Understanding SymbolicPower
Equal temperament produces irrational numbers. 2^(1/12) = 1.0594630943592953… cannot be written as a fraction. This page shows you exactly what RMT Compose does with such a value — where it is exact, where it is not, and how to tell which you are looking at.
Prerequisites: Microtonal Composition, or equivalent comfort with ^ expressions.
Read this before you rely on exactness
The headline you may have heard — "chain twelve semitones and you get a perfect octave" — is not true on the evaluator that ships. It is true for 2^(12/12). It is false for 2^(1/12) * 2^(1/12) * … twelve times. The rest of this page explains the difference, because the difference is the whole story.
Two kinds of power
The ^ operator hands its result to a rational-power check before anything else. That check has two ways to succeed.
1. The exponent resolves to an integer.
2^(12/12) # 12/12 = 1 -> exactly 2
2^3 # exactly 8
2^(-1) # exactly 1/22. The result is a perfect n-th root.
4^(1/2) # exactly 2
8^(1/3) # exactly 2In both cases you get an exact rational. The note is not flagged, not hatched, and its Evaluated: line shows a plain value with no ≈.
Everything else is irrational, and irrational is where the guarantees stop:
2^(1/12) # irrational
3^(1/13) # irrational
2^(7/12) # irrationalWhen ^ yields an irrational, the evaluator does one thing: it converts the result to an approximating fraction and sets a corruption flag on the property. There is no symbolic value sitting on the stack afterwards. The approximation is the value from that point on.
What corruption looks like
| Where | Directly corrupted | Transitively corrupted |
|---|---|---|
| Note rectangle | Crosshatch (two diagonals) | Single diagonal hatch |
Note widget Evaluated: | ≈<value>, italic amber | ≈<value>, italic amber |
"Directly corrupted" means this note's own expression produced the irrational. "Transitively corrupted" means it inherited one from something it references. Both show ≈; only the hatching distinguishes them.
The flag propagates: give note 1 a TET frequency, write note 2 as [1].f * (3/2), and note 2 is hatched too — its value is rational arithmetic performed on an approximation.
How much error, exactly
This is measurable, so measure it. Base frequency 263 Hz:
| Expression | Evaluates to | Exact? | Flagged? |
|---|---|---|---|
base.f * 2^(12/12) | 526 | yes — exactly | no |
base.f * 4^(1/2) | 526 | yes — exactly | no |
base.f * 2^(1/12) * 2^(1/12) * … (×12) | 525.999999999985675 | no | yes |
base.f * 2^(1/12) | 278.6387938165… | no | yes |
Twelve chained semitones land 1.4 × 10⁻¹¹ short of the octave. Each ^ rounds once, and twelve roundings accumulate. It is a small error and it will never be audible — but it is an error, and a page that promises "no drift" is lying to you.
The lesson is not "avoid TET". It is: write the power you mean. 2^(7/12) rounds once. 2^(1/12) seven times rounds seven times. Prefer the closed form.
What the simplifier does — and refuses to do
Saving a variable runs the expression through a simplifier. It has real symbolic algebra in it, and it does merge like bases:
| You type | Saved as |
|---|---|
base.f + base.f | 2 * base.f |
2 * (1/2) * base.f | base.f |
4^(1/2) * base.f | 2 * base.f |
base.f * 2^(1/12) * 2^(1/12) | base.f * 2^(1/6) |
So 2^(1/12) × 2^(1/12) = 2^(1/6) — the like-base merge — is real. It happens at save time, in the simplifier, not during evaluation.
But now try the twelve-fold chain. The simplifier can fold it to base.f * 2… and then throws that result away and keeps what you wrote.
Why: a rewrite is rejected if it flips the corruption flag. The chain is irrational (hatched); base.f * 2 is rational (clean). Silently swapping one for the other would repaint your note and quietly change what the app is telling you about the value. So the simplifier declines.
