41edo: Difference between revisions

(18 intermediate revisions by 8 users not shown)

Line 23:

41edo can be seen as a tuning of the [[garibaldi temperament|garibaldi]] temperament<ref>[http://x31eq.com/schismic.htm Schismic Temperaments] at x31eq.com, the website of [[Graham Breed]]</ref><ref>[http://x31eq.com/decimal_lattice.htm Lattices with Decimal Notation] at x31eq.com</ref>, as well as [[miracle]], [[magic]], [[superkleismic]], and multiple temperaments in the [[tetracot family]].

Various 13-limit [[~~magic family|~~magic extensions]] are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in [[22edo]].

Various 13-limit [[magic extensions]] are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in [[22edo]].

41edo is also a great [[tetracot]] tuning, and works as an alternative to [[34edo]], providing proper approximations to the 7th and 11th harmonic at the cost of the 13th, and supporting [[monkey]], [[bunya]] and [[octacot]] simultaneously. All three of these extend to the [[11-limit]] by way of interpreting the flat [[10/9]] as an [[11/10]] by tempering out [[100/99]]. This equivalence is especially useful in 41edo, wherein this comma-flat whole tone a.k.a. the second of Tetracot[7] can also be more accurately interpreted as [[21/19]]—which is equated with [[32/29]] above [[31/28]] below (both very near)—providing an explanation of the accuracy of primes [[29/1|29]] and [[31/1|31]] so that it is a uniquely good/versatile choice for interpreting the harmony of tetracot.

Line 30:

=== Subsets and supersets ===

41edo is the 13th [[prime edo]], following [[37edo]] and coming before [[43edo]].

41edo is the 13th [[prime edo]], following [[37edo]] and coming before [[43edo]]. It does not contain any nontrivial subset edos, though it contains [[41ed4]].

[[205edo]], which slices each step of 41edo into five, corrects some approximations of 41edo to near-just quality. As such, 41edo forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41edo as the basic [[13-limit]] intervals requiring fine tuning ±1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41edo circle in 205edo.

[[205edo]], which slices each step of 41edo into five, corrects some approximations of 41edo to near-just quality. As such, 41edo forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41edo as the basic [[13-limit]] intervals requiring fine tuning ±1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41edo circle in 205edo. Its step of 1\205 is called a ''mem''.

[[2460edo]] has potential for a 41edo analog to [[Cent|cents]]. It divides the 41edo step into 60 equal parts, and 60 is a highly composite (a.k.a. antiprime) number, so it contains many other multiples of 41edo, including 205edo, and also contains [[12edo]] among other equal tunings. It also accurately represents [[14afdo|mode 14 of the harmonic series]], as it is consistent all the way up to the 27-odd-limit. This allows for precise detunings in a 41-tone framework to approximate pure just intonation more closely, especially for some higher harmonics. Its step of 1\2460 is called a ''mina''.

== Intervals ==

Line 520:

Line 522:

| upmajor 7th

| ^M7

| C#^, vvD

| ^C#, vvD

| supermajor 7th, classic dim 8ve, unter 8ve

| SM7, KKd8, U8

Line 687:

Line 689:

The step size of 41edo is small enough that the smallest interval (the "red/blue unison", seventh-tone, comma, diesis or whatever you want to call it) is actually fairly consonant with most timbres; it resembles a "noticeably out of tune unison" rather than a minor second, and has its own distinct character and appeal.

If "red" is replaced by "up", "blue" by "down", and "neutral" by "mid", and if "gray" is omitted, this notation becomes essentially the same as [[~~Ups_and_Downs_Notation~~|ups and downs notation]]. The only difference is the use of minor tritone and major tritone.

If "red" is replaced by "up", "blue" by "down", and "neutral" by "mid", and if "gray" is omitted, this notation becomes essentially the same as [[Ups and downs notation|ups and downs notation]]. The only difference is the use of minor tritone and major tritone.

