20edo: Difference between revisions

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20edo contains smaller {{EDOs|legend=1| 2, 4, 5, and 10 }} and is part of the 5''n'' family of equal divisions of the octave. It fairly approximates the harmonics 7 (from [[5edo]]), 11, 13 & 15 (from [[10edo]]), 19 & 27 (from [[4edo]]), 29 and 31; as well as the other harmonics more loosely (though to some people, still functionally) approximated. Thus, 20-EDO does a reasonably convincing approximation of harmonics 4:7:11:13:15.

As 7, 11, & 15 are all flat by approximately 10 cents, their flatness cancels out when combined in composite ratios, making an 11:14:15 chord (0-7-9 steps) and its utonal inversion particularly precise. Using 9/20 as the generator and treating these as the primary major and minor triads produces Balzano nonatonic and undecatonic scales, which is probably the clearest arrangement for the black/white keys on a 20 tone keyboard.

As 7, 11, & 15 are all flat by approximately 10 cents, their flatness cancels out when combined in composite ratios, making an 11:14:15 chord (0-7-9 steps) and its utonal inversion particularly precise. Using 9/20 as the generator and treating these as the primary major and minor triads produces [[Balzano]] nonatonic and undecatonic scales, which is probably the clearest arrangement for the black/white keys on a 20 tone keyboard.

Treating the generator as 11\20 creates the same scale, but the primary triads are now 13:16:19 (0-6-11 steps) and its inversion instead. The 11\20 generator is a near-optimal tuning for both [[No-threes subgroup temperaments#Mavericks|Mavericks]] temperament (which has a ~[[19/13]] generator) and [[No-threes subgroup temperaments#Score|Score]] temperament (which has a ~[[16/11]] generator).

Treating the generator as 11\20 creates the same scale, but the primary triads are now 13:16:19 (0-6-11 steps) and its inversion instead.

Thanks to its sevenths, 20edo is an ideal tuning for its size for [[metallic harmony]].

=== RTT ===

''See [[regular temperament]] for more about what all this means and how to use it.''

The 11\20 generator is a near-optimal tuning for both [[No-threes subgroup temperaments#Mavericks|Mavericks]] temperament (which has a ~[[19/13]] generator) and [[No-threes subgroup temperaments#Score|Score]] temperament (which has a ~[[16/11]] generator).

Alternately, 20edo can be used as a tuning of the [[blackwood]] temperament, combining minor and major thirds to generate a highly symmetrical decatonic scale where every note is root to a major or minor triad and 7-limit tetrad that are heavily tempered, but in a useful way, as you can easily modulate to anywhere in the small cycle of 5ths, and build extended chords that use every note in the scale without clashing. Either of these works better than trying to force 20 into a diatonic framework.

20edo also possesses a [[6L 1s]] scale generated using the narrow major second of 3\20 that is probably best interpreted as the sharp extreme of [[Tetracot_family#Tetracot|tetracot]] temperament and a [[3L 5s]] generated by 7/20 that functions as the flat end of [[squares~~]].~~

20edo also possesses a [[6L 1s]] scale generated using the narrow major second of 3\20 that is probably best interpreted as the sharp extreme of [[Tetracot_family#Tetracot|tetracot]] temperament and a [[3L 5s]] generated by 7/20 that functions as the flat end of [[squares]].

~~Thanks to its sevenths, 20edo is an ideal tuning for its size for [[metallic harmony~~]].

=== Odd harmonics ===

@@ Line 10: / Line 10: @@
 edo contains smaller {{EDOs|legend=1| 2, 4, 5, and 10 }} and is part of the 5''n'' family of equal divisions of the octave. It fairly approximates the harmonics 7 (from [[5edo]]), 11, 13 &amp; 15 (from [[10edo]]), 19 &amp; 27 (from [[4edo]]), 29 and 31; as well as the other harmonics more loosely (though to some people, still functionally) approximated. Thus, 20-EDO does a reasonably convincing approximation of harmonics 4:7:11:13:15.
-As 7, 11, & 15 are all flat by approximately 10 cents, their flatness cancels out when combined in composite ratios, making an 11:14:15 chord (0-7-9 steps) and its utonal inversion particularly precise. Using 9/20 as the generator and treating these as the primary major and minor triads produces Balzano nonatonic and undecatonic scales, which is probably the clearest arrangement for the black/white keys on a 20 tone keyboard.
+As 7, 11, & 15 are all flat by approximately 10 cents, their flatness cancels out when combined in composite ratios, making an 11:14:15 chord (0-7-9 steps) and its utonal inversion particularly precise. Using 9/20 as the generator and treating these as the primary major and minor triads produces [[Balzano]] nonatonic and undecatonic scales, which is probably the clearest arrangement for the black/white keys on a 20 tone keyboard.
-Treating the generator as 11\20 creates the same scale, but the primary triads are now 13:16:19 (0-6-11 steps) and its inversion instead. The 11\20 generator is a near-optimal tuning for both [[No-threes subgroup temperaments#Mavericks|Mavericks]] temperament (which has a ~[[19/13]] generator)  and [[No-threes subgroup temperaments#Score|Score]] temperament (which has a ~[[16/11]] generator).
+Treating the generator as 11\20 creates the same scale, but the primary triads are now 13:16:19 (0-6-11 steps) and its inversion instead.
+Thanks to its sevenths, 20edo is an ideal tuning for its size for [[metallic harmony]].
+=== RTT ===
+''See [[regular temperament]] for more about what all this means and how to use it.''
+The 11\20 generator is a near-optimal tuning for both [[No-threes subgroup temperaments#Mavericks|Mavericks]] temperament (which has a ~[[19/13]] generator)  and [[No-threes subgroup temperaments#Score|Score]] temperament (which has a ~[[16/11]] generator).
 Alternately, 20edo can be used as a tuning of the [[blackwood]] temperament, combining minor and major thirds to generate a highly symmetrical decatonic scale where every note is root to a major or minor triad and 7-limit tetrad that are heavily tempered, but in a useful way, as you can easily modulate to anywhere in the small cycle of 5ths, and build extended chords that use every note in the scale without clashing. Either of these works better than trying to force 20 into a diatonic framework.
 edo also possesses a [[6L 1s]] scale generated using the narrow major second of 3\20 that is probably best interpreted as the sharp extreme of [[Tetracot_family#Tetracot|tetracot]] temperament and a [[3L 5s]] generated by 7/20 that functions as the flat end of [[squares]].
-Thanks to its sevenths, 20edo is an ideal tuning for its size for [[metallic harmony]].
 === Odd harmonics ===