This is a feature, not a bug
The app will not launder an approximation into an exact value behind your back. If you want the exact octave, write base.f * 2 or base.f * 2^(12/12) and get a clean note. If you write twelve irrational steps, you keep twelve irrational steps — hatching and all.
The ▲/▼ arrows and TET
The frequency row's arrows multiply by the interval configured in Settings → Arrows (default ×2 up, ×1/2 down). They fold the factor into the expression's rational coefficient rather than stacking a new multiplier on the front:
| Before | After ▲ |
|---|---|
base.f | 2 * base.f |
(1/2) * base.f | base.f |
base.f * 2^(7/12) | 2 * base.f * 2^(7/12) |
Note the third row. The coefficient absorbs the ×2; the power term is left alone. A TET note stays a TET note — the arrows will never quietly rationalise it, and up-then-down returns you to exactly base.f rather than (1/2) * 2 * base.f.
SymbolicPower, the class
The name on this page comes from a class in the evaluator. Its real shape is:
class SymbolicPower {
coefficient // Fraction
powers // Array<{ base: number, exp: Fraction }>
}It models coefficient × base₁^exp₁ × base₂^exp₂ × … × baseₙ^expₙ — a list of power terms, not a single base/exponent pair. That is what lets a value carry a base-2 term and a base-3 term at once, which is what the shipped Mixed-Base scale needs:
[7].f * 2 ^ (-1/12) * 3 ^ (-1/13)The class is not what evaluates your notes
On the evaluator that ships, the stack holds plain rational fractions. ^ reaches SymbolicPower, gets an irrational back, and immediately approximates it to a Fraction. The like-base merging the class advertises is performed by the simplifier at save time, not by the evaluator at run time.
A statement like "the evaluator preserves symbolic form end to end" is false. Do not build on it.
For the class internals and the bytecode, see SymbolicPower (developer) and Binary Evaluator.
Different bases
Any positive integer base works.
2^(1/12) # 12-TET step — octave-based
3^(1/13) # Bohlen-Pierce — tritave-based
5^(1/7) # a 5-based step, as used in Mixed-BaseTwelve fifths, three ways
The circle of fifths is the cleanest demonstration of everything above. Base frequency 263 Hz; seven octaves up is 33664 Hz.
| Written as | Evaluates to | Exact? | Flagged? |
|---|---|---|---|
base.f * 2^(84/12) | 33664 | yes | no |
base.f * 2^(7/12) * 2^(7/12) * … (×12) | 33663.999999999163265 | no | yes |
base.f * (3/2)^12 | 34123.286865234375 | yes (and not an octave) | no |
Row 1 is the closed form: 84/12 reduces to the integer 7, so the power resolves exactly and the note is clean. Row 2 is the same music written as twelve separate irrational steps — twelve roundings, drifting 8 × 10⁻¹⁰ off, permanently hatched. Row 3 is twelve pure fifths, which is exact rational arithmetic that simply does not land on an octave: it overshoots by the Pythagorean comma (531441/524288, about 23.5 cents).
Three different answers, and all three are correct — they are answers to three different questions. Equal temperament exists to make rows 1 and 3 agree; the evaluator's rounding is what separates rows 1 and 2.
Debugging checklist
Is this note exact? Look at the rectangle. Clean = exact rational. Crosshatched = its own expression produced an irrational. Single-diagonal = it inherited one.
Did my power resolve? Write it and read Evaluated:. No ≈ means the rational-power check succeeded and you have an exact value.
Is my chain drifting? Replace 2^(1/12) repeated k times with a single 2^(k/12). One rounding instead of k.
Did my edit save? The Save button next to the Raw: field only appears once you start typing, and the value does not change until you press it. An invalid expression is rejected with its reason shown in red under the Save button.
Next
- Microtonal Composition — put the temperaments to work
- Complex Dependencies — how corruption spreads through a graph
- Expression Compiler — text to bytecode