User:TromboneBoi9: Difference between revisions

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Currently experimenting with anti-diatonic stuff like EDOs 9, 11, 13, 16, and 23.
Currently experimenting with anti-diatonic stuff like EDOs 9, 11, 13, 16, and 23.
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.


'''[https://akahler.w3spaces.com/ I have a website!!]'''
'''[https://akahler.w3spaces.com/ I have a website!!]'''


== An idea for notation I had ==
I also exist on the [https://discord.com/invite/FSF5JFT XA Discord], currently under the alias ''Sir Semiflat''.
Something I noticed in [[Ups and downs notation|regular EDO notation]], relying on Pythagorean names with an extra layer of accidentals, is that the [Pythagorean] major third in a lot of temperaments is no longer the closest the EDO has to 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.
 
==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 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 Pythaogrean names and arrows for edosteps in between (which can get unwieldy in larger EDOs, e.g. [[72edo#Intervals|72-EDO]]), I considered making the arrow represent the [[syntonic comma]] instead. 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 anything for EDOs with a syntonic comma less than or equal to one step, of course, but it could have an effect on even "sharper" systems like 37-EDO.
In essence, instead of solely relying on Pythaogrean names and arrows for edosteps in between (which can get unwieldy in larger EDOs, e.g. [[72edo#Intervals|72-EDO]]), I considered making the arrow represent the [[syntonic comma]] instead. 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 anything for EDOs with a syntonic comma less than or equal to one step, of course, but it could have an effect on even "sharper" systems like 37-EDO.
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|17
|17
|Bbb
|Bbb
|^Ab
| ^Ab
|^G
|^G
|-
|-
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|Cbb
|Cbb
|^^Ab
|^^Ab
|^Ab
| ^Ab
|-
|-
|19
| 19
|Ex
| Ex
|vvG#
|vvG#
|vG#
|vG#
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|Fx
|Fx
|vG#
|vG#
|vA
| vA
|-
|-
|21
|21
|G#
|G#
|G#
| G#
|G#
|G#
|-
|-
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|F#x
|F#x
|vvA#
|vvA#
|^A#
|vA#
|-
|-
|27
|27
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|-
|-
|28
|28
|A#
| A#
|A#
|A#
|A#
|A#
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|2
|2
|Eb
|Eb
|E
| E
|Eb, (D
|Eb, (D
|-
|-
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|F#
|F#
|F#
|F#
|F#, )G
| F#, )G
|-
|-
|5
| 5
|F
|F
|Gb
|Gb
|F, )A
|F, )A
|-
|-
|6
| 6
|G
|G
|G#
|G#
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|8
|8
|B
|B
|A#
| A#
|B, (G
|B, (G
|-
|-
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|Bb
|Bb
|Bb
|Bb
|Bb, (A
| Bb, (A
|-
|-
|10
|10
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''I have also devised custom accidentals for quartertones in both diatonic and anti-diatonic systems, but the image uploading process is being weird so I'll have to figure that out at some point.''
''I have also devised custom accidentals for quartertones in both diatonic and anti-diatonic systems, but the image uploading process is being weird so I'll have to figure that out at some point.''


== Cloudy scales ==
== Cloudy scales==
I don't know about you, but I love the seventh harmonic. These scales are named after the [[cloudy comma]], and use different [[7-limit]] intervals for generators.
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'''''.


=== Cumulus Alpha ===
''That's right. We're comparing JI to EDOs instead of the other way around.''
'''''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'''''.
{| class="wikitable mw-collapsible"
{| class="wikitable mw-collapsible"
!Steps
! Steps
!Ratio
!Ratio
!Cents
!Cents
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|0
|0
|-
|-
|1
| 1
|16807/16384
| 16807/16384
|43.130
|43.130
|1
|1
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|}
|}


=== Cumulus Beta ===
===Cumulus Beta===
'''''Cumulus Beta''''' is an 4L5s MOS with [[7/6]] as the generator and [[2/1]] as the period. It approximates all intervals of [[9edo|9-EDO]] within a cent, proving 9-EDO's place as an exceptional 7-limit approximation.
'''''Cumulus Beta''''' is an 4L5s MOS with [[7/6]] as the generator and [[2/1]] as the period. It approximates all intervals of [[9edo|9-EDO]] within a cent, with a mean difference of about 0.409 cents, proving 9-EDO's place as an exceptional 7-limit approximation.
{| class="wikitable mw-collapsible"
{| class="wikitable mw-collapsible"
!Steps
!Steps
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!9-EDO Difference
!9-EDO Difference
|-
|-
|0
| 0
|1/1
|1/1
|0.000
|0.000
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|}
|}


=== Cumulus Gamma ===
===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 10 cents.
'''''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"
{| class="wikitable mw-collapsible"
!Steps
!Steps

Revision as of 14:41, 22 May 2023

Hello! My name is Andrew and I like screwing around with xenharmony, especially notation.

Currently experimenting with anti-diatonic stuff like EDOs 9, 11, 13, 16, and 23.

At some point I plan to make a piece similar to Easley Blackwood's 12 Etudes or Aaron Andrew Hunt's Equal-Tempered Keyboard, an "album" experimenting with a range of different EDO systems.

I have a website!!

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

An idea for notation I had

Something I noticed in regular EDO notation, relying on Pythagorean names with an extra layer of accidentals, is that the Pythagorean major third in a lot of EDO systems doesn't match the 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 Pythaogrean names and arrows for edosteps in between (which can get unwieldy in larger EDOs, e.g. 72-EDO), I considered making the arrow represent the syntonic comma instead. 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 anything for EDOs with a syntonic comma less than or equal to one step, of course, but it could have an effect on even "sharper" systems like 37-EDO.

For single edosteps, we can instead use a sort of slash-like symbol Bosanquet used in his notation, and perhaps stack them on top of each other to use less horizontal space.

Here's a full example in 37-EDO:

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# vD#
5 Cx vD# vE
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# vF#
13 E# vF# vG
14 F# F# F#
15 G G G
16 Ab Ab Ab
17 Bbb ^Ab ^G
18 Cbb ^^Ab ^Ab
19 Ex vvG# vG#
20 Fx vG# vA
21 G# G# G#
22 A A A
23 Bb Bb Bb
24 Cb ^Bb ^A
25 Dbb ^^Bb vBb
26 F#x vvA# vA#
27 Gx vA# vB
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# vC#
35 B# vC# vD
36 C# C# C#
37 D D D

And for anti-diatonic systems, use ( and ) instead of ^ and v, using harmonic notation.

An example in 13-EDO:

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

I have also devised custom accidentals for quartertones in both diatonic and anti-diatonic systems, but the image uploading process is being weird so I'll have to figure that out at some point.

Cloudy scales

I don't know about you, but I love the seventh harmonic. These 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 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.

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. It approximates all intervals of 9-EDO within a cent, with a mean difference of about 0.409 cents, proving 9-EDO's place as an exceptional 7-limit approximation.

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 11-EDO within 7 cents, with a mean difference of 3.199 cents.

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
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