=== Sagittal notation ===

Line 732:

Line 734:

=== Interval mappings ===

~~=== Zeta peak index ===~~

~~{{ZPI~~

~~| zpi = 184~~

~~| steps = 40.9880783925993~~

~~| step size = 29.2768055263764~~

~~| tempered height = 7.570230~~

~~| pure height = 7.497833~~

~~| integral = 1.423937~~

~~| gap = 17.722623~~

~~| octave = 1200.34902658143~~

~~| consistent = 16~~

~~| distinct = 10~~

}}

== Relationship to 12edo ==

Line 1,468:

Line 1,456:

|

|}

== Octave stretch or compression ==

Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch or compression]] depends on which subgroup we are focusing on.

For the 5-, 7-, and 11-limit, stretch is advised, though in the case of the 11-limit the stretch should be way milder, whereas for the 13-limit and in particular the 17-limit, little to no stretch or even compression may be suitable for balancing out the sharp and flat tuning tendencies, as is demonstrated in tunings such as [[65edt]], [[106ed6]], and [[147ed12]].

Primes 19, 29, and 31 all tend flat, so stretching will serve again as we take that into account, especially if we use the temperament in any no-17 or no-13 no-17 settings.

What follows is a comparison of stretched- and compressed-octave 41edo tunings.

; [[24edf]]

* Step size: 29.248{{c}}, octave size: 1199.17{{c}}

Compressing the octave of 41edo by around 0.8{{c}} results in [[JND|unnoticeably]] improved primes 11, 17 and 23, but unnoticeably worse primes 2, 3, 5, 7, 13 and 19. This approximates all harmonics up to 16 within 7.6{{c}}. The tuning 24edf does this.

{{Harmonics in equal|24|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 24edf}}

{{Harmonics in equal|24|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 24edf (continued)}}

; [[147ed12]] / [[106ed6]] / [[65edt]]

* 65edt — step size: 29.261{{c}}, octave size: 1199.69{{c}}

* 106ed6 — step size: 29.264{{c}}, octave size: 1199.81{{c}}

* 147ed12 — step size: 29.265{{c}}, octave size: 1199.87{{c}}

Compressing the octave of 41edo by around 0.2{{c}} results in [[JND|unnoticeably]] improved primes 3, 11 and 13, but unnoticeably worse primes 2, 5 and 7. This approximates all harmonics up to 16 within 7.6{{c}}. The tunings 147ed12, 106ed6 and 65edt each do this.

{{Harmonics in equal|106|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 106ed6}}

{{Harmonics in equal|106|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 106ed6 (continued)}}

; 41edo

* Step size: 29.268{{c}}, octave size: 1200.00{{c}}

Pure-octaves 41edo approximates all harmonics up to 16 within 8.3{{c}}. The octaves of its 13-limit [[WE]] and [[TE]] tuning differ by less than 0.1{{c}} from pure.

{{Harmonics in equal|41|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41edo}}

{{Harmonics in equal|41|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41edo (continued)}}

; [[zpi|184zpi]] / [[WE|41et, 11-limit WE tuning]]

* Step size: 29.277{{c}}, octave size: 1200.35{{c}}

Stretching the octave of 41edo by around 0.5{{c}} results in [[JND|unnoticeably]] improved primes 5 and 7, but unnoticeably worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 9.6{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this. So does 184zpi, whose octave is identical to WE within 0.02{{c}}.

{{Harmonics in cet|29.277|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 184zpi tuning}}

{{Harmonics in cet|29.277|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 184zpi (continued)}}

; [[WE|41et, 7-limit WE tuning]]

* Step size: 29.288{{c}}, octave size: 1200.81{{c}}

Stretching the octave of 41edo by just under 1{{c}} results in [[JND|just-noticeably]] improved primes 5 and 7, but just-noticeably worse primes 11 and 13. This approximates all harmonics up to 16 within 11.2{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this.

{{Harmonics in cet|29.288|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 7-limit WE tuning}}

{{Harmonics in cet|29.288|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 7-limit WE tuning (continued)}}

== Scales and modes ==

Line 2,036:

Line 2,065:

The first 41edo guitar was probably this one, built by [[Erv Wilson]] in the 1960's:

[[File:Erv Wilson's full-41 guitar 2.jpg|none|thumb]]

[[File:Erv Wilson's full-41 guitar 2.jpg|none|thumb|200px]]

Note the new bridge, several inches below the original bridge. The new bridge increases the scale length and spreads the frets out, making the guitar more playable. Erv numbered the frets as seen here, with the 3-limit dorian scale in enlarged numbers.

