In music theory, an **interval** is a difference in pitch between two sounds.^{ [1] } An interval may be described as **horizontal**, **linear**, or **melodic** if it refers to successively sounding tones, such as two adjacent pitches in a melody, and **vertical** or **harmonic** if it pertains to simultaneously sounding tones, such as in a chord.^{ [2] }^{ [3] }

- Size
- Frequency ratios
- Cents
- Main intervals
- Interval number and quality
- Number
- Quality
- Example
- Shorthand notation
- Inversion
- Classification
- Melodic and harmonic
- Diatonic and chromatic
- Consonant and dissonant
- Simple and compound
- Steps and skips
- Enharmonic intervals
- Minute intervals
- Compound intervals
- Main compound intervals
- Intervals in chords
- Chord qualities and interval qualities
- Deducing component intervals from chord names and symbols
- Size of intervals used in different tuning systems
- Interval root
- Interval cycles
- Alternative interval naming conventions
- Latin nomenclature
- Non-diatonic intervals
- Pitch-class intervals
- Generic and specific intervals
- Comparison
- Generalizations and non-pitch uses
- See also
- Notes
- References
- Sources
- External links

In Western music, intervals are most commonly differences between notes of a diatonic scale. Intervals between successive notes of a scale are also known as scale steps. The smallest of these intervals is a semitone. Intervals smaller than a semitone are called microtones. They can be formed using the notes of various kinds of non-diatonic scales. Some of the very smallest ones are called commas, and describe small discrepancies, observed in some tuning systems, between enharmonically equivalent notes such as C♯ and D♭. Intervals can be arbitrarily small, and even imperceptible to the human ear.

In physical terms, an interval is the ratio between two sonic frequencies. For example, any two notes an octave apart have a frequency ratio of 2:1. This means that successive increments of pitch by the same interval result in an exponential increase of frequency, even though the human ear perceives this as a linear increase in pitch. For this reason, intervals are often measured in cents, a unit derived from the logarithm of the frequency ratio.

In Western music theory, the most common naming scheme for intervals describes two properties of the interval: the quality (perfect, major, minor, augmented, diminished) and number (unison, second, third, etc.). Examples include the minor third or perfect fifth. These names identify not only the difference in semitones between the upper and lower notes but also how the interval is spelled. The importance of spelling stems from the historical practice of differentiating the frequency ratios of enharmonic intervals such as G–G♯ and G–A♭.^{ [4] }

The size of an interval (also known as its width or height) can be represented using two alternative and equivalently valid methods, each appropriate to a different context: frequency ratios or cents.

The size of an interval between two notes may be measured by the ratio of their frequencies. When a musical instrument is tuned using a just intonation tuning system, the size of the main intervals can be expressed by small-integer ratios, such as 1:1 (unison), 2:1 (octave), 5:3 (major sixth), 3:2 (perfect fifth), 4:3 (perfect fourth), 5:4 (major third), 6:5 (minor third). Intervals with small-integer ratios are often called *just intervals*, or *pure intervals*.

Most commonly, however, musical instruments are nowadays tuned using a different tuning system, called 12-tone equal temperament. As a consequence, the size of most equal-tempered intervals cannot be expressed by small-integer ratios, although it is very close to the size of the corresponding just intervals. For instance, an equal-tempered fifth has a frequency ratio of 2^{.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}7⁄12}:1, approximately equal to 1.498:1, or 2.997:2 (very close to 3:2). For a comparison between the size of intervals in different tuning systems, see § Size of intervals used in different tuning systems.

The standard system for comparing interval sizes is with cents. The cent is a logarithmic unit of measurement. If frequency is expressed in a logarithmic scale, and along that scale the distance between a given frequency and its double (also called octave) is divided into 1200 equal parts, each of these parts is one cent. In twelve-tone equal temperament (12-TET), a tuning system in which all semitones have the same size, the size of one semitone is exactly 100 cents. Hence, in 12-TET the cent can be also defined as one hundredth of a semitone.

Mathematically, the size in cents of the interval from frequency *f*_{1} to frequency *f*_{2} is

The table shows the most widely used conventional names for the intervals between the notes of a chromatic scale. A perfect unison (also known as perfect prime)^{ [5] } is an interval formed by two identical notes. Its size is zero cents. A semitone is any interval between two adjacent notes in a chromatic scale, a whole tone is an interval spanning two semitones (for example, a major second), and a tritone is an interval spanning three tones, or six semitones (for example, an augmented fourth).^{ [lower-alpha 1] } Rarely, the term ditone is also used to indicate an interval spanning two whole tones (for example, a major third), or more strictly as a synonym of major third.

Intervals with different names may span the same number of semitones, and may even have the same width. For instance, the interval from D to F♯ is a major third, while that from D to G♭ is a diminished fourth. However, they both span 4 semitones. If the instrument is tuned so that the 12 notes of the chromatic scale are equally spaced (as in equal temperament), these intervals also have the same width. Namely, all semitones have a width of 100 cents, and all intervals spanning 4 semitones are 400 cents wide.

The names listed here cannot be determined by counting semitones alone. The rules to determine them are explained below. Other names, determined with different naming conventions, are listed in a separate section. Intervals smaller than one semitone (commas or microtones) and larger than one octave (compound intervals) are introduced below.

Number of semitones | Minor, major, or perfect intervals | Short | Augmented or diminished intervals | Short | Widely used alternative names | Short | Audio |
---|---|---|---|---|---|---|---|

0 | Perfect unison | P1 | Diminished second | d2 | ^{ ⓘ } | ||

1 | Minor second | m2 | Augmented unison | A1 | Semitone, half tone, half step | S | ^{ ⓘ } |

2 | Major second | M2 | Diminished third | d3 | Tone, whole tone, whole step | T | ^{ ⓘ } |

3 | Minor third | m3 | Augmented second | A2 | ^{ ⓘ } | ||

4 | Major third | M3 | Diminished fourth | d4 | ^{ ⓘ } | ||

5 | Perfect fourth | P4 | Augmented third | A3 | ^{ ⓘ } | ||

6 | Diminished fifth | d5 | Tritone | TT | ^{ ⓘ } | ||

Augmented fourth | A4 | ||||||

7 | Perfect fifth | P5 | Diminished sixth | d6 | ^{ ⓘ } | ||

8 | Minor sixth | m6 | Augmented fifth | A5 | ^{ ⓘ } | ||

9 | Major sixth | M6 | Diminished seventh | d7 | ^{ ⓘ } | ||

10 | Minor seventh | m7 | Augmented sixth | A6 | ^{ ⓘ } | ||

11 | Major seventh | M7 | Diminished octave | d8 | ^{ ⓘ } | ||

12 | Perfect octave | P8 | Augmented seventh | A7 | ^{ ⓘ } |

In Western music theory, an interval is named according to its *number* (also called *diatonic number*) and *quality*. For instance, *major third* (or **M3**) is an interval name, in which the term *major* (**M**) describes the quality of the interval, and *third* (**3**) indicates its number.

The number of an interval is the number of letter names or staff positions (lines and spaces) it encompasses, including the positions of both notes forming the interval. For instance, the interval C–G is a fifth (denoted **P5**) because the notes from C to the G above it encompass five letter names (C, D, E, F, G) and occupy five consecutive staff positions, including the positions of C and G. The table and the figure above show intervals with numbers ranging from 1 (e.g., **P1**) to 8 (e.g., **P8**). Intervals with larger numbers are called compound intervals.

