Xen concepts for beginners: Difference between revisions

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* [[5edo]]: Equalized pentatonic ("Equipentatonic").

* [[7edo]]: Equalized diatonic ("Equiheptatonic").

* [[9edo]]: The simplest edo with a [[2L5s]] MOS (sssLssL).

* [[9edo]]: The simplest edo with a [[2L5s]] MOS (sssLssL). This MOS is of interest because it can be viewed as a tuning of the diatonic scale where whole steps are smaller than half steps.

* [[11edo]]: Stretched 12edo, has [[4L3s]] MOS (LLsLsLs) which is a stretched diatonic.

* [[13edo]]: Compressed 12edo having the [[5L3s]] MOS (LLsLLsLs) which is a compressed version of the diatonic scale.

* [[15edo]]: The smallest edo with a [[5L5s]] MOS (LsLsLsLsLs) commonly called the Blackwood scale.

* [[15edo]]: The smallest edo with a [[5L5s]] MOS (LsLsLsLsLs) commonly called the Blackwood scale. Also the smallest with a [[7L 1s]] MOS (LLLLsLLL). Both scales are known for supporting relatively familiar major and minor chords with relatively unfamiliar melodic structures.

* [[16edo]]: Has 2L5s (sssLssL) and [[7L2s]] (LLLsLLLLs).

* [[17edo]]: The smallest edo after 12edo with a diatonic scale, which ~~can be harmonically very different from~~ 12edo ~~diatonic depending on how you use~~ it. First neutral diatonic edo (providing neutral seconds, thirds, sixths, and sevenths).

* [[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's often said to have a dramatic sound. First neutral diatonic edo (providing neutral seconds, thirds, sixths, and sevenths).

* [[18edo]]: Has two fifths, 733c and 667c, 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. 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. First [[interordinal]] diatonic edo (interordinals are semifourths, semisixths, semitenths, and semitwelfths).

* [[22edo]]: Diatonic MOS with a fifth so sharp 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]] ~~like [[27edo]]~~.

* [[22edo]]: Diatonic MOS with a fifth so sharp 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.

* [[23edo]]: The largest edo without a diatonic, 5edo, or 7edo 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.

* [[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]].

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

* [[29edo]]: ~~Weird flat neogothic~~ edo.

* [[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]]: ~~Often considered~~ the ~~best~~ Meantone ~~edo~~. 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~~, also approximating other JI ratios like [[6/5]] (just minor third), [[7/6]] (septimal subminor third), and [[25/16]] (classical augmented fifth)~~.

* [[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.

* [[34edo]]: Good for the 5-limit (2.3.5), as it doesn't temper out 81/80 and has a good 5/4.

* [[34edo]]: Good for the 5-limit (2.3.5), as it doesn't temper out 81/80 and has a good 5/4. Also contains all notes of 17edo.

* [[36edo]]: Good for primes [[3/2|3]] and [[7/4|7]].

* [[37edo]]: Good for primes [[5/4|5]], [[7/4|7]], [[11/8|11]] and [[13/8|13]], ~~in return for a sharp~~ 3/2.

* [[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]]: Good 3; flat 5 and 7; sharp 11 and 13. Known for the [[Kite guitar]].

* [[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]].

* [[46edo]]: Neogothic 3; sharp 5; flat 7, 11, and 13; good 17. Supports [[Parapyth]]. ~~Some~~ favor ~~it over 41edo~~.

* [[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.~~<!--~~

* [[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.~~-->~~

* [[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.

[[Category:Overview]]

@@ Line 103: / Line 103: @@
 * [[5edo]]: Equalized pentatonic ("Equipentatonic").
 * [[7edo]]: Equalized diatonic ("Equiheptatonic").
-* [[9edo]]: The simplest edo with a [[2L5s]] MOS (sssLssL).
+* [[9edo]]: The simplest edo with a [[2L5s]] MOS (sssLssL). This MOS is of interest because it can be viewed as a tuning of the diatonic scale where whole steps are smaller than half steps.
 * [[11edo]]: Stretched 12edo, has [[4L3s]] MOS (LLsLsLs) which is a stretched diatonic.
 * [[13edo]]: Compressed 12edo having the [[5L3s]] MOS (LLsLLsLs) which is a compressed version of the diatonic scale.
-* [[15edo]]: The smallest edo with a [[5L5s]] MOS (LsLsLsLsLs) commonly called the Blackwood scale.
+* [[15edo]]: The smallest edo with a [[5L5s]] MOS (LsLsLsLsLs) commonly called the Blackwood scale. Also the smallest with a [[7L 1s]] MOS (LLLLsLLL). Both scales are known for supporting relatively familiar major and minor chords with relatively unfamiliar melodic structures.
 * [[16edo]]: Has 2L5s (sssLssL) and [[7L2s]] (LLLsLLLLs).
-* [[17edo]]: The smallest edo after 12edo with a diatonic scale, which can be harmonically very different from 12edo diatonic depending on how you use it. First neutral diatonic edo (providing neutral seconds, thirds, sixths, and sevenths).
+* [[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's often said to have a dramatic sound. First neutral diatonic edo (providing neutral seconds, thirds, sixths, and sevenths).
 * [[18edo]]: Has two fifths, 733c and 667c, 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. 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. First [[interordinal]] diatonic edo (interordinals are semifourths, semisixths, semitenths, and semitwelfths).
-* [[22edo]]: Diatonic MOS with a fifth so sharp 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]] like [[27edo]].
+* [[22edo]]: Diatonic MOS with a fifth so sharp 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.
 * [[23edo]]: The largest edo without a diatonic, 5edo, or 7edo 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.
 * [[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]].
 * [[27edo]]: Even harder diatonic MOS than 22edo; the fifth is approximately about as sharp (by 9.2c) as 26edo's is flat (by 9.6c). It has 12edo's [[5/4]], a near-exact [[7/6]], and an approximate [[16/13]] neutral third.
-* [[29edo]]: Weird flat neogothic edo.
+* [[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]]: Often considered the best Meantone edo. 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, also approximating other JI ratios like [[6/5]] (just minor third), [[7/6]] (septimal subminor third), and [[25/16]] (classical augmented fifth).
+* [[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.
-* [[34edo]]: Good for the 5-limit (2.3.5), as it doesn't temper out 81/80 and has a good 5/4.
+* [[34edo]]: Good for the 5-limit (2.3.5), as it doesn't temper out 81/80 and has a good 5/4. Also contains all notes of 17edo.
 * [[36edo]]: Good for primes [[3/2|3]] and [[7/4|7]].
-* [[37edo]]: Good for primes [[5/4|5]], [[7/4|7]], [[11/8|11]] and [[13/8|13]], in return for a sharp 3/2.
+* [[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]]: Good 3; flat 5 and 7; sharp 11 and 13. Known for the [[Kite guitar]].
+* [[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]].
-* [[46edo]]: Neogothic 3; sharp 5; flat 7, 11, and 13; good 17. Supports [[Parapyth]]. Some favor it over 41edo.
+* [[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.<!--
+* [[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.-->
+* [[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.
 [[Category:Overview]]