Glossary

Acoustic

Acoustics

Ancohemitonic

Theory

Definitions of music theory terms used throughout Harmonious.

A few longer entries that explain some important general principles include Tuning Systems, Twelve-tone Equal Temperament, Intervals, Tonality, and Atonality or Musical Set Theory.

Alphabetical Index ↓

Acoustic

In Harmonious, the acoustic scale or melodic minor scale refers to the heptatonic (seven-note) set class, the unordered collection of notes with seven modes, including the Lydian Dominant (Acoustic) and Melodic Minor modes. More

Acoustics

The study of the science of mechanical vibrations is called acoustics. To help understand music theory, it is useful to understand the relationship between pitch, the fundamental tone, and overtones, which may or may not be harmonics, or integer multiples of the frequency of the fundamental tone. More

Ancohemitonic Set Theory

A set class is ancohemitonic if it does not contain a chromatic cluster, or two semitone intervals in a row, described as chromatic cluster-free in Harmonious. More

Atonal Theory Set Theory

Atonal theory, or musical set theory, as outlined by Allen Forte and others, is useful as a shared vocabulary for talking about harmony outside the common practice and for discussing the edges of tonality, but atonality in general is outside the scope of Harmonious. More

Atritonic Set Theory

A set class is atritonic if it does not contain any tritones. See the Atritonic index.

Augmented

An augmented chord refers to a chord with a major third (four semitones) and an augmented fifth (eight semitones) above the root, for example, the augmented triad, which perfectly divides the octave in three equal pieces and is a mode of limited transposition.

Avoid Note

In jazz theory, each mode may have one or more avoid notes or intolerably dissonant notes, that are known to produce an undesirable effect when sustained as melody notes against the notes of the harmony. For example, in the Key of C Major, the fourth scale degree (or eleventh), the note C, is avoided when improvising or playing a melody over the G Dom 7 chord or G Mixolydian mode. This also tells us which available tensions or extensions can be used to form chords that will be compatible with each mode. More

Bebop

Bebop (also shortened to bop, as in hard bop) refers to a style of jazz music developed in the 1940’s with smaller ensembles of virtuoso instrument players trading improvised solos (usually heavily scale-based) over the chord changes of a song, instead of relying on the pre-composed, large-ensemble approach of big band-era swing music. More

Blues

Blues is a style of music which developed in the United States after the Civil War and into the early twentieth century, with roots that can be traced back through African-American spirituals and work songs to Africa, and which gradually grew into new styles, including rock and roll, rhythm and blues (R&B) and jazz. More

Cardinality Set Theory

In musical set theory, cardinality just means the number of distinct pitch classes—as opposed to the number of notes—in a scale or chord. Harmonious focuses on chords and scales with cardinality three to nine. More

Cardinality Equivalence Set Theory

Cardinality equivalence treats musical objects with duplicates of a given pitch class (note name) and objects with only a singular appearance of that note as if they were both the same object. More

Cent

In the study of different tuning systems, the octave is divided into 1200 cents, so each equal-tempered semitone is exactly 100 cents. A typical listener with a good ear can probably hear tuning problems or “pitchiness” of around 15 to 25 cents, or less.

Chord

Chords are collections of notes, usually played simultaneously, and usually emphasizing a root note either rhythmically or by placing it in the bass, or both. Chords are named for their root and chord type. Chords of the same chord type are related by transposition. More

Chord Formula

A chord formula is just a transposition-equivalent representation of a chord type listing intervals above the root. For example R 3 5 for a Major Triad means root, major third, perfect fifth. (R is used for the root instead of 1, in Harmonious.) More

Chord Type

A chord type is a set of up to twelve transpositions of a chord, ignore transposition. An OTC-equivalent musical object results in a named chord when combined with a root note. More

Chromatic Cluster Set Theory

Set classes with chromatic clusters have runs of two (or more) consecutive semitone intervals (i.e. three notes or more). Chromatic-cluster-containing scales and chords have limited voice-leading possibilities. More

Chromatic Scale

The twelve-tone chromatic scale is a set of pitches one twelfth of an octave apart that forms the basis of Western music. Played as a scale, the chromatic scale is fairly boring and flat sounding, but used for short stretches it can add flair to otherwise tonal pieces. Harmonious explains the chromatic scale’s inherent structure, starting at Pitch & Intervals. More

