35edo: Difference between revisions

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==== Caveats of dual-fifth ====

However, using two mappings of the perfect fifth presents several problems. For example, in JI, there are the [[10:12:15:18]] and [[12:14:18:21]] chords and their inversions, known as [[anomalous saturated suspension]]s, which are dyadically consonant in the 9-odd-limit, even though they are not a subset of the 9-odd-limit otonal or utonal pentad. If we try to map~~, say,~~ the 10:12:15:18 chord with steps 6/5–5/4–6/5–10/9 (closing at the octave) in 35edo, then the 10:12:15 part suggests mapping the fifth above the root at 20\35, while the 10:15:18 part suggests mapping it to 21\35. As such, one of the 6/5–5/4–6/5–10/9 steps must be mapped to its second-best approximation, close to 3/4 of a 35edo step (about 25 cents) off of just. A similar issue occurs with 12:14:18:21, where one of the 7/6–9/7–7/6–8/7 steps must be mapped to its second-best approximation. Many other chords, such as [[8:10:12:15]], also cannot be mapped without a step being close to 3/4 of a 35edo step off.

However, using two mappings of the perfect fifth presents several problems. For example, in JI, there are the [[10:12:15:18]] and [[12:14:18:21]] chords and their inversions, known as [[anomalous saturated suspension]]s, which are dyadically consonant in the 9-odd-limit, even though they are not a subset of the 9-odd-limit otonal or utonal pentad. Their dyadic consonance relies on the compositeness of the number 9 as 3 × 3, and here the mapping breaks down when we try to use two different mappings of harmonic 3. For example, if we try to map the 10:12:15:18 chord with steps 6/5–5/4–6/5–10/9 (closing at the octave) in 35edo, then the 10:12:15 part suggests mapping the fifth above the root at 20\35, while the 10:15:18 part suggests mapping it to 21\35. As such, one of the 6/5–5/4–6/5–10/9 steps must be mapped to its second-best approximation, close to 3/4 of a 35edo step (about 25 cents) off of just. A similar issue occurs with 12:14:18:21, where one of the 7/6–9/7–7/6–8/7 steps must be mapped to its second-best approximation. Many other chords, such as [[8:10:12:15]], also cannot be mapped without a step being close to 3/4 of a 35edo step off.

Additionally, many structures present in systems with a single fifth do not work well in 35edo. For example, the perfect fifth generates several [[mos scale]], such as the traditional [[diatonic]] scale. The diatonic mos scale does not exist in 35edo, with the 20\35 whitewood fifth generating an [[equalized]] version of the scale, while the 21\35 fifth generates a [[collapsed]] version of the scale. Since 35edo does not have a diatonic scale, [[chain-of-fifths notation]] also does not work in 35edo. However, there are scales such as 6 6 2 6 6 6 3 which sound similar to diatonic, and this particular scale can be obtained by alternately stacking 21\35 and 20\35 fifths, or [[Hobbled scale|hobbling]] a [[34edo]] or [[36edo]] diatonic scale.

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 ==== Caveats of dual-fifth ====
-However, using two mappings of the perfect fifth presents several problems. For example, in JI, there are the [[10:12:15:18]] and [[12:14:18:21]] chords and their inversions, known as [[anomalous saturated suspension]]s, which are dyadically consonant in the 9-odd-limit, even though they are not a subset of the 9-odd-limit otonal or utonal pentad. If we try to map, say, the 10:12:15:18 chord with steps 6/5–5/4–6/5–10/9 (closing at the octave) in 35edo, then the 10:12:15 part suggests mapping the fifth above the root at 20\35, while the 10:15:18 part suggests mapping it to 21\35. As such, one of the 6/5–5/4–6/5–10/9 steps must be mapped to its second-best approximation, close to 3/4 of a 35edo step (about 25 cents) off of just. A similar issue occurs with 12:14:18:21, where one of the 7/6–9/7–7/6–8/7 steps must be mapped to its second-best approximation. Many other chords, such as [[8:10:12:15]], also cannot be mapped without a step being close to 3/4 of a 35edo step off.
+However, using two mappings of the perfect fifth presents several problems. For example, in JI, there are the [[10:12:15:18]] and [[12:14:18:21]] chords and their inversions, known as [[anomalous saturated suspension]]s, which are dyadically consonant in the 9-odd-limit, even though they are not a subset of the 9-odd-limit otonal or utonal pentad. Their dyadic consonance relies on the compositeness of the number 9 as 3 × 3, and here the mapping breaks down when we try to use two different mappings of harmonic 3. For example, if we try to map the 10:12:15:18 chord with steps 6/5–5/4–6/5–10/9 (closing at the octave) in 35edo, then the 10:12:15 part suggests mapping the fifth above the root at 20\35, while the 10:15:18 part suggests mapping it to 21\35. As such, one of the 6/5–5/4–6/5–10/9 steps must be mapped to its second-best approximation, close to 3/4 of a 35edo step (about 25 cents) off of just. A similar issue occurs with 12:14:18:21, where one of the 7/6–9/7–7/6–8/7 steps must be mapped to its second-best approximation. Many other chords, such as [[8:10:12:15]], also cannot be mapped without a step being close to 3/4 of a 35edo step off.
 Additionally, many structures present in systems with a single fifth do not work well in 35edo. For example, the perfect fifth generates several [[mos scale]], such as the traditional [[diatonic]] scale. The diatonic mos scale does not exist in 35edo, with the 20\35 whitewood fifth generating an [[equalized]] version of the scale, while the 21\35 fifth generates a [[collapsed]] version of the scale. Since 35edo does not have a diatonic scale, [[chain-of-fifths notation]] also does not work in 35edo. However, there are scales such as 6 6 2 6 6 6 3 which sound similar to diatonic, and this particular scale can be obtained by alternately stacking 21\35 and 20\35 fifths, or [[Hobbled scale|hobbling]] a [[34edo]] or [[36edo]] diatonic scale.