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Eastern Tunings

From Pandit Srinivasa's 17th-century geometric string divisions to Pandit V. N. Bhatkhande's 20th-century 22-Śruti and 10-Thāṭ system: the acoustic science, string geometry, and harmonic ratios of Indian swaras.

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Acoustic wire model · Sa = 240.0 Hz

Drag string vertically to pluck with velocity
सा Madhya Ṣaḍja (Sa) ŚUDDHA
Sounding length: 36.00 units · Formula: 240 * (36 / 36)

Full open wire of the Vīṇā. Foundational home pitch (ādhāra swara).

Active Frequency
240.00 Hz
Sa = 240 Hz
Harmonic Ratio
1 / 1
1.0000x
Cents (Octave)
0.0 ¢
0.0 ¢ vs 12-TET
Sounding Length
36.00 u
36 / 36 units
Comparison: Srinivasa (240.00 Hz) and Bhatkhande (240.00 Hz) are identical on Sa.
🎧 Hear सा (Sa) Across Eastern Tunings:
Custom Length Plucker:

1. The Chronological Evolution of Indian Tunings: Gramas to Thāṭs

The two-millennium journey of Indian tuning: from 22 relative śrutis on ancient harps to fixed brass frets on the Saraswati Vīṇā, the 72 Melakarta matrix, and modern 20th-century standardization.

Unlike Western music—which spent centuries searching for a single compromise temperament (culminating in 12-TET) to allow keyboard modulation across 24 keys—Indian classical music anchored itself to an immovable drone (the Adhāra Ṣaḍja). Over two millennia, Indian tuning evolved through four distinct historical epochs:

1.1 Bharata Muni (c. 200 BCE – 200 CE, Nāṭyaśāstra) — The Grāmas & The Two-Vīṇā Experiment

In the 28th chapter of the Nāṭyaśāstra, Bharata Muni codified the foundational tuning architecture of ancient India. Music was organized around two parent scales called Grāmas: the Ṣaḍja-grāma (starting on Sa) and the Madhyama-grāma (starting on Ma).

The intervals between notes were measured in Śrutis (microtonal units of audibility). Bharata established the famous foundational śruti distribution:

Ṣaḍja-Grāma Śruti Distribution:

Sa (4) — Re (3) — Ga (2) — Ma (4) — Pa (4) — Dha (3) — Ni (2) = 22 Śrutis

In ancient Grāma theory, the swara was located at the last śruti of its interval. Thus, Sa occupied the 4th śruti, Re the 7th, Ga the 9th, Ma the 13th, Pa the 17th, Dha the 20th, and Ni the 22nd.

The Caturguṇā Śaraṇā (The Two-Vīṇā Experiment):
To prove that an octave contains exactly 22 śrutis and discover the elemental unit of pitch, Bharata devised an ingenious acoustic experiment using two identical 7-string harp-like vīṇās tuned to the Ṣaḍja-grāma:
  1. The Invariable Reference (Dhruva Vīṇā): Kept unchanged as a control benchmark.
  2. The Variable Test Harp (Cala Vīṇā): In the first reduction (Prathamā Śaraṇā), the Pa string was tuned down until it matched the Pa of the Madhyama-grāma. This decrement revealed the fundamental acoustic microtone: the Pramāṇa Śruti, which is exactly the Syntonic Comma (81/80 = 21.51 cents)!
  3. Second Reduction: Lowered by another śruti, causing Ga and Ni of the Cala Vīṇā to coincide with Re and Dha of the Dhruva Vīṇā (a Dviśruti interval = 16/15, 111.7 ¢).
  4. Third Reduction: Lowered by a third śruti, causing Re and Dha to merge with Sa and Pa (a Triśruti interval = 10/9, 182.4 ¢).
  5. Fourth Reduction: Lowered by a fourth śruti, causing Sa, Ma, and Pa to merge with Ni, Ga, and Ma of the fixed vīṇā (a Chatuśruti interval = 9/8, 203.9 ¢).

Bharata's 9 Fundamental Swaras (7 Śuddha + 2 Vikrita)

Swara Ancient Name Śruti Span Cumulative Śruti Harmonic Ratio Cents Acoustic Role
सा (Sa) Ṣaḍja 4 śrutis 4 1 / 1 0.0 ¢ Tonic origin of the Ṣaḍja-grāma
रे (Re) Ṛṣabha 3 śrutis 7 10 / 9 182.4 ¢ Triśruti minor whole tone
ग (Ga) Gāndhāra 2 śrutis 9 32 / 27 294.1 ¢ Ancient Śuddha Ga (minor third)
ग॒ (Antara Ga) Antara Gāndhāra (Vikrita) +2 śrutis 11 5 / 4 386.3 ¢ Pure 5-limit Major Third (shifted up 2 śrutis)
म (Ma) Madhyama 4 śrutis 13 4 / 3 498.0 ¢ Pure fourth; immovable consonance (Samvādi)
प (Pa) Pañcama 4 śrutis 17 3 / 2 702.0 ¢ Pure fifth (in Madhyama-grāma, Pa is 16 śrutis = 40/27, 680.4 ¢)
ध (Dha) Dhaivata 3 śrutis 20 5 / 3 884.4 ¢ Triśruti major sixth
नि (Ni) Niṣāda 2 śrutis 22 16 / 9 996.1 ¢ Ancient Śuddha Ni (minor seventh)
नि॒ (Kākalī Ni) Kākalī Niṣāda (Vikrita) +2 śrutis 2 15 / 8 1088.3 ¢ Pure Major Seventh (shifted up 2 śrutis toward Sa)

1.2 Śārṅgadeva (1210–1247 CE, Saṅgītaratnākara) — The 22 Named Śrutis & 19 Swara Varieties

Writing at the court of King Singhana II in modern-day Maharashtra, Śārṅgadeva synthesized the entire ancient musicological corpus in the definitive text Saṅgītaratnākara ("Ocean of Music").

