Glossary of Just Intonation

acute
adj. (of an interval) broadened by a syntonic comma from the just tuning; n. an acute interval.
apotome
The Pythagorean chromatic semitone, with an interval ratio of 2187/2048.
Alsatian 6th chord
Another name for the Swiss 6th chord.
augment
To broaden an interval by a chromatic semitone, indicated in music by adding a ♯ or canceling a ♭.
bender
A common note in a chord sequence that adjusts in pitch due to the tuning of the chord roots to the scale. In a 1-27-1 progression whose roots move an acute major 2nd, the tonic would flat by a septimal comma. See also hanger
chain of fifths
A tonal model that views the notes of the scale as arranged in a sequence of perfect 5ths or 4ths. Downward movement of a 5th is considered the strongest motion for chord roots (c.f. the song Five Foot Two). Two notes 12 links apart in the chain are called enharmonic (e.g. D♯ and E♭)
circle of fifths
An alternate designation of the chain of fifths in equal temperament, based on the fact that the notes repeat themselves every twelve links due to the tempering out of the Pythagorean comma error.
chord ratio
a complex ratio of the frequencies of three or more different pitches, such as 4:5:6 for a just major triad.
chroma
In just intonation, the small semitone resulting from the use of a sharp or flat (nickname for chromatic semitone). Typically this refers to the the 5-limit chroma (25/24).
chromatic
Referring to the chroma; in general to notes and intervals created by sharping or flatting a scale tone.
chromatic inflection
An adjustment of a note by sharps and flats alone (♯ and ♭), thereby preserving the letter-name.
chromatic scale
The 12-note scale consisting of the diatonic scale and the chromatically altered notes. In this study I use an extended chromatic scale of 25 notes centered around a tonic of C, extending in both directions in the chain of fifths to C enharmonics B♯ and D♭♭.
chromatic interval
An interval between two notes more than 6 links apart in the chain of 5ths. They are a diminished or augmented enharmonic of a scalar interval (e.g. augmented 2nd, which is enharmonic to a minor 3rd)
chromatic semitone
The interval signified by two notes of the same letter-name, differing only by the presence of a sharp or flat; augmented prime.
combination tone
extra tones generated by two or more sounding frequencies, which have some kind of mathematical relationship with them, such as a sum or difference tone.
common-tone diminished 7th chord
A diminished 7th chord that resolves to a chord containing at least one of its notes.
comma
In just intonation, a small interval used to make adjustments between different tunings. Unqualified it normally refers to a syntonic comma (81/80).
commatic
Referring to the comma, or to errors or adjustments in pitch by a comma.
commatic inflection
An adjustment of a note by a comma, and preserving the chromatic name (letter and accidental). It is commonly designated by + and - for a syntonic comma or a super- or subscripted number (or ascii equivalent) for higher limit commas.
Dhatuvardhani scale
Another name for the Swiss scale, based on South Indian classical music.
diatonic
Referring in general to notes and intervals that can be made without sharping or flatting.
diatonic scale
A 7-note scale consisting of 5 whole-tones and 2 semi-tones, such as the white keys on the piano in C major.
diatonic semitone
A semi-tone signified by two consecutive letter-names (e.g. E-F, or G-A♭); minor 2nd.
diatonic interval
An interval between two notes 6 or less links apart in the chain of 5ths, classified as major, minor or perfect, except for the augmented 4th and diminished 5th.
difference tone
A combination tone whose frequency is the difference between the frequencies of two sounding notes. The difference tone of two almost identical notes is often sub-sonic and heard as beats. That of two widely separated notes is often a small interval below the upper note. The difference tone of a just major or minor 6th complete a major triad.
diminish
To narrow an interval by a chromatic semitone by adding a ♭ or canceling a ♯.
dim-dom
A nickname for a common-tone Diminished 7th chord that resolves to a dominant 7th chord.
equal temperament
The division of the octave into (usually twelve) equal intervals, enabling equal playability of music in all keys at the expense of perfect tuning. Fixed pitch instruments such as the piano are generally tuned this way.
English 6th chord
Another name for the Swiss 6th chord.
enharmonic
referring to two notes of different letter names chromatically inflected so they sound the same, or nearly the same (e.g. D♯ and E).
Eulenspiegel scale
A scale created to justify the Eulenspiegel 6th among other chords, which goes C D♯ E F G A♭ B C, also called the Gangeyabhushani scale.
Eulenspiegel 6th chord
A chord with a minor 3rd, an augmented 4th and an augmented 6th above the bass, enharmonic to the half-diminished 7th chord.
5-limit
Of any interval ratio with a multiple of 5. All the notes in the just chromatic scale have a limit of 5 or less.
