Lumatone mapping for 49edo: Difference between revisions

(7 intermediate revisions by the same user not shown)

Line 2:

== Diatonic ==

[[Cam Taylor]] demonstrates this mapping in [https://www.youtube.com/watch?v=fns6688IRpg ''49-equal: 7-equal meets superpyth''] (2023).

Since 49edo is a [[superpyth]] temperament, the classic major third ~[[5/4]] is mapped to the interval of an augmented second (e.g. a 5/4 above C is D♯); also, the Pythagorean major second ~[[9/8]] is mapped inconsistently to be the same as the septimal major second ~[[8/7]]. [[Cam Taylor]] demonstrates this mapping in [https://www.youtube.com/watch?v=fns6688IRpg ''49-equal: 7-equal meets superpyth''] (2023).

~~Keep in mind that~~ 49edo is a [[~~superpyth~~]] ~~temperament~~, ~~so 5~~/~~4 is mapped to~~ the ~~interval of an augmented~~ second (~~e.g~~. a 5/~~4 above C is D♯)~~.

== Whitewood + Bohpier ==

Since [[49edo]] is a multiple of [[7edo]], and not too far beyond the highest non-diatonic multiple thereof, [[whitewood]] mappings are legitimate, as [[Bryan Deister]] demonstrates in [https://www.youtube.com/shorts/BcBtD3nuEQs ''49edo groove''] (2026). The sevenths of octaves proceed right, and therefore the octaves slope upwards with this, yielding a range a bit over 4½ octaves. The generator apart from the octave division proceeds down-right; as 6\42, it functions as a somewhat flat Alpharabian tendoneutral second ~[[12/11]] (and if the 49f val is used, also as a rather sharp but still consistent tridecimal neutral second ~[[13/12]]), thus making this also a [[bohpier]] mapping, with a rotated [[1L 7s]] scale having a 7:6 step ratio. Three of these generators make a somewhat sharp septimal major third ~[[9/7]]; four of them make a somewhat sharp lesser septimal tritone ~[[7/5]]; six of them make a slightly flat classic major sixth ~[[5/3]]; and seven of them make a very sharp classic major seventh ~[[9/5]].

== Didacus ==

If you want to access single step movements in a more intuitive way and slightly extend your range, [[Hemimean_clan#Didacus|Didacus]] mappings ~~may be the best~~ compromise. Didacus uses a flat Pythagorean whole tone ~[[9/8]] as its generator, which is the rightward generator in these mappings; it uses the direct approximation of the 9th harmonic instead of building it by stacking the 3rd harmonic. Unfortunately, these mappings put octaves all over the place.

If you want to access single step movements in a more intuitive way and slightly extend your range, [[Hemimean_clan#Didacus|Didacus]] mappings are a good compromise. Didacus uses a flat Pythagorean whole tone ~[[9/8]] as its generator, which is the rightward generator in these mappings; it uses the direct approximation of the 9th harmonic instead of building it by stacking the 3rd harmonic. Unfortunately, these mappings put octaves all over the place.

=== Flipped 6L 1s (add rotated Sevond + Whitewood) ===

Line 17:

Line 19:

Another possibility is a [[3L 5s]] (8:5 step ratio) mapping for [[49edo]], as demonstrated by [[Bryan Deister]] in [https://www.youtube.com/shorts/zb1Z6o-Uvuw ''weathergirl - FLAVOR FOLEY (microtonal cover in 49edo)''] (2025). This mapping lends itself to three rank-2 temperaments, one of which is Didacus as explained above. Another temperament is [[Kleismic_family#Catalan|Catalan]], for which the right + down-right generator 13\49 is a near-just (slightly sharp) classic minor third ~[[6/5]]; this uses the 3rd harmonic, which is quite sharp, but has most of its error canceled by the sharp 5th harmonic. Yet another temperament is [[Kleismic_family#Clyde|Clyde]], for which the right = 2 down-right generator 18\49 is a mildly sharp septimal major third ~[[9/7]]; this also uses the (sharp) 3rd harmonic to make an even sharper patent 9th harmonic, which has most of its error canceled by the sharp 7th harmonic. The range is just over five octaves with no missed notes and a few repeated notes in each octave, but the octaves slope down severely, incurring a vertical wraparound.

