31:T25c0,// Historically important ratios, by Kyle Gann // scraped from https://www.kylegann.com/Octave.html historical = [ {Ratio: "1/1", Cents: 0, Notes: "tonic", Fraction: 1}, {Ratio: "1732/1731",Cents: 0.9998,Notes: "superparticular cent approximation", Fraction: round(1732/1731,3)}, {Ratio: "32805/32768",Cents: 1.954,Notes: "schisma (3 to the 8th/2 to the 12th x 5/8)", Fraction: round(32805/32768,3)}, {Ratio: "81/80", Cents: 21.506, Notes: "syntonic comma", Fraction: round(81/80,3)}, {Ratio: "531441/524288",Cents: 23.460,Notes: "Pythagorean comma (3 to the 12th/2 to the 19th)", Fraction: round(531441/524288,3)}, {Ratio: "74/73", Cents: 23.555, Notes: "superparticular Pythagorean comma approximation", Fraction: round(74/73,3)}, {Ratio: "56/55", Cents: 31.194, Notes: "Ptolemy's enharmonic", Fraction: round(56/55,3)}, {Ratio: "46/45", Cents: 38.051, Notes: "inferior quarter-tone (Ptolemy)", Fraction: round(46/45,3)}, {Ratio: "128/125", Cents: "41.059", Notes: "diminished second (16/15 x 24/25)", Fraction: "round(128/125,3)"}, {Ratio: "525/512", Cents: "43.408", Notes: "enharmonic diesis (Avicenna)", Fraction: round(525/512,3)}, {Ratio: "39/38", Cents: "44.97", Notes: "superior quarter-tone (Eratosthenes)", Fraction: round(39/38,3)}, {Ratio: "36/35", Cents: "48.77", Notes: "superior quarter-tone (Archytas)", Fraction: round(36/35,3)}, {Ratio: "35/34", Cents: "50.184", Notes: "E.T. 1/4-tone approximation", Fraction: round(35/34,3)}, {Ratio: "32/31", Cents: "54.964", Notes: "inferior quarter-tone (Didymus)", Fraction: round(32/31,3)}, {Ratio: "31/30", Cents: "56.767", Notes: "superior quarter-tone (Didymus)", Fraction: round(31/30,3)}, {Ratio: "28/27", Cents: "62.961", Notes: "inferior quarter-tone (Archytas)", Fraction: round(28/27,3)}, {Ratio: "26/25", Cents: "67.9", Notes: "1/3-tone (Avicenna)", Fraction: round(26/25,3)}, {Ratio: "25/24", Cents: "70.672", Notes: "minor 5-limit half-step", Fraction: round(25/24,3)}, {Ratio: "22/21", Cents: "80.537", Notes: "hard 1/2-step (Ptolemy, Avicenna, Safiud)", Fraction: round(22/21,3)}, {Ratio: "21/20", Cents: "84.467", Notes: "septimal semitone", Fraction: round(21/20,3)}, {Ratio: "256/243", Cents: "90.225", Notes: "Pythagorean half-step", Fraction: round(256/243,3)}, {Ratio: "135/128", Cents: "92.179", Notes: "limma ascendant", Fraction: round(135/128,3)}, {Ratio: "18/17", Cents: "98.955", Notes: "E.T. half-step approximation", Fraction: round(18/17,3)}, {Ratio: "2 to the 1/12th", Cents: "100", Notes: "equal-tempered half-step", Fraction: Math.pow(2,round(1/12,3))}, {Ratio: "89/84", Cents: "100.099", Notes: "ET half-step approximation", Fraction: round(89/84,3)}, {Ratio: "17/16", Cents: "104.955", Notes: "overtone half-step, 17th harmonic", Fraction: round(17/16,3)}, {Ratio: "16/15", Cents: "111.731", Notes: "major 5-limit half-step", Fraction: round(16/15,3)}, {Ratio: "15/14", Cents: "119.443", Notes: "Cowell just half-step", Fraction: round(15/14,3)}, {Ratio: "27/25", Cents: "133.238", Notes: "alternate Renaissance half-step", Fraction: round(27/25,3)}, {Ratio: "13/12", Cents: "138.573", Notes: "3/4-tone (Avicenna)", Fraction: round(13/12,3)}, {Ratio: "12/11", Cents: "150.637", Notes: "undecimal median 1/2-step", Fraction: round(12/11,3)}, {Ratio: "10/9", Cents: "182.404", Notes: "minor whole-tone", Fraction: round(10/9,3)}, {Ratio: "2 to the 1/6th", Cents: "200", Notes: "equal-tempered whole-tone", Fraction: Math.pow(2,round(1/6,3))}, {Ratio: "9/8", Cents: "203.91", Notes: "major whole-tone, 9th harmonic", Fraction: round(9/8,3)}, {Ratio: "8/7", Cents: "231.174", Notes: "septimal whole-tone", Fraction: round(8/7,3)}, {Ratio: "144/125", Cents: "244.969", Notes: "diminished third (6/5 x 24/25)", Fraction: round(144/125,3)}, {Ratio: "196/169", Cents: "256.596", Notes: "consonant interval (Avicenna)", Fraction: round(196/169,3)}, {Ratio: "7/6", Cents: "266.871", Notes: "septimal minor third", Fraction: round(7/6,3)}, {Ratio: "75/64", Cents: "274.582", Notes: "augmented second (9/8 x 25/24)", Fraction: round(75/64,3)}, {Ratio: "32/27", Cents: "294.135", Notes: "Pythagorean minor third", Fraction: round(32/27,3)}, {Ratio: "19/16", Cents: "297.513", Notes: "overtone minor third, 19th harmonic", Fraction: round(19/16,3)}, {Ratio: "2 to the 1/4th", Cents: "300", Notes: "equal-tempered minor third", Fraction: Math.pow(2,round(1/4,3))}, {Ratio: "6/5", Cents: "315.641", Notes: "5-limit minor third", Fraction: round(6/5,3)}, {Ratio: "11/9", Cents: "347.408", Notes: "undecimal median third", Fraction: round(11/9,3)}, {Ratio: "8192/6561", Cents: "384.36", Notes: "Pythagorean schismatic third", Fraction: round(8192/6561,3)}, {Ratio: "5/4", Cents: "386.314", Notes: "5-limit major third, 5th harmonic", Fraction: round(5/4,3)}, {Ratio: "2 to the 1/3rd", Cents: "400", Notes: "equal-tempered major third", Fraction: Math.pow(2,round(1/3,3))}, {Ratio: "81/64", Cents: "407.82", Notes: "Pythagorean major third", Fraction: round(81/64,3)}, {Ratio: "32/25", Cents: "427.373", Notes: "diminished fourth", Fraction: round(32/25,3)}, {Ratio: "9/7", Cents: "435.084", Notes: "septimal major third", Fraction: round(9/7,3)}, {Ratio: "125/96", Cents: "456.986", Notes: "augmented third (5/4 x 25/24)", Fraction: round(125/96,3)}, {Ratio: "21/16", Cents: "470.781", Notes: "septimal fourth", Fraction: round(21/16,3)}, {Ratio: "4/3", Cents: "498.045", Notes: "perfect fourth", Fraction: round(4/3,3)}, {Ratio: "2 to the 5/12ths", Cents: "500", Notes: "equal-tempered perfect fourth", Fraction: Math.pow(2,round(5/12,3))}, {Ratio: "11/8", Cents: "551.318", Notes: "undecimal tritone, 11th harmonic", Fraction: round(11/8,3)}, {Ratio: "25/18", Cents: "568.717", Notes: "augmented fourth (4/3 x 25/24)", Fraction: round(25/18,3)}, {Ratio: "7/5", Cents: "582.512", Notes: "septimal tritone", Fraction: round(7/5,3)}, {Ratio: "1024/729", Cents: "588.27", Notes: "low Pythagorean tritone", Fraction: round(1024/729,3)}, {Ratio: "45/32", Cents: "590.224", Notes: "high 5-limit tritone", Fraction: round(45/32,3)}, {Ratio: "Square root of 2", Cents: "600", Notes: "equal-tempered tritone", Fraction: round(Math.sqrt(2),3)}, {Ratio: "64/45", Cents: "609.776", Notes: "low 5-limit tritone", Fraction: round(64/45,3)}, {Ratio: "729/512", Cents: "611.73", Notes: "high Pythagorean tritone", Fraction: round(729/512,3)}, {Ratio: "10/7", Cents: "617.488", Notes: "septimal tritone", Fraction: round(10/7,3)}, {Ratio: "36/25", Cents: "631.283", Notes: "diminished fifth (3/2 x 24/25)", Fraction: round(36/25,3)}, {Ratio: "40/27", Cents: "680.449", Notes: "dissonant wolf 5-limit fifth", Fraction: round(40/27,3)}, {Ratio: "2 to the 7/12ths", Cents: "700", Notes: "equal-tempered perfect fifth", Fraction: Math.pow(2,round(7/12,3))}, {Ratio: "3/2", Cents: "701.955", Notes: "perfect fifth, 3rd harmonic", Fraction: round(3/2,3)}, {Ratio: "192/125", Cents: "743.014", Notes: "diminished sixth (8/5 x 24/25)", Fraction: round(192/125,3)}, {Ratio: "14/9", Cents: "764.916", Notes: "septimal minor sixth", Fraction: round(14/9,3)}, {Ratio: "25/16", Cents: "772.627", Notes: "augmented fifth", Fraction: round(25/16,3)}, {Ratio: "11/7", Cents: "782.492", Notes: "undecimal minor sixth", Fraction: round(11/7,3)}, {Ratio: "128/81", Cents: "792.18", Notes: "Pythagorean minor sixth", Fraction: round(128/81,3)}, {Ratio: "2 to the 2/3rds", Cents: "800", Notes: "equal-tempered minor sixth", Fraction: Math.pow(2,round(2/3,3))}, {Ratio: "8/5", Cents: "813.686", Notes: "5-limit minor sixth", Fraction: round(8/5,3)}, {Ratio: "6561/4096", Cents: "815.64", Notes: "Pythagorean schismatic sixth", Fraction: round(6561/4096,3)}, {Ratio: "13/8", Cents: "840.528", Notes: "overtone sixth, 13th harmonic", Fraction: round(13/8,3)}, {Ratio: "18/11", Cents: "852.592", Notes: "undecimal median sixth", Fraction: round(18/11,3)}, {Ratio: "5/3", Cents: "884.359", Notes: "5-limit major sixth", Fraction: round(5/3,3)}, {Ratio: "2 to the 3/4ths", Cents: "900", Notes: "equal-tempered major sixth", Fraction: Math.pow(2,round(3/4,3))}, {Ratio: "27/16", Cents: "905.865", Notes: "Pythagorean major sixth", Fraction: round(27/16,3)}, {Ratio: "128/75", Cents: "925.418", Notes: "diminished seventh (16/9 x 24/25)", Fraction: round(128/75,3)}, {Ratio: "12/7", Cents: "933.129", Notes: "septimal major sixth", Fraction: round(12/7,3)}, {Ratio: "125/72", Cents: "955.031", Notes: "augmented sixth (5/3 x 25/24)", Fraction: round(125/72,3)}, {Ratio: "7/4", Cents: "968.826", Notes: "septimal minor seventh, 7th harmonic", Fraction: round(7/4,3)}, {Ratio: "16/9", Cents: "996.09", Notes: "Pythagorean small min. seventh", Fraction: round(16/9,3)}, {Ratio: "2 to the 5/6ths", Cents: "1000", Notes: "equal-tempered minor seventh", Fraction: Math.pow(2,round(5/6,3))}, {Ratio: "9/5", Cents: "1017.596", Notes: "5-limit large minor seventh", Fraction: round(9/5,3)}, {Ratio: "11/6", Cents: 1049.363, Notes: "undecimal median seventh", Fraction: round(11/6,3)}, {Ratio: "15/8", Cents: 1088.269, Notes: "5-limit major seventh", Fraction: round(15/8,3)}, {Ratio: "2 to the 11/12ths", Cents: 1100, Notes: "equal-tempered major seventh", Fraction: Math.pow(2,round(11/12,3))}, {Ratio: "243/128", Cents: 1109.775, Notes: "Pythagorean major seventh", Fraction: round(243/128,3)}, {Ratio: "27/14", Cents: 1137.039, Notes: "septimal major seventh", Fraction: round(27/14,3)}, {Ratio: "125/64", Cents: 1158.941, Notes: "augmented seventh", Fraction: round(125/64,3)}, {Ratio: "2/1", Cents: 1200, Notes: "octave", Fraction: round(2/1,3)} ]32:[] 2f:{"title":"Anatomy of an Octave","cells":[{"id":0,"value":"# Anatomy of an Octave\nFormulaic replication of Kyle Gann's 1998 table at https://www.kylegann.com/Octave.html, allowing for musical experimentation in JavaScript.\n\nFrom his website, this includes:\n- All ratios between whole numbers 64 and lower\n- All ratios between 31-limit numbers up to 96 (31-limit meaning that the numbers contain no prime-number factors larger than 31)\n- All ratios between 11-limit numbers up to 128\n- All ratios between 5-limit numbers up to 1024\n- Harmonics up to 255 (each whole number divided by the closest inferior power of 2)\n- Certain historically important (or mathematically interesting) ratios such as the schisma and Pythagorean comma\n\nGiven the limitations of JavaScript for mathematics, this required custom implementations of basic algebraic functions such as rounding and division.","pinned":false,"hidden":false,"mode":"md"},{"id":17,"value":"