[[File:Erv Wilson's full-41 guitar 3.jpg|frameless|~~838x838px~~]]

[[File:Erv Wilson's full-41 guitar 3.jpg|frameless|500px]]

Several more modern guitars:

File:Melleweijters.com 41edo.jpg|[[Melle Weijters]]' 10-string guitar ([https://melleweijters.com Melleweijters.com])

File:41-EDD_elektrische_gitaar.jpg|41edo electric guitar, by [[Gregory Sanchez]].

File:Ron_Sword_with_a_41ET_Guitar.jpg|41edo classical guitar, by [[Ron Sword]].

</gallery>

[[~~File:Melleweijters.com 41edo.jpg|frameless|832x832px~~]]

The [[Kite Guitar]] is a guitar fretting using every other step of 41edo, i.e. 41ed4 or "20½-edo". However, the interval between two adjacent open strings is always an odd number of 41-edosteps. Thus each string only covers half of 41edo, but the full edo can be found on every pair of adjacent strings. Kite-fretting makes 41edo about as playable as 19edo or 22edo, although there are certain trade-offs.

~~''[[Melle Weijters]]' 10-string~~ guitar ~~([https://melleweijters~~.~~com Melleweijters.com])''~~

~~[[File:41-EDD_elektrische_gitaar~~.~~jpg|alt=41~~-~~EDD elektrische gitaar~~.~~jpg|560x745px|~~41-~~EDD elektrische gitaar~~.~~jpg]]~~

''41edo ~~electric guitar~~, ~~by [[Gregory Sanchez]]~~.''

~~[[File:Ron_Sword_with_a_41ET_Guitar.jpg|alt=Ron_Sword_with_a_41ET_Guitar.jpg|Ron_Sword_with_a_41ET_Guitar.jpg]]~~

''41edo ~~classical guitar~~, ~~by [[Ron Sword]]~~.''

The [[Kite Guitar]] (see also [https://kiteguitar.com KiteGuitar.com] and [http://tallkite.com/misc_files/The%20Kite%20Tuning.pdf Kite Tuning]) is a guitar fretting using every other step of 41edo, i.e. 41ed4 or "20½-edo". However, the interval between two adjacent open strings is always an odd number of 41-edosteps. Thus each string only covers half of 41edo, but the full edo can be found on every pair of adjacent strings. Kite-fretting makes 41edo about as playable as 19edo or 22edo, although there are certain trade-offs.

[[File:Caleb's Kite guitar.jpg|none|thumb|200px|Kite guitar]]

[[File:Caleb's Kite guitar.jpg|~~480x640px~~]]

For more photos of Kite guitars, see [[Kite Guitar Photographs]].

Line 2,069:

Line 2,091:

=== Keyboards ===

A possible 41edo keyboard design:

<~~div style~~=~~"display: inline-grid; margin-right: 25px;~~>[[File:41edo keyboard layout.png|~~none|thumb~~|~~484x484px~~]]~~</div>~~

~~<div style="display: inline-grid; margin-right: 25px;">[[~~File:Xenachord with 41edo layout.png~~|left|thumb~~|[https://richiegreene.com/instruments/ Xenachord] with 41edo layout by [[User:Richie|Richie]]]]

File:41edo keyboard layout.png

File:TS41 Microtonal MIDI Keyboard (Prototype).jpg|[[User:Tristanbay|Tristan Bay]]'s prototype TS41 MIDI keyboard, laid out in bosanquet with 41 keys per octave

File:Xenachord with 41edo layout.png|[https://richiegreene.com/instruments/ Xenachord] with 41edo layout by [[User:Richie|Richie]]

</gallery>

See also [[41-edo Keyboards]] for Linnstrument and Harpejji options, as well as DIY options.

