# Śrutis — the 22 microtones

Optional companion deep dive from Theory’s Śruti section — **not** a step on the
numbered home path (Theory → Practice → SOVT → …).

Classical texts divide the octave into **22 śrutis** — pitch steps finer than a
Western semitone. This page is one coherent **hearing model**: start from the seven
swaras, pass through the familiar **12 tone classes**, then keep both the lower and
higher just seats of each movable note. The measuring step is **pramāṇa = 81/80**.

Interactive hearing (pick Sa, synth tones): [`srutis.html`](srutis.html).
Short version: [Theory → Śruti](/theory/eastern/#shruti).

**How to read this.** Indian classical pitch is *relative*: Sa is whatever tonic you
choose. Frequencies and string lengths below are **ratios from Sa**, not fixed hertz.
Where a teaching video uses Sa = 100 Hz, that is only an arithmetic scale.

> Ratios follow a standard modern reconstruction of Bharata’s three śruti sizes
> (pūrṇa · nyūna · pramāṇa) — including the 22-śruti hearing model associated with
> **Dr. Vidyadhar Oke** (and the 22-śruti harmonium). Other models (e.g. equal 22-EDO)
> disagree. Treat this as a clear way to *hear* the lattice, not the only truth.

---

## Hear your śrutis

On [`srutis.html`](srutis.html), choose Sa (same C2–B4 range as Practice). Each ▶
plays that śruti as `Sa × ratio`. Classical lists also give each seat a **proper
name** (Tīvra, Kumudvatī, Mandā…) — those names belong to the *śruti*, not the
sung swara. Sa sits on **Tīvra**; that is not the same word as *tivra Ma*.

**Bold 12-tone rows** are Phase 1 (Sa, Pa, high Sa) and Phase 2 (the ten primary
swaras). Those seats are the **modern 12 notes** (thaat / keyboard / Practice map).
Blank 12-tone cells are Phase 3 — a pramāṇa away, not a thirteenth key. Once you
hit the octave at high Sa, the cycle of 22 śrutis **resets**, which is why row 23
shares the name **Tīvra** with row 1.

Hz below are the textbook column **Sa = A3 = 220 Hz** (`f = 220 × ratio`). The
interactive page also shows live Hz for whatever Sa you pick. Set Sa to **A3** to
hear these numbers.

| # | Śruti | Traditional name | 12-tone | Via | Ratio | Cents | Hz @ A3 |
|---|-------|------------------|---------|-----|-------|------:|--------:|
| 1 | Sa | Tīvra (Teevra) | **Sa** | — | 1/1 | 0.0 | 220.00 |
| 2 | Lower komal Re | Kumudvatī | — | pūrṇa | 256/243 | 90.2 | 231.77 |
| 3 | Higher komal Re | Mandā | **komal Re** | pramāṇa | 16/15 | 111.7 | 234.67 |
| 4 | Lower śuddha Re | Chhandovatī | — | nyūna | 10/9 | 182.4 | 244.44 |
| 5 | Higher śuddha Re | Dayāvatī | **Re** | pramāṇa | 9/8 | 203.9 | 247.50 |
| 6 | Lower komal Ga | Rañjanī | — | pūrṇa | 32/27 | 294.1 | 260.74 |
| 7 | Higher komal Ga | Raktikā | **komal Ga** | pramāṇa | 6/5 | 315.6 | 264.00 |
| 8 | Lower śuddha Ga | Raudrī (Roudri) | **Ga** | nyūna | 5/4 | 386.3 | 275.00 |
| 9 | Higher śuddha Ga | Krodhā | — | pramāṇa | 81/64 | 407.8 | 278.44 |
| 10 | Lower śuddha Ma | Vajrikā | **Ma** | pūrṇa | 4/3 | 498.0 | 293.33 |
| 11 | Higher śuddha Ma | Prasāriṇī | — | pramāṇa | 27/20 | 519.6 | 297.00 |
| 12 | Lower tivra Ma | Prīti | **tivra Ma** | nyūna | 45/32 | 590.2 | 309.38 |
| 13 | Higher tivra Ma | Mārjanī | — | pramāṇa | 729/512 | 611.7 | 313.24 |
| 14 | Pa | Kṣiti (Kshiti) | **Pa** | pūrṇa | 3/2 | 702.0 | 330.00 |
| 15 | Lower komal Dha | Raktā | — | pūrṇa | 128/81 | 792.2 | 347.65 |
| 16 | Higher komal Dha | Sandīpanī | **komal Dha** | pramāṇa | 8/5 | 813.7 | 352.00 |
| 17 | Lower śuddha Dha | Ālāpinī | **Dha** | nyūna | 5/3 | 884.4 | 366.67 |
| 18 | Higher śuddha Dha | Madantī | — | pramāṇa | 27/16 | 905.9 | 371.25 |
| 19 | Lower komal Ni | Rohiṇī | — | pūrṇa | 16/9 | 996.1 | 391.11 |
| 20 | Higher komal Ni | Ramyā | **komal Ni** | pramāṇa | 9/5 | 1017.6 | 396.00 |
| 21 | Lower śuddha Ni | Ugrā | **Ni** | nyūna | 15/8 | 1088.3 | 412.50 |
| 22 | Higher śuddha Ni | Kṣobhinī | — | pramāṇa | 243/128 | 1109.8 | 417.66 |
| 23 | High Sa | Tīvra (Teevra) | **Sȧ** | pūrṇa | 2/1 | 1200.0 | 440.00 |