There is a one-to-one correspondence between staff positions and diatonic-scale degrees (the notes of diatonic scale).^{ [lower-alpha 2] } This means that interval numbers can also be determined by counting diatonic scale degrees, rather than staff positions, provided that the two notes that form the interval are drawn from a diatonic scale. Namely, C–G is a fifth because in any diatonic scale that contains C and G, the sequence from C to G includes five notes. For instance, in the A♭-major diatonic scale, the five notes are C–D♭–E♭–F–G (see figure). This is not true for all kinds of scales. For instance, in a chromatic scale, there are eight notes from C to G: (C–C♯–D–D♯–E–F–F♯–G). This is the reason interval numbers are also called *diatonic numbers*, and this convention is called *diatonic numbering*.

If one adds any accidentals to the notes that form an interval, by definition the notes do not change their staff positions. As a consequence, any interval has the same interval number as the corresponding natural interval, formed by the same notes without accidentals. For instance, the intervals C–G♯ (spanning 8 semitones) and C♯–G (spanning 6 semitones) are fifths, like the corresponding natural interval C–G (7 semitones).

Notice that interval numbers represent an inclusive count of encompassed staff positions or note names, not the difference between the endpoints. In other words, one starts counting the lower pitch as one, not zero. For that reason, the interval C–C, a perfect unison, is called a prime (meaning "1"), even though there is no difference between the endpoints. Continuing, the interval C–D is a second, but D is only one staff position, or diatonic-scale degree, above C. Similarly, C–E is a third, but E is only two staff positions above C, and so on. As a consequence, joining two intervals always yields an interval number one less than their sum. For instance, the intervals C–E and E–G are thirds, but joined together they form a fifth (C–G), not a sixth. Similarly, a stack of three thirds, such as C–E, E–G, and G–B, is a seventh (C–B), not a ninth.

This scheme applies to intervals up to an octave (12 semitones). For larger intervals, see § Compound intervals below.

The name of any interval is further qualified using the terms **perfect** (**P**), major (**M**), minor (**m**), augmented (**A**), and diminished (**d**). This is called its *interval quality*. It is possible to have doubly diminished and doubly augmented intervals, but these are quite rare, as they occur only in chromatic contexts. The quality of a compound interval is the quality of the simple interval on which it is based. Some other qualifiers like *neutral*, *subminor*, and *supermajor* are used for non-diatonic intervals.

Perfect intervals are so-called because they were traditionally considered perfectly consonant,^{ [6] } although in Western classical music the perfect fourth was sometimes regarded as a less than perfect consonance, when its function was contrapuntal.^{[ vague ]} Conversely, minor, major, augmented, or diminished intervals are typically considered less consonant, and were traditionally classified as mediocre consonances, imperfect consonances, or near-dissonances.^{ [6] }

Within a diatonic scale ^{ [lower-alpha 2] } all unisons (**P1**) and octaves (**P8**) are perfect. Most fourths and fifths are also perfect (**P4** and **P5**), with five and seven semitones respectively. One occurrence of a fourth is augmented (**A4**) and one fifth is diminished (**d5**), both spanning six semitones. For instance, in a C-major scale, the **A4** is between F and B, and the **d5** is between B and F (see table).

By definition, the inversion of a perfect interval is also perfect. Since the inversion does not change the pitch class of the two notes, it hardly affects their level of consonance (matching of their harmonics). Conversely, other kinds of intervals have the opposite quality with respect to their inversion. The inversion of a major interval is a minor interval, the inversion of an augmented interval is a diminished interval.

As shown in the table, a diatonic scale ^{ [lower-alpha 2] } defines seven intervals for each interval number, each starting from a different note (seven unisons, seven seconds, etc.). The intervals formed by the notes of a diatonic scale are called diatonic. Except for unisons and octaves, the diatonic intervals with a given interval number always occur in two sizes, which differ by one semitone. For example, six of the fifths span seven semitones. The other one spans six semitones. Four of the thirds span three semitones, the others four. If one of the two versions is a perfect interval, the other is called either diminished (i.e. narrowed by one semitone) or augmented (i.e. widened by one semitone). Otherwise, the larger version is called major, the smaller one minor. For instance, since a 7-semitone fifth is a perfect interval (**P5**), the 6-semitone fifth is called "diminished fifth" (**d5**). Conversely, since neither kind of third is perfect, the larger one is called "major third" (**M3**), the smaller one "minor third" (**m3**).

Within a diatonic scale,^{ [lower-alpha 2] } unisons and octaves are always qualified as perfect, fourths as either perfect or augmented, fifths as perfect or diminished, and all the other intervals (seconds, thirds, sixths, sevenths) as major or minor.

Augmented intervals are wider by one semitone than perfect or major intervals, while having the same interval number (i.e., encompassing the same number of staff positions): they are wider by a chromatic semitone. Diminished intervals, on the other hand, are narrower by one semitone than perfect or minor intervals of the same interval number: they are narrower by a chromatic semitone. For instance, an augmented third such as C–E♯ spans five semitones, exceeding a major third (C–E) by one semitone, while a diminished third such as C♯–E♭ spans two semitones, falling short of a minor third (C–E♭) by one semitone.

The augmented fourth (**A4**) and the diminished fifth (**d5**) are the only augmented and diminished intervals that appear in diatonic scales^{ [lower-alpha 2] } (see table).

Neither the number, nor the quality of an interval can be determined by counting semitones alone. As explained above, the number of staff positions must be taken into account as well.

For example, as shown in the table below, there are four semitones between A♭ and B♯, between A and C♯, between A and D♭, and between A♯ and E , but

- A♭–B♯ is a second, as it encompasses two staff positions (A, B), and it is doubly augmented, as it exceeds a major second (such as A–B) by two semitones.
- A–C♯ is a third, as it encompasses three staff positions (A, B, C), and it is major, as it spans 4 semitones.
- A–D♭ is a fourth, as it encompasses four staff positions (A, B, C, D), and it is diminished, as it falls short of a perfect fourth (such as A–D) by one semitone.
- A♯-E is a fifth, as it encompasses five staff positions (A, B, C, D, E), and it is triply diminished, as it falls short of a perfect fifth (such as A–E) by three semitones.

Number of semitones | Interval name | Staff positions | ||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | ||

4 | doubly augmented second (AA2) | A♭ | B♯ | |||

4 | major third (M3) | A | C♯ | |||

4 | diminished fourth (d4) | A | D♭ | |||

4 | triply diminished fifth (ddd5) | A♯ | E |

Intervals are often abbreviated with a **P** for perfect, **m** for minor, **M** for major, **d** for diminished, **A** for augmented, followed by the interval number. The indications M and P are often omitted. The octave is P8, and a unison is usually referred to simply as "a unison" but can be labeled P1. The tritone, an augmented fourth or diminished fifth is often **TT**. The interval qualities may be also abbreviated with **perf**, **min**, **maj**, **dim**, **aug**. Examples:

- m2 (or min2): minor second,
- M3 (or maj3): major third,
- A4 (or aug4): augmented fourth,
- d5 (or dim5): diminished fifth,
- P5 (or perf5): perfect fifth.

A simple interval (i.e., an interval smaller than or equal to an octave) may be inverted by raising the lower pitch an octave or lowering the upper pitch an octave. For example, the fourth from a lower C to a higher F may be inverted to make a fifth, from a lower F to a higher C.

There are two rules to determine the number and quality of the inversion of any simple interval:^{ [7] }

- The interval number and the number of its inversion always add up to nine (4 + 5 = 9, in the example just given).
- The inversion of a major interval is a minor interval, and vice versa; the inversion of a perfect interval is also perfect; the inversion of an augmented interval is a diminished interval, and vice versa; the inversion of a doubly augmented interval is a doubly diminished interval, and vice versa.

For example, the interval from C to the E♭ above it is a minor third. By the two rules just given, the interval from E♭ to the C above it must be a major sixth.