Clock Diagram Set Theory

In Harmonious, a colored clock diagram represents a pitch class set, an OPC-equivalent object. A black and white clock diagram collects up to twelve transpositions of a given pitch class set into a single visual representation of an OPTC-equivalent object. More

Cluster-free Set Theory

Harmonious focuses on cluster-free set classes, a subset of 124 of the 336 set classes with three to nine notes. Cluster-free set classes do not contain chromatic clusters, or runs of two (or more) consecutive semitone intervals (i.e. three notes or more). More

Cohemitonic Set Theory

A set class is cohemitonic if it contains a chromatic cluster, or a two-or-more-semitone-in-a-row sequence, described as chromatic cluster-containing in Harmonious. More

Common Practice

The common practice period refers to the era of musical composition dominated by the tonal system, roughly 1650 to 1900, from the baroque period through the Classical, Romantic and Impressionist periods, characterized by certain harmonic conventions, generally employing triads and a major tonic or minor tonic.

Compatibility

The heart of jazz theory is chord-scale compatibility, which allows multiple players to improvise heavily scale-based solo parts over chord changes. Chords are grouped into modes with which they are said to be compatible. More

Complement Set Theory

Complements are set classes with the same Lewin-Quinn FC-components, which are not related by transposition or involution. So for example the pentatonic scale and the diatonic scale are complements—the “holes” in one are the set class for the other: the black keys (pentatonic scale) are the holes in the white key (diatonic) scale, and vice versa. More

Consonance

A chord is consonant when it contains mostly consonant intervals, such as the perfect fourth or fifth, and major or minor third, or their inversions. Dissonant intervals include the tritone and the semitone. The whole tone is generally a consonant interval in scales.

Diatonic

The diatonic scale or major scale refers to the set class (unordered collection of notes) with twelve transpositions of the white keys. This scale is common in many cultures and across styles of music. The seven modes of the diatonic scale include the Ionian or Major mode (Do-Re-Mi-Fa-So-La-Ti-Do), the Natural Minor mode, and more. More

Diminished

A diminished chord refers to a chord with a minor third interval and a flat fifth above the root, for example the diminished triad, half-diminished seventh, or diminished seventh chords. A diminished scale such as the Locrian scale, has some of those same intervals above the root. See also the diminished (octatonic) scale.

Double Augmented Hexatonic

The hexatonic (six-note) double augmented scale consists of alternating semitone and minor third intervals. This scale is a mode of limited transposition and can be thought of as two interlocking augmented chords. More

Double Diminished (Octatonic)

The double diminished (octatonic) scale refers to the set class (unordered collection) of eight notes consisting of alternating semitone and whole-tone intervals. This scale is a mode of limited transposition and can be thought of as two interlocking fully-diminished seventh chords. More

Eleventh

The eleventh is a compound interval consisting of an octave plus a fourth and can refer to the perfect eleventh or the augmented eleventh. In the context of chords, the degree is usually notated as an eleventh instead of a fourth. The Lydian augmented eleventh interval is a common use of the term “eleventh.” See Pitch & Intervals.

Enharmonic Equivalent

Two notes are enharmonic equivalent in 12-TET if they share the same pitch or pitch class, but have different note names. More

Evenness Set Theory

Nearly even or maximally even chords and scales are much more common in harmony because they are related to their involutions and transpositions by short voice leadings, whereas perfectly even chords and scales are limited in their voice leading possibilities. More

Fifth

The fifth or perfect fifth is a consonant interval with a ratio very close to 3:2. See also diminished fifth and augmented fifth intervals and Pitch & Intervals. More

Forte Number Set Theory

Allen Forte assigned every set class (sets of chords and scales with the same interval content) its own number (Forte number). So the major and minor triads have Forte number 3-11, the diminished seventh chord 4-28, etc. More

Fourth

The fourth or perfect fourth is a consonant interval with a ratio very close to 4:3. See also diminished fourth and augmented fourth intervals and Pitch & Intervals. In the context of chords, the degree is usually notated as an eleventh. More

