4L 3s: Difference between revisions
→Tuning ranges: being careful about saying that a certain set of JI interpretations "is" a certain temperament (given that many different abstract temperaments use the same name) |
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=== Hypohard === | === Hypohard === | ||
[[Hypohard]] tunings have [[step ratio]]s between 2 and 3, implying a generator sharper than 4\15 = 320¢ and flatter than 3\11 = 327.27¢. The large step tends to approximate [[8/7]], and the major mosfourth (2 large steps + 1 small step) tends to approximate [[11/8]]; [[26edo]] is stellar in both of these approximations. This set of JI approximations is | [[Hypohard]] tunings have [[step ratio]]s between 2 and 3, implying a generator sharper than 4\15 = 320¢ and flatter than 3\11 = 327.27¢. The large step tends to approximate [[8/7]], and the major mosfourth (2 large steps + 1 small step) tends to approximate [[11/8]]; [[26edo]] is stellar in both of these approximations. This set of JI approximations is associated with [[orgone]] temperament. | ||
Hypohard smitonic edos include [[11edo]], [[15edo]], [[26edo]], and [[37edo]]. | Hypohard smitonic edos include [[11edo]], [[15edo]], [[26edo]], and [[37edo]]. | ||
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In [[parahard]] smitonic (step ratio between 3 and 4, thus with generator between 5\19, 315.79¢ and 4\15, 320¢), the generator is close to a pure [[6/5]] minor third, and 6 minor thirds are used to reach a perfect fifth. The parahard range contains very accurate edos such as [[53edo]] and [[72edo]], and has very accurate approximations to many [[low-overtone JI]] intervals, namely basic [[5-limit]] ratios and some ratios involving 13. However, the 7-note MOS only has one perfect fifth, so extending the chain to bigger MOSes, such as the [[4L 7s]] 11-note MOS, is suggested for getting 5-limit harmony. | In [[parahard]] smitonic (step ratio between 3 and 4, thus with generator between 5\19, 315.79¢ and 4\15, 320¢), the generator is close to a pure [[6/5]] minor third, and 6 minor thirds are used to reach a perfect fifth. The parahard range contains very accurate edos such as [[53edo]] and [[72edo]], and has very accurate approximations to many [[low-overtone JI]] intervals, namely basic [[5-limit]] ratios and some ratios involving 13. However, the 7-note MOS only has one perfect fifth, so extending the chain to bigger MOSes, such as the [[4L 7s]] 11-note MOS, is suggested for getting 5-limit harmony. | ||
This set of JI approximations is | This set of JI approximations is associated with [[kleismic]] temperament (we're specifically describing the 2.3.5.13 extension of it called [[Chromatic pairs#Cata|cata]]). | ||
EDOs that have parahard smitonic include [[15edo]], [[19edo]], [[34edo]], and [[53edo]]. | EDOs that have parahard smitonic include [[15edo]], [[19edo]], [[34edo]], and [[53edo]]. | ||