Other tunings

He wrote at the court of King Siṅghaṇa (सिङ्घण) II, in what is now Maharashtra, in the early 1200s. The book is the Saṅgītaratnākara (सङ्गीतरत्नाकर), the ocean of music. A śruti (श्रुति) is the small step Bharata (भरत) asked you to count. Śārṅgadeva (शार्ङ्गदेव) does not change that count. He names the twenty-two. The word in parentheses is a usual translation of the Sanskrit (संस्कृत) name, not a measurement of the step.

Śārṅgadeva (शार्ङ्गदेव) also counts nineteen svaras (स्वर), not only the seven. Śuddha (शुद्ध) means the plain form of a note. Vikṛta (विकृत) means a form that has been moved. Seven are plain, and twelve have moved. Cyuta (च्युत) means slipped: the note has moved by one śruti (श्रुति). Acyuta (अच्युत) means it has not. A sādhāraṇa (साधारण) note sits between the usual forms. Later writers attach ratios to some of these. He is naming the moves. The ratios are their reading.

• Cyuta-ṣaḍja (च्युत-षड्ज): Sa has slipped down one śruti (श्रुति), to make room for the raised Ni, kākalī (काकली).
• Acyuta-ṣaḍja (अच्युत-षड्ज): Sa stays where it was.
• Sādhāraṇa gāndhāra (साधारण गान्धार): A Ga between the usual ones. Later writers give this one the ratio 6/5, about 316 cents.
• Cyuta-madhyama (च्युत-मध्यम): Ma slipped down one śruti (श्रुति) when the raised Ga, antara (अन्तर), is the one being sung.
• Cyuta-pañcama (च्युत-पञ्चम): The lower Pa of the madhyama-grāma (मध्यम-ग्राम). Later writers give it 40/27.
• Kaiśikī niṣāda (कैशिकी निषाद): A Ni one śruti (श्रुति) above the lower Ni. Later writers give it 9/5, about 1018 cents.

Rāmāmātya (रामामात्य) wrote the Svaramelakalānidhi (स्वरमेलकलानिधि) in Vijayanagara in 1550. The older vīṇā (वीणा) on the Eastern page is a harp: you retune a string by moving a bridge. His instrument is the fretted vīṇā (वीणा) still played in South India, often called the Sarasvatī (सरस्वती) vīṇā (वीणा). Acala (अचल) means unmoving. An acala svara (स्वर) is a brass fret fixed in wax. Once it is set, that note stays for the whole performance.

He describes four playing strings, plus drone strings at the side. Mandra (मन्द्र) means the low octave. Anumandra (अनुमन्द्र) is the octave below that. Madhya (मध्य) is the middle octave. A fifth is the gap from Sa to Pa. An octave is the same note, twice as high.

Six frets on those strings give him fourteen notes he treats as usable. He sorts the rāgas (राग) of his time into twenty melas (मेल). A mela (मेल) is a parent scale: a choice of which form of each note a family of rāgas (राग) treats as home. The seventy-two of the next section are the later, fuller map of the same idea.

Veṅkaṭamakhin (वेङ्कटमखिन्) wrote the Caturdaṇḍīprakāśikā (चतुर्दण्डीप्रकाशिका) in Thanjavur, about 1620. A svarasthāna (स्वरस्थान) is one seat: one of twelve places in the octave, the place a fret marks. A scale still needs seven letter names, Sa Re Ga Ma Pa Dha Ni. Some seats are allowed to carry more than one name, and he counts sixteen names across the twelve seats. A melakarta (मेलकर्ता) is one way of choosing seven of those names, with the rule that the next name sits higher than the one before it. The seventy-two are all the choices that rule allows. The pitches are the ones the fretboard already had.

How the 72 are counted

The lower half of the scale is Re and Ga. The upper half is Dha and Ni. In each half, the second note has to sit higher than the first, and there are six ways to do that. Ma has two forms, plain and raised, so the same thirty-six pairs are written twice.

The book is the Saṅgītapārijāta (सङ्गीतपारिजात), from northern India, about 1665. A monochord is one string and a place to stop it. He tells you where to stop it. The sentences in quotation marks are the sense of his rules. The ratios beside them are the reading those stops give. He is talking about lengths.

Those stops are the kind Srinivasa (श्रीनिवास), on the Eastern page, builds into fourteen cuts on one string.

A well temperament spreads the Pythagorean comma, the 23.46 cents by which twelve pure fifths miss seven octaves, across several fifths instead of leaving it in one wolf. Circulating means you can move from key to key without retuning. Andreas Werckmeister published Musicalische Temperatur in Quedlinburg in 1691. His third recipe narrows four fifths, C–G, G–D, D–A, and B–F♯, by a quarter of that comma, and leaves the other eight pure.

J. S. Bach's Well-Tempered Clavier (1722 and 1742) is a prelude and fugue in each of the twelve major keys and each of the twelve minor keys, for a circulating temperament of this kind. Which of Werckmeister's recipes he used is not settled. The cents below are this recipe. The Western page has the short account. This is the full chart.