== Archipelago + Catalan ==

The most efficient mapping for [[49edo]], having a range of 5¼ octaves (which slope down moderately) with no missed notes and no repeated notes, is [[Bryan Deister]]'s [[4L 3s]] (10:3 step ratio) mapping that functions for both a rank-2 or rank-3 [[archipelago]]-related temperament and [[catalan]] temperament, demonstrated in [https://www.youtube.com/shorts/34w7euOF-Ss ''49edo improv''] (2026). The rank-2 archipelago-related temperament (which is not currently on the archipelago page) uses one key right (10\49) as a slightly sharp tridecimal semifourth ~[[15/13]] (in the 49f val) for its generator; two of them make the fairly flat fourth ~[[4/3]]; four of them make the sharp septimal minor seventh ~[[7/4]]; six of them (after octave reduction) make a slightly sharp septimal minor third ~[[7/6]]; thirteen of them (after more octave reduction) make a near-just undecimal minor sixth ~[[11/7]]; and fourteen of them make a sharp classic minor seventh ~[[9/5]]. Although this is a respectable list of intervals, the unmodified rank-2 version of the temperament does not make a very good [[MOS]] scale — instead it makes for scales in which the small interval is only 1\49. Therefore, for facility in making scales, including the aforementioned 4L 3s, a second generator is needed (thus elevating this to a rank-3 temperament that tempers out [[364/363]], [[540/539]], and [[847/845]]); that generator uses one key down-right (3\49) as the slightly sharp classic chromatic semitone ~[[25/24]], which 49edo uses here more like a diatonic semitone (although the actual 49edo diatonic semitone is only 2\49). Catalan uses one key right plus one key down-right (13\49) as a slightly sharp classic minor third for its generator; two of them make a very flat Axirabian paraminor fifth ~[[16/11]]; four of them (after octave reduction) make a slightly sharp classic chromatic semitone ~[[25/24]]; five of them make a mildly sharp classic major third ~[[5/4]]; six of them make a sharp fifth ~[[3/2]]; seven of them make a sharp classic minor seventh ~[[9/5]]; eight of them (after more octave-reduction) make a mildly flat undecimal neutral second ~[[12/11]]; and ten of them make a near-just undecimal minor sixth ~[[11/7]]. Catalan does not need a second generator to produce usable MOS scales, including the aforementioned 4L 3s.

==== Reverse chroma version ====

It is possible to make a reverse chroma version of the above mapping. The range is considerably less, at a bit over 3⅓ octaves, with considerable regions of non-contiguous notes in the upper left and lower right corners, and the octaves slope down, incurring a vertical wraparound; on the other hand, a considerable number of repeated notes are available in each complete octave to mitigate vertical wraparounds. [[Bryan Deister]] has demonstrated this in [https://www.youtube.com/shorts/VmUIxWb8NCY ''49edo prelude''] (2026), although with note 0 set to where note 38 is here.

== Machine ==

It is possible to get closer to level octaves for [[49edo]] with a slight gain in practical efficiency (having three repeated notes in each octave, but no non-contiguous notes in the lower left and upper right corners compared to the the archipelago + catalan mapping) with a machine scale ([[5L 1s]], with a 9:4 step ratio). [[Bryan Deister]] has demonstrated this in [[Bryan Deister]]'s [https://www.youtube.com/shorts/_yNrDI6nS1I ''49edo riff''] (2026). This mapping has a contiguous range of 5⅓ octaves; and the repeated notes may provide a bit of assistance with mitigation of vertical wraparounds. This mapping uses 9\49 (one key right) as its generator, which functions as a very flat septimal major second ~[[8/7]], a very sharp (and inconsistently-mapped) Pythagorean major second ~[[9/8]], and (much more accurately) a near-just undecimal acute whole tone ~[[25/22]] (both the Archytas comma [[64/63]] and the valinorsma [[176/175]] are tempered out). Two of them make a somewhat sharp septimal major third ~[[9/7]]; four of them make a slightly sharp classic major sixth ~[[5/3]]; seven of them (after octave reduction) make a somewhat flat undecimal neutral third ~[[11/9]]; eight of them make a very sharp undecimal major fourth ~[[11/8]]; and nine of them make a near-just undecimal minor sixth ~[[11/7]]. Thus, this mapping favors mostly xenharmonic intervals.