---

## From 7 notes to 22 śrutis

Seven swaras: **Sa Re Ga Ma Pa Dha Ni**.

They become **12** when you add komal Re / Ga / Dha / Ni and tivra Ma. **Sa** and
**Pa** are *achal* (immovable) — they do not take komal/tivra forms in the basic
system. That twelve-tone map is the working alphabet for thaat, harmonium, and most
apps.

**22 śrutis** sit under that map:

- Sa and Pa have **one seat each**.
- Every other named note has a **lower + higher** variant.
- 20 variants + Sa + Pa = **22**.

A **śruti** is a pitch *point*. A **swara** is the sung note — often with *andolan*
(a gentle oscillation) around a chosen śruti. **Nāda** is continuous living sound,
the substance those points sit in.

So: **7** = sung names. **12** = identifiable seats for teaching and keys (the
**12-tone** column in Hear your śrutis). **22** = finer lattice for intonation —
two useful just flavours of the same named note, a pramāṇa apart.

---

## Perfect consonance vs 12-TET

Western equal temperament (12-TET) builds twelve notes by splitting the octave into
**twelve equal ratio steps**. Each semitone is exactly

`2^(1/12) ≈ 1.05946`

**Cents reminder:** 1200 cents = one octave; a 12-TET semitone is **100 cents**;
twelve steps → `2^(12/12) = 2/1`.

That is a brilliant engineering compromise: every key sounds the same, and you can
modulate freely. But none of the “simple” intervals except the octave are pure.

### Harmonic series (Sa = 100 Hz as a teaching scale)

If Sa is taught as 100 Hz, the first useful harmonics are:

| Harmonic | Hz | Swara |
|----------|----|-------|
| 1 | 100 | Sa |
| 2 | 200 | high Sa |
| 3 | 150 | Pa |
| 5 | 125 | Ga |

So **Sa : Ga : Pa = 100 : 125 : 150 = 1/1 : 5/4 : 3/2** — perfect consonance.

12-TET on the same Sa ≈ **100 : 126 : 149.83**. Piano Ga is ~**14 cents sharp** of
just Ga; piano Pa is ~**2 cents flat** of just Pa. To match the drone: flatten piano
Ga ~14¢, raise Pa ~2¢.

A 12-TET fifth is `2^(7/12) ≈ 1.4983`, slightly flat of just Pa `3/2 = 1.5`.
A 12-TET major third is `2^(4/12) = 2^(1/3) ≈ 1.2599`, sharp of just Ga `5/4 = 1.25`.

| | Just / śruti path (this page) | 12-TET (twelfth root of 2) |
|--|--|--|
| **How notes are made** | Stack small-integer ratios (3/2, 4/3, 5/4…) and keep nearby commas | One geometric step: multiply by `2^(1/12)` twelve times |
| **Octave math** | Products of fractions land on `2/1` | `(2^(1/12))^12 = 2` by design |
| **Consonance** | Chosen intervals are acoustically “clean” (simple beat ratios) | Every interval slightly mistuned so all keys match |
| **Density** | More than 12 useful seats (here: 22 steps) because commas are kept | Exactly 12 pitch classes; no room for a separate lower/higher Ga |
| **Trade-off** | Beautiful drones & rāga shades; keys are not all equivalent | Keyboard freedom & harmony in every key; thirds/fifths a bit impure |

