Xenharmonic Wiki

TromboneBoi9

Joined 2 May 2023
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Hello! My name is Andrew and I like screwing around with xenharmony, especially EDOs, JI, and various notations.
Hello! My name is Andrew and I like screwing around with xenharmony, especially [[EDO|EDOs]], free [[Just intonation|JI]], and various [[Musical notation|notations]].


''[https://tilde.town/~tromboneboi9/ Here's my website]...if you could call it that.''
Here's [https://tilde.town/~tromboneboi9/ my website], it's got various things from photos to web-apps as well as scales I've designed.
 
Here's [https://tromboneboi9.github.io my GitHub page], where I might put various web-apps and web development projects.


I also exist on the [https://discord.com/invite/FSF5JFT XA Discord], currently under the alias ''Sir Semiflat''.
I also exist on the [https://discord.com/invite/FSF5JFT XA Discord], currently under the alias ''Sir Semiflat''.
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== Xenharmonic Discography ==
As of mid-November 2024 (non-comprehensive)


At some point I plan to make a piece similar to [[wikipedia:Twelve_Microtonal_Etudes_for_Electronic_Music_Media|Easley Blackwood's 12 Etudes]] or [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard Aaron Andrew Hunt's Equal-Tempered Keyboard], an "album" experimenting with a range of different EDO systems.
* '''''Torn Gamelan''''' for solo piano in [[31edo]], 2023
 
* '''''Apollo's Broken Piano''''' for solo piano in [[7-limit|7-limit just intonation]], 2023
<u>Keep in mind that I have been making regular changes to this page, so I could have made a lot of mistakes here.</u>
* '''''Chicago Olēka''''' for rock band in [[19edo]], 2023
 
* '''''A Harmonization of a Microtonal Etude''''' for string quartet in [[24edo]], 2024
==Extended Ups and Downs ==
-->
Something I noticed in [[Ups and downs notation|regular EDO notation]]--relying on [[Pythagorean]] names with an extra layer of accidentals--is that the [[81/64|Pythagorean major third]] in a lot of EDO systems doesn't match the [[5/4|just major third 5/4]]. So, with some thought from Just Intonation notations, I came up with something that I think might be promising.
 
In essence, instead of solely relying on Pythagorean names and arrows for edosteps in between (which can get unwieldy in larger EDOs, e.g. [[72edo#Intervals|72-EDO]]), I considered giving the [[syntonic comma]] a single symbol instead. That way you can emphasize the [[5-limit]] in your compositions more easily, and it will stay that way when directly read in a different EDO.
 
If your EDO has a different pitch for the just major third and the Pythagorean major third, then of course, it has a syntonic comma that hasn't been tempered out. This won't change much notation-wise for EDOs with a syntonic comma of 0 or 1 step, of course, but it could have an effect on even "sharper" systems like 37-EDO with a larger comma.
 
The syntonic comma can be represented by slashes: '''/''' (pitch up) and '''\''' (pitch down). Single edosteps are still notated with arrows: '''^''' and '''v'''.
 
Here's a full example in [[37edo|37-EDO]] where the syntonic comma is 2 steps large:
{| class="wikitable mw-collapsible"
!Steps
!Pythagorean notation
!Old notation
!New notation
|-
|0
|D
|D
|D
|-
|1
|Eb
|Eb
|Eb
|-
|2
|Fb
|^Eb
|/D
|-
|3
|Gbb
|^^Eb
|/Eb
|-
|4
|Bx
|vvD#
|\D#
|-
|5
|Cx
|vD#
|\E
|-
|6
|D#
|D#
|D#
|-
|7
|E
|E
|E
|-
|8
|F
|F
|F
|-
|9
|Gb
|Gb
|Gb
|-
|10
|Abb
|^Gb
|/F
|-
|11
|Bbbb
|^^Gb
|/Gb
|-
|12
|Dx
|vvF#
|\F#
|-
|13
|E#
|vF#
|\G
|-
|14
|F#
|F#
|F#
|-
|15
|G
|G
|G
|-
|16
|Ab
|Ab
|Ab
|-
|17
|Bbb
| ^Ab
|/G
|-
|18
|Cbb
|^^Ab
| /Ab
|-
| 19
| Ex
|vvG#
|\G#
|-
|20
|Fx
|vG#
| \A
|-
|21
|G#
| G#
|G#
|-
|22
|A
|A
|A
|-
|23
|Bb
|Bb
|Bb
|-
|24
|Cb
|^Bb
|/A
|-
|25
|Dbb
|^^Bb
|\Bb
|-
|26
|F#x
|vvA#
|\A#
|-
|27
|Gx
|vA#
|\B
|-
|28
| A#
|A#
|A#
|-
|29
|B
|B
|B
|-
|30
|C
|C
|C
|-
|31
|Db
|Db
|Db
|-
|32
|Ebb
|^Db
|/C
|-
|33
|Fbb
|^^Db
|/Db
|-
|34
|Ax
|vvC#
|\C#
|-
|35
|B#
|vC#
|\D
|-
|36
|C#
|C#
|C#
|-
|37
|D
|D
|D
|}
For systems with a negative syntonic comma (most often in sub-meantone systems), use ''')''' (pitch up) and '''(''' (pitch down) instead of '''/''' and '''\'''.
 
