Diaschismic and Kleismic family data

Here is my compilation of data for diaschismic and kleismic
temperaments, two of my favourite classes of all limits and my most
favourite in the 5-limit. I did this for the Tonalsoft encyclopedia
because I didn't see data for either of those temperament families at
all there.
I omitted poptimal generators from most of the temperaments listed,
because I lack the tools to calculate them quickly enough, and same goes
for some of the TM basis sets shown here.
If you or I find any errors in the information here, then I will correct
them as soon as I find out.
If you find any poptimal generators or TM basis sets, then I will check
them and add the valid ones to my data file.

~Diaschismic family~
Family name: Diaschismic
Period: Demioctave (Half octave; 45:32)
Generator: Semitone (Half of 9:8; 16:15)

5-limit
Comma: Diaschisma, 2048:2025 |11.-4.-2>
Mapping: [2 | 0][3 | 1][5 | -2]
Poptimal generator: 10/114
Optimal minimax generator: 1/3 of 6:5; approx. 105,21 cents (5:4 and 3:2
are 1/6 diaschisma wide)
TOP period/generator: [599,55 | 104,70] cents
TOP-RMS period/generator: [599,41 | 104,80] cents
Poptimal MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 46 \ 80 \ 114
TOP MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 46 \ 80 \ 126
TOP-RMS MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 46 \ 80 \ 126

7-limit, 9-limit
~Pajara~ 10&12
Other names: "Paultone", "Twintone"
TM Basis: {50:49 \ 64:63}
Comma sequence: [2048:2025 | 50:49]
Wedgie: << 2'-4'-4"-11'-12" 2||
Mapping: [2 | 0][3 | 1][5 | -2][6 | -2]
7-limit poptimal*: 2/22
9-limit poptimal: 5/56
TOP-RMS period and generator: [598,86 | 106,84]
7-limit poptimal MOS cardinalities: 10 \ 12 \ 22
9-limit poptimal MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 56
TOP-RMS MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 56
*This value is based on inclusion of the entire mimimax range, which
includes 2/22.

~Starlidiaschismic~ 12&46
This title comes from the 126:125 unison of Starling temperament
Other names: "Standard diaschismic", or simply "Diaschismic"
TM Basis: {126:125 \ 2048:2025}
Comma sequence: [2048:2025 | 126:125]
Wedgie: << 2'-4'-16"-11'-31"-26||
Mapping: [2 | 0][3 | 1][5 | -2][7 | -8]
TOP-RMS period and generator: [599,45 | 103,59]
TOP-RMS MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 46 \ 58 \ 104 \ 162

~Ragidiaschismic~ 34&46
This title comes from the 4375:4374 unison of this temperament, a.k.a.
the ragisma
Other names: "Spearmint" from one of Gene Ward Smith's lists of linear
temperaments
TM Basis: {2048:2025 \ 4375:4374}
Comma sequence: [2048:2025 | 4375:4374]
Wedgie: << 2'-4'30"-11'42"81||
Mapping: [2 | 0][3 | 1][5 | -2][3 | 15]
TOP-RMS period and generator: [599,41 | 104,71]
TOP-RMS MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 46 \ 80 \ 126

~Hemidiaschismic~ 22&46
Although this is a vague title, "Shrutar" seems to imply a reference to
Indian concept of shruti, which I would rarely consider when using this
temperament
Other names: "Shrutar"
TM Basis: {245:243 \ 2048:2025}
Comma sequence: [2048:2025 | 245:243]
Wedgie: << 4'-8'14"-22'11"55||
Mapping: [2 | 0][3 | 2][5 | -4][5 | 7]
TOP-RMS period and generator: [599,54 | 52,77]
TOP-RMS MOS cardinalities: 22 \ 24 \ 46 \ 68

