Creates a scale table from a product set definition
Usage
ps_scale_table(
ps_def = c("1x3x5", "5x9x11", "1x7x9", "1x3x11", "3x5x9", "1x5x7", "3x9x11", "1x7x11",
"5x7x9", "3x5x11", "1x3x7", "7x9x11", "1x5x9", "3x7x9", "5x7x11", "1x9x11", "3x5x7",
"1x3x9", "1x5x11", "3x7x11"),
root_divisor
)Arguments
- ps_def
the product set scale definition. This is a character vector of products. Each product is a set of any number of integers separated by a lower-case "x". For example, the
ps_defof the 1-3-5-7 Hexany isc("1x3", "1x5", "1x7", "3x5", "3x7", "5x7")The default is the
ps_deffor the 1-3-5-7-9-11 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 given product set definition, re-ordered by the degrees of the resulting scale (character)ratio: the ratio that defines the note, as a number between 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 default yields the 1-3-5-7-9-11 Eikosany
print(eikosany <- ps_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
# Kraig Grady's Eikosany as two complementary extended Dekanies
# See _Microtonality and the Tuning Systems of Erv Wilson_, pages 127 - 131
# for the process used to create these scales
print(grady_a <- ps_scale_table(c(
"1x3x11",
"1x9",
"3x9x11",
"1x7x11",
"1x3x7",
"7x9x11",
"3x7x9",
"1x9x11",
"1x3x9",
"1x7",
"3x7x11",
"1x7x9"
), root_divisor = 33))
#> note_name ratio ratio_frac ratio_cents interval_cents degree
#> 1: 1x3x11 1.000000 1 0.0000 NA 0
#> 2: 1x9 1.090909 12/11 150.6371 150.63706 1
#> 3: 3x9x11 1.125000 9/8 203.9100 53.27294 2
#> 4: 1x7x11 1.166667 7/6 266.8709 62.96090 3
#> 5: 1x3x7 1.272727 14/11 417.5080 150.63706 4
#> 6: 7x9x11 1.312500 21/16 470.7809 53.27294 5
#> 7: 3x7x9 1.431818 63/44 621.4180 150.63706 6
#> 8: 1x9x11 1.500000 3/2 701.9550 80.53704 7
#> 9: 1x3x9 1.636364 18/11 852.5921 150.63706 8
#> 10: 1x7 1.696970 56/33 915.5530 62.96090 9
#> 11: 3x7x11 1.750000 7/4 968.8259 53.27294 10
#> 12: 1x7x9 1.909091 21/11 1119.4630 150.63706 11
#> 13: 1x3x11' 2.000000 2 1200.0000 80.53704 12
print(grady_a_offsets <- offset_matrix(grady_a))
#> C C# D D# E F F# G G# A A# B
#> 1x3x11 0 -100 -200 -300 -400 -500 -600 -700 -800 -900 -1000 -1100
#> 1x9 151 51 -49 -149 -249 -349 -449 -549 -649 -749 -849 -949
#> 3x9x11 204 104 4 -96 -196 -296 -396 -496 -596 -696 -796 -896
#> 1x7x11 267 167 67 -33 -133 -233 -333 -433 -533 -633 -733 -833
#> 1x3x7 418 318 218 118 18 -82 -182 -282 -382 -482 -582 -682
#> 7x9x11 471 371 271 171 71 -29 -129 -229 -329 -429 -529 -629
#> 3x7x9 621 521 421 321 221 121 21 -79 -179 -279 -379 -479
#> 1x9x11 702 602 502 402 302 202 102 2 -98 -198 -298 -398
#> 1x3x9 853 753 653 553 453 353 253 153 53 -47 -147 -247
#> 1x7 916 816 716 616 516 416 316 216 116 16 -84 -184
#> 3x7x11 969 869 769 669 569 469 369 269 169 69 -31 -131
#> 1x7x9 1119 1019 919 819 719 619 519 419 319 219 119 19
print(grady_b <- ps_scale_table(c(
"3x5x11",
"1x5x9",
"3x5x9x11",
"5x7x11",
"3x5x7",
"1x5x11",
"1x3x5",
"5x9x11",
"3x5x9",
"1x5x7",
"3x5x7x11",
"5x7x9"
), root_divisor = 3*5*11))
#> note_name ratio ratio_frac ratio_cents interval_cents degree
#> 1: 3x5x11 1.000000 1 0.0000 NA 0
#> 2: 1x5x9 1.090909 12/11 150.6371 150.63706 1
#> 3: 3x5x9x11 1.125000 9/8 203.9100 53.27294 2
#> 4: 5x7x11 1.166667 7/6 266.8709 62.96090 3
#> 5: 3x5x7 1.272727 14/11 417.5080 150.63706 4
#> 6: 1x5x11 1.333333 4/3 498.0450 80.53704 5
#> 7: 1x3x5 1.454545 16/11 648.6821 150.63706 6
#> 8: 5x9x11 1.500000 3/2 701.9550 53.27294 7
#> 9: 3x5x9 1.636364 18/11 852.5921 150.63706 8
#> 10: 1x5x7 1.696970 56/33 915.5530 62.96090 9
#> 11: 3x5x7x11 1.750000 7/4 968.8259 53.27294 10
#> 12: 5x7x9 1.909091 21/11 1119.4630 150.63706 11
#> 13: 3x5x11' 2.000000 2 1200.0000 80.53704 12
print(grady_b_offsets <- offset_matrix(grady_b))
#> C C# D D# E F F# G G# A A# B
#> 3x5x11 0 -100 -200 -300 -400 -500 -600 -700 -800 -900 -1000 -1100
#> 1x5x9 151 51 -49 -149 -249 -349 -449 -549 -649 -749 -849 -949
#> 3x5x9x11 204 104 4 -96 -196 -296 -396 -496 -596 -696 -796 -896
#> 5x7x11 267 167 67 -33 -133 -233 -333 -433 -533 -633 -733 -833
#> 3x5x7 418 318 218 118 18 -82 -182 -282 -382 -482 -582 -682
#> 1x5x11 498 398 298 198 98 -2 -102 -202 -302 -402 -502 -602
#> 1x3x5 649 549 449 349 249 149 49 -51 -151 -251 -351 -451
#> 5x9x11 702 602 502 402 302 202 102 2 -98 -198 -298 -398
#> 3x5x9 853 753 653 553 453 353 253 153 53 -47 -147 -247
#> 1x5x7 916 816 716 616 516 416 316 216 116 16 -84 -184
#> 3x5x7x11 969 869 769 669 569 469 369 269 169 69 -31 -131
#> 5x7x9 1119 1019 919 819 719 619 519 419 319 219 119 19