SymbolicPower Algebra

Understanding how RMT Compose handles irrational numbers like 2^(1/12) with full algebraic precision.

The Problem with Irrational Numbers

Equal temperament tuning produces irrational numbers. For example:

12-TET semitone = 2^(1/12) = 1.0594630943592953...

This number cannot be exactly represented as a fraction or floating-point number. If we just used floats:

// Float multiplication loses precision
const semitone = Math.pow(2, 1/12)  // 1.0594630943592953
const octave = semitone ** 12       // 1.9999999999999998 (not exactly 2!)

SymbolicPower: Algebraic Precision

RMT Compose uses SymbolicPower to preserve the algebraic structure of expressions.

How It Works

Instead of computing 2^(1/12) as a float, SymbolicPower stores:

• Base: 2
• Exponent: 1/12

This representation is exact and allows perfect algebraic simplification.

Automatic Simplification

// 2^(1/12) × 2^(1/12) = 2^(2/12) = 2^(1/6)
// Exponents add: 1/12 + 1/12 = 2/12 = 1/6

// Full octave: (2^(1/12))^12 = 2^1 = 2 (exactly!)

RMT Compose handles this automatically when you chain TET operations.

Creating SymbolicPower Values

Basic Syntax

// 2 to the power of 1/12
2^(1/12)

// 3 to the power of 1/13 (Bohlen-Pierce)
3^(1/13)

Legacy JavaScript syntax

// 2 to the power of 1/12
new Fraction(2).pow(new Fraction(1, 12))

// 3 to the power of 1/13 (Bohlen-Pierce)
new Fraction(3).pow(new Fraction(1, 13))

Chained Operations

// Major third in 12-TET (4 semitones)
base.f * 2^(4/12)

// Same result, built step by step
base.f * 2^(1/12) * 2^(1/12) * 2^(1/12) * 2^(1/12)

Both produce exactly 2^(4/12) = 2^(1/3).

Legacy JavaScript syntax

// Major third in 12-TET (4 semitones)
module.baseNote.getVariable('frequency')
  .mul(new Fraction(2).pow(new Fraction(4, 12)))

// Same result, built step by step
module.baseNote.getVariable('frequency')
  .mul(new Fraction(2).pow(new Fraction(1, 12)))
  .mul(new Fraction(2).pow(new Fraction(1, 12)))
  .mul(new Fraction(2).pow(new Fraction(1, 12)))
  .mul(new Fraction(2).pow(new Fraction(1, 12)))

Algebraic Rules Applied

Multiplication

When multiplying SymbolicPower values with the same base:

a^m × a^n = a^(m+n)

// Example: 2^(1/12) × 2^(3/12) = 2^(4/12)
2^(1/12) * 2^(3/12)

Division

a^m ÷ a^n = a^(m-n)

// Going down a semitone
frequency / 2^(1/12)
// = frequency × 2^(-1/12)

Legacy JavaScript syntax

frequency.div(new Fraction(2).pow(new Fraction(1, 12)))

Power of Power

(a^m)^n = a^(m×n)

// Octave check: (2^(1/12))^12 = 2^1 = 2

Mixed Operations with Fractions

SymbolicPower values can multiply with regular fractions:

// 440 × 2^(7/12) = 659.25... Hz (perfect fifth in 12-TET)
440 * 2^(7/12)

The system tracks both components:

• Rational part: 440
• Irrational part: 2^(7/12)

Legacy JavaScript syntax

new Fraction(440).mul(new Fraction(2).pow(new Fraction(7, 12)))

The ≈ Display

When a value contains SymbolicPower components, the Variable Widget shows:

≈ 659.26 Hz

The ≈ indicates:

1. The displayed number is an approximation
2. The actual stored value is algebraically exact
3. Further calculations use the exact symbolic form

Why This Matters

Scenario: Building a Full Chromatic Scale

Without SymbolicPower:

// Float accumulation error
let freq = 440
for (let i = 0; i < 12; i++) {
  freq *= 1.0594630943592953
}
// freq = 879.9999999999999 (not exactly 880!)

With SymbolicPower:

// Algebraic precision
// 440 × (2^(1/12))^12 = 440 × 2 = 880 exactly

Scenario: Transposition

When transposing a melody:

// Transpose up a major third (4 semitones)
originalFreq * 2^(4/12)

The system preserves relationships:

If note1 = 440 × 2^(3/12)
And note2 = note1 × 2^(4/12)
Then note2 = 440 × 2^(7/12)  // Exact!

