Creates a scale table from a combination product set definition
Usage
cps_scale_table(harmonics = c(1, 3, 5, 7, 9, 11), choose = 3, root_divisor)Arguments
- harmonics
a vector of the harmonics to use - defaults to the first six odd numbers, the harmonics that define the 1-3-5-7-9-11 Eikosany.
- choose
the number of harmonics to choose for each combination - defaults to 3, the number of harmonics for each combination in the Eikosany.
- root_divisor
a divisor that scales one of the products to 1/1. Most published CPS scales just use the smallest of the products for this, but Erv Wilson used 1x3x11 for the Eikosany, because that maps 1x5x11 to concert pitches for A: 55, 110, 220, 440 etc. There is no default value.
Value
a data.table with six columns:
note_name: the product of harmonics that defines the note (character)ratio: the ratio that defines the note, as a number >= 1 and < 2ratio_frac: the ratio as a vulgar fraction (character)ratio_cents: the ratio in cents (hundredths of a semitone)interval_cents: interval between this note and the previous notedegree: scale degree from zero to (number of notes) - 1
Examples
# the defaults yield the 1-3-5-7-9-11 Eikosany.
# Erv Wilson's design
print(eikosany <- cps_scale_table(root_divisor = 33))
#> note_name ratio ratio_frac ratio_cents interval_cents degree
#> 1: 1x3x11 1.000000 1 0.00000 NA 0
#> 2: 3x5x9 1.022727 45/44 38.90577 38.90577 1
#> 3: 1x5x7 1.060606 35/33 101.86668 62.96090 2
#> 4: 3x9x11 1.125000 9/8 203.91000 102.04332 3
#> 5: 1x7x11 1.166667 7/6 266.87091 62.96090 4
#> 6: 5x7x9 1.193182 105/88 305.77668 38.90577 5
#> 7: 3x5x11 1.250000 5/4 386.31371 80.53704 6
#> 8: 1x3x7 1.272727 14/11 417.50796 31.19425 7
#> 9: 7x9x11 1.312500 21/16 470.78091 53.27294 8
#> 10: 1x5x9 1.363636 15/11 536.95077 66.16987 9
#> 11: 3x7x9 1.431818 63/44 621.41797 84.46719 10
#> 12: 5x7x11 1.458333 35/24 653.18462 31.76665 11
#> 13: 1x9x11 1.500000 3/2 701.95500 48.77038 12
#> 14: 3x5x7 1.590909 35/22 803.82168 101.86668 13
#> 15: 1x3x9 1.636364 18/11 852.59206 48.77038 14
#> 16: 1x5x11 1.666667 5/3 884.35871 31.76665 15
#> 17: 3x7x11 1.750000 7/4 968.82591 84.46719 16
#> 18: 1x3x5 1.818182 20/11 1034.99577 66.16987 17
#> 19: 5x9x11 1.875000 15/8 1088.26871 53.27294 18
#> 20: 1x7x9 1.909091 21/11 1119.46296 31.19425 19
#> 21: 1x3x11' 2.000000 2 1200.00000 80.53704 20
#> note_name ratio ratio_frac ratio_cents interval_cents degree
# The usual public Eikosany
print(eikosany <- cps_scale_table(root_divisor = 15))
#> note_name ratio ratio_frac ratio_cents interval_cents degree
#> 1: 1x3x5 1.000000 1 0.00000 NA 0
#> 2: 5x9x11 1.031250 33/32 53.27294 53.27294 1
#> 3: 1x7x9 1.050000 21/20 84.46719 31.19425 2
#> 4: 1x3x11 1.100000 11/10 165.00423 80.53704 3
#> 5: 3x5x9 1.125000 9/8 203.91000 38.90577 4
#> 6: 1x5x7 1.166667 7/6 266.87091 62.96090 5
#> 7: 3x9x11 1.237500 99/80 368.91423 102.04332 6
#> 8: 1x7x11 1.283333 77/60 431.87513 62.96090 7
#> 9: 5x7x9 1.312500 21/16 470.78091 38.90577 8
#> 10: 3x5x11 1.375000 11/8 551.31794 80.53704 9
#> 11: 1x3x7 1.400000 7/5 582.51219 31.19425 10
#> 12: 7x9x11 1.443750 231/160 635.78514 53.27294 11
#> 13: 1x5x9 1.500000 3/2 701.95500 66.16987 12
#> 14: 3x7x9 1.575000 63/40 786.42219 84.46719 13
#> 15: 5x7x11 1.604167 77/48 818.18885 31.76665 14
#> 16: 1x9x11 1.650000 33/20 866.95923 48.77038 15
#> 17: 3x5x7 1.750000 7/4 968.82591 101.86668 16
#> 18: 1x3x9 1.800000 9/5 1017.59629 48.77038 17
#> 19: 1x5x11 1.833333 11/6 1049.36294 31.76665 18
#> 20: 3x7x11 1.925000 77/40 1133.83013 84.46719 19
#> 21: 1x3x5' 2.000000 2 1200.00000 66.16987 20
#> note_name ratio ratio_frac ratio_cents interval_cents degree