== Infraorwell ==

The septimal minor third ~[[7/6]] (11\49) of [[49edo]] is near-just, which suggests its use as a generator, as in [[Orwell]]; however, it is too close to just for actual Orwell eemperament, and so a stack of it manages to miss all of the intervals characteristic of Orwell, other than a fairly flat classic minor sixth ~[[8/5]] — hence [[Orwellismic_temperaments#Infraorwell|infraorwell]]. It is possible to produce a forward chroma [[4L 5s]] (gramitonic, 6:5 step ratio) infraorwell mapping for 49edo, as demonstrated in [[Bryan Deister]]'s [https://www.youtube.com/watch?v=7pK-JcIrd18 Deltarune – ''Man'' (cover)] (2023) and (very differently) in [https://www.youtube.com/shorts/V8t7MyP2Nuo ''microtonal improv in 49edo''] (2024). This mapping gets all notes in each octave and has a range of a bit over 4 octaves with repeated notes to mitigate vertical wraparounds; the octaves alternate between near/far and middle, with double octaves sloping slightly upwards.

@@ Line 2: / Line 2: @@
 == Diatonic ==
-[[Cam Taylor]] demonstrates this mapping in [https://www.youtube.com/watch?v=fns6688IRpg ''49-equal: 7-equal meets superpyth''] (2023).
+Since 49edo is a [[superpyth]] temperament, the classic major third ~[[5/4]] is mapped to the interval of an augmented second (e.g. a 5/4 above C is D♯); also, the Pythagorean major second ~[[9/8]] is mapped inconsistently to be the same as the septimal major second ~[[8/7]]. [[Cam Taylor]] demonstrates this mapping in [https://www.youtube.com/watch?v=fns6688IRpg ''49-equal: 7-equal meets superpyth''] (2023).
 {{Lumatone EDO mapping|n=49|start=10|xstep=9|ystep=-7}}
-Keep in mind that 49edo is a [[superpyth]] temperament, so 5/4 is mapped to the interval of an augmented second (e.g. a 5/4 above C is D♯).
+== Whitewood + Bohpier ==
+Since [[49edo]] is a multiple of [[7edo]], and not too far beyond the highest non-diatonic multiple thereof, [[whitewood]] mappings are legitimate, as [[Bryan Deister]] demonstrates in [https://www.youtube.com/shorts/BcBtD3nuEQs ''49edo groove''] (2026). The sevenths of octaves proceed right, and therefore the octaves slope upwards with this, yielding a range a bit over 4½ octaves. The generator apart from the octave division proceeds down-right; as 6\42, it functions as a somewhat flat Alpharabian tendoneutral second ~[[12/11]] (and if the 49f val is used, also as a rather sharp but still consistent tridecimal neutral second ~[[13/12]]), thus making this also a [[bohpier]] mapping, with a rotated [[1L&nbsp;7s]] scale having a 7:6 step ratio. Three of these generators make a somewhat sharp septimal major third ~[[9/7]]; four of them make a somewhat sharp lesser septimal tritone ~[[7/5]]; six of them make a slightly flat classic major sixth ~[[5/3]]; and seven of them make a very sharp classic major seventh ~[[9/5]].
+{{Lumatone EDO mapping|n=49|start=34|xstep=7|ystep=-1}}
 == Didacus ==
-If you want to access single step movements in a more intuitive way and slightly extend your range, [[Hemimean_clan#Didacus|Didacus]] mappings may be the best compromise. Didacus uses a flat Pythagorean whole tone ~[[9/8]] as its generator, which is the rightward generator in these mappings; it uses the direct approximation of the 9th harmonic instead of building it by stacking the 3rd harmonic. Unfortunately, these mappings put octaves all over the place.
