24edo: Difference between revisions

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The [[5-limit]] approximations in ~~24-tone equal temperament~~ are the same as those in ~~12-tone equal temperament~~, therefore ~~24-tone equal temperament~~ offers nothing new as far as approximating the 5-limit is concerned.

The [[5-limit]] approximations in 24edo are the same as those in 12edo, therefore 24edo offers nothing new as far as approximating the 5-limit is concerned.

The 7th harmonic-based intervals ([[7/4]], [[7/5]] and [[7/6]]) are almost as bad in ~~24-tET~~ as in ~~12-tET~~. To achieve a satisfactory level of approximation while maintaining the 12 notes of ~~12-tET~~ requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]] or [[156edo|156et]]. However, it should be noted that 24edo, like [[22edo]], ''does'' temper out the [[quartisma]], linking the otherwise sub-par 7-limit harmonies with those of the 11-limit.

The 7th harmonic-based intervals ([[7/4]], [[7/5]] and [[7/6]]) are almost as bad in 24edo as in 12edo. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]] or [[156edo|156et]]. However, it should be noted that 24edo, like [[22edo]], ''does'' temper out the [[quartisma]], linking the otherwise sub-par 7-limit harmonies with those of the 11-limit.

Speaking of 11-limit representation in 24edo, the 11th harmonic, and most intervals derived from it, (11/10, 11/9, 11/8, 11/6, 12/11, 15/11, 16/11, 18/11, 20/11) are very well approximated in this EDO. The 24-tone interval of 550 cents is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half the perfect fifth.

The tunings supplied by 72 cannot be used for all low-limit just intervals, but they can be used on the 17-limit [[k*N_subgroups|3*24 subgroup]] 2.3.125.35.11.325.17 [[just intonation subgroup]], making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11 and 17. Another approach would be to treat 24-EDO as a 2.3.11.17.19 subgroup temperament, on which it is quite accurate.

The tunings supplied by [[72edo]] cannot be used for all low-limit just intervals, but they can be used on the 17-limit [[k*N_subgroups|3*24 subgroup]] 2.3.125.35.11.325.17 [[just intonation subgroup]], making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11 and 17. Another approach would be to treat 24-EDO as a 2.3.11.17.19 [[Just intonation subgroup|subgroup]] temperament, on which it is quite accurate.

24edo is the 6th [[highly composite EDO]].

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=== William Lynch's notation ===

~~24 EDO~~ breaks intervals into two sets of five cartegories. Infra - Minor - Neutral - Major - Ultra for seconds, thirds, sixths, and sevenths; and diminished - narrow - perfect - wide - augmented for fourths, fifths, unison, and octave.

24edo breaks intervals into two sets of five cartegories. Infra - Minor - Neutral - Major - Ultra for seconds, thirds, sixths, and sevenths; and diminished - narrow - perfect - wide - augmented for fourths, fifths, unison, and octave.

For other strange enharmonics, wide and narrow can be used in conjunction with augmented and diminished intervals such as 550 cents being called a narrow diminished fifth and 850 cents being called a wide augmented fifth.

These are the intervals of ~~24 EDO~~ that do not exist in ~~12 EDO~~:

These are the intervals of 24edo that do not exist in 12edo:

{| class="wikitable center-all right-6"

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These chords tend to lack the forcefulness to sound like resolved, tonal sonorities, but can be resolved of that issue by using tetrads in place of triads. For example, the neutral triad can have the neutral 7th added to it to make a full neutral tetrad: 0-7-14-21. However, another option is to replace the neutral third with an 11/8 to produce a sort of 11 limit neutral tetrad: 0-14-21-35 [[William Lynch]] considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality.

~~24 edo~~ also is very good at 15 limit and does 13 quite well allowing barbodos 10:13:15 and barbodos minor triad 26:30:39 to be used as an entirely new harmonic system.

24edo also is very good at 15 limit and does 13 quite well allowing barbodos 10:13:15 and barbodos minor triad 26:30:39 to be used as an entirely new harmonic system.