~Quadrandiaschismic~ 12&56
TM Basis: {2048:2025 \ 3136:3125}
Comma sequence: [2048:2025 | 3136:3125]
Wedgie: << 4'-8'-20"-22'-43"-24||
Mapping: [4 | 0][6 | 1][10 | -2][13 | -5]
TOP-RMS period and generator: [299,69 | 105,25]
TOP-RMS MOS cardinalities: 8 \ 12 \ 20 \ 32 \ 44 \ 56 \ 68 \ 80

11-limit
~Pajara~ 12&22
TM Basis: {50:49 \ 64:63 \ 99:98}
Comma sequence: [2048:2025 | 50:49 | 99:98]
Wedgie: << 2'-4'-4'-12"-11'-12'-26" 2'-14"-20||
Mapping: [2 | 0][3 | 1][5 | -2][6 | -2][8 | -6]
TOP-RMS period and generator: [598,86 | 106,68]
TOP-RMS MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 56

~Starlidiaschismic~ 12&46
TM Basis: {126:125 \ 176:175 \ 896:891}
Comma sequence: [2048:2025 | 126:125 | 176:175]
Wedgie: << 2'-4'-16'-24"-11'-31'-45"-26'-42"-12||
Mapping: [2 | 0][3 | 1][5 | -2][7 | -8][9 | -12]
TOP-RMS period and generator: [599,45 | 103,62]
TOP-RMS MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 46 \ 58 \ 104 \ 162

~Ragidiaschismic~ 46&80
TM Basis: {176:175 \ 896:891 \ 2200:2187} - uncertain about 2200:2187
Comma sequence: [2048:2025 | 4375:4374 | 176:175]
Wedgie: << 2'-4'30'22"-11'42'28"81'65"-42||
Mapping: [2 | 0][3 | 1][5 | -2][3 | 15][5 | 11]
TOP-RMS period and generator: [599,44 | 104,76]
TOP-RMS MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 46 \ 80 \ 126 \ 206

~Hemidiaschismic~ 22&46
TM Basis: {121:120 \ 176:175 \ 245:243}
Comma sequence: [2048:2025 | 245:243 | 121:120]
Wedgie: << 4'-8'14'-2"-22'11'-17"55'23"-54||
Mapping: [2 | 0][3 | 2][5 | -4][5 | 7][7 | -1]
TOP-RMS period and generator: [599,77 | 52,66]
TOP-RMS MOS cardinalities: 22 \ 24 \ 46 \ 68 \ 114

~Kleismic family~
Family name: Kleismic
Period: Octave
Generator: Slightly wide 6:5

5-limit
Comma: Kleisma, 15625:15552 |-6.-5. 6>
Mapping: [1 | 0][0 | 6][1 | 5]
Poptimal generator: 65/246
Optimal minimax generator: 1/6 of 3:1; approx. 317 cents (5:4 and 5:3
are 1/6 kleisma narrow)
TOP-RMS period/generator: [1200,17 | 317,05] cents
MOS cardinalities: 3 \ 4 \ 7 \ 11 \ 15 \ 19 \ 34 \ 53 \ 87 \ 140

7-limit, 9-limit
~Simple Kleismic~ 4&15
Other names: "Keemun"
TM Basis: {49:48 \ 126:125}
Comma sequence: [15625:15552 | 49:48]
Wedgie: << 6' 5' 3"-6'-12"-7||
Mapping: [1 | 0][0 | 6][1 | 5][2 | 3]
7-limit poptimal: 14/53
TOP-RMS period and generator: [1202,65 | 317,17]
MOS cardinalities: 3 \ 4 \ 7 \ 11 \ 15 \ 19 \ 34 \ 53

~Catakleismic~ 19&34
Other names: "Hanson"
TM Basis: {225:224 \ 4375:4374}
Comma sequence: [15625:15552 | 225:224]
Wedgie: << 6' 5'22"-6'18"37||
Mapping: [1 | 0][0 | 6][1 | 5][-3 | 22]
7-limit poptimal: 19/72 ??
TOP-RMS period and generator: [1200,60 | 316,89]
MOS cardinalities: 3 \ 4 \ 7 \ 11 \ 15 \ 19 \ 34 \ 53 \ 72