Line 2,147:

Line 2,171:

* [https://soundcloud.com/floracanou/sets/notes-of-the-generation ''Notes of the Generation''] (2023) – an 8-piece album in 41et

: "Chaotic Witch #1" · "Party Cubes" · "Big Dreamer Pavilion" · "Lost Cyclops" · "Sky Tree" · "Long Night Ahead" · "Fractocraft" · "After the Generation"

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/kLMuRP82bZw ''microtonal dance in 41edo ''] (2023)

* [https://www.youtube.com/shorts/m8X-IqH8tok ''Waltz in 41edo''] (2025)

* [https://www.youtube.com/shorts/Ur--SKiRsY0 ''41edo groove''] (2025)

; [[Francium]]

Line 2,158:

Line 2,187:

; [[L4MPLIGHT]]

* ''Caftaphata'' (2024) – [https://www.youtube.com/watch?v=cMnuMjXeHrY YouTube] | [https://soundcloud.com/l4mplight/caftaphata-microtones-conlang SoundCloud] – also partially in just intonation and 12edo

* ''Yxeni'' (2025) - [https://www.youtube.com/watch?v=PrfAz8V4WNc YouTube]

; [[Ray Perlner]]

Line 2,184:

Line 2,214:

* [https://soundcloud.com/sacred-skeleton/modified-kite-guitar-take-1 ''Modified Kite Guitar Take 1 - Clean'']

* [https://soundcloud.com/sacred-skeleton/modified-kite-guitar-take-2 ''Modified Kite Guitar Take 2 - Fuzz'']

; [[John Platter]]

* [https://johnplatter.bandcamp.com/album/in-the-know In the Know] - Full album recorded using kite guitar & bass.

; [[Pixel Archipelago]]

Line 2,218:

Line 2,251:

[[Category:3-limit record edos]]

[[Category:3-limit record edos|##]]

[[Category:Magic]]

[[Category:Superkleismic]]