Śārṅgadeva assigned evocative Sanskrit names to each of the 22 śrutis. These names reflect the emotional aesthetics and psychoacoustic tension of the microtonal steps:

Śārṅgadeva's 22 Named Śrutis across the 7 Swaras:
  1. Sa (4 śrutis): 1. Tīvrā (sharp), 2. Kumudvatī (lotus-like), 3. Mandā (gentle), 4. Chandovatī (rhythmic)
  2. Re (3 śrutis): 5. Dayāvatī (compassionate), 6. Rañjanī (delighting), 7. Raktikā (passionate)
  3. Ga (2 śrutis): 8. Raudrī (fierce), 9. Krodhā (wrathful)
  4. Ma (4 śrutis): 10. Vajrikā (thunderbolt), 11. Prasāriṇī (expanding), 12. Prīti (loving), 13. Mārjanī (cleansing)
  5. Pa (4 śrutis): 14. Kṣiti (earthly), 15. Raktā (colored), 16. Sandīpanī (illuminating), 17. Ālāpinī (conversational)
  6. Dha (3 śrutis): 18. Madantī (intoxicating), 19. Rohiṇī (ascending), 20. Ramyā (delightful)
  7. Ni (2 śrutis): 21. Ugrā (formidable), 22. Kṣobhiṇī (agitating)

Śārṅgadeva expanded Bharata's system into 19 total swaras (7 Śuddha and 12 Vikrita). Crucially, he recognized shifts of Sa and Ma:

  • Cyuta-Ṣaḍja: When Sa drops 1 śruti to accommodate Kākalī Ni.
  • Acyuta-Ṣaḍja: When Sa holds firm on its 4th śruti.
  • Sādhāraṇa Gāndhāra: Ga shifted up 1 śruti (10th śruti = 6/5, 315.6 ¢).
  • Cyuta-Madhyama: Ma dropped 1 śruti when Antara Ga is sounded.
  • Cyuta-Pañcama: The lowered Pa of Madhyama-grāma (16th śruti, 40/27).
  • Kaisiki Niṣāda: Ni shifted up 1 śruti (21st śruti = 9/5, 1017.6 ¢).

1.3 Rāmāmātya (1550 CE, Svaramelakalānidhi) — The Fixed-Fret Vīṇā Revolution

In 16th-century Vijayanagara, Rāmāmātya revolutionized Indian music by moving away from harp-like open strings with movable bridges to the fretted Vīṇā (Saraswati Vīṇā).

Rāmāmātya eliminated movable tuning during performance by introducing Acala Svaras (fixed brass frets set in wax). He standardized the 4-string playing layout with 3 drone/talam side strings that defines South Indian classical music to this day:

Rāmāmātya's 4 Playing Strings (Tuned in Fifths & Fourths):
  1. Anumandra Sa: Lowest bass drone/bass string ($S_,$)
  2. Mandra Pa: Fifth below middle Sa ($P_,$ = 3/2 down)
  3. Mandra Sa: Octave below middle Sa ($S$)
  4. Madhya Ma (or Pa): Suddha Madhyama ($m$) or Panchama

Across just 6 physical frets on these strings, Rāmāmātya derived 14 functional swaras and classified all contemporary ragas into 20 Melas (parent scales). This was the direct ancestor of modern Carnatic fretboard geometry.

1.4 Veṅkaṭamakhin (1620 CE, Caturdaṇḍī Prakāśikā) — The 72 Melakarta Matrix

In Thanjavur, Veṅkaṭamakhin achieved the greatest combinatorial breakthrough in world musical scale theory: the 72 Melakarta Scheme.

Veṅkaṭamakhin observed that the human ear recognizes 12 physical fret positions (Swarasthanas) per octave, but modal music requires 7 distinct letter names (S, R, G, M, P, D, N). By establishing that certain physical positions can take on alternative theoretical names, he defined 16 nominal swaras across the 12 fret positions:

Fret # Swarasthana (Fret Location) Carnatic Swara Nomenclature Harmonic Ratio Cents Hindustani Equivalent
0 Ṣaḍja Ṣaḍja (Sa) 1 / 1 0.0 ¢ Sa
1 Śuddha Ṛṣabha Śuddha Ṛṣabha (R1) 16 / 15 111.7 ¢ Komal Re
2 Chatuśruti Ṛṣabha / Śuddha Gāndhāra Chatuśruti Ṛṣabha (R2) · Śuddha Gāndhāra (G1) 9 / 8 203.9 ¢ Śuddha Re
3 Ṣaṭśruti Ṛṣabha / Sādhāraṇa Gāndhāra Ṣaṭśruti Ṛṣabha (R3) · Sādhāraṇa Gāndhāra (G2) 6 / 5 (or 75/64) 315.6 ¢ Komal Ga
4 Antara Gāndhāra Antara Gāndhāra (G3) 5 / 4 386.3 ¢ Śuddha Ga
5 Śuddha Madhyama Śuddha Madhyama (M1) 4 / 3 498.0 ¢ Śuddha Ma
6 Prati Madhyama Prati Madhyama (M2) 45 / 32 (or 64/45) 590.2 ¢ Tīvra Ma
7 Pañcama Pañcama (Pa) 3 / 2 702.0 ¢ Pa
8 Śuddha Dhaivata Śuddha Dhaivata (D1) 8 / 5 813.7 ¢ Komal Dha
9 Chatuśruti Dhaivata / Śuddha Niṣāda Chatuśruti Dhaivata (D2) · Śuddha Niṣāda (N1) 5 / 3 884.4 ¢ Śuddha Dha
10 Ṣaṭśruti Dhaivata / Kaiśiki Niṣāda Ṣaṭśruti Dhaivata (D3) · Kaiśiki Niṣāda (N2) 9 / 5 (or 16/9) 1017.6 ¢ Komal Ni
11 Kākalī Niṣāda Kākalī Niṣāda (N3) 15 / 8 1088.3 ¢ Śuddha Ni
12 Tāra Ṣaḍja Tāra Ṣaḍja (S') 2 / 1 1200.0 ¢ Tāra Sa

