# Glossary

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A concise list of essential terms in xenharmonic music theory.

## A

- achiral scale
- A scale whose mirror image (reversing the order of its steps) is a mode of the same scale.

## B

- beat
- An interference pattern between two sounds of slightly different frequencies, perceived as a periodic variation in volume whose rate is the difference of the two frequencies.

- binary scale
- A scale with two different step sizes.

## C

- cent (¢)
- A unit of interval size equal to 1/1200th of an octave, or 1/100th of a 12edo semitone.

- chiral scale
- A scale whose mirror image (reversing the order of its steps) is a different scale (not a mode of the same scale).

- chroma
- An interval equal to the difference between the large step and the small step (L-s) of a binary scale.

- comma
- A small interval that occurs in just intonation between two intervals which are close in pitch.

- comma pump
- A harmonic sequence (or chord progression) wherein the tonic shifts up or down by a comma every time the sequence is traversed.

- complement
- The difference between an interval and the octave or another equivalence interval.

- concordance
- The perceived smoothness of an interval or chord.

- consonance
- The subjective, contextual smoothness of an interval or chord.

- constant structure (CS)
- The property of a scale whose interval classes are distinct.

## D

- delta-N ratio
- A ratio whose difference between its numerator and its denominator is equal to N.

- delta-rational chord (DR chord)
- A chord with at least two dyads which represent frequency differences which have a rational ratio. This property is thought to result in synchronized interference beating between the dyads.

- diatonic scale
- An moment-of-symmetry (MOS) scale with 5 large steps and 2 small steps per octave.

- diesis
- An interval equal to the absolute value of the difference between the large step and twice the small step (|L-2s|) of a binary scale.

- discordance
- The perceived roughness of an interval or chord.

- dissonance
- The subjective, contextual roughness of an interval or chord.

## E

- equal temperament
- A rank-1 temperament. The temperament-agnostic term is
*equal-step tuning*(or equal tuning).

- equave (or interval of equivalence)
- An interval considered to be equivalent to the unison, such as an octave (2/1 or 1200 cents).

- equivalence
- The phenomenon of psychoacoustics that two notes an octave apart, or possibly another interval apart, are considered "(substantially) equivalent" to the brain.

## F

- frequency
- The number of oscillations per unit of time of a vibrating object.

- frequency ratio
- The relationship between the frequencies of the pitches of two or more notes.

## G

- generators
- A set of intervals that generates a larger set through linear combination; e.g. primes {2, 3} generate Pythagorean tuning.

- generator sequence (GS)
- A scale-building procedure based on alternating generators.

## H

- harmonic
- A whole-number multiple of the fundamental frequency of a sound; also implicitly used in the sense of "harmonic partial" or "harmonic overtone".

- harmonic series
- The infinite sequence of tones that are whole-number frequency ratios over a fundamental.

- harmonic (series) segment
- Any finite sequence of consecutive harmonics in the harmonic series. Examples: 3:4:5:6:7, 7:8:9:10:11:12:13:14, 49:50:51:52:53:54. As shorthand, these may be written 3::7, 7::14, and 49::54.

- harmonic (series) mode
- A harmonic segment spanning an outer interval of 1:
*n*and usually taken as constituting a scale with period*n*/1, i.e. one of the form*m*::*mn*. By default*mode m*refers to octave-spanning harmonic modes, i.e.*m*::2*m*. For example, mode 6 is 6:7:8:9:10:11:12, or 6::12.

- harmonic distance
- A measure of complexity for just intonation intervals, associated with concordance and discordance.

- harmonic limit
- See prime limit, odd limit and integer limit.

## I

- integer limit
- The set of all frequency ratios whose terms do not exceed a given maximum integer.

- interval
- The relationship between two pitches, whose main property is interval size.

- interval category
- See interval region.

- interval class
- The set of all intervals in a scale which subtend the same number of scale steps.

- interval of equivalence
- See equave.

- interval quality
- The relative size of an interval compared to similar intervals (of the same interval class or of the same interval region).

- interval region
- A range of intervals of similar size.

- interval size
- The distance between two pitches, usually expressed as a frequency ratio or as a logarithmic measure (e.g. cents).

- interval variety
- The number of different interval qualities available for an interval class in a scale.

## J

- just intonation (JI)
- The set of intervals whose frequency ratios are expressible as rational numbers.

- just intonation subgroup
- The set of JI intervals obtainable by stacking (finitely many) copies of a finite set of JI generators up or down. For example, 7/6 and 49/32 are both in the 2.3.7 subgroup, the set of JI ratios obtained by stacking copies of 2/1, 3/1 and 7/1 up and down.

## K

## L

- limit
- See prime limit, odd limit and integer limit.

- low-complexity just intonation (LCJI)
- An approach to tuning based on a restricted set of simple just intonation intervals.

