Xen concepts for beginners: Difference between revisions

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* [[17edo]]: The smallest edo after 12edo with a diatonic scale, and the smallest after 12edo to provide perfect fifths which are consonant for most purposes. Its major intervals are sharper and its minor intervals flatter than in 12edo, so it is often said to have a dramatic sound. First neutral diatonic edo (providing neutral seconds, thirds, sixths, and sevenths).

* [[18edo]]: Has two fifths, 733{{c}} and 667{{c}}, that are nearly equally off from [[3/2]].

* [[19edo]]: The smallest edo after 12edo which supports [[meantone]]. Just major and minor thirds are better approximated than in 12edo, but perfect fifths are represented significantly worse. First [[interordinal]] diatonic edo (interordinals are semifourths, semisixths, semitenths, and semitwelfths).

* [[19edo]]: The smallest edo after 12edo which supports [[meantone]]. Just major and minor thirds are better approximated than in 12edo, but perfect fifths are represented significantly worse, at 7.2{{c}} flat instead of 2{{c}}. First [[interordinal]] diatonic edo (interordinals are semifourths, semisixths, semitenths, and semitwelfths). Diminished and augmented seconds, thirds, sixths, and sevenths are now distinct intervals with entirely new functions, whereas in 12edo they are conflated with simpler intervals.

* [[22edo]]: Diatonic mos with a fifth significantly sharper than just, so that ~~it has~~ supermajor and subminor thirds (approximately [[9/7]] and [[7/6]]) ~~for its~~ major and minor thirds. Has a 5-limit major third (approximate [[5/4]]) which ''~~cannot~~'' be reached by stacking four fifths. Supports [[superpyth]] and 7L 1s.

* [[22edo]]: Diatonic mos with a fifth significantly sharper than just, so that four fifths and three fourths give supermajor and subminor thirds (approximately [[9/7]] and [[7/6]]) instead of major and minor thirds. Has a 5-limit major third (approximate [[5/4]]) which is ''not'' be reached by stacking four fifths. Supports [[superpyth]] along with [[7L 1s]] and [[7L 8s]] [[Porcupine]] scales.

* [[23edo]]: The largest edo without a diatonic, 5edo, or 7edo fifth. Supports mavila like 9edo and 16edo with the flat fifth.

* [[23edo]]: The largest edo without a diatonic, 5edo, or 7edo fifth. Supports mavila, just like 9edo and 16edo, with the flat fifth.

* [[24edo]]: Has both neutral thirds (and other neutral intervals) and semifourths (and other interordinals), each of these lending itself to different harmony. Has 12edo mos scales as well as new ones.

* [[24edo]]: Has both neutral thirds (and other neutral intervals) and semifourths (and other interordinals), each of these lending itself to different harmony. Has 12edo mos scales as well as new ones, such as [[5L 4s]] and [[4L 3s]].

* [[26edo]]: ~~Even~~ softer diatonic mos than 19edo, so much that the diatonic major third is nearly exactly [[26/21]] and the diatonic minor second is nearly exactly [[13/12]]. The [[7/4]] is also nearly exact, and the edo also has a good [[10/9]], [[14/11]], and [[11/8]].

* [[26edo]]: Has a fifth even flatter than that of 19edo, at 9.6{{c}} flat, and an even softer diatonic mos than 19edo, so much that the diatonic major third is nearly exactly [[26/21]] and the diatonic minor second is nearly exactly [[13/12]]. The [[7/4]] is also nearly exact, and the edo also has a good [[10/9]], [[14/11]], and [[11/8]].

* [[27edo]]: Even harder diatonic mos than 22edo; ~~the fifth is approximately about as sharp (by~~ 9.2{{c}}) as 26edo's is flat ~~(by 9.6{{c}})~~. It has 12edo's [[5/4]], a near-exact [[7/6]], and an approximate [[16/13]] neutral third.

* [[27edo]]: Even harder diatonic mos than 22edo; at 9.2{{c}} sharp of just, the fifth is approximately about as sharp as 26edo's is flat. It has 12edo's [[5/4]], a near-exact [[7/6]], and an approximate [[16/13]] neutral third. Four fifths give a supermajor thirdand three fourths give a subminor third, just like in 22edo.

