Harmony of 23edo: Difference between revisions

Line 1:

If you take a look at the intervals of [[~~23edo|~~23edo]], you'll find that this system does not contain good representations of the harmonics 3, 5, 7, 11, or 13, which appear as central in most just intonation systems. Rather than trivialize 23edo by calling it "atonal" or "inharmonic", you should consider the higher-limit harmonies that could serve as useful sonorities, perhaps even 'consonances', in the context of careful composition. ~~[[23edo|~~23edo]] contains intervals which approach very well the harmonics 9, 17, 21, 23, 33, 35, 55, 79 & 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics (and other intervals):

If you take a look at the intervals of [[23edo]], you'll find that this system does not contain good representations of the harmonics 3, 5, 7, 11, or 13, which appear as central in most just intonation systems. Rather than trivialize 23edo by calling it "atonal" or "inharmonic", you should consider the higher-limit harmonies that could serve as useful sonorities, perhaps even 'consonances', in the context of careful composition. 23edo contains intervals which approach very well the harmonics 9, 17, 21, 23, 33, 35, 55, 79 & 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics (and other intervals):

{| class="wikitable"

|-

| | Degrees

| | Armodue note

| | [[Armodue]] note

| | ~~Cents~~ sizes

| | [[Cent]]s sizes

| ~~| [[nearest_harmonic~~|Nearest ~~Harmonic]]~~

| | Nearest harmonic

| | Cents

| | "Error"

Line 184:

Thus we produce ten triads, five tetrads, & one pentad, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), & I recommend trying different voicings. Moving chord tones up & down by octaves, you can unmuddy a muddy chord.

==Triads==

== Triads ==

'''16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21)'''.

Line 193:

18/17 (98.955, error: +5.393)

===16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21).===

=== 16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21). ===

17/16 (104.955, error -0.607)

Line 200:

21/17 (365.825, error: -0.608)

===16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21).===

=== 16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21). ===

17/16 (104.955, error -0.607)

Line 207:

23/17 (523.319, error: -1.578)

===16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19).===

=== 16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19). ===

18/16 = 9/8 (203.910, error +4.786)

Line 214:

21/18 = 7/6 (266.871, error: -6.001)

===16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19).===

=== 16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19). ===

18/16 = 9/8 (203.910, error +4.786)

Line 221:

23/18 (424.364, error: -6.973)

===16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14).===

=== 16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14). ===

21/16 (470.781, error -1.216)

Line 228:

23/21 (157.493, error: -0.971)

===17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21).===

=== 17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21). ===

18/17 (98.955, error: +5.393)

Line 235:

21/18 = 7/6 (266.871, error: -6.001)

===17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21).===

=== 17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21). ===

18/17 (98.955, error: +5.393)

Line 242:

23/18 (424.364, error: -6.973)

===17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16).===

=== 17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16). ===

21/17 (365.825, error: -0.608)

Line 249:

23/21 (157.493, error: -0.971)

===18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18).===

=== 18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18). ===

21/18 = 7/6 (266.871, error: -6.001)

Line 256:

23/21 (157.493, error: -0.971)

==Tetrads==

== Tetrads ==

'''16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21)'''.

Line 271:

21/18 = 7/6 (266.871, error: -6.001)

===16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21).===

=== 16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21). ===

17/16 (104.955, error -0.607)

Line 284:

23/18 (424.364, error: -6.973)

===16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21).===

=== 16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21). ===

17/16 (104.955, error -0.607)

Line 297:

23/21 (157.493, error: -0.971)

===16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19).===

=== 16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19). ===

18/16 = 9/8 (203.910, error +4.786)

Line 310:

23/21 (157.493, error: -0.971)

===17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21).===

=== 17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21). ===

18/17 (98.955, error: +5.393)

Line 323:

23/21 (157.493, error: -0.971)

==~~Pentad~~==

== Pentads ==

'''16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21)'''.

Line 345:

23/21 (157.493, error: -0.971)

[[Category:23edo]]

[[Category:~~harmony~~]]

[[Category:Harmony]]

[[Category:~~interval~~]]

[[Category:Pentad]]

[[Category:~~pentad~~]]

[[Category:Tetrad]]

[[Category:~~tetrad]]~~

[[Category:Triad]]

~~[[Category:theory]]~~

~~[[Category:triad~~]]