The 72 Melakarta Combinatorial Math

By pairing the permutations of the lower tetrachord (Pūrvāṅga: R and G) with the upper tetrachord (Uttarāṅga: D and N):

  1. Pūrvāṅga combinations: There are 6 valid ordered pairs of (R, G) where G is higher in pitch than R: (R1,G1), (R1,G2), (R1,G3), (R2,G2), (R2,G3), (R3,G3).
  2. Uttarāṅga combinations: There are 6 valid ordered pairs of (D, N) where N is higher in pitch than D: (D1,N1), (D1,N2), (D1,N3), (D2,N2), (D2,N3), (D3,N3).
  3. With Śuddha Madhyama (M1): $6 \text{ (Pūrvāṅga)} \times 6 \text{ (Uttarāṅga)} = \mathbf{36 \text{ Melakartas}}$ (Chakras 1 to 6: Indu, Netra, Agni, Veda, Bana, Rutu).
  4. With Prati Madhyama (M2): Exactly identical $6 \times 6 = \mathbf{36 \text{ Melakartas}}$ (Chakras 7 to 12: Rishi, Vasu, Brahma, Disi, Rudra, Aditya).
  5. Total Mathematical Universe: $36 + 36 = \mathbf{72 \text{ Melakartas}}$!

2. Master Comparison Table: Srinivasa vs. Bhatkhande (Sa = 240 Hz)

Side-by-side comparison of physical string lengths, harmonic ratios, exact vibrational frequencies (āndolana), and cents deviations from 12-tone equal temperament.

Swara Type Srinivasa Length Srinivasa Ratio Srinivasa Hz Bhatkhande Ratio Bhatkhande Hz Delta (¢) Acoustic Relationship
Madhya Ṣaḍja (Sa) Śuddha 36.00 1 / 1 240.00 Hz 1 / 1 240.00 Hz 0.0 ¢ Unison anchor (ādhāra swara).
Komal Ṛṣabha (Re) Komal 33 1/3 (100/3) 27 / 25 259.20 Hz 16 / 15 256.00 Hz -21.5 ¢ Differs by exactly 81/80 (Syntonic Comma).
Śuddha Ṛṣabha (Re) Śuddha 32.00 9 / 8 270.00 Hz 9 / 8 270.00 Hz 0.0 ¢ Identical major tone (Chatuḥśruti Re).
Gāndhāra (Komal Ga) Komal 30.00 6 / 5 288.00 Hz 6 / 5 288.00 Hz 0.0 ¢ Identical pure minor third (6:5).
Tīvra / Śuddha Ga Ga 28 2/3 (86/3) 54 / 43 301.40 Hz 5 / 4 300.00 Hz -8.0 ¢ Srinivasa halfway division vs Bhatkhande pure 5:4 third.
Śuddha Madhyama (Ma) Śuddha 27.00 4 / 3 320.00 Hz 4 / 3 320.00 Hz 0.0 ¢ Identical pure fourth (4:3).
Tīvra Madhyama (Ma) Tīvra 25 1/9 (226/9) 162 / 113 344.07 Hz 45 / 32 337.50 Hz -33.4 ¢ Srinivasa trisection vs Bhatkhande 5-limit tritone.
Pañcama (Pa) Śuddha 24.00 3 / 2 360.00 Hz 3 / 2 360.00 Hz 0.0 ¢ Identical invariant pure fifth (3:2).
Komal Dhaivata (Dha) Komal 22 2/9 (200/9) 81 / 50 388.80 Hz 8 / 5 384.00 Hz -21.5 ¢ Differs by exactly 81/80 (Syntonic Comma).
Śuddha Dhaivata (Dha) Śuddha 21 1/3 (64/3) 27 / 16 405.00 Hz 5 / 3 400.00 Hz -21.5 ¢ Pythagorean 27/16 (3:2 from Re) vs 5-limit 5/3.
Niṣāda (Komal Ni) Komal 20.00 9 / 5 432.00 Hz 9 / 5 432.00 Hz 0.0 ¢ Identical harmonic minor seventh (432 Hz).
Tīvra / Śuddha Ni Ni 19 1/9 (172/9) 81 / 43 452.09 Hz 15 / 8 450.00 Hz -8.0 ¢ Srinivasa trisection vs Bhatkhande pure 15:8 seventh.
Tāra Ṣaḍja (Ṡa) Śuddha 18.00 2 / 1 480.00 Hz 2 / 1 480.00 Hz 0.0 ¢ Identical exact octave (2:1).
The Syntonic Comma (Pramāṇa Śruti) Discovery
Notice that for Komal Re, Komal Dha, and Śuddha Dha, the ratio between Srinivasa and Bhatkhande is precisely 81/80 = 1.0125 (21.51 cents). In Indian musicology, this interval is the ancient Pramāṇa Śruti; in Western acoustics, it is the Syntonic Comma. Srinivasa derived Dha by stacking a pure fifth onto Śuddha Re (9/8 * 3/2 = 27/16), producing the Pythagorean major sixth; Bhatkhande used the 5-limit pure major sixth (5/3). The difference (27/16) / (5/3) is exactly 81/80.

3. Pandit Srinivasa's 14 Physical Steps of String Division

Step-by-step geometric trisections and bisections on a 36-unit wire from Meru (nut = 0) to Ghudach (bridge = 36), authored in Rāgatattvavibodha.