# the 1-3-5-7 Hexany
hexany_harmonics <- c(1, 3, 5, 7)
hexany_choose <- 2
print(hexany <-
cps_scale_table(hexany_harmonics, hexany_choose, 3)
)
#> note_name ratio ratio_frac ratio_cents interval_cents degree
#> 1: 1x3 1.000000 1 0.0000 NA 0
#> 2: 1x7 1.166667 7/6 266.8709 266.87091 1
#> 3: 3x5 1.250000 5/4 386.3137 119.44281 2
#> 4: 5x7 1.458333 35/24 653.1846 266.87091 3
#> 5: 1x5 1.666667 5/3 884.3587 231.17409 4
#> 6: 3x7 1.750000 7/4 968.8259 84.46719 5
#> 7: 1x3' 2.000000 2 1200.0000 231.17409 6
# the 1-7-9-11-13 2)5 Dekany
dekany_harmonics <- c(1, 7, 9, 11, 13)
dekany_choose <- 2
print(dekany <-
cps_scale_table(dekany_harmonics, dekany_choose, 7)
)
#> note_name ratio ratio_frac ratio_cents interval_cents degree
#> 1: 1x7 1.000000 1 0.00000 NA 0
#> 2: 9x13 1.044643 117/112 75.61176 75.61176 1
#> 3: 7x9 1.125000 9/8 203.91000 128.29824 2
#> 4: 11x13 1.276786 143/112 423.01970 219.10970 3
#> 5: 1x9 1.285714 9/7 435.08410 12.06440 4
#> 6: 7x11 1.375000 11/8 551.31794 116.23385 5
#> 7: 1x11 1.571429 11/7 782.49204 231.17409 6
#> 8: 7x13 1.625000 13/8 840.52766 58.03563 7
#> 9: 9x11 1.767857 99/56 986.40204 145.87438 8
#> 10: 1x13 1.857143 13/7 1071.70176 85.29972 9
#> 11: 1x7' 2.000000 2 1200.00000 128.29824 10
# We might want to print out sheet music for a conventional keyboard
# player, since the synthesizer is mapping MIDI note numbers to pitches.
# We assume at least a 37-key synthesizer with middle C on the left,
# so the largest CPS scale we can play is a 35-note "35-any", made from
# seven harmonics taken three at a time.
harmonics_35 <- c(1, 3, 5, 7, 9, 11, 13)
choose_35 <- 3
print(any_35 <-
cps_scale_table(harmonics_35, choose_35, root_divisor = 15)
)
#> note_name ratio ratio_frac ratio_cents interval_cents degree
#> 1: 1x3x5 1.000000 1 0.00000 NA 0
#> 2: 5x9x11 1.031250 33/32 53.27294 53.27294 1
#> 3: 7x11x13 1.042708 1001/960 72.40280 19.12985 2
#> 4: 1x7x9 1.050000 21/20 84.46719 12.06440 3
#> 5: 1x5x13 1.083333 13/12 138.57266 54.10547 4
#> 6: 1x3x11 1.100000 11/10 165.00423 26.43157 5
#> 7: 3x5x9 1.125000 9/8 203.91000 38.90577 6
#> 8: 3x7x13 1.137500 91/80 223.03985 19.12985 7
#> 9: 1x5x7 1.166667 7/6 266.87091 43.83105 8
#> 10: 1x11x13 1.191667 143/120 303.57689 36.70598 9
#> 11: 5x9x13 1.218750 39/32 342.48266 38.90577 10
#> 12: 3x9x11 1.237500 99/80 368.91423 26.43157 11
#> 13: 1x7x11 1.283333 77/60 431.87513 62.96090 12
#> 14: 1x3x13 1.300000 13/10 454.21395 22.33881 13
#> 15: 5x7x9 1.312500 21/16 470.78091 16.56696 14
#> 16: 9x11x13 1.340625 429/320 507.48689 36.70598 15
#> 17: 3x5x11 1.375000 11/8 551.31794 43.83105 16
#> 18: 1x3x7 1.400000 7/5 582.51219 31.19425 17
#> 19: 7x9x11 1.443750 231/160 635.78514 53.27294 18
#> 20: 3x9x13 1.462500 117/80 658.12395 22.33881 19
#> 21: 5x11x13 1.489583 143/96 689.89060 31.76665 20
#> 22: 1x5x9 1.500000 3/2 701.95500 12.06440 21
#> 23: 1x7x13 1.516667 91/60 721.08485 19.12985 22
#> 24: 3x7x9 1.575000 63/40 786.42219 65.33734 23
#> 25: 5x7x11 1.604167 77/48 818.18885 31.76665 24
#> 26: 3x5x13 1.625000 13/8 840.52766 22.33881 25
#> 27: 1x9x11 1.650000 33/20 866.95923 26.43157 26
#> 28: 7x9x13 1.706250 273/160 924.99486 58.03563 27
#> 29: 3x5x7 1.750000 7/4 968.82591 43.83105 28
#> 30: 3x11x13 1.787500 143/80 1005.53189 36.70598 29
#> 31: 1x3x9 1.800000 9/5 1017.59629 12.06440 30
#> 32: 1x5x11 1.833333 11/6 1049.36294 31.76665 31
#> 33: 5x7x13 1.895833 91/48 1107.39857 58.03563 32
#> 34: 3x7x11 1.925000 77/40 1133.83013 26.43157 33
#> 35: 1x9x13 1.950000 39/20 1156.16895 22.33881 34
#> 36: 1x3x5' 2.000000 2 1200.00000 43.83105 35
#> note_name ratio ratio_frac ratio_cents interval_cents degree