Since compound intervals are larger than an octave, "the inversion of any compound interval is always the same as the inversion of the simple interval from which it is compounded".^{ [8] }

For intervals identified by their ratio, the inversion is determined by reversing the ratio and multiplying the ratio by 2 until it is greater than 1. For example, the inversion of a 5:4 ratio is an 8:5 ratio.

For intervals identified by an integer number of semitones, the inversion is obtained by subtracting that number from 12.

Since an interval class is the lower number selected among the interval integer and its inversion, interval classes cannot be inverted.

Intervals can be described, classified, or compared with each other according to various criteria.

An interval can be described as

In general,

- A
*diatonic interval*is an interval formed by two notes of a diatonic scale. - A
*chromatic interval*is a non-diatonic interval formed by two notes of a chromatic scale.

The table above depicts the 56 diatonic intervals formed by the notes of the C major scale (a diatonic scale). Notice that these intervals, as well as any other diatonic interval, can be also formed by the notes of a chromatic scale.

The distinction between diatonic and chromatic intervals is controversial, as it is based on the definition of diatonic scale, which is variable in the literature. For example, the interval B–E♭ (a diminished fourth, occurring in the harmonic C-minor scale) is considered diatonic if the harmonic minor scales are considered diatonic as well.^{ [9] } Otherwise, it is considered chromatic. For further details, see the main article.

By a commonly used definition of diatonic scale^{ [lower-alpha 2] } (which excludes the harmonic minor and melodic minor scales), all perfect, major and minor intervals are diatonic. Conversely, no augmented or diminished interval is diatonic, except for the augmented fourth and diminished fifth.

The distinction between diatonic and chromatic intervals may be also sensitive to context. The above-mentioned 56 intervals formed by the C-major scale are sometimes called *diatonic to C major*. All other intervals are called *chromatic to C major*. For instance, the perfect fifth A♭–E♭ is chromatic to C major, because A♭ and E♭ are not contained in the C major scale. However, it is diatonic to others, such as the A♭ major scale.

Consonance and dissonance are relative terms that refer to the stability, or state of repose, of particular musical effects. Dissonant intervals are those that cause tension and desire to be *resolved* to consonant intervals.

These terms are relative to the usage of different compositional styles.

- In 15th- and 16th-century usage, perfect fifths and octaves, and major and minor thirds and sixths were considered harmonically consonant, and all other intervals dissonant, including the perfect fourth, which by 1473 was described (by Johannes Tinctoris) as dissonant, except between the upper parts of a vertical sonority—for example, with a supporting third below ("6-3 chords").
^{ [10] }In the common practice period, it makes more sense to speak of consonant and dissonant chords, and certain intervals previously considered dissonant (such as minor sevenths) became acceptable in certain contexts. However, 16th-century practice was still taught to beginning musicians throughout this period. - Hermann von Helmholtz (1821–1894) theorised that dissonance was caused by the presence of beats.
^{ [11] }Helmholtz further believed that the beating produced by the upper partials of harmonic sounds was the cause of dissonance for intervals too far apart to produce beating between the fundamentals.^{ [12] }Helmholtz then designated that two harmonic tones that shared common low partials would be more consonant, as they produced less beats.^{ [13] }^{ [14] }Helmholtz disregarded partials above the seventh, as he believed that they were not audible enough to have significant effect.^{ [15] }From this Helmholtz categorises the octave, perfect fifth, perfect fourth, major sixth, major third, and minor third as consonant, in decreasing value, and other intervals as dissonant. - David Cope (1997) suggests the concept of
**interval strength**,^{ [16] }in which an interval's strength, consonance, or stability is determined by its approximation to a lower and stronger, or higher and weaker, position in the harmonic series. See also: Lipps–Meyer law and #Interval root

All of the above analyses refer to vertical (simultaneous) intervals.

A simple interval is an interval spanning at most one octave (see Main intervals above). Intervals spanning more than one octave are called compound intervals, as they can be obtained by adding one or more octaves to a simple interval (see below for details).^{ [17] }

Linear (melodic) intervals may be described as *steps* or *skips*. A *step*, or *conjunct motion*,^{ [18] } is a linear interval between two consecutive notes of a scale. Any larger interval is called a *skip* (also called a *leap*), or *disjunct motion*.^{ [18] } In the diatonic scale,^{ [lower-alpha 2] } a step is either a minor second (sometimes also called *half step*) or major second (sometimes also called *whole step*), with all intervals of a minor third or larger being skips.

For example, C to D (major second) is a step, whereas C to E (major third) is a skip.

More generally, a step is a smaller or narrower interval in a musical line, and a skip is a wider or larger interval, where the categorization of intervals into steps and skips is determined by the tuning system and the pitch space used.

Melodic motion in which the interval between any two consecutive pitches is no more than a step, or, less strictly, where skips are rare, is called *stepwise* or *conjunct* melodic motion, as opposed to *skipwise* or *disjunct* melodic motions, characterized by frequent skips.

Two intervals are considered * enharmonic *, or *enharmonically equivalent*, if they both contain the same pitches spelled in different ways; that is, if the notes in the two intervals are themselves enharmonically equivalent. Enharmonic intervals span the same number of semitones.

For example, the four intervals listed in the table below are all enharmonically equivalent, because the notes F♯ and G♭ indicate the same pitch, and the same is true for A♯ and B♭. All these intervals span four semitones.

Number of semitones | Interval name | Staff positions | |||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | ||

4 | major third | F♯ | A♯ | ||

4 | major third | G♭ | B♭ | ||

4 | diminished fourth | F♯ | B♭ | ||

4 | doubly augmented second | G♭ | A♯ |

When played as isolated chords on a piano keyboard, these intervals are indistinguishable to the ear, because they are all played with the same two keys. However, in a musical context, the diatonic function of the notes these intervals incorporate is very different.

The discussion above assumes the use of the prevalent tuning system, 12-tone equal temperament ("12-TET"). But in other historic meantone temperaments, the pitches of pairs of notes such as F♯ and G♭ may not necessarily coincide. These two notes are enharmonic in 12-TET, but may not be so in another tuning system. In such cases, the intervals they form would also not be enharmonic. For example, in quarter-comma meantone, all four intervals shown in the example above would be different.

There are also a number of minute intervals not found in the chromatic scale or labeled with a diatonic function, which have names of their own. They may be described as microtones, and some of them can be also classified as commas, as they describe small discrepancies, observed in some tuning systems, between enharmonically equivalent notes. In the following list, the interval sizes in cents are approximate.

- A
*Pythagorean comma*is the difference between twelve justly tuned perfect fifths and seven octaves. It is expressed by the frequency ratio 531441:524288 (23.5 cents). - A
*syntonic comma*is the difference between four justly tuned perfect fifths and two octaves plus a major third. It is expressed by the ratio 81:80 (21.5 cents). - A
*septimal comma*is 64:63 (27.3 cents), and is the difference between the Pythagorean or 3-limit "7th" and the "harmonic 7th". - A
*diesis*is generally used to mean the difference between three justly tuned major thirds and one octave. It is expressed by the ratio 128:125 (41.1 cents). However, it has been used to mean other small intervals: see diesis for details. - A
*diaschisma*is the difference between three octaves and four justly tuned perfect fifths plus two justly tuned major thirds. It is expressed by the ratio 2048:2025 (19.6 cents). - A
*schisma*(also skhisma) is the difference between five octaves and eight justly tuned fifths plus one justly tuned major third. It is expressed by the ratio 32805:32768 (2.0 cents). It is also the difference between the Pythagorean and syntonic commas. (A schismic major third is a schisma different from a just major third, eight fifths down and five octaves up, F♭ in C.) - A
*kleisma*is the difference between six minor thirds and one*tritave*or*perfect twelfth*(an octave plus a perfect fifth), with a frequency ratio of 15625:15552 (8.1 cents) (^{ ⓘ }). - A
*septimal kleisma*is the amount that two major thirds of 5:4 and a septimal major third, or supermajor third, of 9:7 exceeds the octave. Ratio 225:224 (7.7 cents). - A
*quarter tone*is half the width of a semitone, which is half the width of a whole tone. It is equal to exactly 50 cents.

