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This is a page where I will draft edits before making them on the actual page. This may possibly include drafting a new page to be created. If you have something to add to any of them, or any concerns, please suggest them on the talk page. If a template is set to debug, make sure to remove that setting when editing the target page.
= 21edo =
= Pajara =
== Theory ==
There are two different mappings of the 11-limit. One is just called ''pajara'' and is slightly more complex but suffers almost no loss of accuracy compared to the 7-limit. It is best tuned flat of 22edo. The other, called ''pajarous'' to avoid confusion, maps the 11th harmonic slightly simpler, but 22edo is the only [[11-odd-limit]] [[diamond monotone]] tuning, where primes [[3/1|3]] and [[5/1|5]] are less accurate than in optimal tunings of canonical 11-limit pajara.
''Notes: Excellent odd harmonics 7, 15, 23, 29, 31, 33, 39, 43, all derived from 84edo; less accurate but still usable 17, 19, 27; also note 3*21 subgroup from 63edo''
21edo contains three [[7edo]] "equiheptatonic" scales, and can be interpreted as 7edo but with the capability to inflect up or down by a quarter-tone. The 7edo subset functions as a basic "diatonic" scale, though maximum-variety-3 options might also be preferable (such as omnidiatonic). In other words, all intervals have "minor", "neutral", and "major" variations, which makes building scales in 21edo rather interesting. If 21edo is analyzed purely diatonically, no chromatic alterations can exist because the chromatic semitone is equal to 0 cents (a fact characteristic of [[whitewood]] temperaments). So, another kind of accidental (such as ups and downs) is usually used instead, though it might be "reskinned" as sharps and flats to aid melodic intuition.
In the following tables, odd harmonics 1–11 and their inverses are in '''bold'''.
21edo supports tertian harmony with both 7edo's neutral chords and inflected major and minor chords. The major third is identical to 12edo's, but is a more extreme third in 21edo due to the flatness of the fifth (which makes the minor third close to subminor), so that the chords might be more comparable to neogothic chords.
In terms of just intonation, outside the 5-limit (where 21edo contains a flat fifth and the familiar but controversial 400c major third), 21edo also closely approximates the harmonics [[7/4]] (a subminor seventh), [[17/16]] (a semitone), [[19/16]] (a minor third), [[23/16]] (a tritone), and [[29/16]] (a minor seventh), with harmonics 7, 23, and 29 being especially accurate (and harmonic 7 being more accurate than in any other edo below 26). The intervals [[16/15]] and [[27/16]], if directly approximated, are also very accurate. 21edo can be liberally treated as a no-11s 29-limit temperament, but treating 21edo as a 2.15.7.23.27.29 subgroup temperament allows for a more accurate JI interpretation of the tuning, since almost every interval in 21edo can be described as a ratio within the 29-odd-limit. 21edo also works well on the 2.9/5.7.11/5.13/5.17/5 subgroup, which is possibly a more sensible way to treat it.
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! rowspan="2" | #
In terms of interval regions, 21edo possesses four types of 2nd (subminor, minor, submajor, and supermajor), three types of 3rd (subminor, neutral, and major), a "third-fourth" (an interval that can function as either a supermajor 3rd or a narrow 4th), a wide (or acute) 4th, and a narrow tritone, as well as the octave-inversions of all of these intervals.
! colspan="2" | Period 0
! colspan="2" | Period 1
Because 21edo is a Fibonacci edo, it contains an approximation to the [[logarithmic phi]] superfifth, which generates golden MOS scales 8L 5s, 5L 3s, and 3L 2s.
Thanks to its sevenths, 21edo is an ideal tuning for its size for [[metallic harmony]].
∗1: based on treating 21edo as a 2.7.15.23.27.29 subgroup temperament
∗2: based on treating 21edo as a 2.9/5.11/5.13/5.17/5.35/5 subgroup temperament
∗3: based on treating 21edo as 13-limit laconic temperament
= Main page =
= Main page =
== Welcome to the Xenharmonic Wiki! ==
== Welcome to the Xenharmonic Wiki! ==
The Xenharmonic Wiki is an open resource dedicated to musical [[tuning system]]s, focusing on [[xenharmonic music]] while also documenting [[historical tunings]] and tuning practices from [[world musical traditions|world traditions]]. It covers the [[theory]] and [[practice|practical applications]] of these systems.
The [[Xenharmonic Wiki]] is an open resource dedicated to musical [[tuning system]]s, focusing on [[xenharmonic music]] while also documenting [[historical tunings]] and tuning practices from [[world musical traditions|world traditions]]. It covers the [[theory]] and [[practice|practical applications]] of these systems.
For a lengthier introduction, see [[Xenharmonic Wiki: Introduction]].
For a lengthier introduction, see [[Xenharmonic Wiki: Introduction]].
Latest revision as of 23:34, 18 February 2026
This is a page where I will draft edits before making them on the actual page. This may possibly include drafting a new page to be created. If you have something to add to any of them, or any concerns, please suggest them on the talk page. If a template is set to debug, make sure to remove that setting when editing the target page.
There are two different mappings of the 11-limit. One is just called pajara and is slightly more complex but suffers almost no loss of accuracy compared to the 7-limit. It is best tuned flat of 22edo. The other, called pajarous to avoid confusion, maps the 11th harmonic slightly simpler, but 22edo is the only 11-odd-limitdiamond monotone tuning, where primes 3 and 5 are less accurate than in optimal tunings of canonical 11-limit pajara.
In the following tables, odd harmonics 1–11 and their inverses are in bold.
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