@@ Line 23: / Line 23: @@
 edo can be seen as a tuning of the [[garibaldi temperament|garibaldi]] temperament<ref>[http://x31eq.com/schismic.htm Schismic Temperaments] at x31eq.com, the website of [[Graham Breed]]</ref><ref>[http://x31eq.com/decimal_lattice.htm Lattices with Decimal Notation] at x31eq.com</ref>, as well as [[miracle]], [[magic]], [[superkleismic]], and multiple temperaments in the [[tetracot family]].
-Various 13-limit [[magic family|magic extensions]] are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in [[22edo]].
+Various 13-limit [[magic extensions]] are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in [[22edo]].
 edo is also a great [[tetracot]] tuning, and works as an alternative to [[34edo]], providing proper approximations to the 7th and 11th harmonic at the cost of the 13th, and supporting [[monkey]], [[bunya]] and [[octacot]] simultaneously. All three of these extend to the [[11-limit]] by way of interpreting the flat [[10/9]] as an [[11/10]] by tempering out [[100/99]]. This equivalence is especially useful in 41edo, wherein this comma-flat whole tone a.k.a. the second of Tetracot[7] can also be more accurately interpreted as [[21/19]]—which is equated with [[32/29]] above [[31/28]] below (both very near)—providing an explanation of the accuracy of primes [[29/1|29]] and [[31/1|31]] so that it is a uniquely good/versatile choice for interpreting the harmony of tetracot.
@@ Line 30: / Line 30: @@
 === Subsets and supersets ===
-edo is the 13th [[prime edo]], following [[37edo]] and coming before [[43edo]].
+edo is the 13th [[prime edo]], following [[37edo]] and coming before [[43edo]]. It does not contain any nontrivial subset edos, though it contains [[41ed4]].
-[[205edo]], which slices each step of 41edo into five, corrects some approximations of 41edo to near-just quality. As such, 41edo forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41edo as the basic [[13-limit]] intervals requiring fine tuning ±1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41edo circle in 205edo.
+[[205edo]], which slices each step of 41edo into five, corrects some approximations of 41edo to near-just quality. As such, 41edo forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41edo as the basic [[13-limit]] intervals requiring fine tuning ±1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41edo circle in 205edo. Its step of 1\205 is called a ''mem''.
+[[2460edo]] has potential for a 41edo analog to [[Cent|cents]]. It divides the 41edo step into 60 equal parts, and 60 is a highly composite (a.k.a. antiprime) number, so it contains many other multiples of 41edo, including 205edo, and also contains [[12edo]] among other equal tunings. It also accurately represents [[14afdo|mode 14 of the harmonic series]], as it is consistent all the way up to the 27-odd-limit. This allows for precise detunings in a 41-tone framework to approximate pure just intonation more closely, especially for some higher harmonics. Its step of 1\2460 is called a ''mina''.
 == Intervals ==
@@ Line 520: / Line 522: @@
 | upmajor 7th
 | ^M7
-| C#^, vvD
+| ^C#, vvD
 | supermajor 7th, classic dim 8ve, unter 8ve
 | SM7, KKd8, U8
@@ Line 687: / Line 689: @@
 The step size of 41edo is small enough that the smallest interval (the "red/blue unison", seventh-tone, comma, diesis or whatever you want to call it) is actually fairly consonant with most timbres; it resembles a "noticeably out of tune unison" rather than a minor second, and has its own distinct character and appeal.
-If "red" is replaced by "up", "blue" by "down", and "neutral" by "mid", and if "gray" is omitted, this notation becomes essentially the same as [[Ups_and_Downs_Notation|ups and downs notation]]. The only difference is the use of minor tritone and major tritone.
+If "red" is replaced by "up", "blue" by "down", and "neutral" by "mid", and if "gray" is omitted, this notation becomes essentially the same as [[Ups and downs notation|ups and downs notation]]. The only difference is the use of minor tritone and major tritone.
 === Sagittal notation ===
@@ Line 732: / Line 734: @@
 === Interval mappings ===
 {{Q-odd-limit intervals|41}}
-=== Zeta peak index ===
-{{ZPI
-| zpi = 184
-| steps = 40.9880783925993
-| step size = 29.2768055263764
-| tempered height = 7.570230
-| pure height = 7.497833
-| integral = 1.423937
-| gap = 17.722623