Guitar

The guitar is a fretted, stringed instrument of medieval and perhaps ancient lineage whose modern forms can be traced back to the Spanish classical guitar of Antonio de Torres. The six-string guitar has standardized on a tuning (E2 A2 D3 G3 B3 E4) that allows the same chord to be played across the fretboard in many ways, yet this tuning also allows solo or lead playing that can rival any other instrument in terms of expressiveness. Some guitarists can play lead and accompaniment at the same time, like on a piano. More

Harmonic Major

The harmonic major scale is a fairly even heptatonic (seven-note) scale consisting of mostly semitone and whole-tone intervals, and one minor third (three-semitone) gap. It is involution-related to the harmonic minor scale. More

Harmonic Minor

The harmonic minor scale is a fairly even heptatonic (seven-note) scale consisting of mostly semitone and whole-tone intervals, and one minor third (three-semitone) gap. It is involution-related to the harmonic major scale. More

Harmonious

adj. 1. having components pleasingly or appropriately combined; symmetrical 2. constituting a consistent or an aesthetically pleasing whole 3. exhibiting equivalence or correspondence among constituents 4. exhibiting harmony or being in harmony 5. congruous; suitable and fitting; concordant; consonant; symphonious; melodious; dulcet; tuneful

Harmony

Harmony is the study of what notes sound good together and how to embellish melodies with more notes to sound more full. Melody and harmony are related, so the study of harmony helps composers write melodies from chord progressions and chord progressions from melodies. More

Interval

Two notes played sequentially or simultaneously form an interval. An interval measures the distance between notes in semitones (one twelfth of an octave.) All chords and scales are formed by two or more intervals (three or more notes), so intervals are as fundamental to harmony as pitch or notes. Some intervals have multiple names (for historical tuning reasons, and for theoretical reasons). More

Interval Class Set Theory

In musical set theory, the octave and unison are not treated as intervals (see octave equivalence) and all possible intervals are grouped by inversion into six interval classes, denoted by colors in the intervals table. More

Interval Content Set Theory

Allen Forte’s interval content vectors nearly capture the internal structure of each set class but Lewin-Quinn FC components capture more information and use complementarity to group musical objects into fewer objects. (See Quinn’s master thesis in References.) More

Inversion

(Not to be confused with the inverse or involution.) Two chords or intervals are related by registral inversion when they contain the same note names, just in a different order. For example, a root position C major triad has the notes C E G in that order, while a first inversion C major has E G C in that order. Results of interactive key search and interactive fret search show all of the inversions for a given chord or a given unordered set of note names. More

Involution Set Theory

Also called the inverse. (Not to be confused with registral inversion.) Two objects are related by involution when they have the same interval content but in a different order. More

Jazz

Jazz is a style of music characterized by certain harmonic conventions (jazz theory), a swung beat, improvisation, and gravitating toward certain instrumental ensembles. Its roots lie partially in the blues and it developed out of African-American communities in certain large US cities.

Jazz Theory

There is no reason the well-established harmonic conventions in a century of jazz theory (or jazz harmony) cannot be divorced from the style of music where it was developed, jazz—with its swung beat and specific instruments, tempos, and moods—and used as a (partial) basis of generalizing harmony beyond the strictures of the common practice. More

Key

Major-minor tonality in the common practice is organized into twelve major keys and twelve minor keys. Each key collects up a set of chords and progressions which emphasizes movement away from and back to the tonic, a chord sharing the name of the key. More

Keyboard

The musical keyboard is the interface to many instruments, including the pipe organ (and modern electronic descendants); the carillon (bell tower); the clavichord and the harpsichord, precursors to the piano; and modern electronic synthesizers and samplers. Though mainly played with the fingers, foot-controlled pedal units (usually for playing in lower registers) are also available, with a similar layout and function. More

Lewin-Quinn FC-components Set Theory

David Lewin (1959) proposed a six-number representation (FC1 through FC6) of the interval content of a set class that measures how unbalanced the set class is relative to each of the six interval classes. More

Limited Transposition Set Theory

Perfectly even chords and scales are limited in their voice leading possibilities partly because they have a limited number of transpositions. The Limited Transposition index lists all possible modes of limited transposition for 12-TET. More

M-Relation Set Theory

Two set classes are M-related if they share the same interval content in the interval classes for whole-tone, minor third, major third, and tritone (equivalent to the FC2, FC3, FC4, FC6 components) but have semitones and fourths swapped. The “M” stands for Multiplication, referring to M₅(y) and M₇(y). More