~Hemikleismic~ (Neutral Seconds) 15&53
TM Basis: {4000:3969 \ 6144:6125} - uncertain about further reduction
Comma sequence: [15625:15552 | 4000:3969]
Wedgie: <<12'10'-9"-12'-48"-49||
Mapping: [1 | 0][0 | 12][1 | 10][4 | -9]
7-limit poptimal: 16/121 ??
TOP-RMS period and generator: [1199,40 | 158,57]
MOS cardinalities: 7 \ 8 \ 15 \ 23 \ 38 \ 53 \ 68 \ 121

~Hemikleismic~ (Semisixths) 19&49
Other names: "Clyde"
TM Basis: {245:243 \ 3136:3125}
Comma sequence: [15625:15552 | 245:243]
Wedgie: <<12'10'25"-12' 6"30||
Mapping: [1 | 0][6 | -12][6 | -10][12 | -25]
TOP-RMS period and generator: [1199,84 | 441,28]
MOS cardinalities: 2 \ 3 \ 5 \ 8 \ 11 \ 19 \ 30 \ 49 \ 68 \ 87

~Countercatakleismic~ (Counter-53) 53&87
Other names: "Countercata"
TM Basis: {5120:5103 \ 15625:15552}
Comma sequence: [15625:15552 | 5120:5103]
Wedgie: << 6' 5'-31"-6'-66"-86||
Mapping: [1 | 0][0 | 6][1 | 5][11 | -31]
TOP-RMS period and generator: [1199,92 | 317,10]
MOS cardinalities: 3 \ 4 \ 7 \ 11 \ 19 \ 34 \ 53 \ 87 \ 140

~Tritikleismic~ 15&57
TM Basis: {1029:1024 \ 15625:15552} - Differ by landscape comma
250047:250000 which sets 63:50 to 1/3-octave
Comma sequence: [15625:15552 | 1029:1024]
Wedgie: <<18'15'-6"-18'-60"-56||
Mapping: [3 | 0][6 | -6][8 | -5][8 | 2]
TOP-RMS period and generator: [400,18 | 83,17]
MOS cardinalities: 12 \ 15 \ 27 \ 42 \ 57 \ 72 \ 87 \ 159

~Quadritikleismic~ 4&68
Other names: "Breedsmikleismic"
TM Basis: {2401:2400 \ 15625:15552} - Differ by 390625:388962 which sets
25:21 to 1/4-octave
Comma sequence: [15625:15552 | 2401:2400]
Wedgie: <<24'20'16"-24'-42"-19||
Mapping: [4 | 0][6 | 6][9 | 5][11 | 4]
TOP-RMS period and generator: [300,05 | 17,00]
MOS cardinalities: 68 \ 72 \ 140 \ 212

11-limit
~Catakleismic~ 53&72
TM Basis: {225:224 \ 385:384 \ 4375:4374} - fully reduced?
Comma sequence: [15625:15552 | 225:224 | 385:384]
Wedgie: << 6' 5'22'21"-6'18'-54"37'-66"-135||
Mapping: [1 | 0][0 | 6][1 | 5][-3 | 22][9 | -21]
TOP-RMS period and generator: [1200,65 | 316,89]
MOS cardinalities: 3 \ 4 \ 7 \ 11 \ 15 \ 19 \ 34 \ 53 \ 72

~Hemikleismic~ (Neutral seconds) 53&68
TM Basis: {121:120 \ 176:175 \ 4000:3969} - fully reduced?
Comma sequence: [15625:15552 | 6144:6125 | 121:120]
Wedgie: <<12'10'-9'11"-12'-48'-24"-49'-9"62||
Mapping: [1 | 0][0 | 12][1 | 10][4 | -9][2 | 11]
TOP-RMS period and generator: [1199,81 | 158,65]
MOS cardinalities: 3 \ 4 \ 7 \ 11 \ 15 \ 19 \ 34 \ 53 \ 72

