Magic: Difference between revisions
From Xenharmonic Reference
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{{adv|Magic divides ~16/15 in half (into two 25/24's), so it easily extends to prime 31 (at +3 generators) by tempering out S31 {{=}} 961/960. Since in 7-limit Magic, the 16/15 is also a 15/14, we can also extend 2.3.5.7.31 Magic to add prime 29 (at +9 generators) by equating 16/15 to 31/29.}} | {{adv|Magic divides ~16/15 in half (into two 25/24's), so it easily extends to prime 31 (at +3 generators) by tempering out S31 {{=}} 961/960. Since in 7-limit Magic, the 16/15 is also a 15/14, we can also extend 2.3.5.7.31 Magic to add prime 29 (at +9 generators) by equating 16/15 to 31/29.}} | ||
This can be seen from: | {{Adv|This can be seen from:}} | ||
3125/3072 | 3125/3072 | ||
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= (S25*S26*S27*S28*S29*S30)^2*S31 | = (S25*S26*S27*S28*S29*S30)^2*S31 | ||
= (S15*S28*S29*S30)^2*S31 | = (S15*S28*S29*S30)^2*S31 | ||
making the above 2.3.5.7.29.31 a [[canonical extension|structurally induced extension]]. | |||
{{cat|Temperaments}}{{Navbox regtemp}} | {{cat|Temperaments}}{{Navbox regtemp}} | ||
Revision as of 21:30, 5 March 2026
| Magic |
225/224, 245/243 (7-limit)
9-odd-limit: 5.9¢
9-odd-limit: 13 notes
Magic is a 2.3.5 temperament that equates a stack of five 5/4 major thirds to one 3/1. It also equates 25/24 to 128/125, shrinking the difference between 5/4 and 6/5.
Extensions
Magic divides ~16/15 in half (into two 25/24's), so it easily extends to prime 31 (at +3 generators) by tempering out S31 = 961/960. Since in 7-limit Magic, the 16/15 is also a 15/14, we can also extend 2.3.5.7.31 Magic to add prime 29 (at +9 generators) by equating 16/15 to 31/29.
This can be seen from:
3125/3072 = (25/24)^2/(16/15) = (25/24)*(25/24)/(32/31*31/30) = (25/24)S25*S26*S27*S28*S29*S30/(32/31) = (S25*S26*S27*S28*S29*S30)^2*S31 = (S15*S28*S29*S30)^2*S31
making the above 2.3.5.7.29.31 a structurally induced extension.
| View • Talk • EditRegular temperaments | |
|---|---|
| Rank-2 | |
| Acot | Blackwood (1/5-octave) • Whitewood (1/7-octave) • Compton (1/12-octave) |
| Monocot | Meantone • Schismic • Gentle-fifth temperaments • Archy |
| Complexity 2 | Diaschismic (diploid monocot) • Pajara (diploid monocot) • Injera (diploid monocot) • Rastmatic (dicot) • Mohajira (dicot) • Intertridecimal (dicot) • Interseptimal (alpha-dicot) |
| Complexity 3 | Augmented (triploid) • Misty (triploid) • Slendric (tricot) • Porcupine (omega-tricot) |
| Complexity 4 | Diminished (tetraploid) • Tetracot (tetracot) • Buzzard (alpha-tetracot) • Squares (beta-tetracot) • Negri (omega-tetracot) |
| Complexity 5-6 | Magic (alpha-pentacot) • Amity (gamma-pentacot) • Kleismic (alpha-hexacot) • Miracle (hexacot) |
| Higher complexity | Orwell (alpha-heptacot) • Sensi (beta-heptacot) • Octacot (octacot) • Wurschmidt (beta-octacot) • Valentine (enneacot) • Ammonite (epsilon-enneacot) • Myna (beta-decacot) • Ennealimmal (enneaploid dicot) |
| Straddle-3 | A-Team (alter-tricot) • Machine (alter-monocot) |
| No-3 | Trismegistus (alpha-triseph) • Orgone (trimech) • Didacus (diseph) |
| No-octaves | Sensamagic (monogem) |
| Exotemperament | Dicot • Mavila • Father |
| Higher-rank | |
| Rank-3 | Hemifamity • Marvel • Parapyth |
