Mathematics of musical scales

Western common practice music usually cannot be played in just intonation, even when it is confined to a single key. This is because the supertonic chord, or ii-chord, which is the most important of the minor triads in a major key, serves to bridge between the dominant and subdominant, having a root at once a minor third below the root of the subdominant triad, and hence sharing two of its notes, and a fifth above the root of the dominant triad or dominant seventh chord. The problem becomes still worse when modulation, the key changes so imp ...

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Mathematics of musical scales, Mathematics of musical scales - Pythagorean tuning, Mathematics of musical scales - Just intonation, Mathematics of musical scales - Temperament, Mathematics of musical scales - Equal temperament, Mathematics of musical scales - Sound samples, Mathematics of musical scales - Source

Read more here: » Mathematics of musical scales: Encyclopedia II - Mathematics of musical scales - Temperament

Harmony is the use and study of pitch simultaneity and chords, actual or implied, in music. It is sometimes referred to as the "vertical" aspect of music, with melody being the "horizontal" aspect. Very often, harmony is a result of counterpoint or polyphony, several melodic lines or motifs being played at once, though harmony may control the counterpoint. The word harmony comes from the Greek ἁρμονία harmonía meaning "a fastening or join". The conce ...

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Vincenzo Galilei (father of Galileo Galilei) may have been the first person to advocate equal temperament (in a 1581 treatise). The first person known to introduce a mathematically accurate specification for equal temperament is probably Chu Tsai-Yu (朱載堉) in the Ming Dynasty, who published a theory of the temperament in 1584. Soon after, European mathematicians Simon Stevin (1585, inspired by V. Galilei) and Marin Mersenne ( ...

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Equal temperament, Equal temperament - Explanation, Equal temperament - History, Equal temperament - Twelve-tone equal temperament, Equal temperament - Cent values of equal temperament, Equal temperament - Non-12 TET, Equal temperament - Sources

Read more here: » Equal temperament: Encyclopedia II - Equal temperament - History

Human beings distinguish sounds on the basis of their frequency. Actually what really matters is the ratio between their frequencies. The natural scale is attributed to the Grecian philosopher Aristoxenus Tarentinus and consists in a succession of notes with increasing frequencies. After fixing the frequency of the first note — the C of the scale — the frequencies of the other notes are determined from the ratios indicated in the following table. On the last C the fol ...

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Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - Evolution of the natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament

Read more here: » Musical acoustics: Encyclopedia II - Musical acoustics - The natural scale

Human beings distinguish sounds on the basis of their frequency. Actually what really matters is the ratio between their frequencies. The natural scale is attributed to the Grecian philosopher Aristoxenus Tarentinus and consists in a succession of notes with increasing frequencies. After fixing the frequency of the first note - the C of the scale - the frequencies of the other notes are determined from the ratios indicated in the following table. On the last C the fol ...

See also:

Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament

Read more here: » Musical acoustics: Encyclopedia II - Musical acoustics - The natural scale

The lowest possible frequency of a harmonic oscillator is called its fundamental frequency. This frequency determines the musical pitch or note that is created by vibration over the full length of the string or air column. In nearly every musical instrument, the fundamental note is always accompanied by other, higher-frequency tones that are generally called overtones. In pitched (i.e., non-percussion) instruments, these shorter, faster waves are reflected between the two ends of the string or air column. As the reflecte ...

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Harmonic series music, Harmonic series music - Description of the harmonic series, Harmonic series music - Terminology, Harmonic series music - Harmonics and tuning, Harmonic series music - Timbre of musical instruments, Harmonic series music - Register and special effects of musical instruments

Read more here: » Harmonic series music: Encyclopedia II - Harmonic series music - Description of the harmonic series

In the natural scale the ratio of the frequencies of two notes which differ for one tone is not always the same. Consequently a certain melody cannot be played starting from a random note of the scale. For instance, a melody starting with the two notes C and D (ratio 9/8) cannot be transposed one tone higher, since the ratio of the frequencies of E and of D is very near ((5/4)/(9/8) = 10/9), but not equal to 9/8. To obviate this inconveniency, we today use the so-called Equal Temperament, which constitutes the compromise adopted in moder ...