* [[29edo]]: First edo with a perfect fifth closer to just intonation than 12edo. The minor third is extremely close to just [[13/11]]. It offers a tuning of 7L 1s with more consonant fifths than 15edo or 22edo before it. Its diatonic scale has similar melodic properties to 17edo, although subtler.

* [[31edo]]: One of the most popular meantone edos. Close to historical [[quarter-comma meantone]]. Not only is its major third close to just [[5/4]], it also matches the harmonic seventh [[7/4]] well.

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* [[37edo]]: Good for primes [[5/4|5]], [[7/4|7]], [[11/8|11]] and [[13/8|13]], but renders 3/2 sharp, even more so than 27edo.

* [[41edo]]: Often considered remarkably good for the primes up to 11. Good 3; flat 5 and 7; sharp 11 and 13. Known for the [[Kite guitar]].

* [[43edo]]: Possibly the most optimal tuning for meantone, with 5 tuned sharp and 3 tuned flat by almost exactly the same amount. Has better approximations of 11 and 13 than 19edo and 31edo, though the 7/4 is noticeably sharp.

* [[46edo]]: Neogothic 3; sharp 5; flat 7, 11, and 13; good 17. Supports [[parapyth]]. Often compared to 41edo; some favor one, some the other.

* [[53edo]]: Is a stack of near-just 3/2's which also approximates primes 5, 7, 13, and 19.

* [[72edo]]: A notable subdivision of 12edo that is a very strong 11-limit (primes 2, 3, 5, 7, 11) temperament for its size.

* [[87edo]]: Even better in 2.3.5.11.13 than 72edo is in the 11-limit, and a consistent and precise edo for approximating harmonics 8 to 16, but ratios with 7 suffer due to the 7 being flat and the 3 being sharp.

* [[311edo]]: An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41. ~~The~~ final boss of RTT edos.

* [[311edo]]: An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41. Essentially, the final boss of RTT edos.

[[Category:Overview]]

