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: v2 · 2026-09-19 · History
Domain-specific #: 3541
Domain group: Arts & Aesthetic Practice
Origin domain: Music & Musicology
Subdomains: Music Theory, Microtonal Commas → Music & Musicology

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.

How would you explain it like I'm…

The Teeny Music Gap

Imagine two different paths of musical steps that should end up at the same note. When you work it out exactly, they land almost in the same place but not quite, off by a teeny, teeny sliver. That tiny sliver is called the breedsma. Some ways of tuning an instrument just pretend the sliver isn't there and use one key for both.

The Almost-Same Note

In music, each interval (the jump between two notes) can be written as a ratio of vibration speeds, like 3:2 for a fifth. If you stack and unstack certain intervals that involve the number 7, you end up with a leftover gap of exactly 2401:2400, which is a super tiny difference in pitch. That leftover is the breedsma. A tuning system 'tempers it out' when it treats the two nearly-equal pitches as the very same note, so they share one step.

The 2401:2400 Septimal Comma

In just intonation, intervals are exact frequency ratios, and combining intervals means multiplying or dividing those ratios. A comma is the small leftover interval you get when two different chains of just intervals almost, but don't quite, reach the same pitch. The breedsma is one specific comma: when you multiply and divide certain septimal intervals (ratios built with the prime 7), the residual is exactly 2401:2400, an extremely small interval. A temperament is a tuning system that maps many just intervals onto a limited set of steps; it 'tempers out' the breedsma if both of those pitches land on the same step. So the breedsma is a precise ratio, not just a vague name for 'a tiny difference.'

The breedsma is a very small comma in just-intonation theory, defined as the exact frequency ratio 2401:2400. It is the residue left when septimal (7-limit) interval ratios are combined by multiplication and division along two routes that nearly coincide. In temperament theory, a temperament tempers out the breedsma when its mapping sends both pitches of the comma to the same scale step, so the comma vanishes in that tuning. Identifying the breedsma is a matter of exact arithmetic: the interval must be computed under a single frequency-ratio convention and equal 2401/2400. A different small comma, or an interval merely described as 'septimal and tiny', is not the breedsma. The concept links just intervals with the mappings that define temperaments, which is why it matters for classifying which temperaments treat which septimal intervals as equivalent.

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.
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) — outgoing typed relations

Breedsma is a kind of Measurement Prime

The interval measures logarithmic pitch separation.

Hierarchy path (1) — routes to 1 parentless root

Breedsma → is a kind of Measurement

Neighborhood in Abstraction Space¶

Breedsma sits in a moderately populated region (40^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25