User:BudjarnLambeth/Tritavesque: Difference between revisions

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-The '''tritavesque intervals'''{{idiosyncratic}} are all those intervals a/b, where b is the largest possible integer that is less than half of a.
+The '''tritavesque intervals'''{{idiosyncratic}} are all those intervals a/b, where b is the largest possible integer that is less than half of a, and shares no common factors with a.
-The tritavesque intervals are:
+The first few tritavesque intervals are:
-[[3/1]], [[4/1]], [[5/2]], 6/2, [[7/3]], [[8/3]], [[9/4]], 10/4, [[11/5]], [[12/5]], [[13/6]], 14/6, [[15/7]], [[16/7]], [[17/8]], 18/8, [[19/9]], [[20/9]], [[21/10]], 22/10, [[23/11]], [[24/11], [[25/12]], ...
+[[3/1]], [[4/1]], [[5/2]], [[6/1]], [[7/3]], [[8/3]], [[9/4]], [[10/3]], [[11/5]]...
-As tritavesque intervals get more complex, they tend closer and closer towards [[2/1]] but never reach it.
+As tritavesque intervals get more complex, they usually but not always get closer to [[2/1]].
-Many of these intervals see use as [[equave]]s for [[edonoi]] and other [[nonoctave]] scales.
+Many of these intervals see use as [[period]]s for [[edonoi]] and other [[nonoctave]] scales. They also see use as ninths of scales in edonoi, especially [[EDT]]s.
-[[Category:Interval category]][[Category:Nonoctave]][[Category:Edonoi]]
+The more complex of these intervals can also be used as [[pseudo-octave]]s.
+== List of tritavesque intervals ==
+The numerators are the integers starting from 3 (3, 4, 5, 6, 7...). The denominators are [https://oeis.org/A116921 OEIS sequence A116921] — ''a(n) = largest integer <= n/2 which is coprime to n'' — but skipping the first entry (0).
+# [[3/1]]
+# [[4/1]]
+# [[5/2]]
+# [[6/1]]
+# [[7/3]]
+# [[8/3]]
+# [[9/4]]
+# [[10/3]]
+# [[11/5]]
+# [[12/5]]
+# [[13/6]]
+# [[14/5]]
+# [[15/7]]
+# [[16/7]]
+# [[17/8]]
+# [[18/7]]
+# [[19/9]]
+# [[20/9]]
+# [[21/10]]
+# [[22/9]]
+# [[23/11]]
+# [[24/11]]
+# [[25/12]]
+# [[26/11]]
+# [[27/13]]
+# [[28/13]]
+# [[29/14]]
+# [[30/13]]
+# [[31/15]]
+# [[32/15]]
+# [[33/16]]
+# [[34/15]]
+# [[35/17]]
+# [[36/17]]
+# [[37/18]]
+# [[38/17]]
+# [[39/19]]
+# [[40/19]]
+And so on...
+== See also ==
+* [[Taxicab distance]]
+[[Category:Lists of intervals]][[Category:Nonoctave]][[Category:Edonoi]]