A compound interval is an interval spanning more than one octave.^{ [17] } Conversely, intervals spanning at most one octave are called simple intervals (see Main intervals below).

In general, a compound interval may be defined by a sequence or "stack" of two or more simple intervals of any kind. For instance, a major tenth (two staff positions above one octave), also called *compound major third*, spans one octave plus one major third.

Any compound interval can be always decomposed into one or more octaves plus one simple interval. For instance, a major seventeenth can be decomposed into two octaves and one major third, and this is the reason why it is called a compound major third, even when it is built by adding up four fifths.

The diatonic number *DN*_{c} of a compound interval formed from *n* simple intervals with diatonic numbers *DN*_{1}, *DN*_{2}, ..., *DN*_{n}, is determined by:

which can also be written as:

The quality of a compound interval is determined by the quality of the simple interval on which it is based. For instance, a compound major third is a major tenth (1+(8−1)+(3−1) = 10), or a major seventeenth (1+(8−1)+(8−1)+(3−1) = 17), and a compound perfect fifth is a perfect twelfth (1+(8−1)+(5−1) = 12) or a perfect nineteenth (1+(8−1)+(8−1)+(5−1) = 19). Notice that two octaves are a fifteenth, not a sixteenth (1+(8−1)+(8−1) = 15). Similarly, three octaves are a twenty-second (1+3×(8−1) = 22), and so on.

Number of semitones | Minor, major, or perfect intervals | Short | Augmented or diminished intervals | Short |
---|---|---|---|---|

12 | Diminished ninth | d9 | ||

13 | Minor ninth | m9 | Augmented octave | A8 |

14 | Major ninth | M9 | Diminished tenth | d10 |

15 | Minor tenth | m10 | Augmented ninth | A9 |

16 | Major tenth | M10 | Diminished eleventh | d11 |

17 | Perfect eleventh | P11 | Augmented tenth | A10 |

18 | Diminished twelfth | d12 | ||

Augmented eleventh | A11 | |||

19 | Perfect twelfth or Tritave | P12 | Diminished thirteenth | d13 |

20 | Minor thirteenth | m13 | Augmented twelfth | A12 |

21 | Major thirteenth | M13 | Diminished fourteenth | d14 |

22 | Minor fourteenth | m14 | Augmented thirteenth | A13 |

23 | Major fourteenth | M14 | Diminished fifteenth | d15 |

24 | Perfect fifteenth or Double octave | P15 | Augmented fourteenth | A14 |

25 | Augmented fifteenth | A15 |

It is also worth mentioning here the major seventeenth (28 semitones)—an interval larger than two octaves that can be considered a multiple of a perfect fifth (7 semitones) as it can be decomposed into four perfect fifths (7 × 4 = 28 semitones), or two octaves plus a major third (12 + 12 + 4 = 28 semitones). Intervals larger than a major seventeenth seldom come up, most often being referred to by their compound names, for example "two octaves plus a fifth"^{ [19] } rather than "a 19th".

Chords are sets of three or more notes. They are typically defined as the combination of intervals starting from a common note called the root of the chord. For instance a major triad is a chord containing three notes defined by the root and two intervals (major third and perfect fifth). Sometimes even a single interval (dyad) is considered a chord.^{ [20] } Chords are classified based on the quality and number of the intervals that define them.

The main chord qualities are major, minor, augmented, diminished, half-diminished, and dominant. The symbols used for chord quality are similar to those used for interval quality (see above). In addition, **+** or **aug** is used for augmented, **°** or **dim** for diminished, ** ^{ø}** for half diminished, and

The main rules to decode chord *names or symbols* are summarized below. Further details are given at Rules to decode chord names and symbols.

- For 3-note chords (triads), major or minor always refer to the interval of the third above the root note, while augmented and diminished always refer to the interval of the fifth above root. The same is true for the corresponding symbols (e.g., Cm means C
^{m3}, and C+ means C^{+5}). Thus, the terms third and fifth and the corresponding symbols 3 and 5 are typically omitted. This rule can be generalized to all kinds of chords,^{ [lower-alpha 3] }provided the above-mentioned qualities appear immediately after the root note, or at the beginning of the chord name or symbol. For instance, in the chord symbols Cm and Cm^{7}, m refers to the interval m3, and 3 is omitted. When these qualities do not appear immediately after the root note, or at the beginning of the name or symbol, they should be considered interval qualities, rather than chord qualities. For instance, in Cm^{M7}(minor major seventh chord), m is the chord quality and refers to the m3 interval, while M refers to the M7 interval. When the number of an extra interval is specified immediately after chord quality, the quality of that interval may coincide with chord quality (e.g., CM^{7}= CM^{M7}). However, this is not always true (e.g., Cm^{6}= Cm^{M6}, C+^{7}= C+^{m7}, CM^{11}= CM^{P11}).^{ [lower-alpha 3] }See main article for further details. - Without contrary information, a major third interval and a perfect fifth interval (major triad) are implied. For instance, a C chord is a C major triad, and the name C minor seventh (Cm
^{7}) implies a minor 3rd by rule 1, a perfect 5th by this rule, and a minor 7th by definition (see below). This rule has one exception (see next rule). - When the fifth interval is diminished, the third must be minor.
^{ [lower-alpha 4] }This rule overrides rule 2. For instance, Cdim^{7}implies a diminished 5th by rule 1, a minor 3rd by this rule, and a diminished 7th by definition (see below). - Names and symbols that contain only a plain interval number (e.g., "seventh chord") or the chord root and a number (e.g., "C seventh", or C
^{7}) are interpreted as follows:- If the number is 2, 4, 6, etc., the chord is a major added tone chord (e.g., C
^{6}= C^{M6}= C^{add6}) and contains, together with the implied major triad, an extra major 2nd, perfect 4th, or major 6th (see names and symbols for added tone chords). - If the number is 7, 9, 11, 13, etc., the chord is dominant (e.g., C
^{7}= C^{dom7}) and contains, together with the implied major triad, one or more of the following extra intervals: minor 7th, major 9th, perfect 11th, and major 13th (see names and symbols for seventh and extended chords). - If the number is 5, the chord (technically not a chord in the traditional sense, but a dyad) is a power chord. Only the root, a perfect fifth and usually an octave are played.

- If the number is 2, 4, 6, etc., the chord is a major added tone chord (e.g., C

The table shows the intervals contained in some of the main chords (**component intervals**), and some of the symbols used to denote them. The interval qualities or numbers in ** boldface ** font can be deduced from chord name or symbol by applying rule 1. In symbol examples, C is used as chord root.

