Diamond monotone: Difference between revisions

In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] — and to some extent on the wiki and in the regular temperament community — this tuning range was referred to simply as the "valid" tuning range.  

== Temperaments without diamond monotone tunings ==

While diamond pure tunings are always guaranteed to occur, diamond monotone tunings are not. For instance, from the tuning map [{{val| 1 0 5 }}, {{val| 0 1 -2 }}] for the temperament tempering out 45/32 we find that all tunings are of the form {{val| 1 0 5 }} + ''a''{{val| 0 1 -2 }} = {{val| 1 ''a'' 5-2''a'' }}. For example, if ''a'' was 7/5, then the map would be {{val|1 (7/5) 5-2(7/5)}} = {{val|1 7/5 25/5-14/5}} = {{val|5 7 11}}, and if ''a'' was 4/3 then the map would be {{val|1 (4/3) 5-2(4/3)}} = {{val|1 4/3 15/3-8/3}} = {{val|3 4 7}}. One way to think about preserving the sorting order of the ''p''-odd limit tonality diamond would be to ensure that none of the intervals between its pitches become negative under this temperament. The sorted pitches of the 5-limit tonality diamond are [1/1, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3], and the intervals between those are [6/5, 25/24, 16/15, 9/8, 16/15, 25/24, 6/5]. We only care about the unique intervals, so we consider [6/5, 25/24, 16/15, 9/8]. In vector form those are [{{monzo|1 1 -1}}, {{monzo|-3 -1 2}}, {{monzo|4 -1 -1}}, {{monzo|-3 2 0}}]. If we map those using {{val| 1 ''a'' 5-2''a'' }} we obtain the tempered sizes [3''a'' - 4, 7 - 5''a'', ''a'' - 1, 2''a'' - 3]. Now we need to make sure each of those are not negative, so we get a set of inequalities: ''a'' ≥ 4/3, ''a'' ≤ 7/5, ''a'' ≥ 1, ''a'' ≥ 3/2. These inequalities have no solution: there's no way ''a'' can be both greater or equal to 1.5 and less than or equal to 1.4. Hence there are no diamond monotone tunings of this temperament.