Lumatone mapping for 89edo: Difference between revisions

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== Diatonic ==

Note that since 89edo is a [[schismatic]] tuning, the best approximation to 5/4 is the diminished fourth.

Note that since 89edo is a [[schismatic]] tuning, the best approximation to ~[[5/4]] is the diminished fourth. That said, since the fifth ~[3/2] is flattened by almost exactly one quarer of Boethius' comma [[513/512]], the meantone major third is almost exactly Boethius' major third ~[[24/19]], and since one third of such a small comma is not very far off from one fourth, the corresponding minor third ~[[19/16]] is also not too far from just. Using 89edo as a quasi-meantone tuning with the extremely accurate ~[[24/19]] substituting for the less accurate ~5/4 will enable many chords to be played despite the missing notes, as demonstrated in [https://www.youtube.com/shorts/mU4KcJd6E7U ''89edo improv''] (2026), and it would be possible to switch between presets for different subsets of 89edo to enable modulation to remote keys.

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== Char-based + Sesquart rank-3 temperament ==

[[Bryan Deister]] has demonstrated a [[10L 3s]] mapping for [[89edo]] in [https://www.youtube.com/watch?v=2JNIeqvXKlM ''microtonal improvisation in 89edo''] (2025). The rightward generator 8\89 functions as both the [[16/15]] (the classical diatonic semitone) and [[17/16]] (the large septendecimal semitone), meaning that the charisma [[256/255]] is tempered out, although 89edo is not listed as a tuning of any of the temperaments on the page for 256/255. Eight of these generators make a near-just Axirabian paraminor fifth ~[[16/11]]. Although being prime, 89edo technically needs no second generator, a second generator helps with reaching consonant intervals other than ~16/11; one obvious choice is the scale chroma (upwards generator) 5\89 which functions as a somewhat sharp small tridecimal third tone ~[[27/26]]; three of these make a near-just Pythagorean whole tone ~[[9/8]]; four of them make a near-just septimal minor third ~[[7/6]]. It is also possible to combine both generators to get 13\89, a slightly sharp undevicesimal supraneutral second ~[[21/19]]; four of these generator combinations make a near-just fifth ~[[3/2]], as in [[Schismatic_family#Sesquiquartififths|Sesquiquartififths]] temperament and its higher-limit [[Schismatic_family#Sesquart|Sesquart]] extension. The range is slightly over 2⅓ octaves with no missed notes and a few repeated notes, and the octaves slope up mildly.

[[Bryan Deister]] has demonstrated a [[10L 3s]] (8:3 step ratio) mapping for [[89edo]] in [https://www.youtube.com/watch?v=2JNIeqvXKlM ''microtonal improvisation in 89edo''] (2025). The rightward generator 8\89 functions as both the [[16/15]] (the classical diatonic semitone) and [[17/16]] (the large septendecimal semitone), meaning that the charisma [[256/255]] is tempered out, although 89edo is not listed as a tuning of any of the temperaments on the page for 256/255. Eight of these generators make a near-just Axirabian paraminor fifth ~[[16/11]]. Although being prime, 89edo technically needs no second generator, a second generator helps with reaching consonant intervals other than ~16/11; one obvious choice is the scale chroma (upwards generator) 5\89 which functions as a somewhat sharp small tridecimal third tone ~[[27/26]]; three of these make a near-just Pythagorean whole tone ~[[9/8]]; four of them make a near-just septimal minor third ~[[7/6]]. It is also possible to combine both generators to get 13\89, a slightly sharp undevicesimal supraneutral second ~[[21/19]]; four of these generator combinations make a near-just fifth ~[[3/2]], as in [[Schismatic_family#Sesquiquartififths|Sesquiquartififths]] temperament and its higher-limit [[Schismatic_family#Sesquart|Sesquart]] extension. The range is slightly over 2⅓ octaves with no missed notes and a few repeated notes, and the octaves slope up mildly.