So the systems answer different questions. Twelfth-root-of-2 asks: *“How do I carve
one octave into twelve equal, interchangeable steps?”* The śruti / just path asks:
*“How do I sit notes on simple vibration ratios from Sa — and what tiny gaps open
when two good paths almost meet?”* Ordinary harmoniums and many apps use 12-TET keys;
the voice can still aim at just placement against the drone.

---

## How a rāga chooses a śruti

Same named notes can sit on **different śrutis**. The rāga picks the flavour.

**Bhoopali vs Deshkar.** Both use Sa Re Ga Pa Dha (no Ma, no Ni). Bhoopali takes the
**lower** Ga and Dha; Deshkar takes the **higher**. Same five names; two different
lattices.

**Todi vs Bhairav.** Both use komal Re. Todi’s Re is the extra-flat **lower** komal;
Bhairav’s Re sits **higher**. Calling both “komal Re” is true and not enough.

A guru often says “a little higher” or “a little lower” rather than naming a ratio.
The reconstruction on this page gives exact fractions for those two seats. In a DAW
you can microtune to the same ratios. **Dr. Vidyadhar Oke** built a 22-śruti
harmonium so the hand can reach the seats the voice already knows.

This is still one hearing model: useful for practice, not a claim that every gharānā
sits every swara on these exact fractions.

---

## The string picture + three gap sizes

Picture a string. Open (full vibrating length) is **Sa** `1/1`. Stop at the
**midpoint** and you get high Sa `2/1`. Pitch in this model settles at **22 points**
between those two.

The tiny gap between the **lower** and **higher** flavour of the *same* named note is
**pramāṇa** `81/80`. On a 1000 mm string, lower śuddha Re vibrates at 900 mm, higher
śuddha Re at 888.89 mm:

`900 / 888.89 = 1.0125 = 81/80`.

Three just step sizes fill Sa → high Sa:

| Size | Ratio | ≈ cents | Role |
|------|-------|---------|------|
| **Pramāṇa** | 81/80 | 21.5 | Measuring step — between lower and higher of the same note |
| **Nyūna** | 25/24 | 70.7 | “Small” chromatic-ish step |
| **Pūrṇa** | 256/243 | 90.2 | “Full” limma (Pythagorean semitone) |

Ten of the twenty-two successive steps are pramāṇa, five are nyūna, seven are pūrṇa.
Their product is exactly **2/1** — one octave.

---

## Placement on a 1000 mm string

Take open-string vibrating length = **1000 mm** at Sa. Then:

- **Vibrating length** = `1000 / ratio`
- **From nut** = `1000 − length` (distance from the Sa end to the stopping point)

| # | Seat | Ratio | Vibrating length (mm) | From nut (mm) |
|---|------|-------|----------------------:|--------------:|
| 1 | Sa | 1/1 | 1000 | 0 |
| 2 | Lower komal Re | 256/243 | 949.22 | 50.78 |
| 3 | Higher komal Re | 16/15 | 937.50 | 62.50 |
| 4 | Lower śuddha Re | 10/9 | 900 | 100 |
| 5 | Higher śuddha Re | 9/8 | 888.89 | 111.11 |
| 6 | Lower komal Ga | 32/27 | 843.75 | 156.25 |
| 7 | Higher komal Ga | 6/5 | 833.33 | 166.67 |
| 8 | Lower śuddha Ga | 5/4 | 800 | 200 |
| 9 | Higher śuddha Ga | 81/64 | 790.12 | 209.88 |
| 10 | Lower śuddha Ma | 4/3 | 750 | 250 |
| 11 | Higher śuddha Ma | 27/20 | 740.74 | 259.26 |
| 12 | Lower tivra Ma | 45/32 | 711.11 | 288.89 |
| 13 | Higher tivra Ma | 729/512 | 702.33 | 297.67 |
| 14 | Pa | 3/2 | 666.67 | 333.33 |
| 15 | Lower komal Dha | 128/81 | 632.81 | 367.19 |
| 16 | Higher komal Dha | 8/5 | 625 | 375 |
| 17 | Lower śuddha Dha | 5/3 | 600 | 400 |
| 18 | Higher śuddha Dha | 27/16 | 592.59 | 407.41 |
| 19 | Lower komal Ni | 16/9 | 562.50 | 437.50 |
| 20 | Higher komal Ni | 9/5 | 555.56 | 444.44 |
| 21 | Lower śuddha Ni | 15/8 | 533.33 | 466.67 |
| 22 | Higher śuddha Ni | 243/128 | 526.75 | 473.25 |
| 23 | High Sa | 2/1 | 500 | 500 |

