Revision as of 20:08, 26 March 2019

Theory

8-edo forms an odd and even pitch set of two diminished seventh chords, which when used in combination yield dissonance. The system has been described as a "barbaric" harmonic system; even so, it does a good job representing the just intonation subgroup 2.11/3.13/5, with good intervals of 13/10 and an excellent version of 11/6.

Another way of looking at 8-EDO is to treat a chord of 0-1-2-3-4 degrees (0-150-300-450-600 cents) as approximating harmonics 10:11:12:13:14 (~0-165-316-454-583 cents), which is not too implausible if you can buy that 12-EDO is a 5-limit temperament. This interpretation would imply that 121/120, 144/143, 169/168, and hence also 36/35 and 66/65, are tempered out.

Intervals

8ed2-001.svg

8-edo can be notated as a subset of 24-edo, using ups and downs. It can also be notated as a subset of 16-edo, but this is a less intuitive notation.

Edostep	Cents	pions	7mus	24-edo subset notation		16-edo subset notation
0	0¢	0π^¢	0	P1	D	P1	D
1	150	159	192 (C0₁₆)	~2	Ev	M2	E
2	300	318	384 (180₁₆)	m3	F	M3	F#
3	450	477	576 (240₁₆)	^M3 / v4	F#^ / Gv	d3 / A4	Fb / G#
4	600	636	768 (300₁₆)	A4 / d5	G# / Ab	d4 / A5	Gb / A#
5	750	780	960 (3C0₁₆)	^5 / vm6	A^ / Bbv	d5 / A6	Ab / B#
6	900	954	1152 (480₁₆)	M6	B	m6	Bb
7	1050	1092	1344 (540₁₆)	~7	C^	m7	C
8	1200	1272	1536 (600₁₆)	P8	D	P8	D

8-edo chords are very ambiguous, with many chord homonyms. Even the major and minor triads are homonyms. Chord components usually default to M2, M3, P4, P5, M6, m7, M9, P11 and M13. Thus D7 has a M3, P5 and m7. 8-edo chord names using 24-edo subset names are greatly simplified by using different defaults: ~2, ^M3, v4, ^5, M6, ~7, ~9, v11 and M13. Thus D7 becomes ^M3, ^5 and ~7.

Chord edosteps	Chord notes	Proper name	Abbreviated name	Homonyms
0 – 3 – 5	D F♯^ A^	D.^(^5)	D	F♯^.m or Gv.m
0 – 2 – 5	D F A^	Dm(^5)	Dm	A^ or B♭v
0 – 3 – 5 – 7	D F♯^ A^ C^	D.^7(^5)	D7	F♯^.m♯11 or Gv.m♯11
0 – 3 – 5 – 6	D F♯^ A^ B	D6(^3,^5)	D6	Bm7 and Gv,♯9
0 – 2 – 5 – 7	D F A^ C^	Dm~7(^5)	Dm7	F6 and B♭v,♯9
0 – 2 – 5 – 6	D F A^ B	Dm6(^5)	Dm6	Bm7(♭5)
0 – 2 – 4 – 7	D F A♭ C^	Ddim~7	Dm7(♭5)	Fm6

Scala scale file

For those who use the new tuner Lingot, which accepts scala files, or for anyone else, here is a .scl file of 8 EDO 08-EDO.scl

0. 1/1 C Unison

1. 150.000 cents C# (50 cents sharper than C# of 12 tet) Neutral 2nd

2. 300.000 cents D# Minor 3rd (identical to 12-tet)

3. 450.000 cents E Sub-4th, Supermajor 3rd (50 cents sharper than E of 12 tet)

4. 600.000 cents F# Tritone (identical to 12-tet)

5. 750.000 cents G Super 5th, Subminor 6th (50 cents sharper than G of 12 tet)

6. 900.000 cents A Major 6th (identical to 12-tet)

7. 1050.000 cents A# Neutral 7th (50 cents sharper than A# of 12 tet)

8. 2/1 C Octave

Compositions

Tenacious Chorale (only movement II is in 8-EDO) by Stephen Weigel

Fantasia in 8ET by Aaron Andrew Hunt

Fugue in 8ET by Aaron Andrew Hunt

Špendlíky play by Milan Guštar

Acoustic Improvisation in 8-edo by Ron Sword (dead link)

Comets Over Flatland 7 by Randy Winchester

Malebolge Alison Jones

News To Me Jake Sherman

Tomorrow Night (pop music from 2045) Jake Sherman

FunkEight 1 by Carlo Serafini (blog entry)