  1. Step 1 — Tāra Ṣaḍja (Tāra Sa): Placed exactly at the midpoint of the wire.
    Distance from Meru = 18 units. Sounding length = 36 - 18 = 18 units.
    Ratio = 36 / 18 = 2/1 (480.00 Hz, 1200.00 cents).
  2. Step 2 — Madhya Ṣaḍja (Sa): The full sounding string from nut to bridge.
    Sounding length = 36 units. Ratio = 36 / 36 = 1/1 (240.00 Hz, 0.00 cents).
  3. Step 3 — Ati-Tāra Ṣaḍja: Midpoint between Tāra Sa and the bridge (Ghudach).
    Sounding length = 9 units. Ratio = 36 / 9 = 4/1 (960.00 Hz, 2400.00 cents).
  4. Step 4 — Śuddha Madhyama (Ma): Placed halfway between Madhya Sa and Tāra Sa.
    Distance from Meru = 18 / 2 = 9 units. Sounding length = 36 - 9 = 27 units.
    Ratio = 36 / 27 = 4/3 (320.00 Hz, 498.05 cents).
  5. Step 5 — Pañcama (Pa): Divide space between Sa and Tāra Sa (18 units) into three equal parts (6 units each); place Pa at 2 parts from Sa (12 units from Meru).
    Sounding length = 36 - 12 = 24 units.
    Ratio = 36 / 24 = 3/2 (360.00 Hz, 701.96 cents).
  6. Step 6 — Gāndhāra (Modern Komal Ga): Placed halfway between Sa (36) and Pa (24).
    Distance from Meru = 12 / 2 = 6 units. Sounding length = 36 - 6 = 30 units.
    Ratio = 36 / 30 = 6/5 (288.00 Hz, 315.64 cents).
  7. Step 7 — Śuddha Ṛṣabha (Modern Śuddha Re): Divide space between Sa (36) and Pa (24) into three equal parts (4 units each); place Re at 1 part from Sa (4 units from Meru).
    Sounding length = 36 - 4 = 32 units.
    Ratio = 36 / 32 = 9/8 (270.00 Hz, 203.91 cents).
  8. Step 8 — Dhaivata (Modern Śuddha Dha): A pure fifth above Śuddha Re using Ṣaḍja-Pañcama-bhāva (Re * 1.5).
    Sounding length = 32 * (2/3) = 21 1/3 units (64/3 units).
    Ratio = 36 / (64/3) = 108/64 = 27/16 (405.00 Hz, 905.87 cents).
  9. Step 9 — Niṣāda (Modern Komal Ni): Divide space between Pa (24) and Tāra Sa (18) into three equal parts (2 units each); place Ni at 2 parts from Pa.
    Distance from Meru = 12 + 4 = 16 units. Sounding length = 36 - 16 = 20 units.
    Ratio = 36 / 20 = 9/5 (432.00 Hz, 1017.60 cents).
  10. Step 10 — Komal Ṛṣabha: Divide space between Sa (36) and Śuddha Re (32) into three equal parts (4/3 units each); place Komal Re at 2 parts from Sa.
    Distance from Meru = 8/3 units = 2 2/3 units. Sounding length = 36 - 8/3 = 33 1/3 units (100/3 units).
    Ratio = 36 / (100/3) = 108/100 = 27/25 (259.20 Hz, 133.24 cents).
  11. Step 11 — Tīvra Gāndhāra (Modern Śuddha Ga): Placed halfway between Sa (36) and Śuddha Dha (21 1/3).
    Sounding length = (36 + 64/3) / 2 = 172/6 = 28 2/3 units (86/3 units).
    Ratio = 36 / (86/3) = 108/86 = 54/43 (301.40 Hz, 394.30 cents).
  12. Step 12 — Tīvratar Madhyama (Modern Tīvra Ma): Divide the interval between Tīvra Ga (28 2/3) and Tāra Sa (18) into three equal parts; place Tīvratar Ma at 1 part from Tīvra Ga.
    Difference = 86/3 - 54/3 = 32/3 units. One third = 32/9 = 3 5/9 units.
    Sounding length = 86/3 - 32/9 = 258/9 - 32/9 = 25 1/9 units (226/9 units).
    Ratio = 36 / (226/9) = 324/226 = 162/113 (344.07 Hz, 623.60 cents).
  13. Step 13 — Komal Dhaivata: A pure fifth above Komal Re using Ṣaḍja-Pañcama-bhāva (Komal Re * 1.5).
    Sounding length = (100/3) * (2/3) = 200/9 = 22 2/9 units.
    Ratio = 36 / (200/9) = 324/200 = 81/50 (388.80 Hz, 835.19 cents).
  14. Step 14 — Tīvra Niṣāda (Modern Śuddha Ni): Divide the interval between Pure Dha (21 1/3) and Tāra Sa (18) into three equal parts; place Tīvra Ni at 2 parts from Pure Dha towards Tāra Sa.
    Difference = 64/3 - 54/3 = 10/3 units. Two thirds = 20/9 = 2 2/9 units.
    Sounding length = 64/3 - 20/9 = 192/9 - 20/9 = 19 1/9 units (172/9 units).
    Ratio = 36 / (172/9) = 324/172 = 81/43 (452.09 Hz, 1096.26 cents).

4. Pandit V. N. Bhatkhande's System: Mathematical Derivation & Acoustic Logic

The architecture of modern Hindustani classical music: shifting to Bilawal Thāṭ, 5-limit Just Intonation, and the step-by-step harmonic derivation of all 12 swaras on a 36-unit wire.

In the early 20th century, Pandit Vishnu Narayan Bhatkhande (1860–1936) revolutionized Indian musicology by bridging ancient theory with contemporary performance practice across his six-volume Hindustani Sangeet Paddhati and Lakṣya-Saṅgītam. Recognizing that centuries of musical evolution had shifted away from the ancient Dorian baseline (Kāfī Thāṭ), Bhatkhande adopted Bilawal Thāṭ (the Western Major scale) as the primary Śuddha reference scale.

Acoustically, Bhatkhande grounded the 12 swaras in 5-limit Just Intonation. While medieval treatises like Srinivasa's Rāgatattvavibodha relied heavily on physical wire bisections and trisections (producing Pythagorean intervals for certain notes), Bhatkhande integrated the pure natural major third (Swayambhū Gāndhāra, ratio 5:4) into the harmonic matrix.