Major

Major can refer to the major third, the major triad (or related chords), or major keys. Sometimes the unordered collection of notes of the diatonic scale is referred to as the major scale, and sometimes the Ionian mode of the diatonic scale is called the major scale. More

Melody

Melody refers to a rhythmic sequence of notes meant to lodge itself in the listener’s mind, perhaps through repetition, perhaps through strangeness, perhaps through utter beauty. Melody can exist without harmony (common in many cultures), but melody and harmony cannot exist without rhythm. More

Minor

Minor can refer to the minor third, the minor triad (or related chords), or minor keys. More

Mode

Modes are named collections of specific notes, usually played sequentially, that emphasize a certain root. The unordered collection of notes in a single scale can have multiple modes with different names, where each note in the scale is the root of a different mode. More

Ninth

The ninth is a compound interval consisting of an octave plus a second. In the context of chords, the degree is usually notated as a ninth instead of a second. See minor ninth, major ninth, and augmented ninth intervals and Pitch & Intervals.

Note

A note played by different instruments can sound very different, but still be considered the same note, since the two sounds share the same fundamental frequency (the same pitch). More

OC-Equivalence Set Theory

In Harmonious, Octave + Cardinality-Equivalence, or OC-Equivalence, refers to when two or more scales or chords have the same notes present (pitch classes) and the same root note, but each note may appear in different registers or a different number of times (but at least once in each chord or scale). More

Octatonic

A scale is octatonic if it contains eight distinct notes. The most common eight-note scale is the double diminished octatonic scale referred to as the diminished (octatonic) scale in Harmonious.

Octave

An octave is the interval between two notes with the same note name, a ratio of 2:1. (See Pitch & Intervals) Two notes that are an octave apart are clearly not the same pitch but sound the same, though one is clearly “higher” and one is “lower.” More

Octave-Equivalence Set Theory

Since the octave is the foundation of human hearing and the foundation of most tuning systems, it makes sense to treat two chords and scales the same when they are played exactly the same but up or down an octave from one another—they have the same name and act the same for analysis, for memorization, etc. More

OPC-Equivalence Set Theory

Octave + Permutation + Cardinality Equivalence or OPC-Equivalence treats musical objects the same if they have the same note names or pitch classes even if the notes come in a different order, in different octave registers, or if the pitch classes occur just once or many times. More

OPTC-Equivalence Set Theory

Octave + Permutation + Transposition + Cardinality Equivalence or OPTC-Equivalence treats musical objects the same if they are transpositions of the same unordered collection of notes (see OPC-equivalence); that is, if they share the same prime form. More

OPTIC-Equivalence Set Theory

Octave + Permutation + Transposition + Involution + Cardinality Equivalence or OPTIC-Equivalence treats musical objects the same if they are transpositions and/or involutions of the same unordered collection of notes (see OPC-equivalence); that is, if they share the same forte number. More

OPTIC/K-Equivalence Set Theory

Octave + Permutation + Transposition + Involution + Cardinality + Complementarity Equivalence or OPTIC/K-Equivalence treats musical objects the same if they are OPTIC-Equivalent and/or complements; that is, if they share the same row of the set classes table. (The K for Complementarity is not standard terminology, it is just used here to make a working acronym.) More

OTC-Equivalence Set Theory

Octave + Transposition + Cardinality Equivalence or OTC-Equivalence treats musical objects the same if they have the same intervals relative to a root. Chord inversion types and mode types are examples of OTC-equivalent objects. More

Other Scales

Harmonious focuses on a small handful of unordered scale types. Other scales are either subsets of these, or less common in jazz theory. More

Parallel Key

The parallel minor key for a major key refers to the minor key with the same tonic note, and vice versa. For example the keys of C major (no sharps or flats) and C minor (three flats) are parallel, or the keys of A Major (three sharps) and A minor (no sharps or flats). More

Pentatonic

Pentatonic refers to any scale or chord with five notes, but usually refers to the pentatonic scale, a very consonant scale that is the complement of the diatonic scale. The scale consists of whole-tone steps and minor thirds. More

Permutation Equivalence Set Theory

Permutation equivalence treats musical objects the same if they have the same note names or pitch classes even if the notes come in a different order. More

