In music theory, an interval is the 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]}
In Western music, intervals are most commonly differences between notes of a diatonic scale. 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), 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^{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 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).^{[a]} 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^{[5]}^{[b]} | P1 | Diminished second | d2 | Play (help·info) | ||
1 | Minor second | m2 | Augmented unison^{[5]}^{[b]} | A1 | Semitone,^{[c]} half tone, half step | S | Play (help·info) |
2 | Major second | M2 | Diminished third | d3 | Tone, whole tone, whole step | T | Play (help·info) |
3 | Minor third | m3 | Augmented second | A2 | Play (help·info) | ||
4 | Major third | M3 | Diminished fourth | d4 | Play (help·info) | ||
5 | Perfect fourth | P4 | Augmented third | A3 | Play (help·info) | ||
6 | Diminished fifth | d5 | Tritone^{[a]} | TT | Play (help·info) | ||
Augmented fourth | A4 | ||||||
7 | Perfect fifth | P5 | Diminished sixth | d6 | Play (help·info) | ||
8 | Minor sixth | m6 | Augmented fifth | A5 | Play (help·info) | ||
9 | Major sixth | M6 | Diminished seventh | d7 | Play (help·info) | ||
10 | Minor seventh | m7 | Augmented sixth | A6 | Play (help·info) | ||
11 | Major seventh | M7 | Diminished octave | d8 | Play (help·info) | ||
12 | Perfect octave | P8 | Augmented seventh | A7 | Play (help·info) |
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 it encompasses or staff positions it encompasses. Both lines and spaces (see figure) are counted, 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 a diatonic scale)^{[d]}. This means that interval numbers can be also 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, the notes from C to G are eight (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.
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. Conversely, minor, major, augmented or diminished intervals are typically considered less consonant, and were traditionally classified as mediocre consonances, imperfect consonances, or dissonances.^{[6]}
Within a diatonic scale^{[d]} 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^{[d]} 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,^{[d]} 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). Diminished intervals, on the other hand, are narrower by one semitone than perfect or minor intervals of the same interval number. 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^{[d]} (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
Number of semitones |
Interval name | Staff positions | ||||
---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | ||
4 | doubly augmented second | A♭ | B♯ | |||
4 | major third | A | C♯ | |||
4 | diminished fourth | A | D♭ | |||
4 | triply diminished fifth | 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:
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]}
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 by 2. 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,
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^{[d]} (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.
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).^{[13]}
Linear (melodic) intervals may be described as steps or skips. A step, or conjunct motion,^{[14]} 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.^{[14]} In the diatonic scale,^{[d]} 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 compound interval is an interval spanning more than one octave.^{[13]} Conversely, intervals spanning at most one octave are called simple intervals (see Main intervals above).
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"^{[15]} 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.^{[16]} 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 dom for dominant (the symbol − alone is not used for diminished).
The main rules to decode chord names or symbols are summarized below. Further details are given at Rules to decode chord names and symbols.
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 minor 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 | 16:15 27:25 |
112 133 |
90 | 117 | 100 |
2 | Major second | 9:8 10:9 |
204 182 |
204 | 193 | 200 |
3 | Minor third | 6:5 32:27 |
316 294 |
294 318 |
310 (wolf) 269 |
300 |
4 | Major third | 5:4 | 386 | 408 384 |
386 (wolf) 427 |
400 |
5 | Perfect fourth | 4:3 27:20 |
498 520 |
498 (wolf) 522 |
503 (wolf) 462 |
500 |
6 | Augmented fourth Diminished fifth |
45:32 25:18 |
590 569 |
612 588 |
579 621 |
600 |
7 | Perfect fifth | 3:2 40:27 |
702 680 |
702 (wolf) 678 |
697 (wolf) 738 |
700 |
8 | Minor sixth | 8:5 | 814 | 792 | 814 | 800 |
9 | Major sixth | 5:3 27:16 |
884 906 |
906 | 890 | 900 |
10 | Minor seventh | 16:9 9:5 |
996 1018 |
996 | 1007 | 1000 |
11 | Major seventh | 15:8 50:27 |
1088 1067 |
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 + 11ε, the wolf fifth or diminished sixth); 8 major thirds have size about 386 cents (400 − 4ε), 4 have size about 427 cents (400 + 8ε, 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. Note that ^{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. Note that 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^{[12]} and Hindemith^{[17]} 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^{[12]} 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.^{[18]}
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, minor semitone |
limma (256:243) | ||
augmented unison or augmented prime |
A1 | chromatic semitone, major 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^{[a]} | ||||
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".^{[19]}^{[20]}^{[21]} 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 |
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.^{[22]}
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.^{[23]}^{[24]}
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.