The Three Foundational Acoustic Generators

Every swara in Bhatkhande's 12-tone system is generated from three fundamental acoustic relationships:

  1. Ṣaḍja-Pañcama-bhāva (The Invariant Fifth = 3/2): The consonant ratio of 3:2 (701.96 cents). Used to generate Pa from Sa, Re from Pa, and Komal Ni from Komal Ga.
  2. Ṣaḍja-Madhyama-bhāva (The Invariant Fourth = 4/3): The consonant ratio of 4:3 (498.05 cents), which is the acoustic inversion of the pure fifth (2 / (3/2) = 4/3). Used to generate Śuddha Ma from Sa, and Komal Dha from Komal Ga.
  3. Swayambhū Gāndhāra (The Pure 5:4 Major Third): The 5th natural harmonic of a vibrating string, lowered two octaves (5 / 4 = 1.25, 386.31 cents). Used to generate Śuddha Ga from Sa, Śuddha Dha from Ma, Śuddha Ni from Pa, and Tīvra Ma from Re.

Step-by-Step Derivation of the 12 Swaras (Sa = 240 Hz, Length = 36 Units)

Using a standard 36-unit monochord wire anchored at Madhya Ṣaḍja (240 Hz), where sounding length l = 36 / Ratio and frequency f = 240 * Ratio:

  1. 1. Madhya Ṣaḍja (Sa — 1/1): The immutable tonic anchor (Ādhāra Swara).
    Ratio = 1/1. Sounding length = 36.00 units. Frequency = 240.00 Hz (0.00 cents).
  2. 2. Pañcama (Pa — 3/2): The pure consonant fifth above Sa (Ṣaḍja-Pañcama-bhāva).
    Ratio = 3/2. Sounding length = 36 * (2/3) = 24.00 units. Frequency = 240 * (3/2) = 360.00 Hz (701.96 cents).
  3. 3. Śuddha Madhyama (Ma — 4/3): The pure consonant fourth above Sa (Ṣaḍja-Madhyama-bhāva), or inversion of the fifth (2 / (3/2)).
    Ratio = 4/3. Sounding length = 36 * (3/4) = 27.00 units. Frequency = 240 * (4/3) = 320.00 Hz (498.05 cents).
  4. 4. Śuddha Gāndhāra (Ga — 5/4): The natural 5th harmonic (Swayambhū Svara) brought into the primary octave.
    Ratio = 5/4. Sounding length = 36 * (4/5) = 28.80 units (144/5 units). Frequency = 240 * (5/4) = 300.00 Hz (386.31 cents).
  5. 5. Śuddha Ṛṣabha (Re — 9/8): Cycle of fifths: a pure fifth above Pañcama (3/2 * 3/2 = 9/4), reduced to the home octave by dividing by 2.
    Ratio = (9/4) / 2 = 9/8 (Major Tone / Chatuḥśruti Re). Sounding length = 36 * (8/9) = 32.00 units. Frequency = 240 * (9/8) = 270.00 Hz (203.91 cents).
  6. 6. Śuddha Dhaivata (Dha — 5/3): A pure major third (5/4) above Śuddha Madhyama (Ma = 4/3): (4/3) * (5/4) = 5/3.
    Ratio = 5/3. Sounding length = 36 * (3/5) = 21.60 units (108/5 units). Frequency = 240 * (5/3) = 400.00 Hz (884.36 cents).
    Note on Srinivasa's difference: Srinivasa used the Pythagorean fifth above Re (9/8 * 3/2 = 27/16 = 405.00 Hz). The ratio (27/16) / (5/3) = 81/80 (the Syntonic Comma = 21.51 cents).
  7. 7. Śuddha Niṣāda (Ni — 15/8): A pure major third (5/4) above Pañcama (Pa = 3/2): (3/2) * (5/4) = 15/8.
    Ratio = 15/8 (Kākalī Niṣāda / pure leading tone). Sounding length = 36 * (8/15) = 19.20 units (96/5 units). Frequency = 240 * (15/8) = 450.00 Hz (1088.27 cents).
  8. 8. Komal Gāndhāra (Ga — 6/5): Pure natural minor third (6:5) above Sa, which is also the inversion of the major third below Pa: (3/2) / (5/4) = 6/5.
    Ratio = 6/5. Sounding length = 36 * (5/6) = 30.00 units. Frequency = 240 * (6/5) = 288.00 Hz (315.64 cents).
  9. 9. Komal Niṣāda (Ni — 9/5): A pure fifth (3/2) above Komal Gāndhāra (6/5): (6/5) * (3/2) = 9/5.
    Ratio = 9/5 (Harmonic minor seventh). Sounding length = 36 * (5/9) = 20.00 units. Frequency = 240 * (9/5) = 432.00 Hz (1017.60 cents).
  10. 10. Komal Dhaivata (Dha — 8/5): Inversion of the pure major third from the upper octave: 2 / (5/4) = 8/5, or a pure fourth above Komal Ga: (6/5) * (4/3) = 8/5.
    Ratio = 8/5 (Pure minor sixth). Sounding length = 36 * (5/8) = 22.50 units (45/2 units). Frequency = 240 * (8/5) = 384.00 Hz (813.69 cents).
    Note on Srinivasa's difference: Srinivasa's Komal Dha was 81/50 (388.80 Hz). The difference (81/50) / (8/5) = 81/80 (Syntonic Comma = 21.51 cents).
  11. 11. Komal Ṛṣabha (Re — 16/15): Inversion of the pure major seventh: 2 / (15/8) = 16/15, or a pure major third below Śuddha Ma: (4/3) / (5/4) = 16/15.
    Ratio = 16/15 (Diatonic semitone). Sounding length = 36 * (15/16) = 33.75 units (135/4 units). Frequency = 240 * (16/15) = 256.00 Hz (111.73 cents).
    Note on Srinivasa's difference: Srinivasa's Komal Re was 27/25 (259.20 Hz). The difference (27/25) / (16/15) = 81/80 (Syntonic Comma = 21.51 cents).
  12. 12. Tīvra Madhyama (Ma — 45/32): A pure major third (5/4) above Śuddha Ṛṣabha (9/8): (9/8) * (5/4) = 45/32.
    Ratio = 45/32 (5-limit Augmented Fourth / Prati Madhyama). Sounding length = 36 * (32/45) = 25.60 units (128/5 units). Frequency = 240 * (45/32) = 337.50 Hz (590.22 cents).
  13. 13. Tāra Ṣaḍja (Ṡa — 2/1): The exact higher octave tonic.
    Ratio = 2/1. Sounding length = 36 * (1/2) = 18.00 units. Frequency = 240 * 2 = 480.00 Hz (1200.00 cents).

