_link" href="/Starling%20temperaments#Valentine%20temperament"&gt;valentine temperament&lt;/a&gt; - aka &lt;a class="wiki_link" href="/Armodue%20theory#Semi-equalized%20Armodue"&gt;semi-equalized Armodue&lt;/a&gt;.&lt;br /&gt;

   

&lt;br /&gt;

  The larger and clunkier of the 31edo semitones. In meantone, it is the &lt;em&gt;diatonic semitone&lt;/em&gt; which appears in the diatonic scale between, for instance, the major third and perfect fourth, and the major seventh and octave. 3\31 stands in for just ratios 16:15 (111.73¢); 15:14 (119.44¢) and others. It is notable that two of these make an 8/7; this implies that the 3\31 is a &lt;em&gt;secor&lt;/em&gt; and generates &lt;a class="wiki_link" href="/Gamelismic%20clan"&gt;miracle temperament&lt;/a&gt;.&lt;br /&gt;

   

&lt;br /&gt;

  Exactly one half of the minor third and twice the minor semitone. 4\31 stands in for 12:11 (150.64¢); 35:32 (155.14¢); 11:10 (165.00¢) and others. Although neutral seconds are typically associated with the 11-limit, 4\31 approximates the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; interval 35/32 quite well, as the 5th harmonic of the 7th harmonic or vice versa, both of which are closely approximated in 31edo. And although 31 is not extremely accurate in the 11-limit, it is notable that since 11 and 3 are both flat, the interval that distinguishes them (12/11) is only about 4.5¢ off. Generates &lt;a class="wiki_link" href="/Starling%20temperaments"&gt;nusecond temperament&lt;/a&gt;.&lt;br /&gt;

   

&lt;br /&gt;

  A rather smallish whole tone. Sometimes called melodically dull. As it falls between (and functions as) just whole tones 9:8 and 10:9, 5\31 is considered a &amp;quot;meantone&amp;quot;. Two meantones make a near-just major third. Perhaps it is worth noting that its relative narrowness (to JI 9/8) makes it easier to distinguish from the 8/7 approximation. And although it is over 10¢ flat of 9/8, 5\31 can function as a somewhat &amp;quot;active&amp;quot; (as opposed to perfectly stable) harmonic ninth, and it can be effective in combination with the also-narrow 11th harmonic. Indeed, the 11/9 approximation is excellent. Try, for instance 31's version of a 4:6:9:11 chord (steps 0-18-36-45). Generates &lt;a class="wiki_link" href="/Gamelismic%20clan"&gt;hemithirds temperament&lt;/a&gt; and &lt;a class="wiki_link" href="/Wuerschmidt%20family"&gt;hermiwuerschmidt temperament&lt;/a&gt;.&lt;br /&gt;

   

&lt;br /&gt;

  Exactly one half of a narrow fourth, twice a major semitone, or thrice a minor semitone. In 7-limit tonal music, 6\31 closely represents 8:7 (231.17¢). In meantone, it is a diminished third, eg. C to Ebb. Generates &lt;a class="wiki_link" href="/Meantone%20family"&gt;mothra temperament&lt;/a&gt;.&lt;br /&gt;

   

&lt;br /&gt;

  Exactly one half of a superfourth (11:8 approximation). In 7-limit tonal music, 7\31 stands in for 7:6 (266.87¢). In meantone temperament, it is an augmented 2nd, eg. C to D#. Generates &lt;a class="wiki_link" href="/Semicomma%20family"&gt;orwell temperament&lt;/a&gt;.&lt;br /&gt;

   

&lt;br /&gt;

&lt;br /&gt;

  A neutral 3rd, about 1¢ away from 11:9 (347.41¢). 9\31 is half a perfect fifth (making it a suitable generator for &lt;a class="wiki_link" href="/mohajira"&gt;mohajira temperament&lt;/a&gt;), and also thrice a major semitone. It is closer in quality to a minor third than a major third, but indeed, it is distinct. It is 11¢ shy of 16/13 (359.47¢), suggesting a &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; interpretation for 31edo. However, its close proximity to 11/9 makes it hard to hear it as 16/13, which in JI has a different quality (and, as a neutral third, is more &amp;quot;major-like&amp;quot; than &amp;quot;minor-like&amp;quot;). Also, its inversion, 22\31 (851.61¢) is wide of the 13th harmonic by about 11¢, while all the lower harmonics are either near-just or narrow. This means the errors can accumulate, for instance, with 13/9 (636.62¢) represented by 17\31 (658.06¢), a good 21.4¢ sharp.&lt;br /&gt;

   

&lt;br /&gt;

  11\31 functions as 14:11 (417.51¢), 23:18 (424.36¢), 32:25 (427.37¢), 9:7 (435.08¢) and others. In meantone temperament, it is a diminished fourth, eg. C to Fb. It is notable as closely approximating an interval of the &lt;a class="wiki_link" href="/23-limit"&gt;23-limit&lt;/a&gt;, suggesting the possibility of treating 16\31 (619.35¢) as a flat version of 23/16 (628.27¢). It is perhaps also notable for being close to 6\17, the bright major third of the ever-popular &lt;a class="wiki_link" href="/17edo"&gt;17edo&lt;/a&gt;. Generates &lt;a class="wiki_link" href="/Meantone%20family"&gt;squares temperament&lt;/a&gt;.&lt;br /&gt;

   

&lt;br /&gt;

&lt;br /&gt;

  Exactly twice a supermajor second, thrice a neutral second, or four times a minor second. In the 7-limit, 12\31 functions as 21:16 (470.78¢). It is also quite close to the &lt;a class="wiki_link" href="/17-limit"&gt;17-limit&lt;/a&gt; interval 17/13 (464.43¢), although 31edo does not offer up reasonable approximations of the 17th or 13th harmonics to help make this identity clear. This interval and its inversion 19\31 (735.48¢, a superfifth) are notable for being the only intervals in the 31edo octave larger than the 3\31 diatonic semitone (and smaller than its inversion, 28\31) that are not 11-limit consonances. Generates &lt;a class="wiki_link" href="/semisept"&gt;semisept&lt;/a&gt; temperament.&lt;br /&gt;

   

&lt;br /&gt;

&lt;br /&gt;

  Exactly twice a subminor third. Functions as both the 11:8 (551.32¢) and 15:11 (536.95¢) undecimal superfourths (121/120 is tempered out). Thus it makes possible a symmetrical tempered version of an 8:11:15 triad. As 11/8, 14\31 is about 9¢ flat; however, it fits nicely with the also-flat 9/8, allowing a near-just 11/9. Nonetheless, most 11-limit chords in 31edo have a somewhat unstable quality which distinguishes them from their just counterparts. Generates &lt;a class="wiki_link" href="/Starling%20temperaments"&gt;casablanca temperament&lt;/a&gt;.&lt;br /&gt;

   

&lt;br /&gt;

&lt;br /&gt;

  In 7-limit tonal music, functions quite well as 7:5 (582.51¢). Exactly thrice a whole tone. Generates &lt;a class="wiki_link" href="/tritonic"&gt;tritonic&lt;/a&gt; temperament.&lt;br /&gt;

   

&lt;!-- ws:start:WikiTextHeadingRule:65:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc32"&gt;&lt;a name="Intervals-Selected just intervals by error-16\31 octave"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:65 --&gt;16\31 octave&lt;/h3&gt;

  The large tritone. Etc.&lt;br /&gt;

&lt;br /&gt;

&lt;!-- ws:start:WikiTextHeadingRule:67:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc33"&gt;&lt;a name="Harmonic Scale"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:67 --&gt;Harmonic Scale&lt;/h1&gt;