-| octave = 1200.34902658143
-| consistent = 16
-| distinct = 10
-}}
 == Relationship to 12edo ==
@@ Line 1,468: / Line 1,456: @@
 |
 |}
+== Octave stretch or compression ==
+Whether there is intonational improvement from [[stretched and compressed tuning|octave stretch or compression]] depends on which subgroup we are focusing on.
+For the 5-, 7-, and 11-limit, stretch is advised, though in the case of the 11-limit the stretch should be way milder, whereas for the 13-limit and in particular the 17-limit, little to no stretch or even compression may be suitable for balancing out the sharp and flat tuning tendencies, as is demonstrated in tunings such as [[65edt]], [[106ed6]], and [[147ed12]].
+Primes 19, 29, and 31 all tend flat, so stretching will serve again as we take that into account, especially if we use the temperament in any no-17 or no-13 no-17 settings.
+What follows is a comparison of stretched- and compressed-octave 41edo tunings.
+; [[24edf]]
+* Step size: 29.248{{c}}, octave size: 1199.17{{c}}
+Compressing the octave of 41edo by around 0.8{{c}} results in [[JND|unnoticeably]] improved primes 11, 17 and 23, but unnoticeably worse primes 2, 3, 5, 7, 13 and 19. This approximates all harmonics up to 16 within 7.6{{c}}. The tuning 24edf does this.
+{{Harmonics in equal|24|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 24edf}}
+{{Harmonics in equal|24|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 24edf (continued)}}
+; [[147ed12]] / [[106ed6]] / [[65edt]]
+* 65edt — step size: 29.261{{c}}, octave size: 1199.69{{c}}
+* 106ed6 — step size: 29.264{{c}}, octave size: 1199.81{{c}}
+* 147ed12 — step size: 29.265{{c}}, octave size: 1199.87{{c}}
+Compressing the octave of 41edo by around 0.2{{c}} results in [[JND|unnoticeably]] improved primes 3, 11 and 13, but unnoticeably worse primes 2, 5 and 7. This approximates all harmonics up to 16 within 7.6{{c}}. The tunings 147ed12, 106ed6 and 65edt each do this.
+{{Harmonics in equal|106|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 106ed6}}
+{{Harmonics in equal|106|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 106ed6 (continued)}}
+; 41edo
+* Step size: 29.268{{c}}, octave size: 1200.00{{c}}
+Pure-octaves 41edo approximates all harmonics up to 16 within 8.3{{c}}. The octaves of its 13-limit [[WE]] and [[TE]] tuning differ by less than 0.1{{c}} from pure.
+{{Harmonics in equal|41|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41edo}}
+{{Harmonics in equal|41|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41edo (continued)}}
+; [[zpi|184zpi]] / [[WE|41et, 11-limit WE tuning]]
+* Step size: 29.277{{c}}, octave size: 1200.35{{c}}
+Stretching the octave of 41edo by around 0.5{{c}} results in [[JND|unnoticeably]] improved primes 5 and 7, but unnoticeably worse primes 2, 3, 11 and 13. This approximates all harmonics up to 16 within 9.6{{c}}. Its 11-limit WE tuning and 11-limit [[TE]] tuning both do this. So does 184zpi, whose octave is identical to WE within 0.02{{c}}.
+{{Harmonics in cet|29.277|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 184zpi tuning}}
+{{Harmonics in cet|29.277|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 184zpi (continued)}}
+; [[WE|41et, 7-limit WE tuning]]
+* Step size: 29.288{{c}}, octave size: 1200.81{{c}}
+Stretching the octave of 41edo by just under 1{{c}} results in [[JND|just-noticeably]] improved primes 5 and 7, but just-noticeably worse primes 11 and 13. This approximates all harmonics up to 16 within 11.2{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this.
+{{Harmonics in cet|29.288|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 41et, 7-limit WE tuning}}
+{{Harmonics in cet|29.288|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 41et, 7-limit WE tuning (continued)}}
 == Scales and modes ==
@@ Line 2,036: / Line 2,065: @@
 The first 41edo guitar was probably this one, built by [[Erv Wilson]] in the 1960's:
-[[File:Erv Wilson's full-41 guitar 2.jpg|none|thumb]]
+[[File:Erv Wilson's full-41 guitar 2.jpg|none|thumb|200px]]
 Note the new bridge, several inches below the original bridge. The new bridge increases the scale length and spreads the frets out, making the guitar more playable. Erv numbered the frets as seen here, with the 3-limit dorian scale in enlarged numbers.
-[[File:Erv Wilson's full-41 guitar 3.jpg|frameless|838x838px]]
+[[File:Erv Wilson's full-41 guitar 3.jpg|frameless|500px]]
