35edo: Difference between revisions

Line 3:

== Theory ==

As 35 is 5 times 7, 35edo allows for mixing the two smallest xenharmonic [[macrotonal edos]]: [[5edo]] and [[7edo]]. A single degree of 35edo represents the difference between 7edo's narrow fifth of 685.71{{c}} and 5edo's wide fifth of 720{{c}}. Because it includes 7edo, 35edo tunes the 29th harmonic with only 1{{c}} of error.

As 35 is 5 times 7, 35edo allows for mixing the two smallest xenharmonic [[macrotonal edos]]: [[5edo]] and [[7edo]]. A single degree of 35edo represents the difference between 7edo's narrow fifth of 685.71{{Cent}} and 5edo's wide fifth of 720{{C}}. Because it includes 7edo, 35edo tunes the 29th harmonic with only 1{{C}} of error.

35edo can also represent the 2.3.5.7.11.17 [[subgroup]] and 2.9.5.7.11.17 subgroup, because of the accuracy of 9 and the flatness of all other subgroup generators (7/5 and 17/11 stand out, having less than 1 cent error). Therefore among [[whitewood]] tunings it is very versatile; you can switch between these different subgroups if you don't mind having to use two different 3/2s to reach the inconsistent 9 (a characteristic of whitewood tunings), and if you ignore [[22edo]]'s more in-tune versions of 35edo MOS's and consistent representation of both subgroups.

Line 13:

=== Dual-fifth harmony ===

35edo has two viable mappings of the perfect fifth, one at 20\35 (4\7), and one at 21\35 (3\5). If one wishes to build a chord with the perfect fifth, one must decide which mapping to use. For example, if one wishes to use the classical major triad [[4:5:6]], then we find that 35edo's best approximation of [[5/4]] is just over 1/4 of a step flat, meaning that the flat mapping of 3/2 should be used in order for [[6/5]] to be tuned accurately. Thus the best approximation of 4:5:6 is 0–11–20 steps (0–377–686{{C}}), and the best approximation of its inverse [[10:12:15|1/(6:5:4)]], the classical minor triad, is 0–9–20 steps (0–309–686{{C}}). Here, the [[5/4]] and [[6/5]] intervals are tuned fairly accurately, being about 7–9{{C}} flat each, while [[3/2]] is more damaged at about 16{{C}} flat of just. However, since 3/2 is a very simple interval, it is recognizable even if heavily detuned. Interestingly, the approximations of 4:5:6 and 10:12:15 are fairly close to [[DR]] tunings as well.

35edo has two viable mappings of the [[3/2|perfect fifth]], one at 20\35 (4\7), and one at 21\35 (3\5). If one wishes to build a chord with the perfect fifth, one must decide which mapping to use. For example, if one wishes to use the classical major triad [[4:5:6]], then we find that 35edo's best approximation of [[5/4]] is just over 1/4 of a step flat, meaning that the flat mapping of 3/2 should be used in order for [[6/5]] to be tuned accurately. Thus the best approximation of 4:5:6 is 0–11–20 steps (0–377–686{{C}}), and the best approximation of its inverse [[10:12:15|1/(6:5:4)]], the classical minor triad, is 0–9–20 steps (0–309–686{{C}}). Here, the [[5/4]] and [[6/5]] intervals are tuned fairly accurately, being about 7–9{{C}} flat each, while [[3/2]] is more damaged at about 16{{C}} flat of just. However, since 3/2 is a very simple interval, it is recognizable even if heavily detuned. Interestingly, the approximations of 4:5:6 and 10:12:15 are fairly close to [[DR]] tunings as well.

