Interval (music)
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In music theory, an interval is the difference between two pitches.^{[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 CTemplate:Music and DTemplate:Music. 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 describe 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-GTemplate:Music and G-ATemplate:Music.^{[4]}
Contents
- 1 Size
- 2 Main intervals
- 3 Interval number and quality
- 4 Shorthand notation
- 5 Inversion
- 6 Classification
- 7 Minute intervals
- 8 Compound intervals
- 9 Intervals in chords
- 10 Size of intervals used in different tuning systems
- 11 Interval root
- 12 Interval cycles
- 13 Alternative interval naming conventions
- 14 Pitch-class intervals
- 15 Generic and specific intervals
- 16 Generalizations and non-pitch uses
- 17 See also
- 18 Notes
- 19 External links
Size
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.
Frequency ratios
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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. To most people, just intervals sound consonant, that is, pleasant and well tuned.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]} }}
Most commonly, however, musical instruments are nowadays tuned using a different tuning system, called 12-tone equal temperament, in which the main intervals are typically perceived as consonant, but none is justly tuned and as consonant as a just interval, except for the unison (1:1) and octave (2:1).{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]} }} 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 section Size in different tuning systems.
Cents
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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
Main intervals
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).^{[6]} 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 FTemplate:Music is a major third, while that from D to GTemplate:Music 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 will also have the same width. Namely, all semitones will have a width of 100 cents, and all intervals spanning 4 semitones will be 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]}^{[7]} | P1 | Diminished second | d2 | Play (help·info) | ||
1 | Minor second | m2 | Augmented unison^{[5]}^{[7]} | A1 | Semitone,^{[8]} 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^{[6]} | 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) |
Interval number and quality
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.
Number
The number of an interval is the number of staff positions it encompasses.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]} }} 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 G 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).^{[9]} This means that interval numbers can be also determined by counting diatonic scale degrees, rather than staff positions, provided that the two notes which 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 ATemplate:Music-major diatonic scale, the five notes are C–DTemplate:Music–ETemplate:Music–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–CTemplate:Music–D–DTemplate:Music–E–F–FTemplate:Music–G). This is the reason interval numbers are also called diatonic numbers, and this convention is called diatonic numbering.
If one takes away any accidentals from the notes which 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–GTemplate:Music (spanning 8 semitones) and CTemplate:Music–G (spanning 6 semitones) are fifths, like the corresponding natural interval C–G (7 semitones).
Interval numbers do not represent exactly interval widths. For instance, 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.
The rule to determine the diatonic number of a compound interval (an interval larger than one octave), based on the diatonic numbers of the simple intervals from which it is built is explained in a separate section.
Quality
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
Perfect intervals are so-called because they were traditionally considered perfectly consonant,^{[10]} although in Western classical music the perfect fourth was sometimes regarded as a less than perfect consonance, when its function was contrapuntal.Template:Vague Conversely, minor, major, augmented or diminished intervals are typically considered to be less consonant, and were traditionally classified as mediocre consonances, imperfect consonances, or dissonances.^{[10]}
Within a diatonic scale^{[9]} all unisons (P1) and octaves (P8) are perfect. Most fourths and fifths are also perfect (P4 and P5), with five and seven semitones respectively. There's one occurrence of a fourth and a fifth which are not perfect, as they both span six semitones: an augmented fourth (A4), and its inversion, a diminished fifth (d5). 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 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.
- Major/minor
As shown in the table, a diatonic scale^{[9]} 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,^{[9]} 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/diminished
Augmented and diminished intervals are so called because they exceed or fall short of either a perfect interval, or a major/minor pair by one semitone, while having the same interval number (i.e., encompassing the same number of staff positions). For instance, an augmented third such as C–ETemplate:Music spans five semitones, exceeding a major third (C–E) by one semitone, while a diminished third such as CTemplate:Music–ETemplate:Music spans two semitones, falling short of a minor third (C–ETemplate:Music) by one semitone.