Carmelo Arden Quin (né Carmelo Heriberto Alves;16 March 1913 – 27 September 2010) was a Uruguayan artist.
FifthFifth is the ordinal form of the number five.
Fifth or The Fifth may refer to:
Fifth Amendment to the United States Constitution, as in the expression "pleading the Fifth"
Fifth column, a political term
Fifth disease, a contagious rash that spreads in school-aged children
Fifth force, a proposed force of nature in addition to the four known fundamental forces
Fifth (Stargate), a robotic character in the television series Stargate SG-1
Fifth (unit), a unit of volume used for distilled beverages in the U.S.
Fifth-generation programming language
The fifth in a series, or four after the first: see ordinal numbers
1st Battalion, 5th Marines
A Fraction (mathematics), such as 1/5
The royal fifth (Spanish and Portuguese), an old royal tax of 20%
ITV SchoolsITV Schools (full name: Independent Television for Schools and Colleges) was the educational television service set up in 1957 by the Independent Television Authority, broadcasting learning programmes for children ages 5 to 18 across ITV-affiliated stations. It was an example of public service broadcasting on a commercial television network (as opposed to the public BBC and their service BBC Schools).
ITV moved its schools programming to Channel 4 in autumn 1987 although ITV continued to produce programmes and the service continued to use the ITV name for another six years. The last ITV Schools programme on Channel 4 aired on Monday 28 June 1993; Channel 4 continued to produce their own schools programmes for many years afterwards.
Index of music articlesThis page is an alphabetized index of articles about music.
John BastonJohn Baston, (fl. 1708–1739) was an English Baroque composer, recorder player and cellist. He performed in his own ‘interval music’ concertos in London; several of these lively pieces were published as Six Concertos in Six Parts for Violins and Flutes (1729).
Jonas BaesJonas Baes is a Philippine composer born in Los Baños, Laguna in 1961. He enrolled in the University of the Philippines' College of Music in 1977 as a student of Ramon P. Santos, and encountered the musical compositions of Jose Maceda, attended several seminar-workshops of visiting lecturers, and did research on the music of the Iraya-Mangyan people of Mindoro, which became the inspiration for his compositions. From 1992-1994, he studied with Mathias Spahlinger in Freiburg, Germany. Baes is known for writing music utilizing "unorthodox" musical instruments like bean-pod rattles, leaves, iron-nail chimes, as well as various Asian instruments such as bamboo scrapers, bamboo flutes, and vocal music using Asian vocal techniques. His early works in the 1980s were influenced by Maceda in the use of large numbers of performers, while in the 1990s he experimented with various methods by which the audience becomes integral in the performance. At the beginning of the new century, Baes experiments with notions of structure-agency integration [after Anthony Giddens] and simulacrum [after Jean Baudrillard]. It is typical for social theory to influence the work of Baes who has made a mark on contemporary music and cultural politics in the Asian region. Jonas Baes is also active as an ethnomusicologist and writer.
Korg WavestationThe Korg Wavestation is a vector synthesis synthesizer first produced in the early 1990s and later re-released as a software synthesizer in 2004. Its primary innovation was Wave Sequencing, a method of multi-timbral sound generation in which different PCM waveform data are played successively, resulting in continuously evolving sounds. The Wavestation's "Advanced Vector Synthesis" sound architecture resembled early vector synths such as the Sequential Circuits Prophet VS.
