Interval ratio
Encyclopedia
In music
, an interval ratio is a ratio
of the frequencies
of the pitches
in a musical interval
. For example, a just
perfect fifth
(for example C to G) is 3:2 , 1.5, and may be approximated by an equal tempered
perfect fifth which is 27/12, 1.498. If the A above middle C
is 440
Hz
, the perfect fifth above it would thus be an E
at (440*1.5=) 660 Hz while the equal tempered E5 is 659.255 Hz.
Ratios, rather than direct frequency measurements, allow musicians to work with relative pitch measurements applicable to many instruments in an intuitive manner, whereas one rarely has the frequencies of fixed pitched instruments memorized and rarely has the capabilities to measure the changes of adjustable pitch instruments (electronic tuner). Ratios have an inverse
relationship to string length, for example stopping a string at two-thirds (2:3) its length produces a pitch one and one-half (3:2) that of the open string (not to be confused with Inversion (music)
).
Intervals may be ranked by relative consonance and dissonance
. As such ratios with lower integers are generally more consonant than intervals with higher integers. For example, 2:1
, 4:3
, 9:8 , 65536:59049
, etc.
Consonance and dissonance may more subtly be defined by limit
, wherein the ratios whose integers, including integer multiples of the limit, are lower are generally more consonant. For example, the 3-limit 128:81
and the 7-limit 14:9 . Despite having larger integers 128:81 is less dissonant than 14:9, as according to limit theory.
For ease of comparison intervals may also be measured in cents
, a logarithmic measurement. For example, the just perfect fifth is 701.955 cents while the equal tempered perfect fifth is 700 cents. Note that ratios of cents may not be considered.
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
, an interval ratio is a ratio
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...
of the frequencies
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
of the pitches
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
in a musical interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
. For example, a just
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
perfect fifth
Perfect fifth
In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones...
(for example C to G) is 3:2 , 1.5, and may be approximated by an equal tempered
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
perfect fifth which is 27/12, 1.498. If the A above middle C
Middle C
C or Do is the first note of the fixed-Do solfège scale. Its enharmonic is B.-Middle C:Middle C is designated C4 in scientific pitch notation because of the note's position as the fourth C key on a standard 88-key piano keyboard...
is 440
A440 (Concert A)
A440 is the musical note A above middle C. It has a frequency of 440 Hz and serves as a general tuning standard for musical pitch.Prior to the standardization on 440 Hz, many countries and organizations followed the Austrian government's 1885 recommendation of 435 Hz...
Hz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....
, the perfect fifth above it would thus be an E
E (musical note)
E or mi is the third note of the solfège.When calculated in equal temperament with a reference of A above middle C as 440 Hz, the frequency of Middle E is approximately 329.628 Hz. See pitch for a discussion of historical variations in frequency.-Designation by octave:...
at (440*1.5=) 660 Hz while the equal tempered E5 is 659.255 Hz.
Ratios, rather than direct frequency measurements, allow musicians to work with relative pitch measurements applicable to many instruments in an intuitive manner, whereas one rarely has the frequencies of fixed pitched instruments memorized and rarely has the capabilities to measure the changes of adjustable pitch instruments (electronic tuner). Ratios have an inverse
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...
relationship to string length, for example stopping a string at two-thirds (2:3) its length produces a pitch one and one-half (3:2) that of the open string (not to be confused with Inversion (music)
Inversion (music)
In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...
).
Intervals may be ranked by relative consonance and dissonance
Consonance and dissonance
In music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance , which is considered to be unstable...
. As such ratios with lower integers are generally more consonant than intervals with higher integers. For example, 2:1
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
, 4:3
Perfect fourth
In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there...
, 9:8 , 65536:59049
Diminished third
In classical music from Western culture, a diminished third is the musical interval produced by narrowing a minor third by a chromatic semitone. For instance, the interval from A to C is a minor third, three semitones wide, and both the intervals from A to C, and from A to C are diminished thirds,...
, etc.
Consonance and dissonance may more subtly be defined by limit
Limit (music)
In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...
, wherein the ratios whose integers, including integer multiples of the limit, are lower are generally more consonant. For example, the 3-limit 128:81
Minor sixth
-Subminor sixth:In music, a subminor sixth or septimal sixth is an interval that is noticeably narrower than a minor sixth but noticeably wider than a diminished sixth.The sub-minor sixth is an interval of a 14:9 ratio or alternately 11:7....
and the 7-limit 14:9 . Despite having larger integers 128:81 is less dissonant than 14:9, as according to limit theory.
For ease of comparison intervals may also be measured in cents
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
, a logarithmic measurement. For example, the just perfect fifth is 701.955 cents while the equal tempered perfect fifth is 700 cents. Note that ratios of cents may not be considered.