**Landmarks from the nut:** Re 100 mm, Ga 200 mm, Ma 250 mm, Dha 400 mm, high Sa
500 mm (the lower śuddha seats, plus the octave).

Same numbering as Hear your śrutis: **1–23** (22 śrutis + high Sa). The 12-tone
seats are the same Phase 1–2 rows as above.

---

## Derive the ratios

In just intonation you **multiply** ratios to stack intervals (“add” them on the
string). Three phases get you from Sa to all 22 seats.

> Quick check: **Pa = 3/2** is right. The partner that completes the octave is
> **Ma = 4/3** (the fourth), not Ga. And yes:
> `Pa × Ma = (3/2)×(4/3) = 2/1` — fifth + fourth = octave.
> Ga’s just seat is the sweeter third `5/4`.

### Phase 1 — anchors (not split)

These three are *achal* in this model: one seat each, never a lower/higher pair.
They are the 12-tone **Sa**, **Pa**, and **Sȧ** rows in Hear your śrutis.

1. **Sa** — home. `1/1`
2. **High Sa** — the octave (double the frequency / half the vibrating length). `2/1`
3. **Pa** — the fifth: the strongest consonant after the octave. `3/2`

### Phase 2 — ten primary swaras

Build the familiar just seats from those anchors (and from the harmonic third).

4. **Ma** — what remains of an octave after a fifth: `2 ÷ 3/2 = 4/3`.
5. **Ga** — just major third (harmonic 5): `5/4`.
6. **Re** — two fifths from Sa, drop an octave:
   `(3/2)×(3/2) = 9/4`, then `(9/4)÷2 = 9/8`.
7. **Dha** — Ga under Ma: `(5/4)×(4/3) = 5/3`.
8. **Ni** — Ga under Pa: `(5/4)×(3/2) = 15/8`.
9. **Komal Ga** — just minor third: `6/5`.
10. **Komal Dha** — `8/5`.
11. **Komal Ni** — `9/5`.
12. **Komal Re** — just minor second: `16/15`.
13. **Tivra Ma** — Ga × Re: `(5/4)×(9/8) = 45/32`.

These ten, plus Sa and Pa, are the “first arrivals” — the rest of the **12-tone**
column in Hear your śrutis. Each movable one still needs its partner a pramāṇa away
(blank 12-tone rows).

### Phase 3 — pramāṇa 81/80 partners

The syntonic comma `81/80` is the gap between two good paths to almost the same note.
Pure fifths also make a Ga at `81/64` (four fifths, folded into the octave). Versus
just Ga `5/4`:

`(81/64) ÷ (5/4) = 81/80`.

Same gap for Re flavours: `(9/8) ÷ (10/9) = 81/80`.

- **Divide by 81/80** to get the **lower** partner:
  `256/243`, `10/9`, `32/27`, `128/81`, `16/9`.
- **Multiply by 81/80** to get the **higher** partner:
  `81/64`, `27/20`, `729/512`, `27/16`, `243/128`.

### Why Re arrives high and Ma arrives low

Fifths **up** from Sa land on the **higher** flavour: Re = `9/8` (two fifths).
Fifths **down** (or “what’s left of the octave”) land on the **lower** flavour:
Ma = `4/3`. So the Phase-2 primary for Re is already the higher seat; the primary
for Ma is already the lower seat. The missing twin is one pramāṇa away in the other
direction. The Ga check is the same story: Pythagorean Ga `81/64` sits a pramāṇa
*above* just Ga `5/4`.