Main chords | Component intervals | |||
---|---|---|---|---|

Name | Symbol examples | Third | Fifth | Seventh |

Major triad | C | M3 | P5 | |

CM, or Cmaj | M3 | P5 | ||

Minor triad | Cm, or Cmin | m3 | P5 | |

Augmented triad | C+, or Caug | M3 | A5 | |

Diminished triad | C°, or Cdim | m3 | d5 | |

Dominant seventh chord | C^{7}, or C^{dom7} | M3 | P5 | m7 |

Minor seventh chord | Cm^{7}, or Cmin^{7} | m3 | P5 | m7 |

Major seventh chord | CM^{7}, or Cmaj^{7} | M3 | P5 | M7 |

Augmented seventh chord | C+^{7}, Caug^{7},C ^{7♯5}, or C^{7aug5} | M3 | A5 | m7 |

Diminished seventh chord | C°^{7}, or Cdim^{7} | m3 | d5 | d7 |

Half-diminished seventh chord | C^{ø}^{7}, Cm^{7♭5}, or Cm^{7dim5} | m3 | d5 | m7 |

Number of semitones | Name | 5-limit tuning (pitch ratio) | Comparison of interval width (in cents) | |||
---|---|---|---|---|---|---|

5-limit tuning | Pythagorean tuning | 1⁄4-comma meantone | Equal temperament | |||

0 | Perfect unison | 1:1 | 0 | 0 | 0 | 0 |

1 | Minor second | 16:1527:25 | 112133 | 90 | 117 | 100 |

2 | Major second | 9:810:9 | 204182 | 204 | 193 | 200 |

3 | Minor third | 6:532:27 | 316294 | 294 318 | 310 (wolf) 269 | 300 |

4 | Major third | 5:4 | 386 | 408 384 | 386(wolf) 427 | 400 |

5 | Perfect fourth | 4:327:20 | 498520 | 498(wolf) 522 | 503 (wolf) 462 | 500 |

6 | Augmented fourth Diminished fifth | 45:3225:18 | 590569 | 612 588 | 579 621 | 600 |

7 | Perfect fifth | 3:240:27 | 702680 | 702(wolf) 678 | 697 (wolf) 738 | 700 |

8 | Minor sixth | 8:5 | 814 | 792 | 814 | 800 |

9 | Major sixth | 5:327:16 | 884906 | 906 | 890 | 900 |

10 | Minor seventh | 16:99:5 | 9961018 | 996 | 1007 | 1000 |

11 | Major seventh | 15:850:27 | 10881067 | 1110 | 1083 | 1100 |

12 | Perfect octave | 2:1 | 1200 | 1200 | 1200 | 1200 |

In this table, the interval widths used in four different tuning systems are compared. To facilitate comparison, just intervals as provided by 5-limit tuning (see symmetric scale n.1) are shown in ** bold ** font, and the values in cents are rounded to integers. Notice that in each of the non-equal tuning systems, by definition the width of *each* type of interval (including the semitone) changes depending on the note that starts the interval. This is the art of just intonation. In equal temperament, the intervals are never precisely in tune with each other. This is the price of using equidistant intervals in a 12-tone scale. For simplicity, for some types of interval the table shows only one value (the most often observed one).

In 1⁄4-comma meantone, by definition 11 perfect fifths have a size of approximately 697 cents (700 − *ε* cents, where *ε* ≈ 3.42 cents); since the average size of the 12 fifths must equal exactly 700 cents (as in equal temperament), the other one must have a size of about 738 cents (700 + 11*ε*, the wolf fifth or diminished sixth); 8 major thirds have size about 386 cents (400 − 4*ε*), 4 have size about 427 cents (400 + 8*ε*, actually diminished fourths), and their average size is 400 cents. In short, similar differences in width are observed for all interval types, except for unisons and octaves, and they are all multiples of ε (the difference between the 1⁄4-comma meantone fifth and the average fifth). A more detailed analysis is provided at 1⁄4-comma meantone Size of intervals. 1⁄4-comma meantone was designed to produce just major thirds, but only 8 of them are just (5:4, about 386 cents).

The Pythagorean tuning is characterized by smaller differences because they are multiples of a smaller *ε* (*ε* ≈ 1.96 cents, the difference between the Pythagorean fifth and the average fifth). Notice that here the fifth is wider than 700 cents, while in most meantone temperaments, including 1⁄4-comma meantone, it is tempered to a size smaller than 700. A more detailed analysis is provided at Pythagorean tuning § Size of intervals.

The 5-limit tuning system uses just tones and semitones as building blocks, rather than a stack of perfect fifths, and this leads to even more varied intervals throughout the scale (each kind of interval has three or four different sizes). A more detailed analysis is provided at 5-limit tuning § Size of intervals. 5-limit tuning was designed to maximize the number of just intervals, but even in this system some intervals are not just (e.g., 3 fifths, 5 major thirds and 6 minor thirds are not just; also, 3 major and 3 minor thirds are wolf intervals).

The above-mentioned symmetric scale 1, defined in the 5-limit tuning system, is not the only method to obtain just intonation. It is possible to construct juster intervals or just intervals closer to the equal-tempered equivalents, but most of the ones listed above have been used historically in equivalent contexts. In particular, the asymmetric version of the 5-limit tuning scale provides a juster value for the minor seventh (9:5, rather than 16:9). Moreover, the tritone (augmented fourth or diminished fifth), could have other just ratios; for instance, 7:5 (about 583 cents) or 17:12 (about 603 cents) are possible alternatives for the augmented fourth (the latter is fairly common, as it is closer to the equal-tempered value of 600 cents). The 7:4 interval (about 969 cents), also known as the harmonic seventh, has been a contentious issue throughout the history of music theory; it is 31 cents flatter than an equal-tempered minor seventh. For further details about reference ratios, see 5-limit tuning § The justest ratios.

In the diatonic system, every interval has one or more * enharmonic equivalents*, such as augmented second for minor third.

Although intervals are usually designated in relation to their lower note, David Cope ^{ [16] } and Hindemith ^{ [21] } both suggest the concept of **interval root**. To determine an interval's root, one locates its nearest approximation in the harmonic series. The root of a perfect fourth, then, is its *top* note because it is an octave of the fundamental in the hypothetical harmonic series. The bottom note of every odd diatonically numbered intervals are the roots, as are the tops of all even numbered intervals. The root of a collection of intervals or a chord is thus determined by the interval root of its strongest interval.

As to its usefulness, Cope^{ [16] } provides the example of the final tonic chord of some popular music being traditionally analyzable as a "submediant six-five chord" (added sixth chords by popular terminology), or a first inversion seventh chord (possibly the dominant of the mediant V/iii). According to the interval root of the strongest interval of the chord (in first inversion, CEGA), the perfect fifth (C–G), is the bottom C, the tonic.

Interval cycles, "unfold [i.e., repeat] a single recurrent interval in a series that closes with a return to the initial pitch class", and are notated by George Perle using the letter "C", for cycle, with an interval-class integer to distinguish the interval. Thus the diminished-seventh chord would be C3 and the augmented triad would be C4. A superscript may be added to distinguish between transpositions, using 0–11 to indicate the lowest pitch class in the cycle.^{ [22] }

As shown below, some of the above-mentioned intervals have alternative names, and some of them take a specific alternative name in Pythagorean tuning, five-limit tuning, or meantone temperament tuning systems such as quarter-comma meantone. All the intervals with prefix *sesqui-* are justly tuned, and their frequency ratio, shown in the table, is a superparticular number (or epimoric ratio). The same is true for the octave.

Typically, a comma is a diminished second, but this is not always true (for more details, see Alternative definitions of comma). For instance, in Pythagorean tuning the diminished second is a descending interval (524288:531441, or about −23.5 cents), and the Pythagorean comma is its opposite (531441:524288, or about 23.5 cents). 5-limit tuning defines four kinds of comma, three of which meet the definition of diminished second, and hence are listed in the table below. The fourth one, called syntonic comma (81:80) can neither be regarded as a diminished second, nor as its opposite. See Diminished seconds in 5-limit tuning for further details.

