Breedsma¶

The microtonal interval with frequency ratio 2401:2400, about 0.72 cents, arising as the difference between particular septimal intervals and tempered out in several equal temperaments.

Version: v1 · 2026-09-08 · History
Domain-specific #: 3541
Origin domain: music theory
Subdomain: microtonal commas

Core Idea¶

The breedsma is a very small just-intonation comma defined by one exact rational ratio. Multiplying and dividing septimal interval ratios leaves the residual 2401:2400; a temperament tempers it out when both pitches map to the same step. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of music theory. It is specific septimal comma linking just intervals and temperament mappings. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition that the interval is calculated under one frequency-ratio convention and equals 2401 divided by 2400 fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test.

Scope of Application¶

Breedsma belongs to music theory and is useful where the analyst can specify two pitches, rational frequency ratio 2401:2400, cents measure, septimal diesis and sixth-tone relations, equal temperament mappings and tempering-out condition, then evaluate the interval is calculated under one frequency-ratio convention and equals 2401 divided by 2400. The scope is broad within that domain but bounded by the need for the interval is calculated under one frequency-ratio convention and equals 2401 divided by 2400. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the interval is calculated under one frequency-ratio convention and equals 2401 divided by 2400 the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Breedsma can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Breedsma. Breedsma compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: two pitches, rational frequency ratio 2401:2400, cents measure, septimal diesis and sixth-tone relations, equal temperament mappings and tempering-out condition. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the interval is calculated under one frequency-ratio convention and equals 2401 divided by 2400 independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of music theory because they reuse two pitches, rational frequency ratio 2401:2400, cents measure, septimal diesis and sixth-tone relations, equal temperament mappings and tempering-out condition, Multiplying and dividing septimal interval ratios leaves the residual 2401:2400; a temperament tempers it out when both pitches map to the same step., and type the carrier, state every parameter and convention in the definition, test that the interval is calculated under one frequency-ratio convention and equals 2401 divided by 2400, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Breedsma Domain-specific

Parents (1) — more general patterns this builds on

Breedsma is a kind of Measurement Prime

The proposed strict upward parent is prime:measurement.

Hierarchy path (1) — routes to 1 parentless root

Breedsma → Measurement

Neighborhood in Abstraction Space¶

Breedsma sits in a moderately populated region (57^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Harmony, Tuning & Musical Form (24 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08