Commas and Generators
Miracle is the temperament with a generator of a neutral semitone, meaning a semitone serving as both a 15/14 and a 16/15, discovered by George Secor; the generator is called a "secor" because of this. While not particularly striking as a 5-limit temperament, it is one of the very best linear temperaments in the 7 and 11 limits. It is well-served in all limits by 72-equal.
Music in Miracle
There is now a small but interesting body of work in the miracle temperament, with which Joe Pehrson has been actively involved and to which I have also contributed. This has focused on the 21-note MOS called Blackjack, but this is hardly the only way of treating miracle.
Rachmaninoff plays Blackjack Stained Glass
Table of Miracles
In the table below, a standard icon means a tuning map for a reduced pair of generators, and poptimal means a poptimal generator. The MOS given are valid for the optimized tunings of the particular version in question.
Period: Octave Generator: Secor (neutral semitone)
5-limit
name: miracle, ampersand, 31&41
comma: ampersand = |-25 7 6>
standard icon: [<1 1 3|, <0 6 -7|]
poptimal: 41/422
TOP period: 1200.631014 generator: 116.720642
MOS: 10, 11, 21, 31, 41, 72, 103, 175
7-limit
name: miracle, 31&41
comma sequence: [ampersand, 225/224] wedgie: <<6 -7 -2 -25 -20 15||
standard icon: [<1 1 3 3|, <0 6 -7 -2|]
7-limit poptimal: 17/175
9-limit poptimal: 32/329
TOP period: 1200.631014 generator: 116.720642
TM basis: {225/224, 1029/1024}
MOS: 10, 11, 21, 31, 41, 72
11-limit
name: miracle, 31&41
comma sequence: [ampersand, 225/224, 385/384] wedgie: <<6 -7 -2 15 -25 -20 3 15 59 49||
mapping: [<1 1 3 3 2|, <0 6 -7 -2 15|]
poptimal: 39/401
TOP period: 1200.631014 mapping: 116.720642
TM basis: {225/224, 243/242, 385/384}
MOS: 10, 11, 21, 31, 41, 72, 113, 185
name: 21&31
comma sequence: [ampersand, 225/224, 385/384] wedgie: <<6 -7 -2 -16 -25 -20 -46 15 -13 -38||
mapping: [<1 1 3 3 5|, <0 6 -7 -2 -16|]
poptimal: 55/568
TOP period: 1201.441965 generator: 116.334833
TM basis: {99/98, 176/175, 225/224, 243/242}
MOS: 10, 11, 21, 31, 41, 72, 103, 134, 165