Amazingly, almost the exact same situation occurs with [[7/4]], for which 35edo's best approximation is also just over 1/4 of a step flat (resulting in a very accurate [[7/5]]). If we wish to use the [[4:6:7]] chord, then just like with 4:5:6, it is best to use the flat mapping of 3/2, resulting in a triad of 0–20–28 steps (0–686–960{{C}}). Its inverse, the [[14:21:24|1/(12:8:7)]] chord, is best mapped to 0–20–27 steps (0–686–926{{C}}). Here the damage is split between [[7/4]] and [[12/7]], with both being around 7–9{{C}} flat of just, which is almost the exact same situation as with 5/4 and 6/5. The 6:7:8 and 4:7:12 voicings of 4:6:7, which split the fourth and the twelfth respectively, are tuned fairly close to DR. The 1/(8:7:6) and 1/(12:7:4) inversions of 1/(12:8:7) are also tuned close to DR, though their delta signatures are significantly more complex. From here, we see that the best approximation of the harmonic seventh chord [[4:5:6:7]] is 0–11–20–28 steps (0–377–686–960{{C}}), while the best approximation of the subharmonic sixth chord [[70:84:105:120|1/(12:10:8:7)]] is 0–9–20–27 steps (0–309–686–926{{C}}).

Line 22:

The best approximation of the harmonic ninth chord [[4:5:6:7:9]] is 0–11–20–28–41 steps (0–377–686–960–1406{{C}}). Here, both mappings of 3/2 are used simultaneously, with the flat mapping occuring at 4:6, and the sharp mapping occuring at 6:9. The mapping of any chord in 35edo can be taken as a subset of the mapping of 4:5:6:7:9, or the mapping of its inverse [[140:180:210:252:315|1/(9:7:6:5:4)]] as 0–13–21–30–41 steps (0–446–720–1029–1406{{C}}), where any interval more complex than the perfect fifth is no more than 11{{C}} out of tune. Additionally, many triads are tuned very close to DR, which may make them sound less out of tune as well.

=== Subsets and supersets ===

Since 35 factors as 5 × 7, its nontrivial subsets are [[5edo]] and [[7edo]]. Its double [[70edo]] corrects the perfect fifth, as well as the [[13/1|13th harmonic]], though the [[5/1|5th]] and [[7/1|7th]] harmonics become relatively inaccurate. The quadruple of 35edo, which is [[140edo]], additionally corrects the mappings of primes 5 and 7, and makes for an excellent [[17-limit]] system and beyond.