@@ Line 118: / Line 118: @@
 * [[17edo]]: The smallest edo after 12edo with a diatonic scale, and the smallest after 12edo to provide perfect fifths which are consonant for most purposes. Its major intervals are sharper and its minor intervals flatter than in 12edo, so it is often said to have a dramatic sound. First neutral diatonic edo (providing neutral seconds, thirds, sixths, and sevenths).
 * [[18edo]]: Has two fifths, 733{{c}} and 667{{c}}, that are nearly equally off from [[3/2]].
-* [[19edo]]: The smallest edo after 12edo which supports [[meantone]]. Just major and minor thirds are better approximated than in 12edo, but perfect fifths are represented significantly worse. First [[interordinal]] diatonic edo (interordinals are semifourths, semisixths, semitenths, and semitwelfths).
+* [[19edo]]: The smallest edo after 12edo which supports [[meantone]]. Just major and minor thirds are better approximated than in 12edo, but perfect fifths are represented significantly worse, at 7.2{{c}} flat instead of 2{{c}}. First [[interordinal]] diatonic edo (interordinals are semifourths, semisixths, semitenths, and semitwelfths). Diminished and augmented seconds, thirds, sixths, and sevenths are now distinct intervals with entirely new functions, whereas in 12edo they are conflated with simpler intervals.
-* [[22edo]]: Diatonic mos with a fifth significantly sharper than just, so that it has supermajor and subminor thirds (approximately [[9/7]] and [[7/6]]) for its major and minor thirds. Has a 5-limit major third (approximate [[5/4]]) which ''cannot'' be reached by stacking four fifths. Supports [[superpyth]] and 7L&nbsp;1s.
+* [[22edo]]: Diatonic mos with a fifth significantly sharper than just, so that four fifths and three fourths give supermajor and subminor thirds (approximately [[9/7]] and [[7/6]]) instead of major and minor thirds. Has a 5-limit major third (approximate [[5/4]]) which is ''not'' be reached by stacking four fifths. Supports [[superpyth]] along with [[7L&nbsp;1s]] and [[7L&nbsp;8s]] [[Porcupine]] scales.
-* [[23edo]]: The largest edo without a diatonic, 5edo, or 7edo fifth. Supports mavila like 9edo and 16edo with the flat fifth.
+* [[23edo]]: The largest edo without a diatonic, 5edo, or 7edo fifth. Supports mavila, just like 9edo and 16edo, with the flat fifth.
-* [[24edo]]: Has both neutral thirds (and other neutral intervals) and semifourths (and other interordinals), each of these lending itself to different harmony. Has 12edo mos scales as well as new ones.
+* [[24edo]]: Has both neutral thirds (and other neutral intervals) and semifourths (and other interordinals), each of these lending itself to different harmony. Has 12edo mos scales as well as new ones, such as [[5L&nbsp;4s]] and [[4L&nbsp;3s]].
-* [[26edo]]: Even softer diatonic mos than 19edo, so much that the diatonic major third is nearly exactly [[26/21]] and the diatonic minor second is nearly exactly [[13/12]]. The [[7/4]] is also nearly exact, and the edo also has a good [[10/9]], [[14/11]], and [[11/8]].
+* [[26edo]]: Has a fifth even flatter than that of 19edo, at 9.6{{c}} flat, and an even softer diatonic mos than 19edo, so much that the diatonic major third is nearly exactly [[26/21]] and the diatonic minor second is nearly exactly [[13/12]]. The [[7/4]] is also nearly exact, and the edo also has a good [[10/9]], [[14/11]], and [[11/8]].
-* [[27edo]]: Even harder diatonic mos than 22edo; the fifth is approximately about as sharp (by 9.2{{c}}) as 26edo's is flat (by 9.6{{c}}). It has 12edo's [[5/4]], a near-exact [[7/6]], and an approximate [[16/13]] neutral third.
+* [[27edo]]: Even harder diatonic mos than 22edo; at 9.2{{c}} sharp of just, the fifth is approximately about as sharp as 26edo's is flat. It has 12edo's [[5/4]], a near-exact [[7/6]], and an approximate [[16/13]] neutral third. Four fifths give a supermajor thirdand three fourths give a subminor third, just like in 22edo.
 * [[29edo]]: First edo with a perfect fifth closer to just intonation than 12edo. The minor third is extremely close to just [[13/11]]. It offers a tuning of 7L&nbsp;1s with more consonant fifths than 15edo or 22edo before it. Its diatonic scale has similar melodic properties to 17edo, although subtler.
 * [[31edo]]: One of the most popular meantone edos. Close to historical [[quarter-comma meantone]]. Not only is its major third close to just [[5/4]], it also matches the harmonic seventh [[7/4]] well.
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 * [[37edo]]: Good for primes [[5/4|5]], [[7/4|7]], [[11/8|11]] and [[13/8|13]], but renders 3/2 sharp, even more so than 27edo.
 * [[41edo]]: Often considered remarkably good for the primes up to 11. Good 3; flat 5 and 7; sharp 11 and 13. Known for the [[Kite guitar]].
+* [[43edo]]: Possibly the most optimal tuning for meantone, with 5 tuned sharp and 3 tuned flat by almost exactly the same amount. Has better approximations of 11 and 13 than 19edo and 31edo, though the 7/4 is noticeably sharp.
 * [[46edo]]: Neogothic 3; sharp 5; flat 7, 11, and 13; good 17. Supports [[parapyth]]. Often compared to 41edo; some favor one, some the other.
 * [[53edo]]: Is a stack of near-just 3/2's which also approximates primes 5, 7, 13, and 19.
 * [[72edo]]: A notable subdivision of 12edo that is a very strong 11-limit (primes 2, 3, 5, 7, 11) temperament for its size.
 * [[87edo]]: Even better in 2.3.5.11.13 than 72edo is in the 11-limit, and a consistent and precise edo for approximating harmonics 8 to 16, but ratios with 7 suffer due to the 7 being flat and the 3 being sharp.
-* [[311edo]]: An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41. The final boss of RTT edos.
+* [[311edo]]: An edo renowned for being a good edo for the whole 41-odd-limit and quite a bit more (mainly composite) harmonics above 41. Essentially, the final boss of RTT edos.
 [[Category:Overview]]