Bhatkhande's 10 Parent Thāṭs

Using these 12 swaras, Bhatkhande classified all standard Hindustani rāgas into 10 parent scales (Thāṭs):

  • Bilawal Thāṭ (Primary Śuddha Scale): S R G M P D N Ṡ (All Śuddha notes — Western Major Scale).
  • Kalyan Thāṭ: S R G M' P D N Ṡ (Tīvra Ma).
  • Khamaj Thāṭ: S R G M P D n Ṡ (Komal Ni).
  • Kafi Thāṭ: S R g M P D n Ṡ (Komal Ga, Komal Ni — the ancient Śuddha scale).
  • Asavari Thāṭ: S R g M P d n Ṡ (Natural Minor Scale).
  • Bhairav Thāṭ: S r G M P d N Ṡ (Komal Re, Komal Dha).
  • Bhairavi Thāṭ: S r g M P d n Ṡ (All 4 Komal swaras).
  • Todi Thāṭ: S r g M' P d N Ṡ (Komal Re, Komal Ga, Tīvra Ma, Komal Dha).
  • Poorvi Thāṭ: S r G M' P d N Ṡ (Komal Re, Tīvra Ma, Komal Dha).
  • Marwa Thāṭ: S r G M' P D N Ṡ (Komal Re, Tīvra Ma, Śuddha Dha).

5. The 22 Śrutis: Microtonal Architecture & Four Harmonic Classes

How 22 microtonal intervals arise from the interaction of whole tones, minor tones, and syntonic commas in live performance.

In Indian classical music, an octave is not merely 12 static notes, but a continuum punctuated by 22 Śrutis (audible microtonal gradations). Vocalists and instrumentalists micro-adjust pitches depending on whether a note is an ascending leading tone, a sustained resting pitch (nyāsa), or oscillating in āndolan.

Acoustically, these 22 intervals do not represent an equal division of the octave. Instead, they are generated by four natural interval sizes:

Śruti Class Sanskrit Name Harmonic Ratio Size in Cents Acoustic Role
Pramāṇa Śruti प्रमाण श्रुति 81 / 80 21.51 ¢ The fundamental microtonal comma (Syntonic Comma). The difference between a Major Tone (9/8) and Minor Tone (10/9).
Laghu Śruti लघु श्रुति 25 / 24 70.67 ¢ Minor chromatic semitone. The distance between Komal Re (16/15) and its lower microtonal inflection.
Khaṇḍa Śruti खण्ड श्रुति 16 / 15 111.73 ¢ Diatonic semitone. The foundational step from Śuddha Ga to Śuddha Ma, and Śuddha Ni to Tāra Sa.
Pūrṇa Śruti पूर्ण श्रुति 9 / 8 203.91 ¢ Major whole tone (Chatuḥśruti). The foundational step from Sa to Śuddha Re, and Pa to Śuddha Dha.

6. Where Did 12-TET Come From: Why & How Equal Temperament Emerged

The mathematical dilemma, the straight frets of lutes and pipa, Prince Zhu Zaiyu's 1584 25-digit calculation, the Western industrial piano, and the 30-year Harmonium Ban on All India Radio.

6.1 The Mathematical Impasse: Why Pure Intervals Cannot Form a Closed Circle

The emergence of 12-Tone Equal Temperament (12-TET) was not an artistic preference for sound; it was an unavoidable mathematical and mechanical compromise.

In nature, the harmonic series produces pure, beatless intervals governed by simple integer ratios: the Octave (2:1), the Perfect Fifth (3:2), and the Major Third (5:4). However, powers of 2 and powers of 3 are mathematically incommensurable:

2^m ≠ (3/2)^n for any positive integers m and n.

If you attempt to complete a "Circle of Fifths" by stacking 12 pure fifths of 3:2, you travel through seven octaves:

  • 12 Pure Fifths: (3/2)^12 = 531,441 / 4,096 ≈ 129.7463...
  • 7 Pure Octaves: 2^7 = 128.0000
  • The Pythagorean Comma: (531,441 / 4,096) / 128 = 531,441 / 524,288 ≈ 1.01364 (23.46 cents).

Stacking 12 pure fifths overshoots the 7th octave by nearly a quarter of a semitone (23.46 cents)! Similarly, four pure fifths (9/8 * 9/8 = 81/64 = 407.82 cents) overshoot the natural pure major third (5/4 = 80/64 = 386.31 cents) by the Syntonic Comma (81/80 = 21.51 cents).

This meant that on any keyboard or instrument tuned to pure natural intervals (Just Intonation or Meantone), playing in the home key sounded heavenly, but modulating or transposing to distant keys resulted in horrific, howling dissonances known as "Wolf Intervals" (e.g. a fifth of 680 cents or 738 cents).

6.2 The Physical Driver: Straight Frets on Stringed Instruments

Keyboard instruments (pipe organs, harpsichords) could survive with unequal temperaments because each key struck a dedicated, independently tunable pipe or string. Players simply avoided remote keys, or instrument builders installed split accidentals (split black keys for D# vs. Eb).