8edo-pre-improv.oggPreliminary noodling in 8 EDO with ZynAddSubFX synth.(by Cenobyte)

darkreflections(sketch).ogg8EDO Ukulele (cenobyte)

skiphop(sketch).ogg8EDO Ukulele (cenobyte)

octo-icy-pensive(sketch).ogg octo-icy-pensive-echo(sketch).ogg2 versions of the same song, one with echo, one without. (cenobyte)

Ear Training

8 EDO ear-training exercises by Alex Ness available here.

Commas

8 EDO tempers out the following commas. (Note: This assumes val < 8 13 19 22 28 30 |). ||~ Comma ||~ Monzo ||~ Cents ||~ Name 1 ||~ Name 2 ||~ Name 3 ||

648/625	\| 3 4 -4 >	62.57	Major Diesis	Diminished Comma
250/243	\| 1 -5 3 >	49.17	Maximal Diesis	Porcupine Comma
78732/78125	\| 2 9 -7 >	13.40	Medium Semi Comma	Sensipent Comma
64/63	\| 6 -2 0 -1 >	27.26	Septimal Comma	Archytas' Comma	Leipziger Komma
875/864	\| -5 -3 3 1 >	21.90	Keema
321489/320000	\| -9 8 -4 2 >	8.04	Varun Isma
6144/6125	\| 11 1 -3 -2 >	5.36	Cornwell
100/99	\| 2 -2 2 0 -1 >	17.40	Ptolemies
121/120	\| -3 -1 -1 0 2 >	14.37	Batista
176/175	\| 4 0 -2 -1 1 >	9.86	Valinor Mo
65536/65219	\| 16 0 0 -2 -3 >	8.39	Organisms
385/384	\| -7 -1 1 1 1 >	4.50	Keenanisma
4000/3993	\| 5 -1 3 0 -3 >	3.03	Wizard Harry

