The twelfth root of two or (or equivalently ) is an algebraic irrational number, approximately equal to 1.0594631. It is most important in Western music theory, where it represents the frequency ratio (musical interval) of a semitone ( i ) in twelve-tone equal temperament. This number was proposed for the first time in relationship to musical tuning in the sixteenth and seventeenth centuries. It allows measurement and comparison of different intervals (frequency ratios) as consisting of different numbers of a single interval, the equal tempered semitone (for example, a minor third is 3 semitones, a major third is 4 semitones, and perfect fifth is 7 semitones). [lower-alpha 1] A semitone itself is divided into 100 cents (1 cent = ).
The twelfth root of two to 20 significant figures is 1.0594630943592952646. [2] Fraction approximations in increasing order of accuracy include 18/17, 89/84, 196/185, 1657/1564, and 18904/17843.
As of December 2013 [update] , its numerical value has been computed to at least twenty billion decimal digits. [3]
A musical interval is a ratio of frequencies and the equal-tempered chromatic scale divides the octave (which has a ratio of 2:1) into twelve equal parts. Each note has a frequency that is 21⁄12 times that of the one below it.[ citation needed ]
Applying this value successively to the tones of a chromatic scale, starting from A above middle C (known as A4) with a frequency of 440 Hz, produces the following sequence of pitches:
Note | Standard interval name(s) relating to A 440 | Frequency (Hz) | Multiplier | Coefficient (to six places) | Just intonation ratio |
---|---|---|---|---|---|
A | Unison | 440.00 | 20⁄12 | 1.000000 | 1 |
A♯/B♭ | Minor second/Half step/Semitone | 466.16 | 21⁄12 | 1.059463 | ≈ 16⁄15 |
B | Major second/Full step/Whole tone | 493.88 | 22⁄12 | 1.122462 | ≈ 9⁄8 |
C | Minor third | 523.25 | 23⁄12 | 1.189207 | ≈ 6⁄5 |
C♯/D♭ | Major third | 554.37 | 24⁄12 | 1.259921 | ≈ 5⁄4 |
D | Perfect fourth | 587.33 | 25⁄12 | 1.334839 | ≈ 4⁄3 |
D♯/E♭ | Augmented fourth/Diminished fifth/Tritone | 622.25 | 26⁄12 | 1.414213 | ≈ 7⁄5 |
E | Perfect fifth | 659.26 | 27⁄12 | 1.498307 | ≈ 3⁄2 |
F | Minor sixth | 698.46 | 28⁄12 | 1.587401 | ≈ 8⁄5 |
F♯/G♭ | Major sixth | 739.99 | 29⁄12 | 1.681792 | ≈ 5⁄3 |
G | Minor seventh | 783.99 | 210⁄12 | 1.781797 | ≈ 16⁄9 |
G♯/A♭ | Major seventh | 830.61 | 211⁄12 | 1.887748 | ≈ 15⁄8 |
A | Octave | 880.00 | 212⁄12 | 2.000000 | 2 |
The final A (A5: 880 Hz) is exactly twice the frequency of the lower A (A4: 440 Hz), that is, one octave higher.
Other tuning scales use slightly different interval ratios:
Since the frequency ratio of a semitone is close to 106% (), increasing or decreasing the playback speed of a recording by 6% will shift the pitch up or down by about one semitone, or "half-step". Upscale reel-to-reel magnetic tape recorders typically have pitch adjustments of up to ±6%, generally used to match the playback or recording pitch to other music sources having slightly different tunings (or possibly recorded on equipment that was not running at quite the right speed). Modern recording studios utilize digital pitch shifting to achieve similar results, ranging from cents up to several half-steps (note that reel-to-reel adjustments also affect the tempo of the recorded sound, while digital shifting does not).
Historically this number was proposed for the first time in relationship to musical tuning in 1580 (drafted, rewritten 1610) by Simon Stevin. [4] In 1581 Italian musician Vincenzo Galilei may be the first European to suggest twelve-tone equal temperament. [1] The twelfth root of two was first calculated in 1584 by the Chinese mathematician and musician Zhu Zaiyu using an abacus to reach twenty four decimal places accurately, [1] calculated circa 1605 by Flemish mathematician Simon Stevin, [1] in 1636 by the French mathematician Marin Mersenne and in 1691 by German musician Andreas Werckmeister. [5]
An equal temperament is a musical temperament or tuning system that approximates just intervals but instead divides an octave into steps such that the ratio of the frequencies of any adjacent pair of notes is the same. This system yields pitch steps perceived as equal in size, due to the logarithmic changes in pitch frequency.
In music, an octave or perfect octave is the interval between one musical pitch and another with 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". The interval between the first and second harmonics of the harmonic series is an octave.
Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are based on the ratio 3:2. This ratio, also known as the "pure" perfect fifth, is chosen because it is one of the most consonant and easiest to tune by ear and because of importance attributed to the integer 3. As Novalis put it, "The musical proportions seem to me to be particularly correct natural proportions." Alternatively, it can be described as the tuning of the syntonic temperament in which the generator is the ratio 3:2, which is ≈ 702 cents wide.
In music theory, an interval is a difference in pitch between two sounds. 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.
