Where does interval collide?

The distance in pitch between two notes, understood as the ratio of their frequencies: a perfect fifth is the ratio 3 to 2, an octave 2 to 1, and the size of any interval can be given as such a ratio or in cents. Zarlino's 1558 Le istitutioni harmoniche gave a systematic Renaissance account of intervals as numerical ratios of consonance, building on the Pythagorean tradition of antiquity, and Rameau's 1722 Traité de l'harmonie later organised intervals into a theory of chords and their roots. The referent is a relation between two sounding pitches, scaled by frequency ratio, and it has nothing to do with a span of elapsed time in a heartbeat, a range that brackets an unknown quantity, or a property of a measurement scale.[1][2]

Unit

Two pitches apart, given as a frequency ratio or in cents

Recovery means

Not applicable

Measured by

Read as a frequency ratio between the two tones, or in semitones and cents on a logarithmic pitch scale

The elapsed time between two defined events in the electrical cycle of the heart, read from the electrocardiogram: the QT interval, for example, runs from the start of ventricular activation to the end of recovery, and its length carries clinical meaning. Bazett's 1920 analysis of the time-relations of electrocardiograms set out how these intervals are measured and how the QT interval shortens as heart rate rises, giving the rate-correction formula still associated with his name. The referent is a duration in milliseconds within one cardiac cycle, and it is unrelated to a ratio between two musical pitches, a range that brackets an unknown parameter, or a level of measurement.[3]

Unit

A duration in milliseconds between two events in the cardiac cycle

Recovery means

Not applicable

Measured by

Measured on the electrocardiogram between defined waveform landmarks, often corrected for heart rate

A range computed from data that is built to cover the true value of an unknown quantity a stated proportion of the time under repeated sampling: a 95 percent confidence interval comes from a procedure that, run again and again on fresh samples, brackets the target in about 95 percent of cases. Neyman's 1937 Outline of a theory of statistical estimation based on the classical theory of probability defined the confidence interval and its coverage property, arguing that such a range is judged by how often the procedure captures the target, not by a probability placed on the fixed unknown. The referent is a random range attached to an estimate, judged by its long-run coverage, not a distance between two pitches, a span of time in a heartbeat, or a rule for spacing a scale.[4]

Unit

A range on the quantity's own scale, with a stated coverage level

Recovery means

Not applicable

Measured by

Computed from the sample and a chosen coverage level, and assessed by its long-run capture rate

A level of measurement in which equal differences between numbers stand for equal amounts of the thing measured, but the zero point is set by convention rather than marking a true absence: temperature in Celsius is the classic case, where the step from 10 to 20 degrees equals the step from 20 to 30, yet zero does not mean no temperature. Stevens's 1946 On the theory of scales of measurement introduced the fourfold split of nominal, ordinal, interval, and ratio scales and fixed the meaning of the interval level. The referent is a property of a measurement scale, one about which arithmetic differences are meaningful, and it is not a musical distance between pitches, a duration in a heartbeat, or a range that brackets an unknown value.[5]

Unit

A property of a scale: equal numeric differences mean equal amounts, zero is arbitrary

Recovery means

Not applicable

Measured by

Established by whether the numbers can be shifted and stretched by a linear rule without changing what the differences mean