35edo: Difference between revisions

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=== Dual-fifth harmony ===

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~~.

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.

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}}).

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. 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}}).

Overall, we find that 35edo's [[patent val]] is surprisingly accurate overall for the [[7-odd-limit]], with 3/2 being the only interval with high damage. However, this mapping does not work well in the [[9-odd-limit]], as [[9/8]] is tuned over 32{{C}} flat of just at 171{{C}}, and thus other intervals of 9 also become very inaccurate. Instead, 35edo has an accurate approximation of 9/8 at 6\35 (206{{C}}), but to reach it, we must stack one 20\35 fifth and one 21\35 fifth. The 21\35 fifth is the [[5edo]] fifth of 720{{C}}, being around 18{{C}} sharp of just. Even though there are two mappings of the perfect fifth, it all fits well in the end, as we will soon see.

For example, suppose we want to use the [[6:7:9]] subminor triad. Then the closest approximation of [[7/6]] is 8 steps, and the closest approximation of [[9/7]] is 13 steps. Stacking these approximations gives the triad 0–8–21 steps (0–274–720{{C}}). Here, we use the sharp fifth instead of the flat one, so that [[7/6]] and [[9/7]] are tuned more accurately, being around 7{{C}} and 11{{C}} sharp of just respectively. The best approximation of the supermajor triad [[14:18:21|1/(9:7:6)]] is 0–13–21 steps (0–446–720{{C}}), which also uses the sharp fifth~~. Similarly to 4:5:6 and 10:12:15, these are also fairly close to DR tunings~~. A similar situation occurs with [[6:9:10]] and its inverse [[10:15:18|1/(9:6:5)]], where the best approximations of [[5/3]] and [[9/5]] are 26\35 and 30\35 respectively, so that the best approximations of 6:9:10 and 1/(9:6:5) are 0–21–26 steps (0–720–891{{C}}) and 0–21–30 steps (0–720–1029{{C}}) respectively, with 5/3 and 9/5 being around 7{{C}} and 11{{C}} sharp respectively. ~~The~~ 9:10~~:12 voicing~~ and ~~especially~~ the 3:5:9 ~~voicing, as well as their inverses, are tuned fairly close to DR~~.

For example, suppose we want to use the [[6:7:9]] subminor triad. Then the closest approximation of [[7/6]] is 8 steps, and the closest approximation of [[9/7]] is 13 steps. Stacking these approximations gives the triad 0–8–21 steps (0–274–720{{C}}). Here, we use the sharp fifth instead of the flat one, so that [[7/6]] and [[9/7]] are tuned more accurately, being around 7{{C}} and 11{{C}} sharp of just respectively. The best approximation of the supermajor triad [[14:18:21|1/(9:7:6)]] is 0–13–21 steps (0–446–720{{C}}), which also uses the sharp fifth. A similar situation occurs with [[6:9:10]] and its inverse [[10:15:18|1/(9:6:5)]], where the best approximations of [[5/3]] and [[9/5]] are 26\35 and 30\35 respectively, so that the best approximations of 6:9:10 and 1/(9:6:5) are 0–21–26 steps (0–720–891{{C}}) and 0–21–30 steps (0–720–1029{{C}}) respectively, with 5/3 and 9/5 being around 7{{C}} and 11{{C}} sharp respectively. This leads to an approximation of the [[6:7:9:10]] harmonic sixth chord (sometimes known as the ''subminor tetrad'') at 0–8–21–26 steps (0–274–720–891{{C}}), and an approximation of the [[70:90:105:126|1/(9:7:6:5)]] subharmonic seventh chord (sometimes called the ''supermajor tetrad'') at 0–13–21–30 steps (0–446–720–1029{{C}}).

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.

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 [[delta-rational]] tunings, which may make them sound less out of tune as well. For examples, the approximations of the triads [[4:5:6]], [[10:12:15|1/(6:5:4)]], [[6:7:9]], and [[14:18:21|1/(9:7:6)]] are very close to DR tunings. Voicings of chords that divide the fourth, those being [[6:7:8]], [[21:24:28|1/(8:7:6)]], [[9:10:12]], and [[15:18:20|1/(12:10:9)]], are also tuned fairly close to DR. The wide voicings that span the twelfth, those being 4:7:12, 1/(12:7:4), 3:5:9, and 1/(9:5:3) are even closer to DR than the voicings that span the fourth, though not as close as the triads spanning the fifth.