+If you want to access single step movements in a more intuitive way and slightly extend your range, [[Hemimean_clan#Didacus|Didacus]] mappings are a good compromise. Didacus uses a flat Pythagorean whole tone ~[[9/8]] as its generator, which is the rightward generator in these mappings; it uses the direct approximation of the 9th harmonic instead of building it by stacking the 3rd harmonic. Unfortunately, these mappings put octaves all over the place.
 === Flipped 6L&nbsp;1s (add rotated Sevond + Whitewood) ===
@@ Line 17: / Line 19: @@
 Another possibility is a [[3L&nbsp;5s]] (8:5 step ratio) mapping for [[49edo]], as demonstrated by [[Bryan Deister]] in [https://www.youtube.com/shorts/zb1Z6o-Uvuw ''weathergirl - FLAVOR FOLEY (microtonal cover in 49edo)''] (2025). This mapping lends itself to three rank-2 temperaments, one of which is Didacus as explained above. Another temperament is [[Kleismic_family#Catalan|Catalan]], for which the right + down-right generator 13\49 is a near-just (slightly sharp) classic minor third ~[[6/5]]; this uses the 3rd harmonic, which is quite sharp, but has most of its error canceled by the sharp 5th harmonic. Yet another temperament is [[Kleismic_family#Clyde|Clyde]], for which the right = 2 down-right generator 18\49 is a mildly sharp septimal major third ~[[9/7]]; this also uses the (sharp) 3rd harmonic to make an even sharper patent 9th harmonic, which has most of its error canceled by the sharp 7th harmonic. The range is just over five octaves with no missed notes and a few repeated notes in each octave, but the octaves slope down severely, incurring a vertical wraparound.
 {{Lumatone EDO mapping|n=49|start=44|xstep=8|ystep=-3}}
+== Archipelago + Catalan ==
+The most efficient mapping for [[49edo]], having a range of 5¼ octaves (which slope down moderately) with no missed notes and no repeated notes, is [[Bryan Deister]]'s [[4L&nbsp;3s]] (10:3 step ratio) mapping that functions for both a rank-2 or rank-3 [[archipelago]]-related temperament and [[catalan]] temperament, demonstrated in [https://www.youtube.com/shorts/34w7euOF-Ss ''49edo improv''] (2026). The rank-2 archipelago-related temperament (which is not currently on the archipelago page) uses one key right (10\49) as a slightly sharp tridecimal semifourth ~[[15/13]] (in the 49f val) for its generator; two of them make the fairly flat fourth ~[[4/3]]; four of them make the sharp septimal minor seventh ~[[7/4]]; six of them (after octave reduction) make a slightly sharp septimal minor third ~[[7/6]]; thirteen of them (after more octave reduction) make a near-just undecimal minor sixth ~[[11/7]]; and fourteen of them make a sharp classic minor seventh ~[[9/5]]. Although this is a respectable list of intervals, the unmodified rank-2 version of the temperament does not make a very good [[MOS]] scale &mdash; instead it makes for scales in which the small interval is only 1\49. Therefore, for facility in making scales, including the aforementioned 4L&nbsp;3s, a second generator is needed (thus elevating this to a rank-3 temperament that tempers out [[364/363]], [[540/539]], and [[847/845]]); that generator uses one key down-right (3\49) as the slightly sharp classic chromatic semitone ~[[25/24]], which 49edo uses here more like a diatonic semitone (although the actual 49edo diatonic semitone is only 2\49). Catalan uses one key right plus one key down-right (13\49) as a slightly sharp classic minor third for its generator; two of them make a very flat Axirabian paraminor fifth ~[[16/11]]; four of them (after octave reduction) make a slightly sharp classic chromatic semitone ~[[25/24]]; five of them make a mildly sharp classic major third ~[[5/4]]; six of them make a sharp fifth ~[[3/2]]; seven of them make a sharp classic minor seventh ~[[9/5]]; eight of them (after more octave-reduction) make a mildly flat undecimal neutral second ~[[12/11]]; and ten of them make a near-just undecimal minor sixth ~[[11/7]]. Catalan does not need a second generator to produce usable MOS scales, including the aforementioned 4L&nbsp;3s.
