Tour of regular temperaments: Difference between revisions

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2011-09-05 22:08:45 UTC</tt>.<br>

: The original revision id was <tt>251021756</tt>.<br>

The magic family includes [[16edo]], [[19edo]], [[22edo]], [[25edo]], and [[41edo]] among its possible tunings, with the latter being near-optimal.

The diaschismic family tempers out 2048/2025, the [[diaschisma]], which tempers things such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include [[12edo]], [[22edo]], [[34edo]], [[46edo]], [[56edo]], [[58edo]] and [[80edo]]. A noted 7-limit extension to diaschismic is[[pajara| pajara]] temperament, where the intervals 50/49 and 64/63 are tempered out, and of which [[22edo]] is an excellent tuning.

This tempers out the pelogic comma, 135/128, also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, the accumulated flatness leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates 2L5s "anti-diatonic" scales. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include [[9edo]], [[16edo]], [[23edo]], and [[25edo]].

The porcupine family tempers out 250/243, the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. Some porcupine temperaments include [[15edo]], [[22edo]], [[37edo]], and [[59edo]].

The wuerschmidt family tempers out Wuerschmidt's comma, 393216/390625 = |17 1 -8&gt;. Wuerschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a perfect 5th two octaves up); that is, (5/4)^8 * (393216/390625) = 6. It tends to generate the same MOS's as [[magic family|magic temperament]], but is tuned slightly more accurately. Both [[31edo]] and [[34edo]] can be used as wuerschmit tunings, as can [[65edo]], which is quite accurate.

The augmented family tempers out the diesis of 128/125, the difference between three 5/4 major thirds and a 2/1 octave, and so identifies the major third with the third-octave. Hence it has the same 400-cent 5/4-approximations as [[12edo]], which is an excellent tuning for augmented. It is the temperament that results in what is commonly called the "augmented scale" (3L3s) in common 12-based music theory, as well as what is commonly called "Tcherepnin's scale" (3L6s).

The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, 25/24 (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2). This temperament hence equates major and minor thirds, evening them out into two neutral-sized intervals that are taken to approximate both. [[7edo]] makes for a "good" dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the "neutral" dicot 3rds span a 3/2. Tunings include [[7edo]], [[10edo]], and [[17edo]].

The Pythagorean family tempers out the Pythagorean comma, |-19 12 0&gt;. Since this is a 3-limit comma, it is also a 5-limit comma and can stand as parent to a 7-limit or higher family, in this case containing compton temperament and catler temperament. Temperaments in this family tend to have a period of 1/12th octave, and the 5-limit compton temperament can be thought of generating as two duplicate

chains of 12-equal, offset from one another justly tuned 5/4.

The magic family includes &lt;a class="wiki_link" href="/16edo"&gt;16edo&lt;/a&gt;, &lt;a class="wiki_link" href="/19edo"&gt;19edo&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;, &lt;a class="wiki_link" href="/25edo"&gt;25edo&lt;/a&gt;, and &lt;a class="wiki_link" href="/41edo"&gt;41edo&lt;/a&gt; among its possible tunings, with the latter being near-optimal.&lt;br /&gt;

  The diaschismic family tempers out 2048/2025, the &lt;a class="wiki_link" href="/diaschisma"&gt;diaschisma&lt;/a&gt;, which tempers things such that 5/4 * 5/4 * 81/64 is taken to equal 2/1. It has a half-octave period and its generator is an approximate 3/2. Diaschismic tunings include &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;, &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt;, &lt;a class="wiki_link" href="/46edo"&gt;46edo&lt;/a&gt;, &lt;a class="wiki_link" href="/56edo"&gt;56edo&lt;/a&gt;, &lt;a class="wiki_link" href="/58edo"&gt;58edo&lt;/a&gt; and &lt;a class="wiki_link" href="/80edo"&gt;80edo&lt;/a&gt;. A noted 7-limit extension to diaschismic is&lt;a class="wiki_link" href="/pajara"&gt; pajara&lt;/a&gt; temperament, where the intervals 50/49 and 64/63 are tempered out, and of which &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt; is an excellent tuning.&lt;br /&gt;

