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==Extended Ups and Downs ==
==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.
See [[User:TromboneBoi9/Extended Ups and Downs]]
 
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|13b-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 ([[10edo|10]], [[17edo|17]], [[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]] with '''+''' for a quartertone sharp and '''d''' for a quartertone flat:
{| 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==
== Scales n' Stuff==

Revision as of 01:20, 14 May 2024

Hello! My name is Andrew and I like screwing around with xenharmony, especially EDOs, JI, and various notations.

Here's my website...if you could call it that.

I also exist on the XA Discord, currently under the alias Sir Semiflat.

Xenharmonic Discography

As of mid-April 2024 (non-comprehensive)

Completed works

  • Torn Gamelan for solo piano in 31edo, 2023
  • Apollo's Broken Piano for solo piano in 7-limit just intonation, 2023
  • Chicago Olēka for rock band in 19edo, 2023
  • A Harmonization of a Microtonal Etude for string quartet in 24edo, 2024
  • Adagio con demenza for solo piano in 16edo, 2023

Incompleted works

  • Keyboard Suite in 20-equal Tuning for solo keyboard in 20edo
  • Treated Four by Four for solo piano in 16edo

Extended Ups and Downs

See User:TromboneBoi9/Extended Ups and Downs

Scales n' Stuff

Website

I have some scales ready in Scala format on my website here.

"Drewnian" Scale System

I recently developed an eleven-tone scale system, similar to Western theory in construction but by no means similar in sound. I like to think it's what Wstern tonality would look like if the 7/4 was prioritized over the 5/4.

It involves two scales, each built with a seven-based tetrachord. One is built with a 8/7, 8/7, 49/48 tetrachord, generating the "major" intervals; and the other is built with a 49/48, 8/7, 8/7 tetrachord, generating the "minor" intervals. As a result, the scale has very large major seconds and very small minor seconds.

Degree Name Ratio Cents
0 P1 1/1 0.000
1 m2 49/48 35.697
2 M2 8/7 231.174
3 m3 7/6 266.871
4 M3 64/49 462.348
5 P4 4/3 498.045
6 P5 3/2 701.955
7 m6 49/32 737.652
8 M6 12/7 933.129
9 m7 7/4 968.826
10 M7 96/49 1164.303
11 P8 2/1 1200.000

The intent with these weird superpyth-like constructions is to make 5-EDO-like scales with some extra small intervals for spice.

It's very easily possible to map this scale to a 12-tone keyboard, in fact it is by design a mimicry of Western tonality. However, there is no tritone. For this interval you can either use the eleventh harmonic 11/8, a septimal whole tone below the "minor sixth". or a septimal whole tone above the "major third".

Enneatonic scale in JI

Supahstar Saga described a scale in 19-EDO in his 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 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?

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 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:

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 supermajor sixths (12/7), of 933.129 cents:

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 phi is somewhat useful in xen areas, as well as other popular irrational numbers, so what would it look like if I extended HEJI (my go-to Just Intonation notation) to support these numbers like factors?

Commas

Golden Ratio

The ratio 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

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

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 3e/8, about 33.1890 cents. I dub this interval the Eulerian comma.

Notation

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 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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