Except for the above-mentioned augmented fourth (A4) and diminished fifth (d5), augmented and diminished intervals do not appear in diatonic scales^{[9]} (see table).
Example
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 ATemplate:Music and BTemplate:Music, between A and CTemplate:Music, between A and DTemplate:Music, and between ATemplate:Music and ETemplate:Music, but
- ATemplate:Music–BTemplate:Music 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–CTemplate:Music is a third, as it encompasses three staff positions (A, B, C), and it is major, as it spans 4 semitones.
- A–DTemplate:Music 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.
- ATemplate:Music-ETemplate:Music 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 | ATemplate:Music | BTemplate:Music | |||
4 | major third | A | CTemplate:Music | |||
4 | diminished fourth | A | DTemplate:Music | |||
4 | triply diminished fifth | ATemplate:Music | ETemplate:Music |
Shorthand notation
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 indication 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.
Inversion
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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:^{[11]}
- 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 ETemplate:Music above it is a minor third. By the two rules just given, the interval from ETemplate:Music 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."^{[12]}
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.
Classification
Intervals can be described, classified, or compared with each other according to various criteria.
Melodic and harmonic
{{#invoke:main|main}} An interval can be described as
- Vertical or harmonic if the two notes sound simultaneously
- Horizontal, linear, or melodic if they sound successively.^{[2]}
Diatonic and chromatic
{{#invoke:main|main}} 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–ETemplate:Music (a diminished fourth, occurring in the harmonic C-minor scale) is considered diatonic if the harmonic minor scales are considered diatonic as well.^{[13]} Otherwise, it is considered chromatic. For further details, see the main article.
By a commonly used definition of diatonic scale^{[9]} (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.
[[Image:Ab major scale.png|right|thumb|The [[A-flat major|ATemplate:Music-major]] scale. Play (help·info)]]
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 ATemplate:Music–ETemplate:Music is chromatic to C major, because ATemplate:Music and ETemplate:Music are not contained in the C major scale. However, it is diatonic to others, such as the ATemplate:Music major scale.
Consonant and dissonant
{{#invoke:main|main}} 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 the Middle Ages, only the unison, octave, perfect fourth, and perfect fifth were considered consonant harmonically.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
{{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]} }}
- 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").^{[14]} In the common practice period, it makes more sense to speak of consonant and dissonant chords, and certain intervals previously thought to be dissonant (such as minor sevenths) became acceptable in certain contexts. However, 16th-century practice continued to be taught to beginning musicians throughout this period.
- Hermann von Helmholtz (1821–1894) defined a harmonically consonant interval as one in which the two pitches have an upper partial (an overtone) in common^{[15]} (specifically excluding the seventh harmonic).{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
{{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]} }} This essentially defines all seconds and sevenths as dissonant, and the above thirds, fourths, fifths, and sixths as consonant.
- Pythagoras defined a hierarchy of consonance based on how small the numbers are that express the ratio.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
{{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]} }} 20th-century composer and theorist Paul Hindemith's system has a hierarchy with the same results as Pythagoras's, but defined by fiat rather than by interval ratios, to better accommodate equal temperament, all of whose intervals (except the octave) would be dissonant using acoustical methods.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]} }}
- 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.
- #Interval root
All of the above analyses refer to vertical (simultaneous) intervals.
Simple and compound
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]}
Steps and skips
{{#invoke:main|main}} 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,^{[9]} 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, with 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.
Enharmonic intervals
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Two intervals are considered to be 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 FTemplate:Music and GTemplate:Music indicate the same pitch, and the same is true for ATemplate:Music and BTemplate:Music. All these intervals span four semitones.
Number of semitones |
Interval name | Staff positions | |||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | ||
4 | major third | FTemplate:Music | ATemplate:Music | ||
4 | major third | GTemplate:Music | BTemplate:Music | ||
4 | diminished fourth | FTemplate:Music | BTemplate:Music | ||
4 | doubly augmented second | GTemplate:Music | ATemplate:Music |
When played on a piano keyboard, these intervals are indistinguishable as they are all played with the same two keys, but in a musical context the diatonic function of the notes incorporated is very different.