Designed as a "pure" synthesizer rather than a music workstation, it lacked an on-board song sequencer, yet the Wavestation, unlike any synthesizer prior to its release, was capable of generating complex, lush timbres and rhythmic sequences that sounded like a complete soundtrack by pressing only one key. Keyboard Magazine readers gave the Wavestation its "Hardware Innovation of the Year" award, and in 1995 Keyboard listed it as one of the "20 Instruments that Shook the World."The Wavestation lineup consisted of four models: the Wavestation and Wavestation EX keyboards, and the Wavestation A/D and Wavestation SR rackmount sound modules.
List of compositions by Arthur BlissThis is a list of compositions by Arthur Bliss.
Bliss’s works have been catalogued by Lewis Foreman, and "F" numbers are now commonly used to identify his music.
List of musical intervalsList of musical intervals may refer to:
Interval (music)#Main intervals as abstract relations between notes in western music theory.
List of pitch intervals as frequency ratios in intonation and tuning of musical instruments and performances.
List of pitch intervalsBelow is a list of intervals expressible in terms of a prime limit (see Terminology), completed by a choice of intervals in various equal subdivisions of the octave or of other intervals.
For commonly encountered harmonic or melodic intervals between pairs of notes in contemporary Western music theory, without consideration of the way in which they are tuned, see Interval (music) § Main intervals.
Music and mathematicsMusic theory has no axiomatic foundation in modern mathematics, yet the basis of musical sound can be described mathematically (in acoustics) and exhibits "a remarkable array of number properties". Elements of music such as its form, rhythm and metre, the pitches of its notes and the tempo of its pulse can be related to the measurement of time and frequency, offering ready analogies in geometry.
The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of set theory, abstract algebra and number theory. Some composers have incorporated the golden ratio and Fibonacci numbers into their work.
RatioIn mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains eight oranges and six lemons, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7).
The numbers in a ratio may be quantities of any kind, such as counts of persons or objects, or such as measurements of lengths, weights, time, etc. In most contexts both numbers are restricted to be positive, and the second to be not zero.
A ratio may be either specified by giving both constituting numbers, written as "a to b" or "a:b", or by giving just the value of their quotient a/b, since the product of the quotient and the second number yields the first, as required by the above definition.
Consequently, a ratio may be considered as an ordered pair of numbers, as a fraction with the first number in the numerator and the second as denominator, or as the value denoted by this fraction. Ratios of counts, given by (non-zero) natural numbers, denote positive rational numbers, and as such may occasionally result in natural numbers.
When two quantities are measured with the same unit, as is often the case, their ratio is a dimensionless number. A quotient of two quantities that are measured with different units is called a rate.
SemitoneA 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. For example, C is adjacent to C♯; the interval between them is a semitone.
In a 12-note approximately equally divided scale, any interval can be defined in terms of an appropriate number of semitones (e.g. a whole tone or major second is 2 semitones wide, a major third 4 semitones, and a perfect fifth 7 semitones.
In music theory, a distinction is made between a diatonic semitone, or minor second (an interval encompassing two different staff positions, e.g. from C to D♭) and a chromatic semitone or augmented unison (an interval between two notes at the same staff position, e.g. from C to C♯). These are enharmonically equivalent when twelve-tone equal temperament is used, but are not the same thing in meantone temperament, where the diatonic semitone is distinguished from and larger than the chromatic semitone (augmented unison). See Interval (music)#Number for more details about this terminology.
In twelve-tone equal temperament all semitones are equal in size (100 cents). In other tuning systems, "semitone" refers to a family of intervals that may vary both in size and name. In Pythagorean tuning, seven semitones out of twelve are diatonic, with ratio 256:243 or 90.2 cents (Pythagorean limma), and the other five are chromatic, with ratio 2187:2048 or 113.7 cents (Pythagorean apotome); they differ by the Pythagorean comma of ratio 531441:524288 or 23.5 cents. In quarter-comma meantone, seven of them are diatonic, and 117.1 cents wide, while the other five are chromatic, and 76.0 cents wide; they differ by the lesser diesis of ratio 128:125 or 41.1 cents. 12-tone scales tuned in just intonation typically define three or four kinds of semitones. For instance, Asymmetric five-limit tuning yields chromatic semitones with ratios 25:24 (70.7 cents) and 135:128 (92.2 cents), and diatonic semitones with ratios 16:15 (111.7 cents) and 27:25 (133.2 cents). For further details, see below.