 Several more modern guitars:
+<gallery widths=300 heights=200>
+File:Melleweijters.com 41edo.jpg|[[Melle Weijters]]' 10-string guitar ([https://melleweijters.com Melleweijters.com])
+File:41-EDD_elektrische_gitaar.jpg|41edo electric guitar, by [[Gregory Sanchez]].
+File:Ron_Sword_with_a_41ET_Guitar.jpg|41edo classical guitar, by [[Ron Sword]].
+</gallery>
-[[File:Melleweijters.com 41edo.jpg|frameless|832x832px]]
+The [[Kite Guitar]] is a guitar fretting using every other step of 41edo, i.e. 41ed4 or "20½-edo". However, the interval between two adjacent open strings is always an odd number of 41-edosteps. Thus each string only covers half of 41edo, but the full edo can be found on every pair of adjacent strings. Kite-fretting makes 41edo about as playable as 19edo or 22edo, although there are certain trade-offs.
-''[[Melle Weijters]]' 10-string guitar ([https://melleweijters.com Melleweijters.com])''
-[[File:41-EDD_elektrische_gitaar.jpg|alt=41-EDD elektrische gitaar.jpg|560x745px|41-EDD elektrische gitaar.jpg]]
-''41edo electric guitar, by [[Gregory Sanchez]].''
-[[File:Ron_Sword_with_a_41ET_Guitar.jpg|alt=Ron_Sword_with_a_41ET_Guitar.jpg|Ron_Sword_with_a_41ET_Guitar.jpg]]
-''41edo classical guitar, by [[Ron Sword]].''
-The [[Kite Guitar]] (see also [https://kiteguitar.com KiteGuitar.com] and [http://tallkite.com/misc_files/The%20Kite%20Tuning.pdf Kite Tuning]) is a guitar fretting using every other step of 41edo, i.e. 41ed4 or "20½-edo". However, the interval between two adjacent open strings is always an odd number of 41-edosteps. Thus each string only covers half of 41edo, but the full edo can be found on every pair of adjacent strings. Kite-fretting makes 41edo about as playable as 19edo or 22edo, although there are certain trade-offs.
+[[File:Caleb's Kite guitar.jpg|none|thumb|200px|Kite guitar]]
-[[File:Caleb's Kite guitar.jpg|480x640px]]
 For more photos of Kite guitars, see [[Kite Guitar Photographs]].
@@ Line 2,069: / Line 2,091: @@
 === Keyboards ===
 A possible 41edo keyboard design:
-<div style="display: inline-grid; margin-right: 25px;>[[File:41edo keyboard layout.png|none|thumb|484x484px]]</div>
+<gallery widths=300 heights=200>
-<div style="display: inline-grid; margin-right: 25px;">[[File:Xenachord with 41edo layout.png|left|thumb|[https://richiegreene.com/instruments/ Xenachord] with 41edo layout by [[User:Richie|Richie]]]]
+File:41edo keyboard layout.png
+File:TS41 Microtonal MIDI Keyboard (Prototype).jpg|[[User:Tristanbay|Tristan Bay]]'s prototype TS41 MIDI keyboard, laid out in bosanquet with 41 keys per octave
+File:Xenachord with 41edo layout.png|[https://richiegreene.com/instruments/ Xenachord] with 41edo layout by [[User:Richie|Richie]]
+</gallery>
 See also [[41-edo Keyboards]] for Linnstrument and Harpejji options, as well as DIY options.
 {{clear}}
@@ Line 2,147: / Line 2,171: @@
 * [https://soundcloud.com/floracanou/sets/notes-of-the-generation ''Notes of the Generation''] (2023) – an 8-piece album in 41et
 : "Chaotic Witch #1" · "Party Cubes" · "Big Dreamer Pavilion" · "Lost Cyclops" · "Sky Tree" · "Long Night Ahead" · "Fractocraft" · "After the Generation"
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/kLMuRP82bZw ''microtonal dance in 41edo ''] (2023)
+* [https://www.youtube.com/shorts/m8X-IqH8tok ''Waltz in 41edo''] (2025)
+* [https://www.youtube.com/shorts/Ur--SKiRsY0 ''41edo groove''] (2025)
 ; [[Francium]]
@@ Line 2,158: / Line 2,187: @@
 ; [[L4MPLIGHT]]
 * ''Caftaphata'' (2024) – [https://www.youtube.com/watch?v=cMnuMjXeHrY YouTube] | [https://soundcloud.com/l4mplight/caftaphata-microtones-conlang SoundCloud] – also partially in just intonation and 12edo
+* ''Yxeni'' (2025) - [https://www.youtube.com/watch?v=PrfAz8V4WNc YouTube]
 ; [[Ray Perlner]]
@@ Line 2,184: / Line 2,214: @@
 * [https://soundcloud.com/sacred-skeleton/modified-kite-guitar-take-1 ''Modified Kite Guitar Take 1 - Clean'']
 * [https://soundcloud.com/sacred-skeleton/modified-kite-guitar-take-2 ''Modified Kite Guitar Take 2 - Fuzz'']
+; [[John Platter]]
+* [https://johnplatter.bandcamp.com/album/in-the-know In the Know] - Full album recorded using kite guitar & bass.
 ; [[Pixel Archipelago]]
@@ Line 2,218: / Line 2,251: @@
 <references/>
-[[Category:3-limit record edos]]
+[[Category:3-limit record edos|##]] <!-- 2-digit number -->
 [[Category:Magic]]
 [[Category:Superkleismic]]