The real mechanical crisis occurred on fretted stringed instruments—most notably the Renaissance lute and viol in Europe, and the pipa and ruan in China:

  • On a lute or viol, frets are tied straight across the fingerboard underneath all six or more strings.
  • Because adjacent strings are tuned in fourths or fifths, every fret position serves multiple strings simultaneously.
  • If a player set fret 4 to produce a pure 5:4 major third (386 cents) on the lowest string, that identical fret position would produce a horribly flat, out-of-tune fourth or fifth on the adjacent string!

Fretted instruments demanded that the octave be partitioned into 12 identical geometric intervals across the entire neck. In 1581, Italian theorist Vincenzo Galilei (father of astronomer Galileo Galilei) published his landmark treatise Dialogo della musica antica, et della moderna, advocating the empirical Rule of 18 (string shortening ratio 18:17 ≈ 1.0588), which created frets of approximately 99 cents—a close practical approximation of equal temperament.

6.3 The 25-Digit Mathematical Breakthrough: Prince Zhu Zaiyu (1584)

While empirical approximations existed, calculating the exact mathematical equal semitone requires computing the twelfth root of two: 2^(1/12). In the 16th century, before the invention of calculus or logarithms, this was an extraordinary mathematical hurdle.

The exact mathematical solution was achieved in 1584 in Ming Dynasty China by Prince Zhu Zaiyu (朱載堉, 1536–1611), an imperial scholar, acoustician, and mathematician:

  • The Problem: Ancient Chinese music relied on the Sanfen Sunyi (三分損益) method (alternately adding and subtracting one-third of the pipe length, generating pure fifths). Like the Pythagorean circle, after 12 steps, it failed to return to the fundamental pitch pipe (Huangzhong / Yellow Bell).
  • The 81-Position Abacus: Using a massive counting board and an 81-position abacus, Prince Zhu executed precise successive square and cube root extractions.
  • The Exact Value: In his 1584 treatise Lüxue Xinshuo (律學新說 - A New Account of the Science of the Pitch Pipes), followed by Lülü Jingyi (律呂精義, 1595/1606), Prince Zhu calculated 2^(1/12) to 25 decimal places:
    1.059463094359295264561825...
  • Physical Realization (Xinlü): He went beyond pure math: recognizing that end-correction and pipe diameter affect pitch, he formulated exact lengths, diameters, and bore volumes for a set of 12 bamboo pitch pipes and a 12-string zither, proving that the 12th step returned perfectly to the fundamental.

In Europe, Dutch polymath Simon Stevin independently computed the equal temperament ratio using decimal fractions around 1585–1605 (published in Vande Spiegheling der Singconst). French friar Marin Mersenne published accurate string lengths for equal temperament in Harmonie Universelle (1636).

6.4 Western Industrialization & The Acoustic Sacrifice

Despite being mathematically understood by 1600, European classical music did not immediately adopt 12-TET. Throughout the 17th and 18th centuries, musicians preferred Well-Temperaments (such as Andreas Werckmeister's Werckmeister III in 1691), which eliminated wolf fifths while preserving distinct "key colors" (Affekt) across all 24 keys—a concept immortalized by J. S. Bach's The Well-Tempered Clavier (1722).

The universal dominance of 12-TET arrived in the 19th century during the Industrial Revolution:

  • Mass production of cast-iron frame pianos by manufacturers like Steinway, Broadwood, and Bösendorfer demanded a single universal factory tuning standard.
  • Romantic composers (Wagner, Liszt, Chopin, Mahler) pushed harmonic modulation to its limits, constantly roving through all 12 key centers within a single piece.
  • The Acoustic Cost: In exchange for total modulatory freedom, 12-TET sacrificed pure harmonic resonance. While 12-TET fifths are only 1.96 cents flat (acoustically imperceptible), 12-TET major thirds are 400.00 cents—a massive 13.69 cents SHARPER than nature's pure 5/4 third (386.31 cents). In sustained chords, 12-TET major thirds produce rapid, audible acoustic beating (~16 beats/sec at C4-E4). Minor thirds are 15.64 cents FLAT (300.00 cents vs pure 315.64 cents).

6.5 The Harmonium Controversy in India: 12-TET vs. 22 Śrutis & The 30-Year AIR Ban

In the mid-19th century, French inventor Alexandre Debain patented the portable reed organ (harmonium). European Christian missionaries brought it to India to accompany church hymns. Indian instrument makers quickly modified it with a hand-pumped bellows so that it could be played while seated cross-legged on the floor.

The harmonium spread rapidly across India because it was loud, portable, easy to learn, and—unlike delicate stringed instruments like the Vīṇā, Sarangi, or Sitar—its fixed metal reeds did not go out of tune in India's intense monsoon humidity.

However, the introduction of the harmonium sparked one of the fiercest cultural and acoustic battles in Indian music history:

  1. The Melodic vs. Harmonic Chasm: Western music is harmonic and polyphonic; it changes keys constantly, making 12-TET an acceptable compromise. Indian classical music is strictly modal, melodic, and anchored to a continuous drone (Tanpura). It never modulates between keys! Because the tonic (Sa) never changes, the absolute acoustic purity of intervals against the drone is paramount.
  2. Destruction of the 22 Śrutis: A fixed 12-TET keyboard has only 12 rigid pitches per octave. It cannot produce the microtonal distinctions essential to rāgas: Komal Re in Rāga Bhairav (Chatuḥśruti Re, 111.7 cents) is acoustically distinct from Komal Re in Rāga Todi (Ati-Komal Re, ~70 cents). On a harmonium, both are forced onto the exact same out-of-tune reed.
  3. Loss of Continuous Glides (Mīṇḍ & Āndolan): The soul of Indian vocal and string music lies between the notes—in continuous glides (mīṇḍ), microtonal oscillations (āndolan), and delicate graces (gamak). A keyed keyboard produces discrete, chopped notes with no capacity for pitch inflection.
  4. Rabindranath Tagore's Opposition: Nobel laureate Rabindranath Tagore was an outspoken critic, famously decrying the harmonium as "the bane of Indian music" and barring it completely from his university at Shantiniketan.
  5. The Historic 31-Year All India Radio Ban (1940–1971): In 1940, Lionel Fielden (Controller of Broadcasting) and English composer John Foulds instituted an official ban prohibiting the harmonium from all All India Radio (AIR) classical broadcasts. Backed by purists and master vocalists, the ban declared that the tempered scale damaged the vocal training of Indian artists and corrupted listener sensitivity to microtonal śrutis. The ban lasted for over three decades until 1971, when it was conditionally relaxed for light vocal accompaniment.
East vs. West Tuning Philosophy
The Western musical trajectory sacrificed the acoustic purity of natural harmonics (giving up pure thirds and distinct key flavors) to achieve infinite horizontal and vertical harmonic modulation across 12 equal keys.
Indian classical music deliberately rejected equal temperament, sacrificing harmonic modulation to preserve infinite vertical depth, pure consonances against the drone, and microtonal expressive nuance across 22 Śrutis.