@@ Line 23: / Line 23: @@
 ! | Edostep
 ! | Cents
+!pions
+!7mus
 ! colspan="2" | 24-edo subset notation
 ! colspan="2" | 16-edo subset notation
@@ Line 28: / Line 30: @@
 | style="text-align:center;" | 0
 | style="text-align:center;" | 0¢
+|0π<sup>¢</sup>
+|0
 | style="text-align:center;" | P1
 | style="text-align:center;" | D
@@ Line 35: / Line 39: @@
 | style="text-align:center;" | 1
 | style="text-align:center;" | 150
+|159
+|192 (C0<sub>16</sub>)
 | style="text-align:center;" | ~2
 | style="text-align:center;" | Ev
@@ Line 42: / Line 48: @@
 | style="text-align:center;" | 2
 | style="text-align:center;" | 300
+|318
+|384 (180<sub>16</sub>)
 | style="text-align:center;" | m3
 | style="text-align:center;" | F
@@ Line 49: / Line 57: @@
 | style="text-align:center;" | 3
 | style="text-align:center;" | 450
+|477
+|576 (240<sub>16</sub>)
 | style="text-align:center;" | ^M3 / v4
 | style="text-align:center;" | F#^ / Gv
@@ Line 56: / Line 66: @@
 | style="text-align:center;" | 4
 | style="text-align:center;" | 600
+|636
+|768 (300<sub>16</sub>)
 | style="text-align:center;" | A4 / d5
 | style="text-align:center;" | G# / Ab
@@ Line 63: / Line 75: @@
 | style="text-align:center;" | 5
 | style="text-align:center;" | 750
+|780
+|960 (3C0<sub>16</sub>)
 | style="text-align:center;" | ^5 / vm6
 | style="text-align:center;" | A^ / Bbv
@@ Line 70: / Line 84: @@
 | style="text-align:center;" | 6
 | style="text-align:center;" | 900
+|954
+|1152 (480<sub>16</sub>)
 | style="text-align:center;" | M6
 | style="text-align:center;" | B
@@ Line 77: / Line 93: @@
 | style="text-align:center;" | 7
 | style="text-align:center;" | 1050
+|1092
+|1344 (540<sub>16</sub>)
 | style="text-align:center;" | ~7
 | style="text-align:center;" | C^
@@ Line 84: / Line 102: @@
 | style="text-align:center;" | 8
 | style="text-align:center;" | 1200
+|1272
+|1536 (600<sub>16</sub>)
 | style="text-align:center;" | P8
 | style="text-align:center;" | D
@@ Line 205: / Line 225: @@
 |-
 | style="text-align:center;" | 648/625
-| | | 3 4 -4 &gt;
+| |<nowiki> | 3 4 -4 </nowiki>&gt;
 | style="text-align:right;" | 62.57
 | style="text-align:center;" | Major Diesis
@@ Line 212: / Line 232: @@
 |-
 | style="text-align:center;" | 250/243
-| | | 1 -5 3 &gt;
+| |<nowiki> | 1 -5 3 </nowiki>&gt;
 | style="text-align:right;" | 49.17
 | style="text-align:center;" | Maximal Diesis
@@ Line 219: / Line 239: @@
 |-
 | style="text-align:center;" | 78732/78125
-| | | 2 9 -7 &gt;
+| |<nowiki> | 2 9 -7 </nowiki>&gt;
 | style="text-align:right;" | 13.40
 | style="text-align:center;" | Medium Semi Comma
@@ Line 226: / Line 246: @@
 |-
 | style="text-align:center;" | 64/63
-| | | 6 -2 0 -1 &gt;
+| |<nowiki> | 6 -2 0 -1 </nowiki>&gt;
 | style="text-align:right;" | 27.26
 | style="text-align:center;" | Septimal Comma
@@ Line 233: / Line 253: @@
 |-
 | style="text-align:center;" | 875/864
-| | | -5 -3 3 1 &gt;
+| |<nowiki> | -5 -3 3 1 </nowiki>&gt;
 | style="text-align:right;" | 21.90
 | style="text-align:center;" | Keema
@@ Line 240: / Line 260: @@
 |-
 | style="text-align:center;" | 321489/320000
-| | | -9 8 -4 2 &gt;
+| |<nowiki> | -9 8 -4 2 </nowiki>&gt;
 | style="text-align:right;" | 8.04
 | style="text-align:center;" | Varun Isma
@@ Line 247: / Line 267: @@
 |-
 | style="text-align:center;" | 6144/6125
-| | | 11 1 -3 -2 &gt;
+| |<nowiki> | 11 1 -3 -2 </nowiki>&gt;
 | style="text-align:right;" | 5.36
 | style="text-align:center;" | Cornwell
@@ Line 254: / Line 274: @@
 |-
 | style="text-align:center;" | 100/99
-| | | 2 -2 2 0 -1 &gt;
+| |<nowiki> | 2 -2 2 0 -1 </nowiki>&gt;
 | style="text-align:right;" | 17.40
 | style="text-align:center;" | Ptolemies
@@ Line 261: / Line 281: @@
 |-
 | style="text-align:center;" | 121/120
-| | | -3 -1 -1 0 2 &gt;
+| |<nowiki> | -3 -1 -1 0 2 </nowiki>&gt;
 | style="text-align:right;" | 14.37
 | style="text-align:center;" | Batista
@@ Line 268: / Line 288: @@
 |-
 | style="text-align:center;" | 176/175
-| | | 4 0 -2 -1 1 &gt;
+| |<nowiki> | 4 0 -2 -1 1 </nowiki>&gt;
 | style="text-align:right;" | 9.86
 | style="text-align:center;" | Valinor Mo
@@ Line 275: / Line 295: @@
 |-
 | style="text-align:center;" | 65536/65219
-| | | 16 0 0 -2 -3 &gt;
+| |<nowiki> | 16 0 0 -2 -3 </nowiki>&gt;
 | style="text-align:right;" | 8.39
 | style="text-align:center;" | Organisms
@@ Line 282: / Line 302: @@
 |-
 | style="text-align:center;" | 385/384
-| | | -7 -1 1 1 1 &gt;
+| |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
 | style="text-align:right;" | 4.50
 | style="text-align:center;" | Keenanisma
@@ Line 289: / Line 309: @@
 |-
 | style="text-align:center;" | 4000/3993
-| | | 5 -1 3 0 -3 &gt;
+| |<nowiki> | 5 -1 3 0 -3 </nowiki>&gt;
 | style="text-align:right;" | 3.03
 | style="text-align:center;" | Wizard Harry

8edo: Difference between revisions

Revision as of 20:08, 26 March 2019

Contents

Theory

Intervals

Scala scale file

Compositions

Ear Training

See also

Commas

Navigation menu

8edo: Difference between revisions

Revision as of 20:08, 26 March 2019

Theory

Intervals

Scala scale file

Compositions

Ear Training

See also

Commas

Navigation menu

Search