36edo: Difference between revisions

Line 9:

=== Harmonics ===

In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to 5:4 is the overly-familiar 400{{c}} major third. However, it excels at the 7th harmonic and intervals involving 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[subgroup]], 36edo's single degree of around 33{{c}} serves a double function as 49:48, the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36{{c}}, and as 64:63, the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27{{c}}. Meanwhile, its second degree functions as 28:27, the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (since {{nowrap|28/27 {{=}} 49/48 × 64/63}}). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].

In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to [[5/4]] is the overly-familiar 400{{c}} major third. However, it excels at the 7th harmonic and intervals involving 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[subgroup]], 36edo's single degree of around 33{{c}} serves a double function as [[49/48]], the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36{{c}}, and as [[64/63]], the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27{{c}}. Meanwhile, its second degree functions as [[28/27]], the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (since {{nowrap|28/27 {{=}} 49/48 × 64/63}}). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].

36edo is also notable for being the smallest multiple of 12edo to be [[distinctly consistent]] in the [[7-odd-limit]] (that is, all 7-odd-limit just intervals are represented by different steps).

Line 418:

The intervals in 36edo are all either the familiar 12edo intervals, or else "red" and "blue" versions of them. In [[24edo]], intervals such as 250 cents (halfway between a tone and a third) and 450 cents (halfway between a fourth and a third) tend to sound genuinely foreign, whereas the new intervals in 36edo are all variations on existing ones. Unlike 24edo, 36edo is also relatively free of what Easley Blackwood called "discordant" intervals. The 5th and 11th harmonics fall almost halfway in between scale degrees of 36edo, and thus intervals containing them can be approximated two different ways, one of which is significantly sharp and the other significantly flat. The 33{{frac|1|3}}{{c}} interval (the "red minor third" or "supraminor third") sharply approximates 6/5 and flatly approximates 11/9, for instance, whereas the sharp 11/9 is 366{{frac|2|3}}{{c}} and the flat 6/5 is 300{{c}}. However, 11/10, 20/11, 15/11, and 22/15 all have accurate and consistent approximations since the errors on the 5th and 11th harmonics cancel out with both tending sharp.

36edo is fairly cosmopolitan because many genres of world music can be played in it. Because of the presence of blue notes, and the closeness with which the 7th harmonic and its intervals are matched, 36edo is an ideal scale to use for African-American genres of music such as blues and jazz, in which septimal intervals are frequently encountered. Indonesian gamelan music using pelog easily adapts to it as well, since 9edo is a subset and can be notated as every fourth note, and Slendro can be approximated in several different ways as well, most notably as a [[1L 4s]] scale. 36edo can therefore function as a "bridge" between these genres and Western music. Arabic and Persian music do not adapt as well, however, since their microtonal intervals consist of mostly quarter tones.

36edo is fairly cosmopolitan because many genres of world music can be played in it. Because of the presence of blue notes, and the closeness with which the 7th harmonic and its intervals are matched, 36edo is an ideal scale to use for African-American genres of music such as blues and jazz, in which septimal intervals are frequently encountered. Indonesian gamelan music using pelog easily adapts to it as well, since 9edo is a subset and can be notated as every fourth note, and Slendro can be approximated in several different ways as well, most notably as a very soft [[1L 4s]] scale. 36edo can therefore function as a "bridge" between these genres and Western music. Arabic and Persian music do not adapt as well, however, since their microtonal intervals consist of mostly quarter tones.

The "red unison" and "blue unison" are in fact the same interval (33.333 cents), which is actually fairly consonant as a result of being so narrow (it is perceived as a unison, albeit noticeably "out of tune", but still not overly unpleasant). In contrast, most people consider 24edo's 50 cent step to sound much more discordant when used as a subminor second.