For anti-diatonic systems, the meantone-favoring flat and sharp symbols are not recommended, but use <u>harmonic notation</u> should they be needed.
 
An example in [[13edo|13-EDO]] where the syntonic comma is -2 steps (technically):
{| class="wikitable mw-collapsible"
!Steps
!Pythagorean/old notation
!26-EDO Subset
!New notation
|-
|0
|D
|D
|D
|-
|1
|E
|Dx, Ebb
|E, )C
|-
|2
|Eb
| E
|Eb, )D
|-
|3
|Fx
|Ex, Fb
|)E, (F
|-
|4
|F#
|F#
| F#, (G
|-
| 5
|F
|Gb
|F, (A
|-
| 6
|G
|G#
|G, (B
|-
|7
|A
|Ab
|A, )F
|-
|8
|B
| A#
|B, )G
|-
|9
|Bb
|Bb
| Bb, )A
|-
|10
|Cx
|B#
|)B, (C
|-
|11
|C#
|C
|C#, (D
|-
|12
|C
|Cx, Dbb
|C, (E
|-
|13
|D
|D
|D
|}
For systems with describable quarter tones, you can optionally use ad-lib quarter tone notation. Though for many systems ([[24edo|24]], [[31edo|31]]), the syntonic comma notation makes it redundant; perhaps it could be of use in larger systems like [[41edo|41]], [[48edo|48]], or [[72edo|72]].
 
An example in [[41edo|41-EDO]]:
{| class="wikitable mw-collapsible"
!Steps
!Pythagorean Notation
!Old Notation
!New Notation
!With Quartertones
|-
|0
|D
|D
|D
|D
|-
|1
|Cx
|^D
|/D
|/D
|-
|2
|Fbb
|^^D, vEb
|\Eb
|D+
|-
|3
|Eb
|vD#, Eb
|Eb
|Eb
|-
|4
|D#
|D#, ^Eb
|D#
|D#
|-
|5
|C#x
|^D#, vvE
|/D#
|Ed
|-
|6
|Fb
|vE
|\E
|\E
|-
|7
|E
|E
|E
|E
|-
|8
|Dx
|^E
|/E
|Fd
|-
|9
|Gbb
|vF
|\F
|E+
|-
|10
|F
|F
|F
|F
|-
|11
|E#
|^F
|/F
|/F
|-
|12
|Abbb
|^^F, vGb
|\Gb
|F+
|-
|13
|Gb
|vF#, Gb
|Gb
|Gb
|-
|14
|F#
|F#, ^Gb
|F#
|F#
|-
|15
|Ex
|^F#, vvG
|/F#
|Gd
|-
|16
|Abb
|vG
|\G
|\G
|-
|17
|G
|G
|G
|G
|-
|18
|Fx
|^G
|/G
|/G
|-
|19
|Bbbb
|^^G, vAb
|\Ab
|G+
|-
|20
|Ab
|vG#, Ab
|Ab
|Ab
|-
|21
|G#
|G#, ^Ab
|G#
|G#
|-
|22
|F#x
|^G#, vvA
|/G#
|Ad
|-
|23
|Bbb
|vA
|\A
|\A
|-
|24
|A
|A
|A
|A
|-
|25
|Gx
|^A
|/A
|/A
|-
|26
|Cbb
|^^A, vBb
|\Bb
|A+
|-
|27
|Bb
|vA#, Bb
|Bb
|Bb
|-
|28
|A#
|A#, ^Bb
|A#
|A#
|-
|29
|G#x
|^A#, vvB
|/A#
|Bd
|-
|30
|Cb
|vB
|\B
|\B
|-
|31
|B
|B
|B
|B
|-
|32
|Ax
|^B
|/B
|Cd
|-
|33
|Dbb
|vC
|\C
|B+
|-
|34
|C
|C
|C
|C
|-
|35
|B#
|^C
|/C
|/C
|-
|36
|Ebbb
|^^C, vDb
|\Db
|C+
|-
|37
|Db
|vC#, Db
|Db
|Db
|-
|38
|C#
|C#, ^Db
|C#
|C#
|-
|39
|Bx
|C#^, vvD
|/C#
|Dd
|-
|40
|Ebb
|vD
|\D
|\D
|-
|41
|D
|D
|D
|D
|}
 
== Scales n' Stuff==
 
=== Website ===
I have some scales ready in Scala format on my website [https://tilde.town/~tromboneboi9/scales.html here].
 
=== Blues scale in 10-EDO ===
I kinda like the '''3 1 1 1 2 2''' scale in 10-EDO, it works alright as a Blues scale. I think the second degree (3\10) is a bit sharper than it should, in fact a lot of intervals are "stretched out" in comparison to the Blues scale in 12, but before I didn't have very many scales in 10 under my belt except for the equipentatonic scale.
 
I dunno, I stick to theory more often than I should; I use ''theoretical'' diatonic intervals/scales more often than intervals/scales that actually ''sound'' diatonic. Luckily I've been experimenting with 14-EDO recently, and I think it's good territory to fix that.
 
