72edo: Difference between revisions

| en = 72edo

{{Wikipedia|72 equal temperament}}

Each step of 72edo is called a ''[[morion]]'' (plural ''moria)''. This produces a twelfth-tone tuning, with the whole tone measuring 200{{c}}, the same as in [[12edo]]. 72edo is also a superset of [[24edo]], a common and standard tuning of [[Arabic, Turkish, Persian music|Arabic music]], and has itself been used to tune Turkish music.

Composers that used 72edo include [[Ivan Wyschnegradsky]], [[Julián Carrillo]] (who is better associated with [[96edo]]), [[Georg Friedrich Haas]], [[Ezra Sims]], [[Rick Tagawa]], [[James Tenney]], and the jazz musician [[Joe Maneri]].

72edo approximates [[11-limit]] [[just intonation]] exceptionally well. It is the second edo (after [[58edo|58]]) to be [[consistent]] in the [[17-odd-limit]], and the second edo (also after 58) to be [[consistency|distinctly consistent]] in the [[11-odd-limit]], but it is the first edo to be [[consistency|consistent to distance 2]] in the 11-odd-limit, meaning every interval in the 11-odd-limit is approximated with less than 25% [[relative interval error|relative error]] (about 4 cents). It also has pretty good accuracy for the [[19-limit]], being almost consistent to the entire [[21-odd-limit]] with the only inconsistency occurring at [[19/13]] and its [[octave complement]]. It is the ninth [[zeta integral edo]].

The octave, fifth and fourth are the same size as they would be in 12edo, 72, 42 and 30 steps respectively, but the classic major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12edo while minor ones are one step sharp. The septimal minor seventh ([[7/4]]) is 58 steps, while the undecimal semiaugmented fourth ([[11/8]]) is 33.

The [[octave reduction|octave reduced]] [[13/1|13th harmonic]] is mapped on 50\72, an interval inherited from [[36edo]] (25\36) that is a very close approximation to [[acoustic phi]], and the [[17/1|17th]] and [[19/1|19th harmonics]] come from 12edo.  

72edo is the smallest multiple of 12edo that (just barely) has another diatonic fifth, 43\72, an extremely hard diatonic fifth suitable for a 5edo [[circulating temperament]].

=== Prime harmonics ===

{{Harmonics in equal|72|columns=11}}

{{Harmonics in equal|72|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}}

=== As a tuning of other temperaments ===

72et is the only 11-limit regular temperament which treats harmonics 24 to 28 as being equidistant in pitch, splits [[25/24]] into two equal [[49/48]][[~]][[50/49]]'s, and splits [[28/27]] into two equal [[55/54]]~[[56/55]]'s (144edo is enfactored in the 11-limit with 72edo, so it is already covered here). It is also an excellent tuning for [[miracle]] temperament, especially the 11-limit version, and the related rank-3 temperament [[prodigy]], and is a good tuning for other temperaments and scales, including [[wizard]], [[harry]], [[catakleismic]], [[compton]], [[unidec]] and [[tritikleismic]].

=== Subsets and supersets ===

Since 72 factors into primes as {{nowrap| 2³ × 3² }}, 72edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 18, 24, and 36 }}. [[144edo]], which doubles it, provides a possible correction to its approximate harmonic 13, though unlike 72 it is not consistent to the [[13-odd-limit]].

| [[10/9]]

| 216.7

| [[17/15]], [[25/22]]

=== Proposed interval names and solfèges ===

! colspan="3" | [[SKULO interval names|SKULO interval names and notation]]

| 33.3

| 50.0

| ^³1, v³m2

| ^³D, v³Eb

| 133.3

| 150.0

| v³E

| UEb/uE

| UEb/uE

| 183.3

| kM2

| classic/comma-narrow major 2nd

| ^³M2, <br>v³m3

| trupmajor 2nd,<br>trudminor 3rd

| ^³E, <br>v³F

| 333.3

| 350.0

| ^³F

| UF/uF#

| UF/uF#

|-

|-

|-

|-

|-

| 450.0

| ^³M3, v³4

| ^³F#, v³G

| 516.7

| ^³G

| U4/N4

| uber 4th / neutral 4th

| SG#, (kkG#), sAb

| v³A

| u5/N5

| unter 5th / neutral 5th

| 733.3

| 750.0

| ^³5, v³m6

| ^³A, v³Bb

| 850.0

| v³B

| 950.0

| ^³M6, <br>v³m7

| ^³B, <br>v³C

| 983.3

| 1000.0

| classic/comma-wide minor 7th

| ^³C

| UC/uC#, hDb

| UC/uC#, uDb

|-

|-

| 1083.3

| vM7

| kC#, lDb

| ^³C#, v³D