Minute intervals
[[Image:Pythagorean comma on C.png|thumb|right|Pythagorean comma on C. Play (help·info). The note depicted as lower on the staff (B[[semitone#Just intonation|Template:Music]]+++) is slightly higher in pitch (than CTemplate:Music).]] {{#invoke:main|main}} 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) ( Play (help·info)).
- A septimal kleisma is six major thirds up, five fifths down and one octave up, with 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.
Compound intervals
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 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.
Main compound intervals
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 which 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 need to be spoken of, most often being referred to by their compound names, for example "two octaves plus a fifth"^{[19]} rather than "a 19th".
Intervals in chords
{{#invoke:main|main}} 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 to be a chord.^{[20]} Chords are classified based on the quality and number of the intervals which define them.
Chord qualities and interval qualities
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, Template:Music for half diminished, and dom for dominant (the symbol − alone is not used for diminished).
Deducing component intervals from chord names and symbols
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 Cm3, 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,^{[21]} 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 Cm7, 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., CM7 = CM/M7). However, this is not always true (e.g., Cm6 = Cm/M6, C+7 = C+/m7, CM11 = CM/P11).^{[21]} 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 (Cm7) 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.^{[22]} This rule overrides rule 2. For instance, Cdim7 implies a diminished 5th by rule 1, a minor 3rd by this rule, and a diminished 7th by definition (see below).
- Names and symbols which contain only a plain interval number (e.g., “Seventh chord”) or the chord root and a number (e.g., “C seventh”, or C7) are interpreted as follows:
- If the number is 2, 4, 6, etc., the chord is a major added tone chord (e.g., C6 = CM6 = Cadd6) 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., C7 = Cdom7) 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.
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 | maj3 | perf5 | |
CM, or Cmaj | maj3 | perf5 | ||
Minor triad | Cm, or Cmin | min3 | perf5 | |
Augmented triad | C+, or Caug | maj3 | aug5 | |
Diminished triad | C°, or Cdim | min3 | dim5 | |
Dominant seventh chord | C7, or Cdom7 | maj3 | perf5 | min7 |
Minor seventh chord | Cm7, or Cmin7 | min3 | perf5 | min7 |
Major seventh chord | CM7, or Cmaj7 | maj3 | perf5 | maj7 |
Augmented seventh chord | C+7, Caug7, C7^{Template:Music5}, or C7^{aug5} |
maj3 | aug5 | min7 |
Diminished seventh chord | C°7, or Cdim7 | min3 | dim5 | dim7 |
Half-diminished seventh chord | CTemplate:Music7, Cm7^{Template:Music5}, or Cmin7^{dim5} | min3 | dim5 | min7 |
Size of intervals used in different tuning systems
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 | 112 | 90 | 117 | 100 |
2 | Major second | 9:8 10:9 |
204 182 |
204 | 193 | 200 |
3 | Minor third | 6:5 75:64 32:27 |
316 (wolf) 275 294 |
294 318 |
310 (wolf) 269 |
300 |
4 | Major third | 5:4 512:405 32:25 81:64 |
386 406 (wolf) 427 408 |
408 384 |
386 (wolf) 427 |
400 |
5 | Perfect fourth | 4:3 675:512 27:20 |
498 478 520 |
498 (wolf) 522 |
503 (wolf) 462 |
500 |
6 | Augmented fourth Diminished fifth |
45:32 64:45 |
590 610 |
612 588 |
579 621 |
600 |
7 | Perfect fifth | 3:2 40:27 1024:675 |
702 680 722 |
702 (wolf) 678 |
697 (wolf) 738 |
700 |
8 | Minor sixth | 8:5 | 814 | 792 | 814 | 800 |
9 | Major sixth | 5:3 | 884 | 906 | 890 | 900 |
10 | Minor seventh | 16:9 9:5 |
996 1018 |
996 | 1007 | 1000 |
11 | Major seventh | 15:8 | 1088 | 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 from which the interval starts. This is the price paid for seeking just intonation. However, for the sake of 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. SomeTemplate:Who assert the 7:4 is one of the blue notes used in jazz. 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.