## M

- maximum variety (MV)
- The maximum interval variety from all interval classes of a periodic scale.

- mediant
- An binary operation on ratios (frequency ratios or other) where the numerator of the result is the sum of the ratios' numerators, and the denominator of the result is the sum of the ratios' denominators.

- mode
- A particular rotation of a scale, e.g. Ionian, Dorian etc. for the diatonic scale.

- monzo
- A representation of a just intonation interval using the exponents in its prime factorization; e.g. 81/80 = [-4 4 -1⟩.

- moment-of-symmetry scale (MOS scale)
- A type of scale that generalizes the diatonic scale by preserving some of its properties, including maximum-variety 2 and rank 2.

- Myhill's property
- The property of a scale with maximum variety 2.

## N

- Neji
- A near-equal just intonation (NEJI) scale.

## O

- octave reduction
- An operation on intervals consisting in dividing or multiplying by 2 repeatedly until the result is greater than or equal to the unison and less than the octave.

- odd limit
- The set of all frequency ratios whose terms, once all powers of 2 are removed, do not exceed a given maximum odd number.

- overtone
- A partial above a fundamental frequency, often assumed to be a harmonic partial, or an independent pitch corresponding to that partial.

## P

- partial
- A sine wave component of a periodic sound.

- period (or interval of repetition)
- The interval that spans the smallest step pattern that can "tile" a scale.

- periodic scale
- A scale that repeats at a period, often assumed to be the octave or other equave.

- pitch
- The property of some sounds that allows them to be ordered on a one-dimensional 'pitch axis'. Often measured by the frequency (in hertz) of a sine wave having the same pitch.

- polymicrotonal (or polysystemic)
- The use of multiple tuning systems in the same piece.

- prime limit
- The set of all frequency ratios whose terms' prime factors do not exceed a maximum prime number.

- Pythagorean tuning
- A tuning system based on stacking the interval 3/2 repeatedly, octave-reducing as necessary.

## Q

- quality
- See interval quality.

## R

- rank
- The number of generators of a set of intervals, i.e. the set's dimensionality (not to be confused with the dimensionality of the temperament itself). For example, 12edo is rank-1 because it can be generated by the semitone; Pythagorean tuning is rank-2 because it can be generated by the primes {2, 3}.

- ratio
- See interval ratio and step ratio. May also be used to designate other mathematical ratios.

- reference tone (or reference pitch)
- An agreed-upon fundamental or dominant frequency that musical instruments will reference when adjusting tuning.

- region
- See interval region.

- regular temperament
- A mapping from a domain (such as a prime limit or just intonation subgroup) to a set of intervals with fewer generators. Expressible as a mapping matrix whose columns are generators of the just intonation and whose rows are generators of the temperament. In particular, the row vectors are called vals or maps.

## S

- scale
- A sequence of pitches (sometimes expressed as a sequence of intervals relative to the tonic); usually contains every pitch of a given musical phrase or passage. Not to be confused with "tuning system".

- scale signature
- An expression using the step sizes of a scale, where the coefficient in front of each step tells you how many of that step size the scale has. A mos scale has a scale signature of aL bs for step sizes L > s. Other scales may have more complex signatures such as 5L 2M 3s.

- size
- See interval size.

- 1, 2, 3...-step
- A 1, 2, 3, ...-step interval in any scale.

- step pattern
- The sequence of steps that occurs in a scale, usually written using letters to represent relative interval sizes (L for large, M for medium, s for small, etc.).

- step ratio
- The ratio between the interval sizes of a scale's steps.

- strict variety (SV)
- The interval variety of all interval classes of a periodic scale, when all interval classes have the same interval variety.

- subharmonic
- A unit fraction of the fundamental frequency of a sound; also implicitly used in the sense of "subharmonic partial" or "subharmonic undertone".

- subharmonic series
- A sequence of tones generated by unit-fraction frequency ratios under a fundamental.

- subtend
- Of an interval size, to occur as a certain number of steps in a given scale.
*In the diatonic scale, the perfect fifth always subtends 4 steps.*

## T

- temperament
- Historically, a tuning system that slightly compromises the pure intervals of (low-complexity) just intonation to meet other requirements. Modern use of the term usually implies "regular temperament".

- ternary scale
- A scale with three different step sizes.

- timbre
- The perceived sound quality of a musical note, which allows a listener to tell different instruments apart.

- tone
- An individual pitch from a chord, scale, tuning system, etc.

- tuning system
- A set of pitches (sometimes expressed as a set of intervals relative to the tonic); contains every pitch of a given musical instrument or piece. Not to be confused with "scale".

## U

- undertone
- A partial below a fundamental frequency, often assumed to be a subharmonic partial, or an independent pitch corresponding to that partial.

- unit of interval size
- An interval used to measure the size of other intervals.

## V

- variety
- See interval variety.