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More on tunnel commas

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

10/29/2006 12:47:16 AM

John Chalmers suggested I show how many generators of a fifth it takes
to get to the consonant interval of the tunnel comma. Below I show the
comma, normalized so that the exponent of 3 is positive, that
exponent, which is the number of generator steps, the interval class
reached, and the comma's monzo.

63/64: 2, 8/7, |-6 2 0 1>
405/416: 4, 13/10, |-5 4 1 0 0 -1>
39/40: 1, 20/13, |-3 1 -1 0 0 1>
891/896: 4, 14/11, |-7 4 0 -1 1>
351/352: 3, 22/13, |-5 3 0 0 -1 1>
5103/5120: 6, 10/7, |-10 6 -1 1>
32805/32768: 8, 8/5, |-15 8 1>
81/80: 4, 5/4, |-4 4 -1>
1701/1664: 5, 13/7, |-7 5 0 1 0 -1>
1053/1024: 4, 16/13, |-10 4 0 0 0 1>
45/44: 2, 11/10, |-2 2 1 0 -1>

Note that four fifths can get you to 16/13, 5/4, 14/11 or 13/10
depending on the approximations used and how flat or sharp you want to
get. Six fifths to 10/7 and eight fifths to 8/5 are important for more
accurate temperaments.