  This tempers out the pelogic comma, 135/128, also known as the major chroma or major limma. These temperaments are notable for having 3/2's tuned so flat that four of them, when stacked together, the accumulated flatness leads you to 6/5 + 2 octaves instead of 5/4 + 2 octaves, and one consequence of this is that it generates 2L5s &amp;quot;anti-diatonic&amp;quot; scales. Mavila and Armodue are some of the most notable temperaments associated with the pelogic comma. Tunings include &lt;a class="wiki_link" href="/9edo"&gt;9edo&lt;/a&gt;, &lt;a class="wiki_link" href="/16edo"&gt;16edo&lt;/a&gt;, &lt;a class="wiki_link" href="/23edo"&gt;23edo&lt;/a&gt;, and &lt;a class="wiki_link" href="/25edo"&gt;25edo&lt;/a&gt;.&lt;br /&gt;

  The porcupine family tempers out 250/243, the difference between three 10/9's (1000/729) and 4/3, known as the maximal diesis or porcupine comma. It subdivides the fourth into three equal parts, each taken as an approximated 10/9, of which two approximate 6/5. It also manifests itself as the difference between three 6/5's and 16/9, as the difference between 10/9 and 27/25, and as the difference between 81/80 and 25/24. Some porcupine temperaments include &lt;a class="wiki_link" href="/15edo"&gt;15edo&lt;/a&gt;, &lt;a class="wiki_link" href="/22edo"&gt;22edo&lt;/a&gt;, &lt;a class="wiki_link" href="/37edo"&gt;37edo&lt;/a&gt;, and &lt;a class="wiki_link" href="/59edo"&gt;59edo&lt;/a&gt;.&lt;br /&gt;

  The wuerschmidt family tempers out Wuerschmidt's comma, 393216/390625 = |17 1 -8&amp;gt;. Wuerschmidt itself has a generator of a major third, eight of which give a 6/1 (the 6th harmonic, or a perfect 5th two octaves up); that is, (5/4)^8 * (393216/390625) = 6. It tends to generate the same MOS's as &lt;a class="wiki_link" href="/magic%20family"&gt;magic temperament&lt;/a&gt;, but is tuned slightly more accurately. Both &lt;a class="wiki_link" href="/31edo"&gt;31edo&lt;/a&gt; and &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt; can be used as wuerschmit tunings, as can &lt;a class="wiki_link" href="/65edo"&gt;65edo&lt;/a&gt;, which is quite accurate.&lt;br /&gt;

  The augmented family tempers out the diesis of 128/125, the difference between three 5/4 major thirds and a 2/1 octave, and so identifies the major third with the third-octave. Hence it has the same 400-cent 5/4-approximations as &lt;a class="wiki_link" href="/12edo"&gt;12edo&lt;/a&gt;, which is an excellent tuning for augmented. It is the temperament that results in what is commonly called the &amp;quot;augmented scale&amp;quot; (3L3s) in common 12-based music theory, as well as what is commonly called &amp;quot;Tcherepnin's scale&amp;quot; (3L6s).&lt;br /&gt;

  The dicot family is a low-accuracy family of temperaments which temper out the chromatic semitone, 25/24 (the difference between 5/4 and 6/5, or alternatively the difference between two 5/4's and 3/2 OR two 6/5's and 3/2). This temperament hence equates major and minor thirds, evening them out into two neutral-sized intervals that are taken to approximate both. &lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt; makes for a &amp;quot;good&amp;quot; dicot tuning, although it is questionable whether this temperament bears any actual resemblance to 5-limit harmony. Two of the &amp;quot;neutral&amp;quot; dicot 3rds span a 3/2. Tunings include &lt;a class="wiki_link" href="/7edo"&gt;7edo&lt;/a&gt;, &lt;a class="wiki_link" href="/10edo"&gt;10edo&lt;/a&gt;, and &lt;a class="wiki_link" href="/17edo"&gt;17edo&lt;/a&gt;.&lt;br /&gt;

  The Pythagorean family tempers out the Pythagorean comma, |-19 12 0&amp;gt;. Since this is a 3-limit comma, it is also a 5-limit comma and can stand as parent to a 7-limit or higher family, in this case containing compton temperament and catler temperament. Temperaments in this family tend to have a period of 1/12th octave, and the 5-limit compton temperament can be thought of generating as two duplicate&lt;br /&gt;

chains of 12-equal, offset from one another justly tuned 5/4.&lt;br /&gt;