More good chords in ~~24-tET~~:

More good chords in 24edo:

* 0-4-8-11-14 ("major" chord with a 9:8 and a 11:8 above the root)

@@ Line 16: / Line 16: @@
 {{Odd harmonics in edo|edo=24}}
-The [[5-limit]] approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned.
+The [[5-limit]] approximations in 24edo are the same as those in 12edo, therefore 24edo offers nothing new as far as approximating the 5-limit is concerned.
-The 7th harmonic-based intervals ([[7/4]], [[7/5]] and [[7/6]]) are almost as bad in 24-tET as in 12-tET. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]] or [[156edo|156et]].  However, it should be noted that 24edo, like [[22edo]], ''does'' temper out the [[quartisma]], linking the otherwise sub-par 7-limit harmonies with those of the 11-limit.
+The 7th harmonic-based intervals ([[7/4]], [[7/5]] and [[7/6]]) are almost as bad in 24edo as in 12edo. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]] or [[156edo|156et]].  However, it should be noted that 24edo, like [[22edo]], ''does'' temper out the [[quartisma]], linking the otherwise sub-par 7-limit harmonies with those of the 11-limit.
 Speaking of 11-limit representation in 24edo, the 11th harmonic, and most intervals derived from it, (11/10, 11/9, 11/8, 11/6, 12/11, 15/11, 16/11, 18/11, 20/11) are very well approximated in this EDO. The 24-tone interval of 550 cents is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half the perfect fifth.
-The tunings supplied by 72 cannot be used for all low-limit just intervals, but they can be used on the 17-limit [[k*N_subgroups|3*24 subgroup]] 2.3.125.35.11.325.17 [[just intonation subgroup]], making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11 and 17. Another approach would be to treat 24-EDO as a 2.3.11.17.19 subgroup temperament, on which it is quite accurate.
+The tunings supplied by [[72edo]] cannot be used for all low-limit just intervals, but they can be used on the 17-limit [[k*N_subgroups|3*24 subgroup]] 2.3.125.35.11.325.17 [[just intonation subgroup]], making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11 and 17. Another approach would be to treat 24-EDO as a 2.3.11.17.19 [[Just intonation subgroup|subgroup]] temperament, on which it is quite accurate.
 edo is the 6th [[highly composite EDO]].
@@ Line 297: / Line 297: @@
 === William Lynch's notation ===
-EDO breaks intervals into two sets of five cartegories. Infra - Minor - Neutral - Major - Ultra for seconds, thirds, sixths, and sevenths; and diminished - narrow - perfect - wide - augmented for fourths, fifths, unison, and octave.
+edo breaks intervals into two sets of five cartegories. Infra - Minor - Neutral - Major - Ultra for seconds, thirds, sixths, and sevenths; and diminished - narrow - perfect - wide - augmented for fourths, fifths, unison, and octave.
 For other strange enharmonics, wide and narrow can be used in conjunction with augmented and diminished intervals such as 550 cents being called a narrow diminished fifth and 850 cents being called a wide augmented fifth.
-These are the intervals of 24 EDO that do not exist in 12 EDO:
+These are the intervals of 24edo that do not exist in 12edo:
 {| class="wikitable center-all right-6"
@@ Line 596: / Line 596: @@
 These chords tend to lack the forcefulness to sound like resolved, tonal sonorities, but can be resolved of that issue by using tetrads in place of triads. For example, the neutral triad can have the neutral 7th added to it to make a full neutral tetrad: 0-7-14-21. However, another option is to replace the neutral third with an 11/8 to produce a sort of 11 limit neutral tetrad: 0-14-21-35 [[William Lynch]] considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality.
 edo also is very good at 15 limit and does 13 quite well allowing barbodos 10:13:15 and barbodos minor triad 26:30:39 to be used as an entirely new harmonic system.
-More good chords in 24-tET:
+More good chords in 24edo:
 * 0-4-8-11-14 ("major" chord with a 9:8 and a 11:8 above the root)