Number of semitones | Generic names | Specific names | |||||
---|---|---|---|---|---|---|---|

Quality and number | Other naming convention | Pythagorean tuning | 5-limit tuning | 1⁄4-comma meantone | |||

Full | Short | ||||||

0 | perfect unison or perfect prime | P1 | |||||

diminished second | d2 | descending Pythagorean comma (524288:531441) | lesser diesis (128:125) | ||||

diaschisma (2048:2025) greater diesis (648:625) | |||||||

1 | minor second | m2 | semitone, half tone, half step | diatonic semitone, major semitone | limma (256:243) | ||

augmented unison or augmented prime | A1 | chromatic semitone, minor semitone | apotome (2187:2048) | ||||

2 | major second | M2 | tone, whole tone, whole step | sesquioctavum (9:8) | |||

3 | minor third | m3 | sesquiquintum (6:5) | ||||

4 | major third | M3 | sesquiquartum (5:4) | ||||

5 | perfect fourth | P4 | sesquitertium (4:3) | ||||

6 | diminished fifth | d5 | tritone ^{ [lower-alpha 1] } | ||||

augmented fourth | A4 | ||||||

7 | perfect fifth | P5 | sesquialterum (3:2) | ||||

12 | perfect octave | P8 | duplex (2:1) |

Additionally, some cultures around the world have their own names for intervals found in their music. For instance, 22 kinds of intervals, called shrutis, are canonically defined in Indian classical music.

Up to the end of the 18th century, Latin was used as an official language throughout Europe for scientific and music textbooks. In music, many English terms are derived from Latin. For instance, semitone is from Latin * semitonus*.

The prefix semi- is typically used herein to mean "shorter", rather than "half".^{ [23] }^{ [24] }^{ [25] } Namely, a semitonus, semiditonus, semidiatessaron, semidiapente, semihexachordum, semiheptachordum, or semidiapason, is shorter by one semitone than the corresponding whole interval. For instance, a semiditonus (3 semitones, or about 300 cents) is not half of a ditonus (4 semitones, or about 400 cents), but a ditonus shortened by one semitone. Moreover, in Pythagorean tuning (the most commonly used tuning system up to the 16th century), a semitritonus (d5) is smaller than a tritonus (A4) by one Pythagorean comma (about a quarter of a semitone).

Number of semitones | Quality and number | Short | Latin nomenclature |
---|---|---|---|

0 | Perfect unison | P1 | unisonus |

1 | Minor second | m2 | semitonus |

Augmented unison | A1 | unisonus superflua | |

2 | Major second | M2 | tonus |

Diminished third | d3 | ||

3 | Minor third | m3 | semiditonus |

Augmented second | A2 | tonus superflua | |

4 | Major third | M3 | ditonus |

Diminished fourth | d4 | semidiatessaron | |

5 | Perfect fourth | P4 | diatessaron |

Augmented third | A3 | ditonus superflua | |

6 | Diminished fifth | d5 | semidiapente, semitritonus |

Augmented fourth | A4 | tritonus | |

7 | Perfect fifth | P5 | diapente |

Diminished sixth | d6 | semihexachordum | |

8 | Minor sixth | m6 | hexachordum minus, semitonus maius cum diapente, tetratonus |

Augmented fifth | A5 | diapente superflua | |

9 | Major sixth | M6 | hexachordum maius, tonus cum diapente |

Diminished seventh | d7 | semiheptachordum | |

10 | Minor seventh | m7 | heptachordum minus, semiditonus cum diapente, pentatonus |

Augmented sixth | A6 | hexachordum superflua | |

11 | Major seventh | M7 | heptachordum maius, ditonus cum diapente |

Diminished octave | d8 | semidiapason | |

12 | Perfect octave | P8 | diapason |

Augmented seventh | A7 | heptachordum superflua |

Intervals in non-diatonic scales can be named using analogs of the diatonic interval names, by using a diatonic interval of similar size and distinguishing it by varying the quality, or by adding other modifiers. For example, the just interval 7/6 may be referred to as a *subminor third*, since it is ~267 cents wide, which is narrower than a minor third (300 cents in 12-TET, ~316 cents for the just interval 6/5), or as the * septimal minor third *, since it is a 7-limit interval. These names refer just to the individual interval's size, and the interval number need not correspond to the number of scale degrees of a (heptatonic) scale. This naming is particularly common in just intonation and microtonal scales.^{ [26] }

The most common of these extended qualities are a **neutral** interval, in between a minor and major interval; and **subminor** and **supermajor** intervals, respectively narrower than a minor or wider than a major interval. The exact size of such intervals depends on the tuning system, but they often vary from the diatonic interval sizes by about a quarter tone (50 cents, half a chromatic step). For example, the neutral second, the characteristic interval of Arabic music, in 24-TET is 150 cents, exactly halfway between a minor second and major second. Combined, these yield the progression *diminished, subminor, minor, neutral, major, supermajor, augmented* for seconds, thirds, sixths, and sevenths. This naming convention can be extended to unisons, fourths, fifths, and octaves with *sub* and *super*, yielding the progression *diminished, sub, perfect, super, augmented*. This allows one to name all intervals in 24-TET or 31-TET, the latter of which was used by Adriaan Fokker. Various further extensions are used in Xenharmonic music.^{ [26] }

In post-tonal or atonal theory, originally developed for equal-tempered European classical music written using the twelve-tone technique or serialism, integer notation is often used, most prominently in musical set theory. In this system, intervals are named according to the number of half steps, from 0 to 11, the largest interval class being 6.

In atonal or musical set theory, there are numerous types of intervals, the first being the ordered pitch interval, the distance between two pitches upward or downward. For instance, the interval from C upward to G is 7, and the interval from G downward to C is −7. One can also measure the distance between two pitches without taking into account direction with the unordered pitch interval, somewhat similar to the interval of tonal theory.

The interval between pitch classes may be measured with ordered and unordered pitch-class intervals. The ordered one, also called directed interval, may be considered the measure upwards, which, since we are dealing with pitch classes, depends on whichever pitch is chosen as 0. For unordered pitch-class intervals, see interval class.^{ [27] }

In diatonic set theory, specific and generic intervals are distinguished. Specific intervals are the interval class or number of semitones between scale steps or collection members, and generic intervals are the number of diatonic scale steps (or staff positions) between notes of a collection or scale.

Notice that staff positions, when used to determine the conventional interval number (second, third, fourth, etc.), are counted including the position of the lower note of the interval, while generic interval numbers are counted excluding that position. Thus, generic interval numbers are smaller by 1, with respect to the conventional interval numbers.

Specific interval | Generic interval | Diatonic name | |
---|---|---|---|

Number of semitones | Interval class | ||

0 | 0 | 0 | Perfect unison |

1 | 1 | 1 | Minor second |

2 | 2 | 1 | Major second |

3 | 3 | 2 | Minor third |

4 | 4 | 2 | Major third |

5 | 5 | 3 | Perfect fourth |

6 | 6 | 3 4 | Augmented fourth Diminished fifth |

7 | 5 | 4 | Perfect fifth |

8 | 4 | 5 | Minor sixth |

9 | 3 | 5 | Major sixth |

10 | 2 | 6 | Minor seventh |

11 | 1 | 6 | Major seventh |

12 | 0 | 7 | Perfect octave |

The term "interval" can also be generalized to other music elements besides pitch. David Lewin's *Generalized Musical Intervals and Transformations* uses interval as a generic measure of distance between time points, timbres, or more abstract musical phenomena.^{ [28] }^{ [29] }

For example, an interval between two bell-like sounds, which have no pitch salience, is still perceptible. When two tones have similar acoustic spectra (sets of partials), the interval is just the distance of the shift of a tone spectrum along the frequency axis, so linking to pitches as reference points is not necessary. The same principle naturally applies to pitched tones (with similar harmonic spectra), which means that intervals can be perceived "directly" without pitch recognition. This explains in particular the predominance of interval hearing over absolute pitch hearing.^{ [30] }^{ [31] }