Pitch is a perceptual property of sounds that allows their ordering on a frequency-related scale, or more commonly, pitch is the quality that makes it possible to judge sounds as "higher" and "lower" in the sense associated with musical melodies. Pitch is a major auditory attribute of musical tones, along with duration, loudness, and timbre.
In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so.
In musical tuning, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as C and B♯, or D♭ and C♯. It is equal to the frequency ratio (1.5)12⁄27 = 531441⁄524288 ≈ 1.01364, or about 23.46 cents, roughly a quarter of a semitone (in between 75:74 and 74:73). The comma that musical temperaments often "temper" is the Pythagorean comma.
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. Typically, cents are used to express small intervals, or to compare the sizes of comparable intervals in different tuning systems, and in fact the interval of one cent is too small to be perceived between successive notes.
The Bohlen–Pierce scale is a musical tuning and scale, first described in the 1970s, that offers an alternative to the octave-repeating scales typical in Western and other musics, specifically the equal-tempered diatonic scale.
In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave position." Important to musical set theory, a pitch class is "all pitches related to each other by octave, enharmonic equivalence, or both." Thus, using scientific pitch notation, the pitch class "C" is the set
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale. For example, C is adjacent to C♯; the interval between them is a semitone.
Scientific pitch notation (SPN), also known as American standard pitch notation (ASPN) and international pitch notation (IPN), is a method of specifying musical pitch by combining a musical note name and a number identifying the pitch's octave.
This is a list of the fundamental frequencies in Hertz (cycles per second) of the keys of a modern 88-key standard or 108-key extended piano in twelve-tone equal temperament, with the 49th key, the fifth A (called A4), tuned to 440 Hz (referred to as A440). Every octave is made of twelve steps called semitones. A jump from the lowest semitone to the highest semitone in one octave doubles the frequency (for example, the fifth A is 440 Hz and the sixth A is 880 Hz). The frequency of a pitch is derived by multiplying (ascending) or dividing (descending) the frequency of the previous pitch by the twelfth root of two (approximately 1.059463). For example, to get the frequency one semitone up from A4 (A♯4), multiply 440 Hz by the twelfth root of two. To go from A4 up two semitones (one whole tone) to B4, multiply 440 twice by the twelfth root of two (or once by the sixth root of two, approximately 1.122462). To go from A4 up three semitones to C5 (a minor third), multiply 440 Hz three times by the twelfth root of two (or once by the fourth root of two, approximately 1.189207). For other tuning schemes, refer to musical tuning.
In music theory, a comma is a very small interval, the difference resulting from tuning one note two different ways. Strictly speaking, there are only two kinds of comma, the syntonic comma, "the difference between a just major 3rd and four just perfect 5ths less two octaves", and the Pythagorean comma, "the difference between twelve 5ths and seven octaves". The word comma used without qualification refers to the syntonic comma, which can be defined, for instance, as the difference between an F♯ tuned using the D-based Pythagorean tuning system, and another F♯ tuned using the D-based quarter-comma meantone tuning system. Intervals separated by the ratio 81:80 are considered the same note because the 12-note Western chromatic scale does not distinguish Pythagorean intervals from 5-limit intervals in its notation. Other intervals are considered commas because of the enharmonic equivalences of a tuning system. For example, in 53TET, B♭ and A♯ are both approximated by the same interval although they are a septimal kleisma apart.
Quarter-comma meantone, or 1⁄4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma (81 : 80), with respect to its just intonation used in Pythagorean tuning ; the result is 3/2 × 1⁄4 = 4√5 ≈ 1.49535, or a fifth of 696.578 cents. This fifth is then iterated to generate the diatonic scale and other notes of the temperament. The purpose is to obtain justly intoned major thirds. It was described by Pietro Aron in his Toscanello de la Musica of 1523, by saying the major thirds should be tuned to be "sonorous and just, as united as possible." Later theorists Gioseffo Zarlino and Francisco de Salinas described the tuning with mathematical exactitude.
12 equal temperament (12-ET) is the musical system that divides the octave into 12 parts, all of which are equally tempered on a logarithmic scale, with a ratio equal to the 12th root of 2. That resulting smallest interval, 1⁄12 the width of an octave, is called a semitone or half step.
In musical tuning theory, a Pythagorean interval is a musical interval with a frequency ratio equal to a power of two divided by a power of three, or vice versa. For instance, the perfect fifth with ratio 3/2 (equivalent to 31/ 21) and the perfect fourth with ratio 4/3 (equivalent to 22/ 31) are Pythagorean intervals.
Music theory analyzes the pitch, timing, and structure of music. It uses mathematics to study elements of music such as tempo, chord progression, form, and meter. 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.
A regular diatonic tuning is any musical scale consisting of "tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same size, with the 'S's being smaller than the 'T's. In such a tuning, then the notes are connected together in a chain of seven fifths, all the same size which makes it a Linear temperament with the tempered fifth as a generator.
Five-limit tuning, 5-limit tuning, or 5-prime-limit tuning (not to be confused with 5-odd-limit tuning), is any system for tuning a musical instrument that obtains the frequency of each note by multiplying the frequency of a given reference note (the base note) by products of integer powers of 2, 3, or 5 (prime numbers limited to 5 or lower), such as 2−3·31·51 = 15/8.