The twelve notes

Tracing the circle of fifths step-by-step from C4 = 0.00 ¢ (261.63 Hz):

1. C4: Tonic reference = 0.00 ¢ → 261.63 Hz.
2. G4 (Tempered Fifth 1): 0 + 696.09 = 696.09 ¢ → 391.10 Hz. (Narrowed by 5.9 ¢).
3. D4 (Tempered Fifth 2): 696.09 + 696.09 - 1200 = 192.18 ¢ → 292.28 Hz. (Narrowed by 5.9 ¢).
4. A4 (Tempered Fifth 3): 192.18 + 696.09 = 888.27 ¢ → 437.01 Hz. (Narrowed by 5.9 ¢).
5. E4 (pure fifth from A): 888.27 + 701.96 − 1200 = 390.22 ¢ → 327.77 Hz. C–G, G–D, and D–A were each narrowed, so the third C–E lands at 390.22 cents, about 4 cents wider than the pure 5/4 at 386.31.
6. B4 (Pure Fifth from E): 390.22 + 701.96 = 1092.18 ¢ → 491.64 Hz.
7. F#4 (Tempered Fifth 4): 1092.18 + 696.09 - 1200 = 588.27 ¢ → 367.51 Hz. (Final tempered fifth).
8. C#4 (Pure Fifth from F#): 588.27 + 701.96 - 1200 = 90.22 ¢ → 275.60 Hz.
9. G#4 (Pure Fifth from C#): 90.22 + 701.96 = 792.18 ¢ → 413.31 Hz.
10. Eb4 / D#4 (Pure Fifth from G#): 792.18 + 701.96 - 1200 = 294.13 ¢ → 310.08 Hz.
11. Bb4 (Pure Fifth from Eb): 294.13 + 701.96 = 996.09 ¢ → 465.11 Hz.
12. F4 (Pure Fifth from Bb): 996.09 + 701.96 - 1200 = 498.05 ¢ → 348.83 Hz.
13. Check: F to C: 498.05 + 701.96 = 1200.00 ¢. An exact pure fifth of 701.96 cents closes the circle!

The thirds are not all the same size

Equal means every step is the same size. Fifty-three of them fill one octave, so each is 1200 / 53, about 22.64 cents. That step is called Holder's comma, after William Holder, who wrote about it in the seventeenth century. Nicolaus Mercator had already noticed that fifty-three equal steps come close to a chain of pure fifths. In the 1910s Ernest Clements and K. B. Deval applied the same grid to Indian notes, in The Indian Temperament and Deval's book on the twenty-two śrutis (श्रुति).

The commas from the Western page sit near one of these steps. Near is not the same as equal. Bharata (भरत) did not write 81/80. Later writers use that fraction for his unit, and one step of this grid is about a cent wider than 81/80.

1 step = 1200 / 53 ≈ 22.64 cents

• 81/80, the syntonic comma: 21.51 cents. One step of this grid is 22.64, about 1.1 cents wider.
• Pythagorean comma: 23.46 cents, about 0.8 cents wider than one step.
• A pure fifth, 3/2: 701.96 cents. Thirty-one of these steps are 701.89 cents, about 0.07 cents flat.
• A pure major third, 5/4: 386.31 cents. Seventeen steps are 384.91 cents, about 1.4 cents flat.

Clements and Deval then line Bharata's (भरत) gap sizes up with whole numbers of these steps. The pure ratios in parentheses are the later reading, and they miss the equal steps by about a cent. The count of steps does fill the octave: three gaps of 4, two of 3, and two of 2, written as 9, 8, and 5 of these small steps.

A harmonium is a keyboard of metal reeds. The usual one has one reed for each of twelve names, tuned in equal steps. A tanpura (तानपुरा) is the drone. Its Pa and its Ga ring at the pure ratios, and a reed tuned 14 cents away beats against them. That clash, and the fact that a reed cannot glide, is the story on Eastern tunings. From 1940 to 1971 All India Radio kept the harmonium off classical broadcasts.

In 2007 Vidyadhar Oke, a physician in Mumbai, patented an instrument he called the 22-śruti (श्रुति) melodium. Sa and Pa keep one reed each. Each of the other ten keys has two reeds and a lever that chooses the lower or the higher. Ten keys times two, plus Sa and Pa, is twenty-two reeds. They are chosen ratios, a lower one and a higher one for each name. They are not Bharata's (भरत) twenty-two counted gaps, one reed for each gap.

A minor third is three piano steps, 300 cents, or 316 cents if it is the pure 6/5. A major third is four piano steps, 400 cents, or 386 if it is the pure 5/4. The note associated with the eighth-century lute player Manṣūr Zalzal sits between them, near 350 cents. Writers have suggested the ratios 11/9 and 27/22 for it. It is not one of those fractions in the way 3/2 is a fifth. Ears in this music treat it as its own note. Maqām Rast and maqām Bayati are two melody types that use it.

In the thirteenth century the Persian writer Ṣafī al-Dīn al-Urmawī built a scale of seventeen notes from pure fifths. In the nineteenth century Mīkhāʾīl Mishāqa, and then the 1932 Cairo congress, wrote Arab practice as twenty-four equal steps in the octave. A quartertone is half a piano step, 50 cents. Twenty-four of them fill 1200 cents. That grid is a modern way to write the older note down. It is not what Zalzal was counting.

A gamelan is that ensemble. A string's overtones fall at 2, 3, 4 times the note. A bronze bar does not. Its tuning is chosen for the bar, and two ensembles in the same town need not share one pitch. There are two scale patterns.

Just intonation means the notes are whole-number ratios. 5-limit, on the Western page, stops at the prime 5. 11-limit means a ratio may also use 7 and 11. Harry Partch sets this out in Genesis of a Music (1949). The 7/4 is a flat minor seventh, about 969 cents. The 11/8 is about 551 cents. He called the diagram of these ratios a tonality diamond. The instruments he built for them include a reed organ, the Chromelodeon, and two large struck and plucked ones, the Kithara and the Quadrangularis Reversum.

Heinz Bohlen, Kees van Prooijen, and John Pierce each described it, in the 1970s and 1980s. They call the 3/1 a tritave. Thirteen equal steps fill it. 1200 × log2(3) is about 1902 cents, and 1902 / 13 is about 146 cents a step. It is meant for a sound whose overtones are the odd numbers, 1, 3, 5, 7. A clarinet is the usual example, because its even overtones are weak. A piano string, with both even and odd overtones, is the wrong sound for it.