7. Primary Sources & References

Historical Sanskrit treatises, Ming Dynasty records, European acoustic texts, and foundational musicological literature.

Primary Sanskrit Treatises

  1. Pandit Srinivasa: Rāgatattvavibodha (रागत्तत्वविबोध, Late 17th Century).
    Particularly Chapter 3 (Vīṇāvādanādhyāya). Critical Sanskrit edition with commentary edited by Pandit Vibhukumar S. Shastri, Gaekwad's Oriental Series, Baroda. First rigorous mathematical formulation of the 14-step geometric string division on a 36-unit Vīṇā wire.
  2. Pandit Ahobala: Saṅgītapārijāta (सङ्गीतपारिजात, c. 1665 CE).
    The pioneer Sanskrit work establishing physical string lengths on the Vīṇā for twelve swaras; translated into Persian by Dinānātha in 1724 CE. Edited by Kalinda, Hathras: Sangeet Karyalaya.
  3. Pandit Vishnu Narayan Bhatkhande: Hindustani Sangeet Paddhati (हिन्दुस्तानी सङ्गीत पद्धति, 4 Vols., 1910–1932, Bombay).
    The definitive modern codification of North Indian classical music, establishing the 10 Thāṭs, 22 Śrutis, 5-limit Just Intonation swara derivations, and the adoption of Bilawal as the primary Śuddha scale.
  4. Pandit Vishnu Narayan Bhatkhande: Śrīmāl-Lakṣya-Saṅgītam (लक्ष्यसङ्गीतम्, 1910) and A Comparative Study of Some of the Leading Music Systems of the 15th, 16th, 17th and 18th Centuries (1914).
  5. Bharata Muni: Nāṭyaśāstra (नाट्यशास्त्र, c. 200 BCE – 200 CE).
    Chapter 28 (Jāti-Vikalpa and Śruti-Nidarśana). The foundational codification of the 22 Śrutis and the two-harp demonstration (Dvāviṁśati-Śruti Calā-Dhruvā Vīṇā).
  6. Śārṅgadeva: Saṅgītaratnākara (सङ्गीतरत्नाकर, c. 1210–1247 CE).
    Chapter 1 (Svarādhyāya). Adyar Library critical edition edited by Pandit S. Subrahmanya Sastri.

Treatises on 12-TET & Western Temperament History

  1. Prince Zhu Zaiyu (朱載堉): Lüxue Xinshuo (律學新說 - A New Account of the Science of the Pitch Pipes, 1584) and Lülü Jingyi (律呂精義, 1595/1606).
    The world's first exact calculation of the twelfth root of two (2^(1/12)) to 25 decimal digits using an 81-position counting abacus. See Joseph Needham, Science and Civilisation in China, Vol. 4, Part 1 (Cambridge University Press, 1962), and Kenneth Robinson, A Critical Study of Chu Tsai-yü's Contribution to the Theory of Equal Temperament in Chinese Music (Franz Steiner Verlag, 1980).
  2. Simon Stevin: Vande Spiegheling der Singconst (c. 1585–1605).
    First European decimal calculation of equal temperament ratios; edited by D. Bierens de Haan (Amsterdam, 1884).
  3. Vincenzo Galilei: Dialogo della musica antica, et della moderna (Florence, 1581).
    Pioneering formulation of the "Rule of 18" (18:17 ratio) for fretting lutes and viols to approximate equal temperament.
  4. Marin Mersenne: Harmonie Universelle (Paris, 1636).
    Formulation of string vibration physics and precise calculation of equal-tempered string lengths.
  5. Andreas Werckmeister: Musicalische Temperatur (Frankfurt & Leipzig, 1691).
    Codification of circulating well-temperaments that paved the way toward equal temperament without destroying key character.

Modern Acoustic & Musicological Scholarship

  1. Dr. B. Chaitanya Deva: The Music of India: A Scientific Study (Munshiram Manoharlal, New Delhi, 1981).
    Exhaustive acoustic analysis of string physics, formants, frequency ratios, and the physical realization of śrutis on stringed instruments.
  2. Alain Daniélou: The Ragas of Northern Indian Music (Barrie and Rockliff, London, 1968; reprinted Munshiram Manoharlal, 2010).
    Detailed mathematical treatment of 22-śruti frequency ratios, cyclic comma divisions, and cent measurements.
  3. Prof. P. Sambamoorthy: South Indian Music (Vols. I–VI, The Indian Music Publishing House, Chennai).
    Essential reference for comparative microtonal intervals, string geometry, and Indian tuning systems.
  4. Dr. H. V. Modak: Propounding the Musical Scale of Pandit Ahobala (Sangeet Natak Akademi Journal, Vol. 13, 1969).
    Mathematical reconstruction and physical verification of medieval Indian wire measurements.
  5. Dr. S. K. Saxena: The Winged Form: Avestan and Indian Music (Sangeet Natak Akademi, New Delhi).
    Historical documentation of the harmonium's introduction into India, aesthetic controversies, and the All India Radio 1940–1971 broadcast ban.