Bird's eye view of temperaments by accuracy: Difference between revisions
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Srutal archagall is the natural extension of 5-limit [[diaschismic]] to prime 17 by interpreting the generator as a near-just [[17/16]] and the period as [[~]][[24/17]][[~]][[17/12]]. | Srutal archagall is the natural extension of 5-limit [[diaschismic]] to prime 17 by interpreting the generator as a near-just [[17/16]] and the period as [[~]][[24/17]][[~]][[17/12]]. | ||
It is notable as preserving a lot of intuitions of 12edo like [[9/8]] as two semitones, [[6/5]] as three semitones, [[5/4]] as a semitone less than [[4/3]] which is itself a semitone less than half an octave. It essentially "doubles up" on familiar categories by: two major seconds, [[~]][[10/9]] and [[~]][[9/8]][[~]][[17/15]], two minor thirds, [[~]][[20/17]] and [[~]][[6/5]], and two major thirds, [[~]][[5/4]] and [[~]][[51/40]][[~]][[32/25]]. The distance between these pairs is an exaggerated syntonic comma ([[81/80]]) making it useful as a | It is notable as preserving a lot of intuitions of 12edo like [[9/8]] as two semitones, [[6/5]] as three semitones, [[5/4]] as a semitone less than [[4/3]] which is itself a semitone less than half an octave. It essentially "doubles up" on familiar categories by: two minor seconds, [[~]][[25/24]] and [[~]][[18/17]][[~]][[17/16]][[~]][[16/15]], two major seconds, [[~]][[10/9]] and [[~]][[9/8]][[~]][[17/15]], two minor thirds, [[~]][[20/17]] and [[~]][[6/5]], and two major thirds, [[~]][[5/4]] and [[~]][[51/40]][[~]][[32/25]]. Furthermore, major and minor intervals are separated by a semitone of [[17/16]], though interestingly this results in alternating the tuning of the otherwise-familiar category, e.g. [[20/17]] = ([[5/4]])/([[17/16]])). The distance between these pairs of familiar intervals is an exaggerated syntonic comma ([[81/80]]) which is also equal to a flattened diesis ([[128/125]]), making it useful as a more accurate alternative to meantone. Notably the intervals of 5 require using the period offset to reach, so that the minor third reached by the circle of fifths is actually [[20/17]]. | ||
For abundant options, one might prefer a 22-note MOS over a 12-note one, so that srutal archagall can be seen as a detempering of [[22edo]], but the 12-note MOS is likely the easiest and most intuitive to approach for a beginner. [[34edo]] is a good tuning for optimizing the 2.3.5.17 subgroup. | For abundant options, one might prefer a 22-note MOS over a 12-note one, so that srutal archagall can be seen as a detempering of [[22edo]], but the 12-note MOS is likely the easiest and most intuitive to approach for a beginner. [[34edo]] is a good tuning for optimizing the 2.3.5.17 subgroup. | ||