~Countercatakleismic~ 53&87
TM Basis: {385:384 \ 2200:2187 \ 3388:3375}
Comma sequence: [15625:15552 | 5120:5103 | 385:384]
Wedgie: << 6' 5'-31'32"-6'-66'30"-86'57"197||
Mapping: [1 | 0][0 | 6][1 | 5][11 | -31][-5 | 32]
TOP-RMS period and generator: [1200,10 | 317,19]
MOS cardinalities: 3 \ 4 \ 7 \ 11 \ 15 \ 19 \ 34 \ 53 \ 87 \ 140

~Tritikleismic~ 72&87
TM Basis: {385:384 \ 441:440 \ 4000:3993}
Comma sequence: [15625:15552 | 1029:1024 | 385:384]
Wedgie: <<18'15'-6' 9"-18'-60'-48"-56'-31"46||
Mapping: [3 | 0][6 | -6][8 | -5][8 | 2][11 | -3]
TOP-RMS period and generator: [400,16 | 83,15]
MOS cardinalities: 12 \ 15 \ 27 \ 42 \ 57 \ 72 \ 87 \ 159

~Quadritikleismic~ 68&72
TM Basis: {385:384 \ 1375:1372 \ 6250:6237} ??
Comma sequence: [15625:15552 | 2401:2400 | 385:384]
Wedgie: <<24'20'16'-12"-24'-42'-102"-19'43"23||
Mapping: [4 | 0][6 | 6][9 | 5][11 | 4][14 | -3]
TOP-RMS period and generator: [300,10 | 16,93]
MOS cardinalities: 68 \ 72 \ 140 \ 212

Hi aerisselicious,

Was your message below truncated? It's missing an endquote
apparently. Also, who are you?

-Carl

At 04:57 PM 8/1/2009, you wrote:
>Here is my compilation of data for diaschismic and kleismic temperaments, two of my favourite classes of all limits and my most favourite in the 5-limit. I did this for the Tonalsoft encyclopedia because I didn't see data for either of those temperament families at all there.
>I omitted poptimal generators from most of the temperaments listed, because I lack the tools to calculate them quickly enough, and same goes for some of the TM basis sets shown here.
>If you or I find any errors in the information here, then I will correct them as soon as I find out.
>If you find any poptimal generators or TM basis sets, then I will check them and add the valid ones to my data file.
>
>~Diaschismic family~
>Family name: Diaschismic
>Period: Demioctave (Half octave; 45:32)
>Generator: Semitone (Half of 9:8; 16:15)
>
>5-limit
>Comma: Diaschisma, 2048:2025 |11.-4.-2>
>Mapping: [2 | 0][3 | 1][5 | -2]
>Poptimal generator: 10/114
>Optimal minimax generator: 1/3 of 6:5; approx. 105,21 cents (5:4 and 3:2 are 1/6 diaschisma wide)
>TOP period/generator: [599,55 | 104,70] cents
>TOP-RMS period/generator: [599,41 | 104,80] cents
>Poptimal MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 46 \ 80 \ 114
>TOP MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 46 \ 80 \ 126
>TOP-RMS MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 46 \ 80 \ 126
>
>7-limit, 9-limit
>~Pajara~ 10&12
>Other names: "Paultone", "Twintone"
>TM Basis: {50:49 \ 64:63}
>Comma sequence: [2048:2025 | 50:49]
>Wedgie: << 2'-4'-4"-11'-12" 2||
>Mapping: [2 | 0][3 | 1][5 | -2][6 | -2]
>7-limit poptimal*: 2/22
>9-limit poptimal: 5/56
>TOP-RMS period and generator: [598,86 | 106,84]
>7-limit poptimal MOS cardinalities: 10 \ 12 \ 22
>9-limit poptimal MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 56
>TOP-RMS MOS cardinalities: 10 \ 12 \ 22 \ 34 \ 56
>*This value is based on inclusion of the entire mimimax range, which includes 2/22.
>
>~Starlidiaschismic~ 12&46
>This title comes from the 126:125 unison of Starling temperament
>Other names: "Stand