Xenharmonic Wiki

TromboneBoi9

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


''Check it out, I have a website! [https://tilde.town/~tromboneboi9/ Here she is!]''
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 constantly been making 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
==An idea for notation I had ==
-->
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: '''/''' and '''\'''. Single edosteps are still notated with arrows: '''^''' and '''v'''.
 
Here's a full example in [[37edo|37-EDO]]:
{| 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 ''')''' and '''(''' instead of '''/''' and '''\'''.
 
Use <u>harmonic notation</u> for anti-diatonic systems.
 
An example in [[13edo|13-EDO]]:
{| 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 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]].
 
== Scales n' Stuff==
 
=== Cumulus scales ===
I don't know about you, but I love the seventh harmonic. These [[MOS scale|MOS scales]] are named after the [[cloudy comma]], and use different [[7-limit]] intervals for generators.
 
====Cumulus Alpha====
'''''Cumulus Alpha''''' is a 5L6s MOS with [[7/4]] as the generator and [[2/1]] as the period. This appears to approximate a subset of [[26edo|26-EDO]]; it approximates the whole of 26-EDO when extended to a 5L21s MOS, which I dub '''''Cumulus Alpha Holo'''''.
 
''That's right. We're comparing JI to EDOs instead of the other way around.''
{| class="wikitable mw-collapsible"
! Steps
!Ratio
!Cents
!Approx. 26-EDO Degree
|-
|0
|1/1
|0.000
|0
|-
| 1
| 16807/16384
|43.130
|1
|-
|2
|8/7
|231.174
|5
|-
|3
|2401/2048
|275.304
|6
|-
|4
|64/49
|462.348
|10
|-
|5
|343/256
|506.478
|11
|-
|6
|512/343
|693.522
|15
|-
|7
|49/32
|737.652
|16
|-
|8
|4096/2401
|924.696
|20
|-
|9
|7/4
|968.826
|21
|-
|10
|32768/16807
|1155.870
|25
|-
|11
|2/1
|1200.000
|26
|}
 
====Cumulus Beta====
'''''Cumulus Beta''''' is an 4L5s MOS with [[7/6]] as the generator and [[2/1]] as the period. Amazingly, it approximates all intervals of [[9edo|9-EDO]] within a cent, with a mean difference of about 0.409 cents.
{| class="wikitable mw-collapsible"
!Steps
!Ratio
!Cents
!9-EDO Difference
|-
| 0
|1/1
|0.000
|0.000
|-
|1
|2592/2401
|132.516
| -0.817
|-
|2
|7/6
|266.871
|0.204
|-
|3
|432/343
|399.387
| -0.613
|-
|4
|49/36
|533.742
|0.409
|-
|5
|72/49
|666.258
| -0.409
|-
|6
|343/216
|800.613
|0.613
|-
|7
|12/7
|933.129
| -0.204
|-
|8
|2401/1296
|1067.484
|0.817
|-
|9
|7/4
|1200.000
|0.000
|}
 
====Cumulus Gamma====
'''''Cumulus Gamma''''' is an 3L8s MOS with [[9/7]] as the generator and [[2/1]] as the period. It approximates all intervals of [[11edo|11-EDO]] within 7 cents, with a mean difference of 3.199 cents.
{| class="wikitable mw-collapsible"
!Steps
!Ratio
!Cents
!11-EDO Difference
|-
|0
|1/1
|0.000
|0.000
|-
|1
|729/686
|105.252
|3.839
|-
|2
|67228/59049
|224.580
| -6.398
|-
|3
|98/81
|329.832
| -2.559
|-
|4
|9/7
|435.084
|1.280
|-
|5
|6561/4802
|540.336
|5.119
|-
|6
|9604/6561
|659.664
| -5.119
|-
|7
|14/9
|764.916
| -1.280
|-
|8
|81/49
|870.168
|2.559
|-
|9
|59049/33614
|975.420
|6.398
|-
|10
|1372/729
|1094.748
| -3.839
|-
|11
|2/1
|1200.000
|0.000
|}
 
=== 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 "3-limit" ===
[[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:
 
Since 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. What would that sound like?
{| class="wikitable"
!Degree
!Ratio
!Cents
|-
|1
|1/1
|0.000
|-
|2
|9/8
|203.910
|-
|3
|81/64
|407.820
|-
|4
|3√3/4
|452.933
|-
|5
|27√3/32
|656.843
|-
|6
|3/2
|701.955
|-
|7
|27/16
|905.865
|-
|8
|√3/1
|950.978
|-
|9
|9√3/8
|1154.888
|-
|10
|2/1
|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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