Piano

The piano is a musical instrument invented around the year 1700 by Bartolomeo Cristofori—and refined significantly in the next few centuries—in which the strings are struck by mechanical hammers triggered by the player’s fingers striking the keys of the keyboard. The term piano is short for pianoforte, referring to the ability of the player to play both soft and loud, in contrast with the dominant keyboard instrument of the time, the harpsichord, which had no dynamic volume capability. More

Pitch

Pitch describes the frequency of notes, where quicker vibrations are higher frequency or higher in pitch (tighter, shorter strings, shorter pipes and smaller instruments) and slower vibrations are lower frequency or lower in pitch (semi-truck driving by, looser, longer strings, large instruments). More

Pitch Class Set Theory

A pitch class is the collection of all notes with the same name (or enharmonic equivalent in 12-TET) ignoring octave, and ignoring repeats of that note. There are twelve pitch classes, conventionally named with numbers starting from zero. More

Playing Outside

In jazz theory, playing outside refers to an atonal or polytonal technique where a soloist purposely plays notes outside of the expected notes for the current chord or scale, (usually from a very different mode or scale) that sound “wrong,” usually for a brief period of time, to create tension and then a feeling of resolution when suddenly the “right notes” are again played over the chord changes.

Prime Form Set Theory

A prime form is a transposable, numerical representation of a clock diagram, reduced to its lowest (numerical) form, and where t and e mean ten and eleven in base twelve. Prime forms collect up two or more transpositions (usually twelve) into a single object. More

Quartal Set Theory

A chord or scale is quartal if it can be constructed by repeatedly stacking fourths (or fifths, since they share an interval class). Quartal harmony refers to harmony that is not tertiary (thirds-based) but based on perfect fourths and nearby intervals (major thirds, tritones). More

Reharmonization

In jazz theory, reharmonization refers to the substitution of modes or chords for similar modes or chords. More specifically reharmonization would take an existing tune or set of chords changes and swap out one mode for a similar mode (closely related by voice leading) or chords that are compatible with a different mode. More

Relative Key

The relative minor key for a major key refers to the minor key with the same parent scale, and vice versa. For example the keys of C major (no sharps or flats) and A minor (no sharps or flats) are relative, or the keys of E♭ Major (three flats) and C minor (three flats). More

Rhythm

Rhythm is the throbbing heart of music, since rhythm can exist without melody (and does in many cultures), and melody can exist without harmony (again, common in many cultures), but melody and harmony cannot exist without rhythm. More

Roman Numeral Function

Roman numeral function or functional analysis is a representation of a piece of music that shows how the theory works the same for all keys (see transposition). Within a given key, we can convert from chord names or mode names to Roman numerals, then change the key, then convert the numerals back to chords or modes, and now we have transposed a piece while maintaining all of the the relationships between all of the chords, keeping the structure of the piece the same, just moving all of the notes up or down by a set number of semitones. (See Interactive Key Slider and Keys for a breakdown of all chords in all keys.) More

Root

Instead of numbering a note with “1” for a mode formula or chord formula, we can use R, for root, so named because other notes are measured by their distance above this note, and because modes and chords will be named after the root. (See Diatonic Modes & Chords for a detailed explanation and example of this.) More

Scale

Scales (at least in Harmonious) are generalized to mean unordered collections of 5, 6, 7, or 8 notes, usually played sequentially, and from which chords and modes are built. More

Second

A second can refer to multiple types of intervals: either a minor second or flat second, equal to a semitone; or major second, equal to a whole tone. In scales, the term second may be used, whereas in chords, the intervals would usually be referred to as ninths. See Pitch & Intervals.

Semitone

A semitone or half-step (also minor second) is the smallest interval or difference in pitch between two notes in the twelve tone system, and divides the octave into twelve equal pieces. All intervals can be divided into a whole number of semitones. (See also Whole tone.) More

Set Class Set Theory

A set class is an unordered collection of notes without regard to which octave the notes are in, or what order they are played in, reduced to its prime form by transposition. Every possible collection of 3 to 9 distinct notes (whether considered a scale or chord or both) is included on the Set Classes page. More

Seventh

A seventh can refer to multiple types of intervals: either a diminished seventh, a minor seventh, or a major seventh. See Pitch & Intervals. More

Sixth

A sixth can refer to multiple types of intervals: either a minor sixth, or a major sixth. In the context of chords, the degree is usually notated as a sixth for major and minor chords, and a thirteenth for everything else. See Pitch & Intervals. More

Slash Chords

Slash chords describe how to re-formulate a given complicated chord (usually more than three notes) in terms of a simpler chord with fewer notes, breaking the chord into “Chord (Over) Root” notation. More

Suspended

A suspended chord may refer to a suspended second or suspended fourth chord, a triad that is neither major nor minor, or a dominant seventh suspended fourth chord.