\n## Table\nTry searching for _fifth, tritone, 2 to the, Pythagorean, harmonic, /3, 1.2..._","pinned":false,"hidden":false,"mode":"md"},{"id":408,"value":"viewof search = Inputs.search(octave, {placeholder: \"Look for a number or ratio\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":406,"value":"viewof rows = Inputs.table(search,{\n format: {\n Viz: x => sparkseesaw(x)\n },\n sort: \"Cents\",\n //reverse: true\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":562,"value":"
\n## Hear a selected ratio\n\nGeneral audiences can distinguish 5 cents; with training, some musicians can distinguish to the cent.","pinned":false,"hidden":false,"mode":"md"},{"id":579,"value":"viewof freq = Inputs.range([40, 1000], {label: \"Base frequency:\", step: 1, value: 440})","pinned":false,"hidden":false,"mode":"ojs"},{"id":565,"value":"viewof selectedRatio = Inputs.select(octave, {label: \"Select a ratio\", format: x=>x.Ratio, value: octave.find(t=>t.Ratio === 3/2)})","pinned":false,"hidden":false,"mode":"ojs"},{"id":583,"value":"pluck2stereo(freq,round(freq*selectedRatio.Fraction,1),3)","pinned":false,"hidden":false,"mode":"ojs"},{"id":575,"value":"viewof selectedCents = Inputs.select(octave, {label: \"Select cents\", format: x=>x.Cents, sort: true, keyof: x=>x.Cents, value: octave.find(t=>t.Cents === 104.955)})","pinned":false,"hidden":false,"mode":"ojs"},{"id":595,"value":"pluck2stereo(freq,round(freq*selectedCents.Fraction,1),3)","pinned":false,"hidden":false,"mode":"ojs"},{"id":516,"value":"
\n## Calculator\nThe ratio of ${tex`\\frac{${demo1}}{${demo2}}`} is ${round(demo1/demo2,3)}, or ${Math.abs(round(Math.log10(demo1/demo2)*1200/Math.log10(2),3))} cents.","pinned":false,"hidden":false,"mode":"md"},{"id":518,"value":"viewof demo1 = Inputs.range([2,10000], {label: \"Frequency or number 1:\", value: 440, step:1});","pinned":false,"hidden":false,"mode":"ojs"},{"id":520,"value":"viewof demo2 = Inputs.range([1,demo1], {label: \"Frequency or number 2:\", value: 220, step:1});","pinned":false,"hidden":false,"mode":"ojs"},{"id":24,"value":"
\n## Components","pinned":false,"hidden":false,"mode":"md"},{"id":68,"value":"// All ratios between whole numbers 64 and lower\nsixtyfour = quartermaster(plusOne(64))","pinned":false,"hidden":false,"mode":"ojs"},{"id":39,"value":"// All ratios between 31-limit numbers up to 96\nthirtyone = quartermaster(toTheLimit(1,31,96))","pinned":false,"hidden":false,"mode":"ojs"},{"id":43,"value":"// All ratios between 11-limit numbers up to 128\neleven = quartermaster(toTheLimit(1,11,128))","pinned":false,"hidden":false,"mode":"ojs"},{"id":45,"value":"// All ratios between 5-limit numbers up to 1024\nfive = quartermaster(toTheLimit(1,5,1024))","pinned":false,"hidden":false,"mode":"ojs"},{"id":41,"value":"// Harmonics up to 255\nharmonics = musicalHarmonics(1,255)","pinned":false,"hidden":false,"mode":"ojs"},{"id":47,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":32,"value":"// ANATOMY OF AN OCTAVE\n// formulaic replication of work by Kyle Gann 1998\n// https://www.kylegann.com/Octave.html\n// implemented in JavaScript by Alessandro G. Magnasco 2023\n\n// deduplication of arrays modified from Nick Parsons 