@@ Line 1: / Line 1: @@
-If you take a look at the intervals of [[23edo|23edo]], you'll find that this system does not contain good representations of the harmonics 3, 5, 7, 11, or 13, which appear as central in most just intonation systems. Rather than trivialize 23edo by calling it "atonal" or "inharmonic", you should consider the higher-limit harmonies that could serve as useful sonorities, perhaps even 'consonances', in the context of careful composition. [[23edo|23edo]] contains intervals which approach very well the harmonics 9, 17, 21, 23, 33, 35, 55, 79 &amp; 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics (and other intervals):
+If you take a look at the intervals of [[23edo]], you'll find that this system does not contain good representations of the harmonics 3, 5, 7, 11, or 13, which appear as central in most just intonation systems. Rather than trivialize 23edo by calling it "atonal" or "inharmonic", you should consider the higher-limit harmonies that could serve as useful sonorities, perhaps even 'consonances', in the context of careful composition. 23edo contains intervals which approach very well the harmonics 9, 17, 21, 23, 33, 35, 55, 79 &amp; 117. Let's compare the cents values to see how close 23edo intervals come to these harmonics (and other intervals):
 {| class="wikitable"
 |-
 | | Degrees
-| | Armodue note
+| | [[Armodue]] note
-| | Cents sizes
+| | [[Cent]]s sizes
-| | [[nearest_harmonic|Nearest Harmonic]]
+| | Nearest harmonic
 | | Cents
 | | "Error"
@@ Line 184: / Line 184: @@
 Thus we produce ten triads, five tetrads, &amp; one pentad, 16 chords, which, with their inversions (given), doubles to 32 chords. I've written then in a closed position (within one octave), &amp; I recommend trying different voicings. Moving chord tones up &amp; down by octaves, you can unmuddy a muddy chord.
-==Triads==
+== Triads ==
 '''16:17:18, degrees 0, 2, 4 (inversion 0, 19, 21)'''.
@@ Line 193: / Line 193: @@
 /17 (98.955, error: +5.393)
-===16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21).===
+=== 16:17:21, degrees 0, 2, 9 (inversion 0, 14, 21). ===
 /16 (104.955, error -0.607)
@@ Line 200: / Line 200: @@
 /17 (365.825, error: -0.608)
-===16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21).===
+=== 16:17:23, degrees 0, 2, 12 (inversion 0, 11, 21). ===
 /16 (104.955, error -0.607)
@@ Line 207: / Line 207: @@
 /17 (523.319, error: -1.578)
-===16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19).===
+=== 16:18:21, degrees 0, 4, 9 (inversion 0, 14, 19). ===
 /16 = 9/8 (203.910, error +4.786)
@@ Line 214: / Line 214: @@
 /18 = 7/6 (266.871, error: -6.001)
-===16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19).===
+=== 16:18:23, degrees 0, 4, 12 (inversion 0, 11, 19). ===
 /16 = 9/8 (203.910, error +4.786)
@@ Line 221: / Line 221: @@
 /18 (424.364, error: -6.973)
-===16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14).===
+=== 16:21:23, degrees 0, 9, 12 (inversion 0, 11, 14). ===
 /16 (470.781, error -1.216)
@@ Line 228: / Line 228: @@
 /21 (157.493, error: -0.971)
-===17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21).===
+=== 17:18:21, degrees 0, 2, 7 (inversion 0, 16, 21). ===
 /17 (98.955, error: +5.393)
@@ Line 235: / Line 235: @@
 /18 = 7/6 (266.871, error: -6.001)
-===17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21).===
+=== 17:18:23, degrees 0, 2, 10 (inversion 0, 13, 21). ===
 /17 (98.955, error: +5.393)
@@ Line 242: / Line 242: @@
 /18 (424.364, error: -6.973)
-===17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16).===
+=== 17:21:23, degrees 0, 7, 10 (inversion 0, 13, 16). ===
 /17 (365.825, error: -0.608)
@@ Line 249: / Line 249: @@
 /21 (157.493, error: -0.971)
-===18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18).===
+=== 18:21:23, degrees 0, 5, 8 (inversion 0, 15, 18). ===
 /18 = 7/6 (266.871, error: -6.001)
@@ Line 256: / Line 256: @@
 /21 (157.493, error: -0.971)
-==Tetrads==
+== Tetrads ==
 '''16:17:18:21, degrees 0, 2, 4, 9 (inversion 0, 14, 19, 21)'''.
@@ Line 271: / Line 271: @@
 /18 = 7/6 (266.871, error: -6.001)
-===16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21).===
+=== 16:17:18:23, degrees 0, 2, 4, 12 (inversion 0, 11 19, 21). ===
 /16 (104.955, error -0.607)
@@ Line 284: / Line 284: @@
 /18 (424.364, error: -6.973)
-===16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21).===
+=== 16:17:21:23, degrees 0, 2, 9, 12 (inversion 0, 11, 14, 21). ===
 /16 (104.955, error -0.607)
@@ Line 297: / Line 297: @@
 /21 (157.493, error: -0.971)
-===16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19).===
+=== 16:18:21:23, degrees 0, 4, 9, 12 (inversion 0, 11, 14, 19). ===
 /16 = 9/8 (203.910, error +4.786)
@@ Line 310: / Line 310: @@
 /21 (157.493, error: -0.971)
-===17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21).===
+=== 17:18:21:23, degrees 0, 2, 7, 10 (inversion 0, 13, 16, 21). ===
 /17 (98.955, error: +5.393)
@@ Line 323: / Line 323: @@
 /21 (157.493, error: -0.971)
-==Pentad==
+== Pentads ==
 '''16:17:18:21:23, degrees 0, 2, 4, 9, 12 (inversion 0, 11, 14, 19, 21)'''.
@@ Line 345: / Line 345: @@
 /21 (157.493, error: -0.971)
 [[Category:23edo]]
-[[Category:harmony]]
+[[Category:Harmony]]
-[[Category:interval]]
+[[Category:Pentad]]
-[[Category:pentad]]
+[[Category:Tetrad]]
-[[Category:tetrad]]
+[[Category:Triad]]
-[[Category:theory]]
-[[Category:triad]]