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 == Diatonic ==
-Note that since 89edo is a [[schismatic]] tuning, the best approximation to 5/4 is the diminished fourth.
+Note that since 89edo is a [[schismatic]] tuning, the best approximation to ~[[5/4]] is the diminished fourth. That said, since the fifth ~[3/2] is flattened by almost exactly one quarer of Boethius' comma [[513/512]], the meantone major third is almost exactly Boethius' major third ~[[24/19]], and since one third of such a small comma is not very far off from one fourth, the corresponding minor third ~[[19/16]] is also not too far from just. Using 89edo as a quasi-meantone tuning with the extremely accurate ~[[24/19]] substituting for the less accurate ~5/4 will enable many chords to be played despite the missing notes, as demonstrated in [https://www.youtube.com/shorts/mU4KcJd6E7U ''89edo improv''] (2026), and it would be possible to switch between presets for different subsets of 89edo to enable modulation to remote keys.
 {{Lumatone EDO mapping|n=89|start=2|xstep=15|ystep=-8}}
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 == Char-based + Sesquart rank-3 temperament ==
-[[Bryan Deister]] has demonstrated a [[10L&nbsp;3s]] mapping for [[89edo]] in [https://www.youtube.com/watch?v=2JNIeqvXKlM ''microtonal improvisation in 89edo''] (2025). The rightward generator 8\89 functions as both the [[16/15]] (the classical diatonic semitone) and [[17/16]] (the large septendecimal semitone), meaning that the charisma [[256/255]] is tempered out, although 89edo is not listed as a tuning of any of the temperaments on the page for 256/255. Eight of these generators make a near-just Axirabian paraminor fifth ~[[16/11]]. Although being prime, 89edo technically needs no second generator, a second generator helps with reaching consonant intervals other than ~16/11; one obvious choice is the scale chroma (upwards generator) 5\89 which functions as a somewhat sharp small tridecimal third tone ~[[27/26]]; three of these make a near-just Pythagorean whole tone ~[[9/8]]; four of them make a near-just septimal minor third ~[[7/6]]. It is also possible to combine both generators to get 13\89, a slightly sharp undevicesimal supraneutral second ~[[21/19]]; four of these generator combinations make a near-just fifth ~[[3/2]], as in [[Schismatic_family#Sesquiquartififths|Sesquiquartififths]] temperament and its higher-limit [[Schismatic_family#Sesquart|Sesquart]] extension. The range is slightly over 2⅓ octaves with no missed notes and a few repeated notes, and the octaves slope up mildly.
+[[Bryan Deister]] has demonstrated a [[10L&nbsp;3s]] (8:3 step ratio) mapping for [[89edo]] in [https://www.youtube.com/watch?v=2JNIeqvXKlM ''microtonal improvisation in 89edo''] (2025). The rightward generator 8\89 functions as both the [[16/15]] (the classical diatonic semitone) and [[17/16]] (the large septendecimal semitone), meaning that the charisma [[256/255]] is tempered out, although 89edo is not listed as a tuning of any of the temperaments on the page for 256/255. Eight of these generators make a near-just Axirabian paraminor fifth ~[[16/11]]. Although being prime, 89edo technically needs no second generator, a second generator helps with reaching consonant intervals other than ~16/11; one obvious choice is the scale chroma (upwards generator) 5\89 which functions as a somewhat sharp small tridecimal third tone ~[[27/26]]; three of these make a near-just Pythagorean whole tone ~[[9/8]]; four of them make a near-just septimal minor third ~[[7/6]]. It is also possible to combine both generators to get 13\89, a slightly sharp undevicesimal supraneutral second ~[[21/19]]; four of these generator combinations make a near-just fifth ~[[3/2]], as in [[Schismatic_family#Sesquiquartififths|Sesquiquartififths]] temperament and its higher-limit [[Schismatic_family#Sesquart|Sesquart]] extension. The range is slightly over 2⅓ octaves with no missed notes and a few repeated notes, and the octaves slope up mildly.
 {{Lumatone EDO mapping|n=89|start=4|xstep=8|ystep=-5}}
 {{Navbox Lumatone}}