=== Subsets and supersets ===

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 === Dual-fifth harmony ===
-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.
+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.
-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}}).
+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. 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}}).
 Overall, we find that 35edo's [[patent val]] is surprisingly accurate overall for the [[7-odd-limit]], with 3/2 being the only interval with high damage. However, this mapping does not work well in the [[9-odd-limit]], as [[9/8]] is tuned over 32{{C}} flat of just at 171{{C}}, and thus other intervals of 9 also become very inaccurate. Instead, 35edo has an accurate approximation of 9/8 at 6\35 (206{{C}}), but to reach it, we must stack one 20\35 fifth and one 21\35 fifth. The 21\35 fifth is the [[5edo]] fifth of 720{{C}}, being around 18{{C}} sharp of just. Even though there are two mappings of the perfect fifth, it all fits well in the end, as we will soon see.
-For example, suppose we want to use the [[6:7:9]] subminor triad. Then the closest approximation of [[7/6]] is 8 steps, and the closest approximation of [[9/7]] is 13 steps. Stacking these approximations gives the triad 0–8–21 steps (0–274–720{{C}}). Here, we use the sharp fifth instead of the flat one, so that [[7/6]] and [[9/7]] are tuned more accurately, being around 7{{C}} and 11{{C}} sharp of just respectively. The best approximation of the supermajor triad [[14:18:21|1/(9:7:6)]] is 0–13–21 steps (0–446–720{{C}}), which also uses the sharp fifth. Similarly to 4:5:6 and 10:12:15, these are also fairly close to DR tunings. A similar situation occurs with [[6:9:10]] and its inverse [[10:15:18|1/(9:6:5)]], where the best approximations of [[5/3]] and [[9/5]] are 26\35 and 30\35 respectively, so that the best approximations of 6:9:10 and 1/(9:6:5) are 0–21–26 steps (0–720–891{{C}}) and 0–21–30 steps (0–720–1029{{C}}) respectively, with 5/3 and 9/5 being around 7{{C}} and 11{{C}} sharp respectively. The 9:10:12 voicing and especially the 3:5:9 voicing, as well as their inverses, are tuned fairly close to DR.
+For example, suppose we want to use the [[6:7:9]] subminor triad. Then the closest approximation of [[7/6]] is 8 steps, and the closest approximation of [[9/7]] is 13 steps. Stacking these approximations gives the triad 0–8–21 steps (0–274–720{{C}}). Here, we use the sharp fifth instead of the flat one, so that [[7/6]] and [[9/7]] are tuned more accurately, being around 7{{C}} and 11{{C}} sharp of just respectively. The best approximation of the supermajor triad [[14:18:21|1/(9:7:6)]] is 0–13–21 steps (0–446–720{{C}}), which also uses the sharp fifth. A similar situation occurs with [[6:9:10]] and its inverse [[10:15:18|1/(9:6:5)]], where the best approximations of [[5/3]] and [[9/5]] are 26\35 and 30\35 respectively, so that the best approximations of 6:9:10 and 1/(9:6:5) are 0–21–26 steps (0–720–891{{C}}) and 0–21–30 steps (0–720–1029{{C}}) respectively, with 5/3 and 9/5 being around 7{{C}} and 11{{C}} sharp respectively. This leads to an approximation of the [[6:7:9:10]] harmonic sixth chord (sometimes known as the ''subminor tetrad'') at 0–8–21–26 steps (0–274–720–891{{C}}), and an approximation of the [[70:90:105:126|1/(9:7:6:5)]] subharmonic seventh chord (sometimes called the ''supermajor tetrad'') at 0–13–21–30 steps (0–446–720–1029{{C}}).
-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.
+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 [[delta-rational]] tunings, which may make them sound less out of tune as well. For examples, the approximations of the triads [[4:5:6]], [[10:12:15|1/(6:5:4)]], [[6:7:9]], and [[14:18:21|1/(9:7:6)]] are very close to DR tunings. Voicings of chords that divide the fourth, those being [[6:7:8]], [[21:24:28|1/(8:7:6)]], [[9:10:12]], and [[15:18:20|1/(12:10:9)]], are also tuned fairly close to DR. The wide voicings that span the twelfth, those being 4:7:12, 1/(12:7:4), 3:5:9, and 1/(9:5:3) are even closer to DR than the voicings that span the fourth, though not as close as the triads spanning the fifth.
 === Subsets and supersets ===