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Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - Evolution of the natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament

Read more here: » Musical acoustics: Encyclopedia II - Musical acoustics - The equal tempered scale

If two notes are simultaneously played, with frequency ratios that are simple fractions (e.g. 2/1, 3/2 or 5/4), then the composite wave will still be periodic with a short period, and the combination will sound consonant. For instance, a note vibrating at 200 Hz and a note vibrating at 300 Hz (a perfect fifth, or 3/2 ratio, above 200 Hz) will add together to make a wave that repeats at 100 Hz: every 1/100 of a second, the 300 Hz wave will repeat thrice and the 200 Hz wave will repeat twice. Note that the total wave repeats at 100 Hz, but there is not ac ...

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Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - Evolution of the natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament

Read more here: » Musical acoustics: Encyclopedia II - Musical acoustics - Harmony

When a periodic wave is composed of a fundamental and only odd harmonics (f, 3f, 5f, 7f, ...), the summed wave is half-wave symmetric; it can be inverted and phase shifted and be exactly the same. If the wave has any even harmonics (0f, 2f, 4f, 6f, ...), it will be asymmetrical; the top half will not be a mirror image of the bottom. The opposite is also true. A system which changes the shape of the wave (beyond simple scaling or shifting) creates additional harmonics (harmonic distortion). This is called a non-linear system< ...

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Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - Evolution of the natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament

Read more here: » Musical acoustics: Encyclopedia II - Musical acoustics - Harmonics and non-linearities

Five and seven tone equal temperament, with 240 and 171 cent steps relatively, seem the most common outside of 12-tET. A Thai xylophone measured by Morton (1974) "varied only plus or minus 5 cents," from 7-tET. A Ugandan Chop xylophone measured by Haddon (1952) also tuned to 171 cent steps. Gamelans are tuned to 5-tET according to Kunst (1949), but according to Hood (1966) and McPhee (1966) their tuning varies widely and according to Tenzer (2000) contain stretched octaves. It is now well-accepted that of the two primary tuning systems in Ga ...

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Equal temperament, Equal temperament - Explanation, Equal temperament - History, Equal temperament - Twelve-tone equal temperament, Equal temperament - Cent values of equal temperament, Equal temperament - Non-12 TET, Equal temperament - Sources

Read more here: » Equal temperament: Encyclopedia II - Equal temperament - Non-12 TET

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. Today, the natural sciences, engineering, economics, and medici ...

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The fundamental is the frequency at which the entire wave vibrates. Overtones are other sinusoidal components present at frequencies above the fundamental. All of the frequency components that make up the total waveform, including the fundamental and the overtones, are called partials. Overtones which are perfect integer multiples of the fundamental are called harmonics. When an overtone is near to being harmonic, but not exact, it is sometimes called a harmonic partial, although they are often referred to simply as harmonics. Sometimes overtones are created that are not anywhere ...

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Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - Evolution of the natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament

Read more here: » Musical acoustics: Encyclopedia II - Musical acoustics - Harmonics partials and overtones

The ratio between two adjacent semitones can be found with a few steps: 1. Let an be the frequency of a tone n, with a12 an octave above a0. This creates twelve tones for each octave. 2. Since the frequency ratio of a tone from one octave to the next is 2:1, the ratio of the frequency of one tone (a12) to the frequency of a tone an octave lower (a0) is 2:1 as well, so < ...

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Equal temperament, Equal temperament - Explanation, Equal temperament - History, Equal temperament - Twelve-tone equal temperament, Equal temperament - Cent values of equal temperament, Equal temperament - Non-12 TET, Equal temperament - Sources

Read more here: » Equal temperament: Encyclopedia II - Equal temperament - Twelve-tone equal temperament

The distance between each step and the next is aurally the same for any two adjacent steps; though, because steps form a geometric sequence, the difference in frequency increases from one to the next. A linear sequence of one frequency difference would create ever smaller intervals (ratios), such as the harmonic series. See also logarithmic scale. Equal temperaments allow the use of integer notation; a single integer can be used to represent the pitch. The pitch classes can then be expressed in terms of modular arithmetic modulo the number of divisions of the octave, and this expedites mathematic ...