Symmetry Set Theory

A set class is symmetric if the involution of the set class is equal to itself (up to transposition). More

Tenth

The tenth is a compound interval consisting of an octave plus a third. In the context of chords, the term “tenth” is rarely used, and is usually notated in terms of thirds.

Tertiary

Tertiary harmony refers to chords built from stacked major and minor thirds. In order to complete the octave, sometimes a perfect fourth must be employed (for example, major and minor triads). More

Third

A third can refer to multiple types of intervals: either a minor third or flat third, or a major third. See Pitch & Intervals.

Thirteenth

The thirteenth is a compound interval consisting of an octave plus a sixth. In the context of chords, the degree is usually notated as a sixth for major and minor chords, and a thirteenth for everything else. See Pitch & Intervals.

Tonality

Used narrowly, the term tonality refers to tonal music written in the tonal system of the common practice period (roughly 1650 to 1900), or any music that uses the same restrictive harmonic conventions, generally employing triads and a major tonic or minor tonic. Used more broadly, tonal music could be considered anything not atonal. More

Tonic

The tonic chord of a major key (the I chord) (and the i chord in a minor key) shares a root with the name of the key and is considered a restful “home” place for starting and ending many phrases of chord progressions. The dominant (V or V⁷) chord tonicizes or prepares the tonic in many progressions.

Transposition

Musical objects that are related by transposition have the same interval content and typically the same feeling and function. For example C major triad and D major triad are related by transposition, since moving every note in C major by the same number of semitones (two in this case) results in D major. More

Triad

Triads are chords with three distinct pitch classes, such as the major triad, the minor triad, the diminished triad, and the augmented triad.

Tritone

A tritone is the interval between one note and a note three whole tones higher (or lower, also six semitones), with a ratio of √2:1, dividing the octave perfectly in half. More

Tritonic Set Theory

A set class is tritonic if it does contains any tritones. See the Tritonic index.

Tuning Systems

Historically, other tuning systems besides 12-tone equal temperament have been popular. A few important tuning systems are discussed briefly, with pointers outside of Harmonious where you can learn more. More

Twelfth

The twelfth is a compound interval consisting of an octave plus a fifth. In the context of chords, the term “twelfth” is rarely used, and is usually notated in terms of fifths.

Twelve-tone Equal Temperament

Twelve-tone equal temperament or 12-TET is the ubiquitous modern tuning system in Western music that makes certain tradeoffs compared to its tuning-system predecessors, eliminating one major restriction (limited keys per piece) by redistributing impurity more evenly to all intervals and all keys, making major and minor third intervals more impure and making the equal-tempered semitone and tritone very impure (see table). More

Unison

One or more instruments or voices playing the same note at the same time constitute a unison (assuming they are in tune). This the closest thing in harmony to an interval of a “first,” which is not a real term in harmony; the smallest interval is the minor second. In musical set theory unison intervals and octave intervals are treated as equivalent since they are octave equivalent.

Voice Leading

Voice leading in western polyphonic music of the common practice refers to the motion of one or more notes or voices moving by (usually) small distances when one chord (formed by the notes of one or more instruments) “changes” into another chord. More

Whole Tone

A whole tone or whole-step (also major second) is a two-semitone interval or difference in pitch between two notes in the twelve tone system, which divides the octave into six equal pieces. (See also Semitone.) More

Whole-Tone Scale

The whole-tone scale is the unordered set of notes (set class) consisting of six whole tone intervals. The scale is a mode of limited transposition. More

Z-Relation Set Theory

Two set classes are Z-related when they share the same interval content but are not related by involution. Allen Forte assigned Forte numbers which included a Z to indicate which set classes had Z-relations. (According to Forte, the letter Z stands for “zygotic.”) More