@@ Line 9: / Line 9: @@
 === Harmonics ===
-In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to 5:4 is the overly-familiar 400{{c}} major third. However, it excels at the 7th harmonic and intervals involving 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[subgroup]], 36edo's single degree of around 33{{c}} serves a double function as 49:48, the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36{{c}}, and as 64:63, the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27{{c}}. Meanwhile, its second degree functions as 28:27, the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (since {{nowrap|28/27 {{=}} 49/48 × 64/63}}). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].
+In the 5-limit, 36edo offers no improvement over 12edo, since its nearest approximation to [[5/4]] is the overly-familiar 400{{c}} major third. However, it excels at the 7th harmonic and intervals involving 7. As a 3 and 7 tuning, or in other words as a tuning for the 2.3.7 [[subgroup]], 36edo's single degree of around 33{{c}} serves a double function as [[49/48]], the so-called [https://en.wikipedia.org/wiki/Septimal_diesis Slendro diesis] of around 36{{c}}, and as [[64/63]], the so-called [https://en.wikipedia.org/wiki/Septimal_comma septimal comma] of around 27{{c}}. Meanwhile, its second degree functions as [[28/27]], the so-called [https://en.wikipedia.org/wiki/Septimal_third-tone Septimal third-tone] (since {{nowrap|28/27 {{=}} 49/48 × 64/63}}). The 2.3.7 subgroup can be extended to the [[k*N_subgroups|2*36 subgroup]] 2.3.25.7.55.13.17, and on this subgroup it tempers out the same commas as [[72edo]] does in the full [[17-limit]].
 edo is also notable for being the smallest multiple of 12edo to be [[distinctly consistent]] in the [[7-odd-limit]] (that is, all 7-odd-limit just intervals are represented by different steps).
@@ Line 418: / Line 418: @@
 The intervals in 36edo are all either the familiar 12edo intervals, or else "red" and "blue" versions of them. In [[24edo]], intervals such as 250 cents (halfway between a tone and a third) and 450 cents (halfway between a fourth and a third) tend to sound genuinely foreign, whereas the new intervals in 36edo are all variations on existing ones. Unlike 24edo, 36edo is also relatively free of what Easley Blackwood called "discordant" intervals. The 5th and 11th harmonics fall almost halfway in between scale degrees of 36edo, and thus intervals containing them can be approximated two different ways, one of which is significantly sharp and the other significantly flat. The 33{{frac|1|3}}{{c}} interval (the "red minor third" or "supraminor third") sharply approximates 6/5 and flatly approximates 11/9, for instance, whereas the sharp 11/9 is 366{{frac|2|3}}{{c}} and the flat 6/5 is 300{{c}}. However, 11/10, 20/11, 15/11, and 22/15 all have accurate and consistent approximations since the errors on the 5th and 11th harmonics cancel out with both tending sharp.
-edo is fairly cosmopolitan because many genres of world music can be played in it. Because of the presence of blue notes, and the closeness with which the 7th harmonic and its intervals are matched, 36edo is an ideal scale to use for African-American genres of music such as blues and jazz, in which septimal intervals are frequently encountered. Indonesian gamelan music using pelog easily adapts to it as well, since 9edo is a subset and can be notated as every fourth note, and Slendro can be approximated in several different ways as well, most notably as a [[1L&nbsp;4s]] scale. 36edo can therefore function as a "bridge" between these genres and Western music. Arabic and Persian music do not adapt as well, however, since their microtonal intervals consist of mostly quarter tones.
+edo is fairly cosmopolitan because many genres of world music can be played in it. Because of the presence of blue notes, and the closeness with which the 7th harmonic and its intervals are matched, 36edo is an ideal scale to use for African-American genres of music such as blues and jazz, in which septimal intervals are frequently encountered. Indonesian gamelan music using pelog easily adapts to it as well, since 9edo is a subset and can be notated as every fourth note, and Slendro can be approximated in several different ways as well, most notably as a very soft [[1L&nbsp;4s]] scale. 36edo can therefore function as a "bridge" between these genres and Western music. Arabic and Persian music do not adapt as well, however, since their microtonal intervals consist of mostly quarter tones.
 The "red unison" and "blue unison" are in fact the same interval (33.333 cents), which is actually fairly consonant as a result of being so narrow (it is perceived as a unison, albeit noticeably "out of tune", but still not overly unpleasant). In contrast, most people consider 24edo's 50 cent step to sound much more discordant when used as a subminor second.