== Intervals ==

@@ Line 3: / Line 3: @@
 == Theory ==
-As 35 is 5 times 7, 35edo allows for mixing the two smallest xenharmonic [[macrotonal edos]]: [[5edo]] and [[7edo]]. A single degree of 35edo represents the difference between 7edo's narrow fifth of 685.71{{c}} and 5edo's wide fifth of 720{{c}}. Because it includes 7edo, 35edo tunes the 29th harmonic with only 1{{c}} of error.
+As 35 is 5 times 7, 35edo allows for mixing the two smallest xenharmonic [[macrotonal edos]]: [[5edo]] and [[7edo]]. A single degree of 35edo represents the difference between 7edo's narrow fifth of 685.71{{Cent}} and 5edo's wide fifth of 720{{C}}. Because it includes 7edo, 35edo tunes the 29th harmonic with only 1{{C}} of error.
 edo can also represent the 2.3.5.7.11.17 [[subgroup]] and 2.9.5.7.11.17 subgroup, because of the accuracy of 9 and the flatness of all other subgroup generators (7/5 and 17/11 stand out, having less than 1 cent error). Therefore among [[whitewood]] tunings it is very versatile; you can switch between these different subgroups if you don't mind having to use two different 3/2s to reach the inconsistent 9 (a characteristic of whitewood tunings), and if you ignore [[22edo]]'s more in-tune versions of 35edo MOS's and consistent representation of both subgroups.
@@ Line 13: / Line 13: @@
 === Dual-fifth harmony ===
-edo has two viable mappings of the perfect fifth, one at 20\35 (4\7), and one at 21\35 (3\5). If one wishes to build a chord with the perfect fifth, one must decide which mapping to use. For example, if one wishes to use the classical major triad [[4:5:6]], then we find that 35edo's best approximation of [[5/4]] is just over 1/4 of a step flat, meaning that the flat mapping of 3/2 should be used in order for [[6/5]] to be tuned accurately. Thus the best approximation of 4:5:6 is 0–11–20 steps (0–377–686{{C}}), and the best approximation of its inverse [[10:12:15|1/(6:5:4)]], the classical minor triad, is 0–9–20 steps (0–309–686{{C}}). Here, the [[5/4]] and [[6/5]] intervals are tuned fairly accurately, being about 7–9{{C}} flat each, while [[3/2]] is more damaged at about 16{{C}} flat of just. However, since 3/2 is a very simple interval, it is recognizable even if heavily detuned. Interestingly, the approximations of 4:5:6 and 10:12:15 are fairly close to [[DR]] tunings as well.
+edo has two viable mappings of the [[3/2|perfect fifth]], one at 20\35 (4\7), and one at 21\35 (3\5). If one wishes to build a chord with the perfect fifth, one must decide which mapping to use. For example, if one wishes to use the classical major triad [[4:5:6]], then we find that 35edo's best approximation of [[5/4]] is just over 1/4 of a step flat, meaning that the flat mapping of 3/2 should be used in order for [[6/5]] to be tuned accurately. Thus the best approximation of 4:5:6 is 0–11–20 steps (0–377–686{{C}}), and the best approximation of its inverse [[10:12:15|1/(6:5:4)]], the classical minor triad, is 0–9–20 steps (0–309–686{{C}}). Here, the [[5/4]] and [[6/5]] intervals are tuned fairly accurately, being about 7–9{{C}} flat each, while [[3/2]] is more damaged at about 16{{C}} flat of just. However, since 3/2 is a very simple interval, it is recognizable even if heavily detuned. Interestingly, the approximations of 4:5:6 and 10:12:15 are fairly close to [[DR]] tunings as well.
 Amazingly, almost the exact same situation occurs with [[7/4]], for which 35edo's best approximation is also just over 1/4 of a step flat (resulting in a very accurate [[7/5]]). If we wish to use the [[4:6:7]] chord, then just like with 4:5:6, it is best to use the flat mapping of 3/2, resulting in a triad of 0–20–28 steps (0–686–960{{C}}). Its inverse, the [[14:21:24|1/(12:8:7)]] chord, is best mapped to 0–20–27 steps (0–686–926{{C}}). Here the damage is split between [[7/4]] and [[12/7]], with both being around 7–9{{C}} flat of just, which is almost the exact same situation as with 5/4 and 6/5. The 6:7:8 and 4:7:12 voicings of 4:6:7, which split the fourth and the twelfth respectively, are tuned fairly close to DR. The 1/(8:7:6) and 1/(12:7:4) inversions of 1/(12:8:7) are also tuned close to DR, though their delta signatures are significantly more complex. From here, we see that the best approximation of the harmonic seventh chord [[4:5:6:7]] is 0–11–20–28 steps (0–377–686–960{{C}}), while the best approximation of the subharmonic sixth chord [[70:84:105:120|1/(12:10:8:7)]] is 0–9–20–27 steps (0–309–686–926{{C}}).
@@ Line 22: / Line 22: @@
 The best approximation of the harmonic ninth chord [[4:5:6:7:9]] is 0–11–20–28–41 steps (0–377–686–960–1406{{C}}). Here, both mappings of 3/2 are used simultaneously, with the flat mapping occuring at 4:6, and the sharp mapping occuring at 6:9. The mapping of any chord in 35edo can be taken as a subset of the mapping of 4:5:6:7:9, or the mapping of its inverse [[140:180:210:252:315|1/(9:7:6:5:4)]] as 0–13–21–30–41 steps (0–446–720–1029–1406{{C}}), where any interval more complex than the perfect fifth is no more than 11{{C}} out of tune. Additionally, many triads are tuned very close to DR, which may make them sound less out of tune as well.
+=== Subsets and supersets ===
+Since 35 factors as 5 × 7, its nontrivial subsets are [[5edo]] and [[7edo]]. Its double [[70edo]] corrects the perfect fifth, as well as the [[13/1|13th harmonic]], though the [[5/1|5th]] and [[7/1|7th]] harmonics become relatively inaccurate. The quadruple of 35edo, which is [[140edo]], additionally corrects the mappings of primes 5 and 7, and makes for an excellent [[17-limit]] system and beyond.
 == Intervals ==