Generalize Diamond-mos notation to absurd scales

ref #27

src/__tests__/notation.spec.ts

@@ -1,8 +1,24 @@
 import {describe, expect, it} from 'vitest';
-import {generateNotation} from '../notation';
+import {generateNotation, nthNominal} from '../notation';
 import {dot} from 'xen-dev-utils';
 
-describe('Diamond mos notation generator', () => {
+describe('Generalized Diamond-mos nominals', () => {
+  it('has 17 standard nominals', () => {
+    expect([...Array(17).keys()].map(nthNominal).join('')).toBe(
+      'JKLMNOPQRSTUVWXYZ'
+    );
+  });
+  it('has 289 two-character nominals', () => {
+    for (let i = 0; i < 289; ++i) {
+      expect(nthNominal(17 + i)).toHaveLength(2);
+    }
+  });
+  it('has JJJ as the first three-character nominal', () => {
+    expect(nthNominal(17 + 17 * 17)).toBe('JJJ');
+  });
+});
+
+describe('Diamond-mos notation generator', () => {
   it('generates the config for diatonic major', () => {
     const notation = generateNotation('LLsLLLs');
     const basic = [2, 1];
@@ -141,8 +157,13 @@ describe('Diamond mos notation generator', () => {
     expect(scale.has('Z')).toBe(true);
   });
 
-  it('rejects above nominal Z', () => {
-    expect(() => generateNotation('LsLsLsLsLsLsLsLsLs')).toThrow();
+  it('accepts above nominal Z', () => {
+    const {scale} = generateNotation('LsLsLsLsLsLsLsLsLs');
+    expect(scale.has('J')).toBe(true);
+    expect(scale.has('Z')).toBe(true);
+    expect(scale.has('JJ')).toBe(true);
+    expect(scale.has('JK')).toBe(false);
+    expect(scale.has('KJ')).toBe(false);
   });
 
   it('it rejects all L', () => {

src/notation.ts

@@ -4,7 +4,7 @@ import {mosGeneratorMonzo} from './helpers';
 /**
  * Valid nominals for absolute pitches in [Diamond mos notation](https://en.xen.wiki/w/Diamond-mos_notation).
  */
-export type DiamondMosNominal =
+export type DiamondMosAlphabet =
   | 'J'
   | 'K'
   | 'L'
@@ -53,7 +53,7 @@ export type DiamondMosNotation = {
   /**
    * Counts of [L, s] steps for every available nominal with J at unison. Add equaves to reach other octaves.
    */
-  scale: Map<DiamondMosNominal, MosMonzo>;
+  scale: Map<string, MosMonzo>;
   /**
    * Interval of equivalence / octave.
    */
@@ -72,6 +72,45 @@ export type DiamondMosNotation = {
   brightGenerator: MosMonzo;
 };
 
+/** Single characters of valid nominals. */
+export const DIAMOND_MOS_ALPHABET: DiamondMosAlphabet[] = [
+  'J',
+  'K',
+  'L',
+  'M',
+  'N',
+  'O',
+  'P',
+  'Q',
+  'R',
+  'S',
+  'T',
+  'U',
+  'V',
+  'W',
+  'X',
+  'Y',
+  'Z',
+];
+
+/**
+ * Obtain the 0-indexed nth generalized Diamond-mos nominal.
+ * @param n Index of the nominal.
+ * @returns A single character from J through Z or a multi-character string like 'JJ' starting from n = 17.
+ */
+export function nthNominal(n: number): string {
+  if (n < 0 || !Number.isInteger(n)) {
+    throw new Error('Invalid nominal index');
+  }
+  if (n >= DIAMOND_MOS_ALPHABET.length) {
+    return (
+      nthNominal(Math.floor(n / DIAMOND_MOS_ALPHABET.length) - 1) +
+      DIAMOND_MOS_ALPHABET[n % DIAMOND_MOS_ALPHABET.length]
+    );
+  }
+  return DIAMOND_MOS_ALPHABET[n];
+}
+
 /**
  * Generate configuration for [Diamond mos notation](https://en.xen.wiki/w/Diamond-mos_notation).
  *
@@ -80,13 +119,13 @@ export type DiamondMosNotation = {
  * @returns Configuration for notation software.
  */
 export function generateNotation(mode: string): DiamondMosNotation {
-  const scale = new Map<DiamondMosNominal, MosMonzo>();
-  let code = 'J'.charCodeAt(0);
+  const scale = new Map<string, MosMonzo>();
+  let i = 0;
   const monzo: MosMonzo = [0, 0];
   let hasLarge = false;
   let hasSmall = false;
   for (const character of mode) {
-    scale.set(String.fromCharCode(code++) as DiamondMosNominal, [...monzo]);
+    scale.set(nthNominal(i++), [...monzo]);
     if (character === 'L') {
       monzo[0]++;
       hasLarge = true;
@@ -100,9 +139,6 @@ export function generateNotation(mode: string): DiamondMosNotation {
   if (!hasLarge || !hasSmall) {
     throw new Error("The scale must contain both 'L' and 's' steps.");
   }
-  if (code > 'Z'.charCodeAt(0) + 1) {
-    throw new Error('Out of Diamond mos nominals.');
-  }
   const equave: MosMonzo = [...monzo];
   const numPeriods = gcd(equave[0], equave[1]);
   const period: MosMonzo = [equave[0] / numPeriods, equave[1] / numPeriods];