2021\n// https://stackoverflow.com/questions/66523595/merge-array-of-objects-and-get-unique-values\n\noctave = Array.from(\n [...sixtyfour, ...thirtyone, ...eleven, ...five, ...harmonics, ...historical] // latter overwrite former\n .reduce((acc, {Ratio=null, Notes=null, Viz=null, Cents=null, CentsStd=null, Savarts=null, ...rest}) => {\n const key = Object.entries(rest).join(\"-\"); // get a key based on the values (excluding those listed)\n return acc.set(key, {Cents, Ratio, Notes, Viz, ...rest, Savarts});\n }, new Map).values())","pinned":false,"hidden":false,"mode":"ojs"},{"id":494,"value":"### Use:\n~~~js\nimport {octave} from \"@amagnasco/anatomy-of-an-octave\"\n~~~","pinned":false,"hidden":false,"mode":"md"},{"id":21,"value":"
\n## Appendix ","pinned":false,"hidden":false,"mode":"md"},{"id":5,"value":"import {round} from \"@amagnasco/equal-temperament-generator\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":14,"value":"// ported by me to JS and tweaked\n// original python by Sheshank S. 2019\n// https://stackoverflow.com/questions/57258804/divide-each-element-of-an-array-by-each-element-of-an-other-array\n\n// requires round due to JS rounding errors\n\ndivideArray = (array1, array2, precision) => {\n var newArray = []\n for (var i1=0; i1 < array1.length; i1++){\n for (var i2=0; i2 < array2.length; i2++){\n newArray.push(round(array1[i1]/array2[i2], precision))\n }\n }\n return newArray\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":216,"value":"import {multiplyArray} from \"@amagnasco/cartesian-limits\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":62,"value":"import {plusOne} from \"@amagnasco/cartesian-limits\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":72,"value":"// tweaked from vitaly-t, 2021\n// https://stackoverflow.com/questions/17389350/prime-numbers-javascript\n\n// to do: add memoization ?\nfunction* primesGenerator(n) {\n let value = 2, i = 2;\n while (i++ < n) {\n if (value>2) {\n let k, q;\n do {\n k=3;\n value += 2;\n q = Math.floor(Math.sqrt(value));\n while (k <= q && value % k) {\n k += 2;\n }\n } while (k <= q);\n } else {\n value = value === 2 ? 3 : 2;\n }\n yield value;\n }\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":93,"value":"import {factorial} from \"@amagnasco/quartermaster\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":334,"value":"import {ratioToCents} from \"@amagnasco/quartermaster\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":337,"value":"import {ratioToSavarts} from \"@amagnasco/quartermaster\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":91,"value":"import {quartermaster} from \"@amagnasco/quartermaster\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":191,"value":"import {toTheLimit} from \"@amagnasco/cartesian-limits\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":448,"value":"import {musicalHarmonics} from \"@amagnasco/harmonic\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":587,"value":"### Optional imports for visualization and audio","pinned":false,"hidden":false,"mode":"md"},{"id":413,"value":"// optional cute visualization that can be called with\n// Format: {Viz: x => sparkseesaw(x)}\n// fractions between 0 and 1 only\nimport {sparkseesaw} from \"@hamzaamjad/tiny-charts\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":589,"value":"import {pluck2stereo} from \"@amagnasco/pluck-wa\"","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W32","stdlib":"1","titling":"auto"}