Interval root
Although intervals are usually designated in relation to their lower note, David Cope^{[16]} and Hindemith^{[23]} 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 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
{{#invoke:main|main}} 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.^{[24]}
Alternative interval naming conventions
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^{[6]} | ||||
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.
Latin nomenclature
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".^{[25]}^{[26]}^{[27]} 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).
Pitch-class intervals
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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.^{[28]}
Generic and specific intervals
{{#invoke:main|main}} 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.
Comparison
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 |
Generalizations and non-pitch uses
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.^{[29]}^{[30]}
See also
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- Music and mathematics
- Circle of fifths
- List of musical intervals
- List of pitch intervals
- List of meantone intervals
- Ear training
- Pseudo-octave
- Regular temperament
Notes
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}
- ↑ ^{2.0} ^{2.1} Template:GroveOnline
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}
- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}
- ↑ ^{5.0} ^{5.1} ^{5.2} "Prime (ii). See Unison" (from Prime. Grove Music Online. Oxford University Press. Accessed August 2013. Template:Subscription required)
- ↑ ^{6.0} ^{6.1} ^{6.2} The term Tritone is sometimes used more strictly as a synonym of augmented fourth (A4).
- ↑ ^{7.0} ^{7.1} The perfect and the augmented unison are also known as perfect and augmented prime.
- ↑ The minor second (m2) is sometimes called diatonic semitone, while the augmented unison (A1) is sometimes called chromatic semitone.
- ↑ ^{9.0} ^{9.1} ^{9.2} ^{9.3} ^{9.4} ^{9.5} ^{9.6} 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).
- ↑ ^{10.0} ^{10.1} Definition of Perfect consonance in Godfrey Weber's General music teacher, by Godfrey Weber, 1841.
- ↑ Kostka, Stephen; 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, Mich.: Scholarly Press, 1970), p. 10. ISBN 0-403-00326-1.
- ↑ See for example William Lovelock, The Rudiments of Music, 1971.Template:Full
- ↑ Drabkin, William (2001). "Fourth". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers.
- ↑ Helmholtz, Hermann L. F. On the Sensations of Tone as a Theoretical Basis for the Theory of Music Second English Edition translated by Ellis, Alexander J. (1885) reprinted by Dover Publications with new introduction (1954) ISBN 0-486-60753-4, page 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."
- ↑ ^{16.0} ^{16.1} ^{16.2} Cope, David (1997). Techniques of the Contemporary Composer, pp. 40–41. New York, New York: Schirmer Books. ISBN 0-02-864737-8.
- ↑ ^{17.0} ^{17.1} {{#invoke:citation/CS1|citation |CitationClass=book }}
- ↑ ^{18.0} ^{18.1} 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, Otto (1965), Introducing Music, p. 63. Hammondsworth (England), and New York: Penguin Books. ISBN 0-14-020659-0.
- ↑ ^{21.0} ^{21.1}
General rule 1 achieves consistency in the interpretation of symbols such as CM7, Cm6, and C+7. Some musicians legitimately prefer to think that, in CM7, 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 Cm6 and C+7 (in Cm6, 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., CM7 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 CM7 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.
- ↑ 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.
- ↑ Template:GroveOnline
- ↑ 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..."
Gardner, Carl E. (1912) - Essentials of Music Theory, p. 38, http://ia600309.us.archive.org/23/items/essentialsofmusi00gard/essentialsofmusi00gard.pdf
External links
- Encyclopaedia Britannica, Interval
- Morphogenesis of chords and scales Chords and scales classification
- Lissajous Curves: Interactive simulation of graphical representations of musical intervals, beats, interference, vibrating strings
- Elements of Harmony: Vertical Intervals
- Visualisation of musical intervals interactive
- How intervals work, colored music notation.
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