=== Enneatonic scale in JI ===
[[User:SupahstarSaga|Supahstar Saga]] described a scale in [[19-EDO]] in his [https://www.youtube.com/playlist?list=PLha3CFvr8SzwlDpGL9MrJcoN8xOHyowsw ''Exploring 19-TET'' YouTube series] called the Enneatonic scale:
 
In 19-tone, the third harmonic is an even number of steps in 19 (30 steps), splitting evenly into two harmonic (subminor) sevenths. If you take a major pentatonic scale, put a harmonic seventh in between each fifth, and reduce the whole scale into an octave, you get a nine-tone scale somewhat similar to the [[wikipedia:Double_harmonic_scale|double harmonic scale]] in 12.
 
My thought was, if you use pure 3-limit Just Intonation, you can split the third harmonic into two "ratios" of √3, measuring around 950.978 cents. What would that sound like?
{| class="wikitable"
!Degree
!Ratio
!Decimal
!Cents
|-
|1
|1/1
|1.0000
|0.000
|-
|2
|9/8
|1.1250
|203.910
|-
|3
|81/64
|1.2656
|407.820
|-
|4
|3√3/4
|1.2990
|452.933
|-
|5
|27√3/32
|1.4614
|656.843
|-
|6
|3/2
|1.5000
|701.955
|-
|7
|27/16
|1.6875
|905.865
|-
|8
|√3/1
|1.7321
|950.978
|-
|9
|9√3/8
|1.9486
|1154.888
|-
|10
|2/1
|2.0000
|1200.000
|}
But of course Saga wasn't looking for a √3 interval, he meant to use the [[7/4|harmonic (subminor) seventh]]. The two intervals are rather close though: the seventh is about 17.848 cents sharper.
 
Using harmonic sevenths of 968.826 cents:
{| class="wikitable"
!Degree
!Ratio
!Decimal
!Cents
|-
|1
|1/1
|1.0000
|0.000
|-
|2
|9/8
|1.1250
|203.910
|-
|3
|81/64
|1.2656
|407.820
|-
|4
|21/16
|1.3125
|470.781
|-
|5
|189/128
|1.4766
|674.691
|-
|6
|3/2
|1.5000
|701.955
|-
|7
|27/16
|1.6875
|905.865
|-
|8
|7/4
|1.7500
|968.826
|-
|9
|63/32
|1.9486
|1172.736
|-
|10
|2/1
|2.0000
|1200.000
|}
Using harmonic sevenths inverted around the third harmonic, or just [[12/7|supermajor sixths (12/7)]], of 933.129 cents:
{| class="wikitable"
!Degree
!Ratio
!Decimal
!Cents
|-
|1
|1/1
|1.0000
|0.000
|-
|2
|9/8
|1.1250
|203.910
|-
|3
|81/64
|1.2656
|407.820
|-
|4
|9/7
|1.2857
|435.084
|-
|5
|81/56
|1.4464
|638.9941
|-
|6
|3/2
|1.5000
|701.955
|-
|7
|27/16
|1.6875
|905.865
|-
|8
|12/7
|1.7143
|933.129
|-
|9
|27/14
|1.9286
|1137.039
|-
|10
|2/1
|2.0000
|1200.000
|}
 
== Irrational HEJI Extensions ==
I've heard [[Golden ratio|phi]] is somewhat useful in xen areas, as well as other popular irrational numbers, so what would it look like if I extended [[Helmholtz-Ellis notation|HEJI]] (my go-to Just Intonation notation) to support these numbers like factors?
 
===Commas===
 
====Golden Ratio====
The ratio [[Acoustic phi|phi]] adds up to 833.0903 cents, a sharp minor sixth. The Pythagorean minor sixth is [[128/81]], about 792.1800 cents. This leaves a comma of 81ϕ/128, about 40.9103 cents. I dub this interval the '''Golden quartertone'''.


====Pi====
== Pages I've contributed to ==
The ratio π/2 adds up to 781.7954 cents, an okay minor sixth. The Pythagorean minor sixth is [[128/81]], about 792.1800 cents. This leaves a comma of 256/81π, about 10.3846 cents. I dub this interval the '''Circular comma'''.


====Euler's constant====
* [[Harmonic Scale]]
The ratio ''e''/2 adds up to 531.2340 cents, a pretty sharp fourth. The Pythagorean perfect fourth is, of course, [[4/3]], 498.0450 cents. This leaves a comma of 3''e''/8, about 33.1890 cents. I dub this interval the '''Eulerian comma'''.
* [[HEJI]]


===Notation===
== Subpages ==
For the golden quartertone, I plan to use the symbol Blackwood used in his microtonal notation, because it already resembles a phi symbol (ϕ). For pi, I designed a symbol similar to [https://en.xen.wiki/images/c/cf/Sagittal_sharp_kao.png the 55-comma symbol in Sagittal], but the "arrowhead" is replaced with a circular cap, making the symbol resemble a ''J'' with an extra shaft.


I'm yet to design a symbol for e.
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