Implement algorithms for finding concordance shells

Update the Equalize modifier with the minimax error model.
TODO: Shell generation modal.

ref #471

src/__tests__/analysis.spec.ts

@@ -1,7 +1,16 @@
 import { describe, it, expect } from "vitest";
-import { arraysEqual } from "xen-dev-utils";
+import { arraysEqual, valueToCents } from "xen-dev-utils";
 
-import { otonalFundamental, utonalFundamental } from "../analysis";
+import {
+  alignValues,
+  concordanceShell,
+  minimaxConcordanceShells,
+  misalignment,
+  otonalFundamental,
+  utonalFundamental,
+} from "../analysis";
+
+const EPSILON = 1e-4;
 
 describe("Otonal balancer", () => {
   it("can figure out that the major chord in 12edo approximates 4:5:6", () => {
@@ -28,3 +37,74 @@ describe("Utonal balancer", () => {
     ).toBeTruthy();
   });
 });
+
+describe("Equal-division deviation minimizer", () => {
+  it("can figure out the optimal alignment of 4:5:6 on 12edo", () => {
+    const minimumAchievableError = alignValues([4, 5, 6], 100.0).error;
+    expect(minimumAchievableError).closeTo(7.8206, EPSILON);
+
+    // Attempt (and fail) to find a counter-example
+    const pitches = [4, 5, 6].map(valueToCents);
+    for (let i = 0; i < 100; ++i) {
+      const offset = Math.random() * 1200;
+      const candidate = pitches.map((pitch) => pitch + offset);
+      expect(misalignment(candidate, 100.0)).toBeGreaterThanOrEqual(
+        minimumAchievableError
+      );
+    }
+  });
+});
+
+describe("Concordance shell finder", () => {
+  it("finds the 37 : 44 : 66 : 70 : 83 : 93 : 99 : 111 : 157 : 187 : 197 : 209 : 235 : 249 shell in 12edo", () => {
+    const shell = concordanceShell(37, 255, 5.0, 12);
+    expect(shell.harmonics).toEqual([
+      37, 44, 66, 70, 83, 93, 99, 111, 157, 187, 197, 209, 235, 249,
+    ]);
+  });
+});
+
+describe("Minimaxing concordance shell finder", () => {
+  it("finds more variants for the shell rooted at 37 in 12edo", () => {
+    const shells = minimaxConcordanceShells(37, 127, 3, 12);
+
+    // Just some way to create a deterministic order.
+    function cmp(a: number[], b: number[]) {
+      if (!a.length && b.length) {
+        return -1;
+      } else if (a.length && !b.length) {
+        return +1;
+      } else if (!a.length && !b.length) {
+        return 0;
+      }
+      if (a[0] < b[0]) {
+        return -1;
+      } else if (a[0] > b[0]) {
+        return +1;
+      }
+      return cmp(a.slice(1), b.slice(1));
+    }
+    shells.sort((a, b) => cmp(a.harmonics, b.harmonics));
+    for (const shell of shells) {
+      shell.error = Math.round(shell.error * 1000) / 1000;
+      delete shell.scaleDegrees;
+    }
+    expect(shells).toEqual([
+      {
+        harmonics: [37, 44, 66, 70, 83, 99, 111],
+        error: 2.59,
+        degrees: [0, 3, 10, 11, 14, 17, 19],
+      },
+      {
+        harmonics: [37, 44, 66, 70, 99, 105, 111],
+        error: 2.888,
+        degrees: [0, 3, 10, 11, 17, 18, 19],
+      },
+      {
+        harmonics: [37, 44, 83, 93],
+        error: 2.177,
+        degrees: [0, 3, 14, 16],
+      },
+    ]);
+  });
+});

src/analysis.ts

@@ -1,4 +1,5 @@
 import type { Scale } from "scale-workshop-core";
+import { arraysEqual, mmod, valueToCents } from "xen-dev-utils";
 
 const EPSILON = 1e-6;
 