Legacy JavaScript syntax

originalFreq.mul(new Fraction(2).pow(new Fraction(4, 12)))

Advanced: Different Bases

Bohlen-Pierce (Base 3)

// Tritave (3:1) divided into 13 parts
3^(1/13)

SymbolicPower handles any integer base:

• Base 2: Standard octave-based temperaments
• Base 3: Tritave-based scales like Bohlen-Pierce
• Other bases: Experimental tuning systems

Legacy JavaScript syntax

new Fraction(3).pow(new Fraction(1, 13))

Combining Different Bases

When multiplying different bases, the system tracks them separately:

// 2^(1/12) × 3^(1/13) - tracked as compound symbolic value

Internal Representation

For developers, here's how SymbolicPower works internally:

Structure

{
  type: 'symbolic_power',
  base: Fraction(2),      // The base number
  exponent: Fraction(1, 12),  // The exponent
  coefficient: Fraction(440)  // Optional rational multiplier
}

Evaluation Pipeline

1. Parse: Expression text → AST
2. Compile: AST → Bytecode (SymbolicPower opcodes)
3. Evaluate: Stack VM evaluates, preserving symbolic structure
4. Display: Evaluate to float only for final display

Bytecode Operations

The binary evaluator has specific opcodes for symbolic operations:

• POW: Creates SymbolicPower from base and exponent
• MUL_SYM: Multiplies preserving symbolic structure
• DIV_SYM: Divides preserving symbolic structure

Practical Examples

12-TET Scale with Perfect Octave

// Root
frequency: base.f

// Each subsequent note adds one semitone
// Note N: frequency × 2^(N/12)

// After 12 notes, we get:
frequency: base.f * 2^(12/12)
// = baseFreq × 2 (exactly one octave, no drift!)

Legacy JavaScript syntax

// Root
frequency: module.baseNote.getVariable('frequency')

// Each subsequent note adds one semitone
// Note N: frequency × 2^(N/12)

// After 12 notes, we get:
frequency: module.baseNote.getVariable('frequency')
  .mul(new Fraction(2).pow(new Fraction(12, 12)))
// = baseFreq × 2 (exactly one octave, no drift!)

Stacking Fifths

// Perfect fifth in 12-TET = 7 semitones
// Stack 12 fifths: (2^(7/12))^12 = 2^7 = 128 (7 octaves exactly)

// But musically, 12 pure fifths = (3/2)^12 = 129.746...
// The Pythagorean comma! 12-TET eliminates this by distributing the error.

Verification

Test that 12 semitones = 1 octave:

// Build note 12 semitones up
base.f * 2^(12/12)

// This simplifies to:
// baseFreq × 2^1 = baseFreq × 2

// The evaluated value will be exactly 2× the base frequency

Legacy JavaScript syntax

const twelveUp = module.baseNote.getVariable('frequency')
  .mul(new Fraction(2).pow(new Fraction(12, 12)))

Limitations

What SymbolicPower Handles

• Powers of integers: 2^(1/12), 3^(1/13)
• Products of symbolic powers
• Mixed rational and irrational values

What Requires Approximation

• Addition of irrational values: 2^(1/12) + 2^(1/19) (no closed form)
• Transcendental operations on symbolic values
• Nested irrational exponents

For these cases, the system falls back to high-precision float approximation.

Debugging Tips

Check Symbolic Preservation

1. Create a note with TET expression
2. Look for ≈ in the display
3. Chain 12 semitones and verify exact octave

Verify Algebraic Simplification

// This should display exactly 2× the base frequency
frequency: base.f * 2^(1/12) * 2^(1/12) * ... // (repeat 12 times)
// Or simply:
frequency: base.f * 2^(12/12)

Legacy JavaScript syntax

frequency: module.baseNote.getVariable('frequency')
  .mul(new Fraction(2).pow(new Fraction(1, 12)))
  .mul(new Fraction(2).pow(new Fraction(1, 12)))
  // ... (repeat 12 times)

Next Steps

• Complex Dependencies - Build sophisticated note relationships
• Microtonal Composition - Apply symbolic algebra to microtonal music
• Expression Compiler - Technical deep dive