+{{Lumatone EDO mapping|n=49|start=4|xstep=10|ystep=-7}}
+==== Reverse chroma version ====
+It is possible to make a reverse chroma version of the above mapping. The range is considerably less, at a bit over 3⅓ octaves, with considerable regions of non-contiguous notes in the upper left and lower right corners, and the octaves slope down, incurring a vertical wraparound; on the other hand, a considerable number of repeated notes are available in each complete octave to mitigate vertical wraparounds. [[Bryan Deister]] has demonstrated this in [https://www.youtube.com/shorts/VmUIxWb8NCY ''49edo prelude''] (2026), although with note 0 set to where note 38 is here.
+{{Lumatone EDO mapping|n=49|start=6|xstep=3|ystep=7}}
+== Machine ==
+It is possible to get closer to level octaves for [[49edo]] with a slight gain in practical efficiency (having three repeated notes in each octave, but no non-contiguous notes in the lower left and upper right corners compared to the the archipelago + catalan mapping) with a machine scale ([[5L&nbsp;1s]], with a 9:4 step ratio). [[Bryan Deister]] has demonstrated this in [[Bryan Deister]]'s [https://www.youtube.com/shorts/_yNrDI6nS1I ''49edo riff''] (2026). This mapping has a contiguous range of 5⅓ octaves; and the repeated notes may provide a bit of assistance with mitigation of vertical wraparounds. This mapping uses 9\49 (one key right) as its generator, which functions as a very flat septimal major second ~[[8/7]], a very sharp (and inconsistently-mapped) Pythagorean major second ~[[9/8]], and (much more accurately) a near-just undecimal acute whole tone ~[[25/22]] (both the Archytas comma [[64/63]] and the valinorsma [[176/175]] are tempered out). Two of them make a somewhat sharp septimal major third ~[[9/7]]; four of them make a slightly sharp classic major sixth ~[[5/3]]; seven of them (after octave reduction) make a somewhat flat undecimal neutral third ~[[11/9]]; eight of them make a very sharp undecimal major fourth ~[[11/8]]; and nine of them make a near-just undecimal minor sixth ~[[11/7]]. Thus, this mapping favors mostly xenharmonic intervals.
+{{Lumatone EDO mapping|n=49|start=4|xstep=9|ystep=-5}}
+== Infraorwell ==
+The septimal minor third ~[[7/6]] (11\49) of [[49edo]] is near-just, which suggests its use as a generator, as in [[Orwell]]; however, it is too close to just for actual Orwell eemperament, and so a stack of it manages to miss all of the intervals characteristic of Orwell, other than a fairly flat classic minor sixth ~[[8/5]] &mdash; hence [[Orwellismic_temperaments#Infraorwell|infraorwell]]. It is possible to produce a forward chroma [[4L&nbsp;5s]] (gramitonic, 6:5 step ratio) infraorwell mapping for 49edo, as demonstrated in [[Bryan Deister]]'s [https://www.youtube.com/watch?v=7pK-JcIrd18 Deltarune – ''Man'' (cover)] (2023) and (very differently) in [https://www.youtube.com/shorts/V8t7MyP2Nuo ''microtonal improv in 49edo''] (2024). This mapping gets all notes in each octave and has a range of a bit over 4 octaves with repeated notes to mitigate vertical wraparounds; the octaves alternate between near/far and middle, with double octaves sloping slightly upwards.
+{{Lumatone EDO mapping|n=49|start=33|xstep=6|ystep=-1}}
 {{Navbox Lumatone}}