@@ -112,3 +113,257 @@ export function intervalMatrix(scale: Scale) {
   }
   return result;
 }
+
+/**
+ * Calculate the maximum deviation of a set of pitches from an equal grid in pitch-space.
+ * @param pitches Array of pitches measured in cents.
+ * @param gridCents The distance in cents between two grid lines of the equal division e.g. `100.0`.
+ * @returns The maximum distance in cents from a grid "line" in the set.
+ */
+export function misalignment(pitches: number[], gridCents: number) {
+  const gridPitches = pitches.map(
+    (pitch) => Math.round(pitch / gridCents) * gridCents
+  );
+  let error = 0;
+  for (let i = 0; i < pitches.length; ++i) {
+    error = Math.max(error, Math.abs(pitches[i] - gridPitches[i]));
+  }
+  return error;
+}
+
+/**
+ * Align a set of pitches on an equal division of pitch-space such that the maximum absolute error is minimized.
+ * @param pitches An array of pitches measured in cents.
+ * @param gridCents The distance in cents between two grid lines of the equal division e.g. `100.0`.
+ * @returns The minimum misalignment achievable measured in cents and the pitches snapped to the grid.
+ */
+export function alignCents(pitches: number[], gridCents: number) {
+  // The error function we're trying to optimize is piecewise linear.
+
+  // Find the segment where the global optimum lies.
+  let optimalPitches: number[];
+  let minError = Infinity;
+  for (let i = 0; i < pitches.length; ++i) {
+    const aligned = pitches.map((pitch) => pitch - pitches[i]);
+    const error = misalignment(aligned, gridCents);
+    if (error < minError) {
+      optimalPitches = aligned;
+      minError = error;
+    }
+  }
+
+  // Calculate the shape of the segment.
+  const degrees = optimalPitches!.map((pitch) => Math.round(pitch / gridCents));
+  let minOffset = Infinity;
+  let maxOffset = -Infinity;
+  let root = Infinity;
+  for (let i = 0; i < degrees.length; ++i) {
+    const offset = optimalPitches![i] - degrees[i] * gridCents;
+    minOffset = Math.min(minOffset, offset);
+    maxOffset = Math.max(maxOffset, offset);
+    if (degrees[i] < root) {
+      root = degrees[i];
+    }
+  }
+
+  // Calculate minimum achievable absolute error.
+  const error = 0.5 * Math.abs(minOffset) + 0.5 * Math.abs(maxOffset);
+  // Move root to grid origin.
+  for (let i = 0; i < degrees.length; ++i) {
+    degrees[i] -= root;
+  }
+  return {
+    error,
+    degrees,
+  };
+}
+
+/**
+ * Align a set of ratios in frequency-space on an equal division of pitch-space such that the maximum absolute error is minimized.
+ * @param ratios An array of frequency ratios.
+ * @param gridCents The distance in cents between two grid lines of the equal division e.g. `100.0`.
+ * @returns The minimum misalignment achievable measured in cents and the pitches snapped to the grid.
+ */
+export function alignValues(ratios: number[], gridCents: number) {
+  return alignCents(ratios.map(valueToCents), gridCents);
+}
+
+function circleEqual(
+  a: number,
+  b: number,
+  equaveCents: number,
+  tolerance = EPSILON
+) {
+  return (
+    Math.abs(mmod(a - b + equaveCents * 0.5, equaveCents) - equaveCents * 0.5) <
+    tolerance
+  );
+}
+
+export type Shell = {
+  harmonics: number[];
+  error: number;
+  degrees: number[];
+  scaleDegrees: Set<number>;
+};
+
+/**
+ * Find the concordance shell centered around a root harmonic.
+ * @param rootHarmonic Root harmonic.
+ * @param maxHarmonic Maximum harmonic to consider for inclusion.
+ * @param tolerance Maximum deviation in cents from the equal division of pitch-space.
+ * @param division Number of equal divisions of the equave.
+ * @param equave The size of the interval of equivalence in cents.
+ * @returns An objects with
+ *  harmonics of the shell as an array of integers,
+ *  the harmonics snapped to the grid as an array of cents,
+ *  the minimum achievable misalignment of the harmonics within the grid measured in cents.
+ *  the shell as a reduced repeating scale of degrees (with only the zero endpoint included)
+ */
+export function concordanceShell(
+  rootHarmonic: number,
+  maxHarmonic: number,
+  tolerance: number,
+  divisions: number,
+  equaveCents = 1200.0
+): Shell {
+  const gridCents = equaveCents / divisions;
+  const harmonics = [rootHarmonic];
+  const degrees = [0];