- 1 2 The term tritone is sometimes used more strictly as a synonym of augmented fourth (A4).
- 1 2 3 4 5 6 7 The expression diatonic scale is herein strictly defined as a 7-tone scale, which is either a sequence of successive natural notes (such as the C-major scale, C–D–E–F–G–A–B, or the A-minor scale, A–B–C–D–E–F–G) or any transposition thereof. In other words, a scale that can be written using seven consecutive notes without accidentals on a staff with a conventional key signature, or with no signature. This includes, for instance, the major and the natural minor scales, but does not include some other seven-tone scales, such as the melodic minor and the harmonic minor scales (see also Diatonic and chromatic).
- 1 2 General rule 1 achieves consistency in the interpretation of symbols such as CM
^{7}, Cm^{6}, and C+^{7}. Some musicians legitimately prefer to think that, in CM^{7}, M refers to the seventh, rather than to the third. This alternative approach is legitimate, as both the third and seventh are major, yet it is inconsistent, as a similar interpretation is impossible for Cm^{6}and C+^{7}(in Cm^{6}, m cannot possibly refer to the sixth, which is major by definition, and in C+^{7}, + cannot refer to the seventh, which is minor). Both approaches reveal only one of the intervals (M3 or M7), and require other rules to complete the task. Whatever is the decoding method, the result is the same (e.g., CM^{7}is always conventionally decoded as C–E–G–B, implying M3, P5, M7). The advantage of rule 1 is that it has no exceptions, which makes it the simplest possible approach to decode chord quality.According to the two approaches, some may format the major seventh chord as CM^{7}(general rule 1: M refers to M3), and others as C^{M7}(alternative approach: M refers to M7). Fortunately, even C^{M7}becomes compatible with rule 1 if it is considered an abbreviation of CM^{M7}, in which the first M is omitted. The omitted M is the quality of the third, and is deduced according to rule 2 (see above), consistently with the interpretation of the plain symbol C, which by the same rule stands for CM. - ↑ All triads are tertian chords (chords defined by sequences of thirds), and a major third would produce in this case a non-tertian chord. Namely, the diminished fifth spans 6 semitones from root, thus it may be decomposed into a sequence of two minor thirds, each spanning 3 semitones (m3 + m3), compatible with the definition of tertian chord. If a major third were used (4 semitones), this would entail a sequence containing a major second (M3 + M2 = 4 + 2 semitones = 6 semitones), which would not meet the definition of tertian chord.

**Pythagorean tuning** is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈ 702 cents wide.

**Meantone temperaments** are musical temperaments, that is a variety of tuning systems, obtained by narrowing the fifths so that their ratio is slightly less than 3:2, in order to push the thirds closer to pure. Meantone temperaments are constructed similarly to Pythagorean tuning, as a stack of equal fifths, but they are *temperaments* in that the fifths are not pure.

The **chromatic scale** is a set of twelve pitches used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce the chromatic scale, while other instruments capable of continuously variable pitch, such as the trombone and violin, can also produce microtones, or notes between those available on a piano.

In music theory, the **tritone** is defined as a musical interval spanning three adjacent whole tones. For instance, the interval from F up to the B above it is a tritone as it can be decomposed into the three adjacent whole tones F–G, G–A, and A–B.

In music, two written notes have **enharmonic equivalence** if they produce the same pitch but are notated differently. Similarly, written intervals, chords, or key signatures are considered **enharmonic** if they represent identical pitches that are notated differently. The term derives from Latin *enharmonicus*, in turn from Late Latin *enarmonius*, from Ancient Greek ἐναρμόνιος, from ἐν ('in') and ἁρμονία ('harmony').

In music theory, a **perfect fifth** is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so.

In music theory, the **wolf fifth** is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament. More broadly, it is also used to refer to similar intervals produced by other tuning systems, including Pythagorean and most meantone temperaments.

In musical tuning, the **Pythagorean comma** (or **ditonic comma**), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B♯, or D♭ and C♯. It is equal to the frequency ratio (1.5)^{12}⁄2^{7} = 531441⁄524288 ≈ 1.01364, or about 23.46 cents, roughly a quarter of a semitone (in between 75:74 and 74:73). The comma that musical temperaments often "temper" is the Pythagorean comma.

In music theory, the **circle of fifths** is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths.. If C is chosen as a starting point, the sequence is: C, G, D, A, E, B, F♯, C♯, A♭, E♭, B♭, F. Continuing the pattern from F returns the sequence to its starting point of C. This order places the most closely related key signatures adjacent to one another. It is usually illustrated in the form of a circle.

A **semitone**, also called a **half step** or a **half tone**, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale, visually seen on a keyboard as the distance between two keys that are adjacent to each other. For example, C is adjacent to C♯; the interval between them is a semitone.

In classical music, a **third** is a musical interval encompassing three staff positions, and the **major third** is a third spanning four half steps or two whole steps. Along with the minor third, the major third is one of two commonly occurring thirds. It is qualified as *major* because it is the larger interval of the two: the major third spans four semitones; the minor third, three. For example, the interval from C to E is a major third, as the note E lies four semitones above C, and there are three staff positions from C to E. Diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones.

The intervals from the tonic (keynote) in an upward direction to the second, to the third, to the sixth, and to the seventh scale degrees of a major scale are called major.

In music theory, a **minor third** is a musical interval that encompasses three half steps, or semitones. Staff notation represents the minor third as encompassing three staff positions. The minor third is one of two commonly occurring thirds. It is called *minor* because it is the smaller of the two: the major third spans an additional semitone. For example, the interval from A to C is a minor third, as the note C lies three semitones above A. Coincidentally, there are three staff positions from A to C. Diminished and augmented thirds span the same number of staff positions, but consist of a different number of semitones. The minor third is a skip melodically.

In music theory, a **comma** is a very small interval, the difference resulting from tuning one note two different ways. Strictly speaking, there are only two kinds of comma, the syntonic comma, "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma, "the difference between twelve 5ths and seven octaves". The word *comma* used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F♯ tuned using the D-based Pythagorean tuning system, and another F♯ tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B♭ and A♯ are both approximated by the same interval although they are a septimal kleisma apart.

**Quarter-comma meantone**, or ** 1 / 4 -comma meantone**, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma ( 81 : 80 ), with respect to its just intonation used in Pythagorean tuning ; the result is 3 / 2 × [ 80 / 81 ]^{ 1 / 4} = ^{4}√5 ≈ 1.49535, or a fifth of 696.578 cents. This fifth is then iterated to generate the diatonic scale and other notes of the temperament. The purpose is to obtain justly intoned major thirds. It was described by Pietro Aron in his *Toscanello de la Musica* of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible." Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.

In music, **53 equal temperament**, called **53 TET**, **53 EDO**, or **53 ET**, is the tempered scale derived by dividing the octave into 53 equal steps. Each step represents a frequency ratio of 2^{1⁄53}, or 22.6415 cents, an interval sometimes called the **Holdrian comma**.

In modern Western tonal music theory, a **diminished second** is the interval produced by narrowing a minor second by one chromatic semitone. In twelve-tone equal temperament, it is enharmonically equivalent to a perfect unison.; therefore, it is the interval between notes on two adjacent staff positions, or having adjacent note letters, altered in such a way that they have no pitch difference in twelve-tone equal temperament. An example is the interval from a B to the C♭ immediately above; another is the interval from a B♯ to the C immediately above.

In classical music from Western culture, a **diminished third** is the musical interval produced by narrowing a minor third by a chromatic semitone. For instance, the interval from A to C is a minor third, three semitones wide, and both the intervals from A♯ to C, and from A to C♭ are diminished thirds, two semitones wide. Being diminished, it is considered a dissonant interval.

**Diatonic** and **chromatic** are terms in music theory that are most often used to characterize scales, and are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, especially when applied to contrasting features of the common practice music of the period 1600–1900.

