kb2_118

From: Gene Ward Smith (2003-05-24)
Subject: 118 et version of Kirnberger 2

One way to treat this temperament is to retemper it in a schismic
system, since it uses a couple of schismic appoximations anyway. We
then need a good representation of the odd man out, given in cents. 53
is terrible for this, 94 isn't bad, but 88 out of 118 is very close.
Retempered to 118 et, Kirnberger 2 becomes

[9, 20, 29, 38, 49, 58, 69, 78, 88, 98, 107, 118]

in terms of scale degrees, and 

[9, 11, 9, 9, 11, 9, 11, 9, 10, 10, 9, 11]

in terms of step sizes. While there isn't a very good 5-limit version
of 88 out of 118, in the 7-limit it makes a nice 42/25.

Here's a Scala file:

! kb2_118.scl
! [9, 11, 9, 9, 11, 9, 11, 9, 10, 10, 9, 11]
! [9, 20, 29, 38, 49, 58, 69, 78, 88, 98, 107, 118]
118 equal version of Kirnberger 2
 12
!
91.525424
203.389831
294.915254
386.440678
498.305085
589.830509
701.694915
793.220339
894.915254
996.610170
1088.135593
2/1

Scala gives the following information:

|
Scala version 2.06b  Copyright E.F. Op de Coul, the Netherlands, 2003
|
118 equal version of Kirnberger 2
|
Number of notes                      : 12
-- Interval properties --
Smallest one step interval           : 91.525 cents
Average step (divided formal octave) : 100.000 cents
Average / Smallest step              : 1.092593
Largest one step interval            : 111.864 cents
Largest / Average step               : 1.118644
Largest / Smallest step              : 1.222222
Linear approximation average step    : 99.3950 cents
Number of one step interval sizes    : 3
Median interval of one step          : 96.610 cents
Most common interval of one step     : 91.525 cents, amount: 6
Scale is strictly proper
Scale is a mode of a 118-tone equal temperament with octave  2/1
Degrees: 9 20 29 38 49 58 69 78 88 98 107 118
Number of contiguous one-step segments: 0
Least number of segments generator: 640.678 cents and inv.
Number of contiguous generator circle segments: 1
Interval pattern: ABAABABACCAB
Scale is a well-temperament. It has 2 different sizes of fifth
Scale has monotonic third-sizes over circle of fifths
Scale is a Constant Structure, by a margin of 71.186 cents
Number of different intervals        : 31 = 2.81818 / class
Interval vector is [ 0 0 0 0 0 0 0 0 6 2 4 0 0 0 0 0 0 1 2 9 0 0 0 0 0
0 0 0 8 2 2 0 0 0 0 0 0 3 2 7 0 0 0 0 0 0 0 0 10 2 0 0 0 0 0 0 0 5 1 ]
Smallest interval difference         : 10.169 cents
Most common intervals                : 498.305 cents & inv., amount: 10
Most common triad is 0.0 498.305 701.695 cents, amount: 9
Number of recognisable fifths        : 12, average 700.000 cents
Rothenberg stability                 : 1.000000 = 1
Lumma stability                      : 0.830508
Limited transpositions with margin 20.3390 cents :
 5 7 9
Inversional symmetry on degrees      :
 3 9

From: Gene Ward Smith (2003-05-24)
Subject: 118 et version of Kirnberger 2

One way to treat this temperament is to retemper it in a schismic
system, since it uses a couple of schismic appoximations anyway. We
then need a good representation of the odd man out, given in cents. 53
is terrible for this, 94 isn't bad, but 88 out of 118 is very close.
Retempered to 118 et, Kirnberger 2 becomes

[9, 20, 29, 38, 49, 58, 69, 78, 88, 98, 107, 118]

in terms of scale degrees, and 

[9, 11, 9, 9, 11, 9, 11, 9, 10, 10, 9, 11]

in terms of step sizes. While there isn't a very good 5-limit version
of 88 out of 118, in the 7-limit it makes a nice 42/25.

Here's a Scala file:

! kb2_118.scl
! [9, 11, 9, 9, 11, 9, 11, 9, 10, 10, 9, 11]
! [9, 20, 29, 38, 49, 58, 69, 78, 88, 98, 107, 118]
118 equal version of Kirnberger 2
 12
!
91.525424
203.389831
294.915254
386.440678
498.305085
589.830509
701.694915
793.220339
894.915254
996.610170
1088.135593
2/1

Scala gives the following information:

|
Scala version 2.06b  Copyright E.F. Op de Coul, the Netherlands, 2003
|
118 equal version of Kirnberger 2
|
Number of notes                      : 12
-- Interval properties --
Smallest one step interval           : 91.525 cents
Average step (divided formal octave) : 100.000 cents
Average / Smallest step              : 1.092593
Largest one step interval            : 111.864 cents
Largest / Average step               : 1.118644
Largest / Smallest step              : 1.222222
Linear approximation average step    : 99.3950 cents
Number of one step interval sizes    : 3
Median interval of one step          : 96.610 cents
Most common interval of one step     : 91.525 cents, amount: 6
Scale is strictly proper
Scale is a mode of a 118-tone equal temperament with octave  2/1
Degrees: 9 20 29 38 49 58 69 78 88 98 107 118
Number of contiguous one-step segments: 0
Least number of segments generator: 640.678 cents and inv.
Number of contiguous generator circle segments: 1
Interval pattern: ABAABABACCAB
Scale is a well-temperament. It has 2 different sizes of fifth
Scale has monotonic third-sizes over circle of fifths
Scale is a Constant Structure, by a margin of 71.186 cents
Number of different intervals        : 31 = 2.81818 / class
Interval vector is [ 0 0 0 0 0 0 0 0 6 2 4 0 0 0 0 0 0 1 2 9 0 0 0 0 0
0 0 0 8 2 2 0 0 0 0 0 0 3 2 7 0 0 0 0 0 0 0 0 10 2 0 0 0 0 0 0 0 5 1 ]
Smallest interval difference         : 10.169 cents
Most common intervals                : 498.305 cents & inv., amount: 10
Most common triad is 0.0 498.305 701.695 cents, amount: 9
Number of recognisable fifths        : 12, average 700.000 cents
Rothenberg stability                 : 1.000000 = 1
Lumma stability                      : 0.830508
Limited transpositions with margin 20.3390 cents :
 5 7 9
Inversional symmetry on degrees      :
 3 9