+  const scaleDegrees = new Set([0]);
+  const fingerprints = [0];
+  let error = 0;
+  search: for (
+    let harmonic = rootHarmonic + 1;
+    harmonic <= maxHarmonic;
+    ++harmonic
+  ) {
+    const pitch = valueToCents(harmonic / rootHarmonic);
+    const degree = Math.round(pitch / gridCents);
+    const scaleDegree = mmod(degree, divisions);
+    if (scaleDegrees.has(scaleDegree)) {
+      for (const existing of fingerprints) {
+        if (circleEqual(pitch, existing, equaveCents)) {
+          continue search;
+        }
+      }
+    }
+    const deviation = Math.abs(pitch - degree * gridCents);
+    if (deviation < tolerance) {
+      harmonics.push(harmonic);
+      degrees.push(degree);
+      scaleDegrees.add(scaleDegree);
+      fingerprints.push(pitch);
+      error = Math.max(error, deviation);
+    }
+  }
+  return {
+    harmonics,
+    error,
+    degrees,
+    scaleDegrees,
+  };
+}
+
+function* subshells(shell: number[]) {
+  if (shell.length <= 1) {
+    return;
+  }
+  for (let i = 1; i < shell.length; ++i) {
+    const subshell = [...shell];
+    subshell.splice(i, 1);
+    yield subshell;
+  }
+}
+
+function subsetOf(a: number[], b: number[]) {
+  for (const value of a) {
+    if (!b.includes(value)) {
+      return false;
+    }
+  }
+  return true;
+}
+
+/**
+ * Find concordance shells based on the given harmonic.
+ * @param rootHarmonic Root harmonic.
+ * @param maxHarmonic Maximum harmonic to consider for inclusion.
+ * @param tolerance Maximum deviation in cents from the equal division of pitch-space.
+ * @param division Number of equal divisions of the equave.
+ * @param equave The size of the interval of equivalence in cents.
+ * @param minSize Minimum size of the shell to search for.
+ * @param maxShells Maximum number of shells to discover.
+ * @returns An array of objects with
+ *  harmonics of the shell as an array of integers,
+ *  the harmonics snapped to the grid as an array of cents,
+ *  the minimum achievable misalignment of the harmonics within the grid measured in cents.
+ *  the shell as a reduced repeating scale of degrees (with only the zero endpoint included)
+ */
+export function minimaxConcordanceShells(
+  rootHarmonic: number,
+  maxHarmonic: number,
+  tolerance: number,
+  divisions: number,
+  equaveCents = 1200.0,
+  minSize = 3,
+  maxShells = 100
+): Shell[] {
+  const gridCents = equaveCents / divisions;
+  // Calculate the largest possible shell if everything aligns perfectly.
+  const supershell = concordanceShell(
+    rootHarmonic,
+    maxHarmonic,
+    2 * tolerance,
+    divisions,
+    equaveCents
+  );
+
+  if (supershell.harmonics.length < minSize) {
+    return [];
+  }
+
+  const superAlignment = alignValues(supershell.harmonics, gridCents);
+  // Not gonna happen, but let's pretend to be hopeful.
+  if (superAlignment.error <= tolerance) {
+    return [{ ...supershell, ...superAlignment }];
+  }
+
+  // Start breaking the super-shell into smaller and smaller sub-shells that fit within the tolerance.
+  const result: Shell[] = [];
+  const badShells = [supershell.harmonics];
+  while (badShells.length) {
+    const shell = badShells.shift()!; // Break bigger shells first.
+    search: for (const subshell of subshells(shell)) {
+      for (const existing of result) {
+        if (subsetOf(subshell, existing.harmonics)) {
+          continue search;
+        }
+      }
+      for (const existing of badShells) {
+        if (arraysEqual(existing, subshell)) {
+          continue search;
+        }
+      }
+
+      const alignment = alignValues(subshell, gridCents);
+      if (alignment.error <= tolerance) {
+        const scaleDegrees = new Set(
+          alignment.degrees.map((degree) => mmod(degree, divisions))
+        );
+        result.push({
+          harmonics: subshell,
+          scaleDegrees,
+          ...alignment,
+        });
+        if (result.length >= maxShells) {
+          return result;
+        }
+      } else if (subshell.length > minSize) {
+        badShells.push(subshell);
+      }
+    }
+  }
+
+  return result;
+}

src/components/ScaleBuilder.vue

@@ -769,6 +769,7 @@ function copyToClipboard() {
       "
       @cancel="showEqualizeModal = false"
       :scale="scale"
+      :centsFractionDigits="centsFractionDigits"
     />
 
     <TemperModal