A **regular diatonic tuning** is any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's. In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size which makes it a Linear temperament with the tempered fifth as a generator.

**Five-limit tuning**, **5-limit tuning**, or **5-prime-limit tuning** (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2^{−3}·3^{1}·5^{1} = 15/8.

- ↑ Prout, Ebenezer (1903), "I-Introduction",
*Harmony, Its Theory and Practice*(30th edition, revised and largely rewritten ed.), London: Augener; Boston: Boston Music Co., p. 1, ISBN 978-0781207836 - 1 2 Lindley, Mark; Campbell, Murray; Greated, Clive (2001). "Interval". In Sadie, Stanley; Tyrrell, John (eds.).
*The New Grove Dictionary of Music and Musicians*(2nd ed.). London: Macmillan Publishers. ISBN 978-1-56159-239-5. - ↑ Aldwell, E.; Schachter, C.; Cadwallader, A. (11 March 2010), "Part 1: The Primary Materials and Procedures, Unit 1",
*Harmony and Voice Leading*(4th ed.), Schirmer, p. 8, ISBN 978-0495189756 - ↑ Duffin, Ross W. (2007), "3. Non-keyboard tuning",
*How Equal Temperament Ruined Harmony (and Why You Should Care)*(1st ed.), W. W. Norton, ISBN 978-0-393-33420-3 - ↑ "Prime (ii). See Unison",
*Grove Music Online*. Oxford University Press. Accessed August 2013. (subscription required)) - 1 2 Definition of
*perfect consonance*from Weber, Godfrey (1841).*General Music Teacher*.perfect concord

- ↑ Kostka, Stefan; Payne, Dorothy (2008).
*Tonal Harmony*, p. 21. First edition, 1984. - ↑ Prout, Ebenezer (1903).
*Harmony: Its Theory and Practice*, 16th edition. London: Augener & Co. (facsimile reprint, St. Clair Shores, Michigan: Scholarly Press, 1970), p. 10. ISBN 0-403-00326-1. - ↑ See for example William Lovelock,
*The Rudiments of Music*(New York: St Martin's Press; London: G. Bell, 1957):^{[ page needed ]}, reprinted 1966, 1970, and 1976 by G. Bell, 1971 by St Martins Press, 1981, 1984, and 1986 London: Bell & Hyman. ISBN 9780713507447 (pbk). ISBN 9781873497203 - ↑ Drabkin, William (2001). "Fourth".
*The New Grove Dictionary of Music and Musicians*, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan. - ↑ Helmholtz 1895, p. 172: "The roughness from sounding two tones together depends... the number of beats produced in a second.".
- ↑ Helmholtz 1895, p. 178: "The cause of this phenomenon must be looked for in the beats produced by the high upper partials of such compound tones".
- ↑ Helmholtz 1895, p. 182.
- ↑ Helmholtz, Hermann L. F.
*On the Sensations of Tone as a Theoretical Basis for the Theory of Music*, 2nd English edition, translated by Ellis, Alexander J. (1885) reprinted by Dover Publications with new introduction (1954) ISBN 0-486-60753-4, p. 182d: "Just as the coincidences of the two first upper partial tones led us to the natural consonances of the Octave and Fifth, the coincidences of higher upper partials would lead us to a further series of natural consonances." - ↑ Helmholtz 1895, p. 183: "Here I have stopped, because the 7th partial tone is entirely eliminated, or at least much weakened.".
- 1 2 3 Cope, David (1997).
*Techniques of the Contemporary Composer*, pp. 40–41. New York, New York: Schirmer Books. ISBN 0-02-864737-8. - 1 2 Wyatt, Keith; Schroeder, Carl (1998).
*Harmony & Theory*. Hal Leonard Corporation. p. 77. ISBN 9780793579914. - 1 2 Bonds, Mark Evan (2006).
*A History of Music in Western Culture*, p.123. 2nd ed. ISBN 0-13-193104-0. - ↑ Aikin, Jim (2004).
*A Player's Guide to Chords and Harmony: Music Theory for Real-World Musicians*, p. 24. ISBN 0-87930-798-6. - ↑ Károlyi, Ottó (1965),
*Introducing Music*, p. 63. Hammondsworth (England), and New York: Penguin Books. ISBN 0-14-020659-0. - ↑ Hindemith, Paul (1934).
*The Craft of Musical Composition*. New York: Associated Music Publishers. Cited in Cope (1997), p. 40–41. - ↑ Perle, George (1990).
*The Listening Composer*, p. 21. California: University of California Press. ISBN 0-520-06991-9. - ↑ Gioseffo Zarlino,
*Le Istitutione harmoniche ... nelle quali, oltre le materie appartenenti alla musica, si trovano dichiarati molti luoghi di Poeti, d'Historici e di Filosofi, si come nel leggerle si potrà chiaramente vedere*(Venice, 1558): 162. - ↑ J. F. Niermeyer ,
*Mediae latinitatis lexicon minus: Lexique latin médiéval–français/anglais: A Medieval Latin–French/English Dictionary*, abbreviationes et index fontium composuit C. van de Kieft, adiuvante G. S. M. M. Lake-Schoonebeek (Leiden: E. J. Brill, 1976): 955. ISBN 90-04-04794-8. - ↑
*Robert De Handlo: The Rules, and Johannes Hanboys, The Summa: A New Critical Text and Translation*, edited and translated by Peter M. Lefferts. Greek & Latin Music Theory 7 (Lincoln: University of Nebraska Press, 1991): 193fn17. ISBN 0803279345. - 1 2 "Extended-diatonic interval names".
*Xenharmonic wiki*. - ↑ Roeder, John (2001). "Interval Class". In Sadie, Stanley; Tyrrell, John (eds.).
*The New Grove Dictionary of Music and Musicians*(2nd ed.). London: Macmillan Publishers. ISBN 978-1-56159-239-5. - ↑ Lewin, David (1987).
*Generalized Musical Intervals and Transformations*, for example sections 3.3.1 and 5.4.2. New Haven: Yale University Press. Reprinted Oxford University Press, 2007. ISBN 978-0-19-531713-8 - ↑ Ockelford, Adam (2005).
*Repetition in Music: Theoretical and Metatheoretical Perspectives*, p. 7. ISBN 0-7546-3573-2. "Lewin posits the notion of musical 'spaces' made up of elements between which we can intuit 'intervals'....Lewin gives a number of examples of musical spaces, including the diatonic gamut of pitches arranged in scalar order; the 12 pitch classes under equal temperament; a succession of time-points pulsing at regular temporal distances one time unit apart; and a family of durations, each measuring a temporal span in time units....transformations of timbre are proposed that derive from changes in the spectrum of partials..." - ↑ Tanguiane (Tangian), Andranick (1993).
*Artificial Perception and Music Recognition*. Lecture Notes in Artificial Intelligence. Vol. 746. Berlin-Heidelberg: Springer. ISBN 978-3-540-57394-4. - ↑ Tanguiane (Tangian), Andranick (1994). "A principle of correlativity of perception and its application to music recognition".
*Music Perception*.**11**(4): 465–502. doi:10.2307/40285634. JSTOR 40285634.

- Helmholtz, H. L. F. von (1895) [1877].
*On the Sensations of Tone as a Theoretical Basis for the Theory of Music*. Translated by Alexander John Ellis (3rd English ed.). Longmans, Green, and Co.

- Gardner, Carl E. (1912):
*Essentials of Music Theory*, p. 38 - "Interval",
*Encyclopædia Britannica* - Lissajous Curves: Interactive simulation of graphical representations of musical intervals, beats, interference, vibrating strings
- Elements of Harmony: Vertical Intervals
- Just intervals, from the unison to the octave, played on a drone note on YouTube

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