---

## Building the octave

Another way to the same 22 seats: start at Sa = `1/1` and multiply by the next step
size in this fixed order (pūrṇa / pramāṇa / nyūna from the reconstruction):

```
P · Pr · N · Pr · P · Pr · N · Pr · P · Pr · N · Pr · P · P · Pr · N · Pr · P · Pr · N · Pr · P
```

(P = pūrṇa, Pr = pramāṇa, N = nyūna.)

After 22 multiplications you land on taar Sa. Common just swaras fall out naturally —
each with its nearby shade a pramāṇa or nyūna away.

### Full pitch table (ratios from Sa)

| # | Via | Ratio | ≈ cents | Seat |
|---|-----|-------|---------|------|
| 0 | — | 1/1 | 0.0 | **Sa** |
| 1 | pūrṇa | 256/243 | 90.2 | Lower komal Re |
| 2 | pramāṇa | 16/15 | 111.7 | Higher komal Re |
| 3 | nyūna | 10/9 | 182.4 | Lower śuddha Re |
| 4 | pramāṇa | 9/8 | 203.9 | Higher śuddha Re |
| 5 | pūrṇa | 32/27 | 294.1 | Lower komal Ga |
| 6 | pramāṇa | 6/5 | 315.6 | Higher komal Ga |
| 7 | nyūna | 5/4 | 386.3 | Lower śuddha Ga |
| 8 | pramāṇa | 81/64 | 407.8 | Higher śuddha Ga |
| 9 | pūrṇa | 4/3 | 498.0 | Lower śuddha Ma |
| 10 | pramāṇa | 27/20 | 519.6 | Higher śuddha Ma |
| 11 | nyūna | 45/32 | 590.2 | Lower tivra Ma |
| 12 | pramāṇa | 729/512 | 611.7 | Higher tivra Ma |
| 13 | pūrṇa | 3/2 | 702.0 | **Pa** |
| 14 | pūrṇa | 128/81 | 792.2 | Lower komal Dha |
| 15 | pramāṇa | 8/5 | 813.7 | Higher komal Dha |
| 16 | nyūna | 5/3 | 884.4 | Lower śuddha Dha |
| 17 | pramāṇa | 27/16 | 905.9 | Higher śuddha Dha |
| 18 | pūrṇa | 16/9 | 996.1 | Lower komal Ni |
| 19 | pramāṇa | 9/5 | 1017.6 | Higher komal Ni |
| 20 | nyūna | 15/8 | 1088.3 | Lower śuddha Ni |
| 21 | pramāṇa | 243/128 | 1109.8 | Higher śuddha Ni |
| 22 | pūrṇa | 2/1 | 1200.0 | **Sȧ** |

---

## Reading the tables

**Hearing table** (Hear your śrutis — one merged table)

- **#** — 1 = Sa, 23 = octave (same as the 1000 mm table).
- **Śruti** — lower / higher seat in this reconstruction.
- **Traditional name** — classical śruti name (Tīvra, Kumudvatī…). Sa’s name Tīvra ≠ tivra Ma. Row 23 repeats Tīvra because the 22-cycle resets.
- **12-tone** — the matching modern / thaat note (Phase 1–2 only). Bold / leaf-green rows are those 12 keys; blank rows are Phase 3 commas.
- **Via** — which size brought you here from the previous pitch.
- **Ratio / cents** — exact just ratio and cents above Sa.
- **Hz** — frequency for your chosen Sa (updates live on the HTML page).
- **Hz @ A3** — textbook column: Sa = A3 = 220 Hz (`f = 220 × ratio`). Set Sa to A3 to hear those numbers.

**1000 mm table**

- **Vibrating length** — `1000 / ratio`.
- **From nut** — how far from the open-string Sa end you have stopped the string.

Day-to-day riyaz still uses the twelve-tone map on
[Practice](practice.html). Use this page when you want to *hear* why classical
writers said the octave was denser than twelve keys, and why a rāga might want the
lower Ga or the higher Re.

---

## See also

- [/theory/eastern/#shruti](/theory/eastern/#shruti) — Pitch stage overview (short teaser)
- [index.html](index.html) — numbered home path (this page is not a step on it)
- [practice.html](practice.html) — daily riyaz on the 12-tone map
- [Ear training](/tools/ear-training.html) — mystery swaras against Sa