Major Limma¶
A just-intonation interval of frequency ratio 135:128, approximately 92.18 cents, derivable as the difference between a 9:8 ditone and a 6:5 minor third.
Core Idea¶
The Major Limma is a musical interval with the just-intonation frequency ratio 135:128. It spans approximately 92.18 cents. Its identity is the exact ratio, while the cents value is a logarithmic approximation useful for comparing it with intervals in other tuning systems. If the lower pitch has frequency \(f\), the upper pitch has frequency \(135f/128\).
Scope of Application¶
The Major Limma is a tuning-theory identity bounded by the directed pitch-frequency ratio 135:128. Its literal habitats are musical settings that preserve that ratio exactly or declare a measured or tempered approximation to it; a bare numerical ratio outside pitch organization is not the interval.
- Just-intonation analysis. The interval is located in 5-limit tuning through its factorization and can be derived as an 81:64 ditone divided by a 6:5 minor third.
- Interval arithmetic. Tuning calculations compose, invert, and octave-reduce the ratio, with cents supplying an additive comparison while the exact rational identity remains primary.
- Microtonal composition. Composers can use 135:128 as a directed pitch adjustment, harmonic relation, or fine structural difference when the score or tuning plan declares the intended ratio.
- Microtonal notation. Johnston's or another declared notation system may encode the interval, but the symbol qualifies only through an explicit mapping to 135:128 rather than by its graphic form alone.
Clarity¶
Naming the major limma as 135:128 removes an ambiguity that the word limma alone cannot. It distinguishes this exact 5-limit interval from the Pythagorean limma (256:243), the 100-cent twelve-tone semitone, and any performed pitch that merely falls near 92 cents.
Manages Complexity¶
Major limma packages a recurring tuning discrepancy into the exact unit 135:128, so a theorist need not rederive the relation from every scale spelling or frequency pair. The analyst tracks the directed ratio, octave reduction, and the tuning system in which intervals are composed.
Abstract Reasoning¶
Reasoning with the major limma starts from an exact relation between two pitches and carries it through interval arithmetic. Given a lower frequency (f), the upper frequency is (135f/128); given a chain of intervals, multiplying or dividing their ratios determines whether the chain contains the same interval.
Knowledge Transfer¶
Within tuning theory, the major limma transfers literally across just-intonation analysis, microtonal composition, notation systems, instrument tuning, and performance measurement whenever the intended directed pitch ratio is 135:128. Its exact ratio, inversion, 5-limit factorization, derivation by interval division, and approximate 92.18-cent representation carry unchanged. So do the principal diagnostics and interventions: state the notation-to-tuning mapping, compare measured or tempered realization with the rational target, quantify cent error, and distinguish the unqualified limma, Pythagorean 256:243, and the 100-cent equal-tempered semitone.
Relationships to Other Abstractions¶
Current abstraction Major Limma Domain-specific
Parents (1) — more general patterns this builds on
-
Major Limma is a kind of Ratio Prime
For an upward interval, the upper pitch frequency is the numerator and the nonzero lower pitch frequency is the denominator; ordered division fixes their quotient at exactly
135/128.
Hierarchy path (1) — routes to 1 parentless root
- Major Limma → Ratio → Comparison → Self Checking
Neighborhood in Abstraction Space¶
Major Limma sits in a moderately populated region (43rd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Musical Intervals & Voice Texture (8 abstractions)
Nearest neighbors
- Musical Interval — 0.92
- Consonance — 0.89
- Atonality — 0.86
- Transposition (Music) — 0.86
- Musical Key — 0.86
Computed from structural-signature embeddings · 2026-10-08