French 6th chord
A chord with a major 3rd, an augmented 4th and an augmented 6th above the bass.
frequency
The pitch of a note in terms of its vibration rate, usually measured in cycles per second, inversely proportional to the wavelength and period.
fundamental
The unit frequency at the bottom of the harmonic series, of which every note in the series is a multiple; also the similar unit frequency at the top of the subharmonic series.
Gangeyabhushani scale
Another name for the Eulenspiegel scale, based on South Indian classical music.
German 6th chord
A chord with a major 3rd, a perfect 5th and an augmented 6th above the bass, enharmonic to the dominant 7th chord.
grave
adj. (of an interval) narrowed by a syntonic comma from the just tuning; n. a grave interval.
hanger
A common note in a chord sequence that is fixed in frequency by adjusting the root movement of the chords. In a 1-27-1 progression the tonic hanger is preserved by moving the roots a septimal major 2nd. Typically a hanger is used when it is a held note (pedaltone), like in a typical barbershop tag. See also, bender
harmonic
n. A note that is a simple multiple of a fundamental note; adj. of an interval or chord tuned to simple multiples of a fundamental, so as to create a consonant sound.
harmonic series
The series of notes whose frequencies are simple multiples of a unit frequency called the fundamental.
imperfect
Referring to intervals, generally 2nds, 3rds, 6ths or 7ths, which have major and minor variants.
inflection
A small adjustment in a note that preserves something of the notes identity.
interval ratio
a fraction representing the ratio of the frequencies of two different pitches, such as 5/4 for a just major 3rd.
Italian 6th chord
A chord with a major 3rd and an augmented 6th above the bass.
just intonation
The tuning of the scale and of chords so that the ratio of frequencies between pitches are small integral values, thereby producing a clean, expanded sound.
just major scale
A major scale tuned according to just intonation; officially, the scale with the ratio 24:27:30:32:36:40:45:48. This tuning results from flatting the 3rd, 6th and 7th degrees of the Pythagorean scale by one syntonic comma.
just chromatic scale
A chromatic scale formed by flatting every four notes in the ascending chain of fifths by an additional syntonic comma, with the four notes around the tonic (scale degrees 4, 1, 5 and 2) tuned the same as the Pythagorean scale.
lattice
a multi-dimensional arrangement of pitches so that notes are separated by a different interval in each axis. An 5-limit example is a lattice of notes vertically separated by perfect 5ths and horizontally by major 3rds.
limit
The highest prime factor of the values in an interval ratio. This can also refer to the highest odd factor.
link
The perfect 5th interval separating any two consecutive notes in the chain of 5ths.
major
Referring to the greater of two imperfect intervals of the same number of scale tones. In 5-limit, an interval ratio with a 5+1 as a factor.
matrix ratio
A single two-dimensional ratio of the frequencies of all the notes of a chord sequence collectively, e.g. 18:21:24:30-16:20:24:32 for a simple 57-1 cadence.
meantone temperament
A scale tuning where each succeeding note in the ascending chain of fifths is flatted by a fixed comma to give some other interval a just tuning. Typically this refers to quarter meantone, where 5ths are flatted by a quarter of a syntonic comma, resulting in just major 3rds.
minor
Referring to the lesser of two imperfect intervals of the same number of scale tones. In 5-limit, an interval ratio with a 5-1 as a factor.
Muszourski Chords
The motion of a half-diminished 7th chord to a German 6th chord by moving the root and 7th down a semitone, named after its use in Pictures at an Exhibition.
normalize
To halve or double the value of an interval ratio so that the value is between 1 and 2 (corresponding to the interval of a scale tone to the nearest tonic note below it), and reduce to lowest terms. 8/3 and 2/3 would both normalize to 4/3.
octave
The interval with the ratio 2/1 - an eighth. Both notes are the same on the scale.
octave equivalence
The principle that the octave of a note is insignificant, allowing for normalization of interval ratios. By this, a perfect 4th (4/3), a perfect 11th (8/3), and a descending 5th (2/3) would all be treated as equivalent.
overtone
A frequency componant of a note that is above the fundamental, usually a harmonic, but in some cases (like a bell) is not.
perfect
Referring to a simple 3-limit interval, generally a prime, 4th, 5th or an octave, not altered by augmentation or diminution.
period
The amount of time consumed by one cycle of a soundwave, inversely proportional to the frequency; also, the amount of time consumed by the wave pattern of two or more tones before it repeats itself, equal to the least common multiple of the periods of the component frequencies.
prime
The interval of a note with itself; a "first".
Pythagorean
A tuning system based on 3-limit intervals, also known as the chain of fifths. Many intervals, particularly 3rds and 6ths, sound out of tune in this system, and in just intonation are converted to 5-limit intervals by inflection with the syntonic comma.
Pythagorean comma
The small interval (531441/524288) that differentiates a given note from the note 12 links removed from it in the chain of fifths; a Pythagorean diminished 2nd.