The condition of having semitones is called hemitonia; that of having no semitones is anhemitonia. A musical scale or chord containing semitones is called hemitonic; one without semitones is anhemitonic.
Seventh (chord)In music, the seventh factor of a chord is the note or pitch seven scale degrees above the root or tonal center. When the seventh is the bass note, or lowest note, of the expressed chord, the chord is in third inversion Play .
Conventionally, the seventh is fourth in importance to the root, fifth, and third, with third inversion being the third strongest inversion and the seventh variably minor or major.
SixthSix is the ordinal form of the number six.
The Sixth Amendment, to the U.S. ConstitutionA keg of beer, equal to 5 U.S. gallons or 1/6 barrelA fraction (mathematics), such as ⅙
The Beyoncé ExperienceThe Beyoncé Experience was the third concert tour by American recording artist Beyoncé. It was launched in support of her second studio album, B'Day (2006). The Beyoncé Experience consisted of ninety-six shows in 2007 in five legs with concerts in Asia, Australia, North America, Europe and Africa. Fan-club tickets and VIP packages were on sale in early April and tickets sold officially on April 20, 2007. The tour began that month and finished that November. Knowles collaborated with America's Second Harvest on a campaign for people who fought with famine, holding pre-concert food drives at every stop during the tour asking from fans to bring food.
Rehearsals for the tour began in March 2007 and the performances featured music performed by an all-female band called Suga Mama chosen by Beyoncé Knowles and her father Mathew Knowles during auditions held before the release of B'Day. The set list of the concerts included songs from B'Day, Dangerously in Love, ten shortened versions of Destiny's Child's songs as well as "Listen", originally recorded by Knowles for the soundtrack of Dreamgirls (2006). Ten instrumentalists, three harmony vocalists and ten dancers backed Knowles onstage which also had disco balls hanging from the ceilings and color-changing stairs. For the wardrobe of the tour, Knowles collaborated with several designers; it consisted of silvery and sparkly dresses which received praise from critics.
The tour received positive reviews from music critics who praised Knowles' vocal abilities, dancing, choreography and energy on stage during the shows performed simultaneously. The performances of the slow-tempo songs and Destiny's Child's material during the concerts were highlighted by critics. Twenty-nine shows were submitted to Billboard's boxscore grossing $24.9 million, with 272,521 fans attending the performances. The tour grossed $90 million from 96 shows in total. Throughout the tour several incidents happened; the pyrotechnics meant for the stage injured two people in the audience, Knowles fell down a flight of stairs during a performance of "Ring the Alarm" and had a wardrobe malfunction. A show in Kuala Lumpur, Malaysia on November 1, 2007 was relocated to Jakarta, Indonesia due to Malaysia's strict rules surrounding the clothing worn by female performers. The performance at the Staples Center in Los Angeles was recorded on September 2, 2007 and was released on November 16, 2007 as a live DVD titled The Beyoncé Experience Live.
TwelfthTwelfth can mean:
The Twelfth Amendment to the United States Constitution
The Twelfth, a Protestant celebration originating in IrelandIn mathematics:
12th, an ordinal number; as in the item in an order twelve places from the beginning, following the eleventh and preceding the thirteenth
1/12, a vulgar fraction, one part of a unit divided equally into twelve parts
Wayne MardleWayne Elliot Mardle (born 10 May 1973) is an English retired professional darts player for the Professional Darts Corporation events, and former player in the BDO. He now lives in Romford. Before turning professional, he worked for the Association of Accounting Technicians.
He has been known as "Hawaii 501" since 2000, a play on the title of the popular television series, Hawaii Five-O, due to the Hawaiian shirts he started wearing in 1998 for a bet, and 501 being the start score of a leg of darts. This dress sense coupled with his crowd-pleasing onstage activity (such as dancing to the interval music) saw him become one of the most popular players on the circuit.
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