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Equal temperament, Equal temperament - Explanation, Equal temperament - History, Equal temperament - Twelve-tone equal temperament, Equal temperament - Cent values of equal temperament, Equal temperament - Non-12 TET, Equal temperament - Sources

Read more here: » Equal temperament: Encyclopedia II - Equal temperament - Explanation

Variations in air pressure against the ear drum, and the subsequent physical and neurological processing and interpretation, give rise to the experience called "sound". Most sound that people recognize as "musical" is dominated by periodic or regular vibrations rather than non-periodic ones (called a definite pitch), and we refer to the transmission mechanism as a "sound wave". In a very simple case, the sound of a sine wave, which is considered to be the most basic model of a sound waveform, causes the air pressure to increase and decrease ...

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Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - Evolution of the natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament

Read more here: » Musical acoustics: Encyclopedia II - Musical acoustics - Sound waves

If the first 15 harmonics are transposed into the span of one octave, they approximate some of the notes in what the West has adopted as the chromatic scale based on the fundamental tone. The Western chromatic scale has been modified into twelve equal semitones, and in relation to that scale, many of the harmonics are slightly out of tune, and the 7th, 11th, and 13th harmonics are significantly so. In the late 1930s, composer Paul Hindemith ranked musical intervals according to their relative ...

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Harmonic series music, Harmonic series music - Description of the harmonic series, Harmonic series music - Terminology, Harmonic series music - Harmonics and tuning, Harmonic series music - Timbre of musical instruments, Harmonic series music - Register and special effects of musical instruments

Read more here: » Harmonic series music: Encyclopedia II - Harmonic series music - Harmonics and tuning

Harmonic vs. partial. Harmonics are often called partials. In some contexts, "partial" may refer to an overtone that is not an integer multiple of the fundamental frequency, but this can be confusing in wire-stringed instruments where, due to inharmonicity, none of the harmonics vibrate at exact integer multiples of the fundamental. In music, and especially among tuning professionals, the words "h ...

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Harmonic series music, Harmonic series music - Description of the harmonic series, Harmonic series music - Terminology, Harmonic series music - Harmonics and tuning, Harmonic series music - Timbre of musical instruments, Harmonic series music - Register and special effects of musical instruments

Read more here: » Harmonic series music: Encyclopedia II - Harmonic series music - Terminology

In wind instruments, which produce sounds with a resonating air column, the lowest possible note is the fundamental resonance of the entire instrument. For a given length of resonator, only notes in the harmonic series of the resonator can be played clearly: higher notes are played by exciting higher harmonics, which is accomplished by changing the vibration at the reed or mouthpiece. Notes that are not in the harmonic series are played by changing the effect ...

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Harmonic series music, Harmonic series music - Description of the harmonic series, Harmonic series music - Terminology, Harmonic series music - Harmonics and tuning, Harmonic series music - Timbre of musical instruments, Harmonic series music - Register and special effects of musical instruments

Read more here: » Harmonic series music: Encyclopedia II - Harmonic series music - Register and special effects of musical instruments

The relative amplitudes of the various harmonics primarily determine the timbre of different instruments and sounds, though formants also have a role. For example, the clarinet and saxophone have similar mouthpieces and reeds, and both produce sound through resonance of air inside a chamber whose mouthpiece end is considered closed. Because the clarinet's resonator is cylindrical, the even-numbered harmonics are suppressed, which produces a purer tone. The saxophone's resonator is conical, which allows the even-numbered harmonics to sound mo ...

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Harmonic series music, Harmonic series music - Description of the harmonic series, Harmonic series music - Terminology, Harmonic series music - Harmonics and tuning, Harmonic series music - Timbre of musical instruments, Harmonic series music - Register and special effects of musical instruments

Read more here: » Harmonic series music: Encyclopedia II - Harmonic series music - Timbre of musical instruments

Variations in air pressure against the ear drum, and the subsequent physical and neurological processing and interpretation, give rise to the experience called "sound". Most sound that people recognize as "musical" is dominated by periodic or regular vibrations rather than non-periodic ones (called a definite pitch), and we refer to the transmission mechanism as a "sound wave". In a very simple case, the sound of a sine wave, which is considered to be the most basic model of a sound waveform, causes the air pressure to increase and decrease ...

See also:

Musical acoustics, Musical acoustics - Methods and fields of study, Musical acoustics - Sound waves, Musical acoustics - Harmonics partials and overtones, Musical acoustics - Harmonics and non-linearities, Musical acoustics - Harmony, Musical acoustics - The natural scale, Musical acoustics - The equal tempered scale, Musical acoustics - Cent values of equal temperament

Read more here: » Musical acoustics: Encyclopedia II - Musical acoustics - Sound waves