Pythagorean chromatic scale
A 3-limit scale tuning based on arranging 12 (or more) notes in a sequence of just perfect 5ths (3/2). The sequence centered around a tonic of C would be G♭, D♭, A♭, E♭, B♭, F, C, G, D, A, E, B, F♯. Theoretically the sequence may be extended indefinitely in both directions, and in this site's discussion is extended to 25 notes. The 13 notes above are the basis of the commonly used key signatures.
Pythagorean major scale
A 3-limit major scale made by arranging 7 notes in a sequence of just perfect 5ths (3/2). The second note in the sequence would function as the tonic (the notes F, C, G, D, A, E, B would form the C major scale).
quartal
Based on 4ths; of chords that are constructed with 4ths.
quintal
Based on 5ths; of chords that are constructed with 5ths.
ragtime progression
A tonic chord followed series of secondary dominant 7th chords descending in a circle of fifths to the tonic. Typically refers to the 4-chord version starting on the 6th degree.
septendecimal
Involving number 17; of or containing an interval ratio with a multiple of 17, also known as 17-limit. The just dinimished 7th chord uses this tuning.
septendecimal chroma
The chromatic semitone differentiating the diminished 7th chord from the half-diminished 7th chord (18/17).
septimal
Involving number 7; of or containing an interval ratio with a multiple of 7, also known as 7-limit. The just dominant 7th chord uses this tuning.
septimal chroma
The chromatic semitone differentiating the dominant 7th chord from the major 7th chord (15/14).
septimal comma
The small interval, with a interval ratio of 64/63, which differentiates the harmonic 7th (7/4) from the 3-limit minor 7th (16/9), used to fine-tune the 7th of the dominant chord in the just major scale, indicated by putting 7 after the note (a < may be used in Ascii text environments).
subharmonic
n. A note that is a simple submultiple of a fundamental note; adj. of an interval or chord tuned to simple submultiples of a fundamental.
subharmonic series
The series of notes whose frequencies are simple submultiples of a unit frequency, considered a hypothetical inverse of the harmonic series because it contains the same intervals but extended downward. It can be described as all the notes of which the fundamental is a harmonic.
summation tone
A combination tone whose frequency is the sum of the frequencies of two sounding notes. It is debatable whether such a note can actually be heard.
Swiss scale
A scale created to justify the Swiss 6th among other chords, which goes C D♯ E F♯ G A♭ B C, also called the Dhatuvardhani scale.
Swiss 6th chord
A chord with a major 3rd, a doubly-augmented 4th and an augmented 6th above the bass, enharmonic to the dominant 7th chord. Also called the English 6th or Alsatian 6th.
syntonic comma
The small interval (81/80) which differentiates a 5-limit interval from its 3-limit equivalent.
tertian
Based on 3rds; of chords that are constructed with 3rds. This is the standard way chords are formed in the western tradition since the common practice period.
3-limit
Of any interval ratio with a multiple of 3, Pythagorean.
tredecimal
Involving number 13; of or containing an interval ratio with a multiple of 13, also known as 13-limit.
tredecimal comma
The small interval, with a frequency ratio of 40/39, which differentiates the 13th harmonic (13/8) from the major 6th (5/3), indicated by putting 13 after the note.
Tristan 6th chord
An chord with an augmented 2nd, an augmented 4th and an augmented 6th above the bass, enharmonic to the half-diminished 7th chord, also called the minor-diminished 7th chord.
Tristan mordent
The motion from a German 6th chord to a Tristan 6th chord by moving the 3rd and 5th down a minor 2nd.
tritone complement
(of a chord) a chord rooted a tritone away from a given chord; (of a progression) a progression using such a chord.
tritone substitution
Replacing a chord in a progression, usually a dominant or half-diminished 7th, with a like chord shifted a tritone.
undecimal
Involving number 11; of or containing an interval ratio with a multiple of 11, also known as 11-limit.
undecimal comma
The small interval with a frequency ratio of 33/32, which differentiates the 11th harmonic (11/8) from the perfect 4th (4/3), indicated by putting 11 after the note (or ^ in strict Ascii environments). The debatable choice of basing this adjustment on the sharping of the perfect 4th degree instead of flatting the raised 4th degree seems to be based on the the simplification of notating the inflection (i.e. F11 instead of F♯11).
undertone
A frequency componant of a note that is below the fundamental, which can be a subharmonic or a combination tone.
wavelength
The distance between the crests of two consecutive waves of a vibration, inversly proportional to the frequency.

Scales
Index of Intervals
Chord Tunings
Chord Progressions
Glossary of Just Intonation
Tables of Pitch Bends
Fun with Vowel Formants
Just Intonation home page
Billy's home page