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51-424: The Western musical convention up to this day divides an octave into twelve different pitches that can be arranged or tempered in different intervals . Carrillo termed his new system Sonido 13, which is Spanish for "Thirteenth Sound" or Sound 13, because it enabled musicians to go beyond the twelve notes that comprise an octave in conventional Western music . Julián Carrillo wrote: "The thirteenth sound will be

102-406: A "tonic" diatonic scale and modulate to the "dominant" scale a fifth above. In the 19th century (to a certain extent), but more in the 20th century, additional types of scales were explored: A large variety of other scales exists, some of the more common being: Scales such as the pentatonic scale may be considered gapped relative to the diatonic scale. An auxiliary scale is a scale other than

153-576: A binary system of twelve zeros or ones to represent each of the twelve notes of a chromatic scale . The most common binary numbering scheme defines lower pitches to have lower numeric value (as opposed to low pitches having a high numeric value). Thus a single pitch class n in the pitch class set is represented by 2^n. This maps the entire power set of all pitch class sets in 12-TET to the numbers 0 to 4095. The binary digits read as ascending pitches from right to left, which some find discombobulating because they are used to low to high reading left to right, as on

204-554: A composition, such as in Claude Debussy 's L'Isle Joyeuse . To the right, the first scale is a whole-tone scale, while the second and third scales are diatonic scales. All three are used in the opening pages of Debussy's piece. Scales in traditional Western music generally consist of seven notes and repeat at the octave. Notes in the commonly used scales (see just below) are separated by whole and half step intervals of tones and semitones. The harmonic minor scale includes

255-422: A note between G and G ♯ or a note moving between both. In blues, a pentatonic scale is often used. In jazz, many different modes and scales are used, often within the same piece of music. Chromatic scales are common, especially in modern jazz. In Western music, scale notes are often separated by equally tempered tones or semitones, creating 12 intervals per octave. Each interval separates two tones;

306-449: A piano keyboard. In this scheme, the major scale is 101010110101 = 2741. This binary representation permits easy calculation of interval vectors and common tones, using logical binary operators. It also provides a perfect index for every possible combination of tones, as every scale has its own number. Scales may also be shown as semitones from the tonic. For instance, 0 2 4 5 7 9 11 denotes any major scale such as C–D–E–F–G–A–B, in which

357-501: A pleasing sound to music. The interval is so natural to humans that when men and women are asked to sing in unison, they typically sing in octave. For this reason, notes an octave apart are given the same note name in the Western system of music notation —the name of a note an octave above A is also A. This is called octave equivalence , the assumption that pitches one or more octaves apart are musically equivalent in many ways, leading to

408-400: A semitone. The blue note is an interval that is technically neither major nor minor but "in the middle", giving it a characteristic flavour. A regular piano cannot play blue notes, but with electric guitar , saxophone , trombone and trumpet , performers can "bend" notes a fraction of a tone sharp or flat to create blue notes. For instance, in the key of E, the blue note would be either

459-599: A third (in this case a major third); D and F also create a third (in this case a minor third). A single scale can be manifested at many different pitch levels. For example, a C major scale can be started at C4 (middle C; see scientific pitch notation ) and ascending an octave to C5; or it could be started at C6, ascending an octave to C7. Scales may be described according to the number of different pitch classes they contain: Scales may also be described by their constituent intervals, such as being hemitonic , cohemitonic , or having imperfections. Many music theorists concur that

510-711: A three-semitone step; the anhemitonic pentatonic includes two of those and no semitones. Western music in the Medieval and Renaissance periods (1100–1600) tends to use the white-note diatonic scale C–D–E–F–G–A–B. Accidentals are rare, and somewhat unsystematically used, often to avoid the tritone . Music of the common practice periods (1600–1900) uses three types of scale: These scales are used in all of their transpositions. The music of this period introduces modulation, which involves systematic changes from one scale to another. Modulation occurs in relatively conventionalized ways. For example, major-mode pieces typically begin in

561-425: A tritone), and one without tritones is atritonic . A scale or chord that contains semitones is called hemitonic, and without semitones is anhemitonic . Scales can be abstracted from performance or composition . They are also often used precompositionally to guide or limit a composition. Explicit instruction in scales has been part of compositional training for many centuries. One or more scales may be used in

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612-445: Is C–B–A–G–F–E–D–[C], with the bracket indicating an octave lower than the first note in the scale. The distance between two successive notes in a scale is called a scale step . The notes of a scale are numbered by their steps from the first degree of the scale. For example, in a C major scale the first note is C, the second D, the third E and so on. Two notes can also be numbered in relation to each other: C and E create an interval of

663-477: Is D–E–F ♯ in Chromatic transposition). Since the steps of a scale can have various sizes, this process introduces subtle melodic and harmonic variation into the music. In Western tonal music, the simplest and most common type of modulation (or changing keys) is to shift from one major key to another key built on the first key's fifth (or dominant) scale degree. In the key of C major, this would involve moving to

714-417: Is a diminished octave (d8). The use of such intervals is rare, as there is frequently a preferable enharmonically -equivalent notation available ( minor ninth and major seventh respectively), but these categories of octaves must be acknowledged in any full understanding of the role and meaning of octaves more generally in music. Octaves are identified with various naming systems. Among the most common are

765-445: Is a part of most advanced musical cultures, but is far from universal in "primitive" and early music . The languages in which the oldest extant written documents on tuning are written, Sumerian and Akkadian , have no known word for "octave". However, it is believed that a set of cuneiform tablets that collectively describe the tuning of a nine-stringed instrument, believed to be a Babylonian lyre , describe tunings for seven of

816-400: Is an octave. In Western music notation , notes separated by an octave (or multiple octaves) have the same name and are of the same pitch class . To emphasize that it is one of the perfect intervals (including unison , perfect fourth , and perfect fifth ), the octave is designated P8. Other interval qualities are also possible, though rare. The octave above or below an indicated note

867-522: Is defined by its characteristic interval pattern and by a special note, known as its first degree (or tonic ). The tonic of a scale is the note selected as the beginning of the octave, and therefore as the beginning of the adopted interval pattern. Typically, the name of the scale specifies both its tonic and its interval pattern. For example, C major indicates a major scale with a C tonic. Scales are typically listed from low to high pitch. Most scales are octave -repeating , meaning their pattern of notes

918-628: Is no limit to how many notes can be injected within any given musical interval. A measure of the width of each scale step provides a method to classify scales. For instance, in a chromatic scale each scale step represents a semitone interval, while a major scale is defined by the interval pattern W–W–H–W–W–W–H, where W stands for whole step (an interval spanning two semitones, e.g. from C to D), and H stands for half-step (e.g. from C to D ♭ ). Based on their interval patterns, scales are put into categories including pentatonic , diatonic , chromatic , major , minor , and others. A specific scale

969-402: Is sometimes abbreviated 8 or 8 ( Italian : all'ottava ), 8 bassa ( Italian : all'ottava bassa , sometimes also 8 ), or simply 8 for the octave in the direction indicated by placing this mark above or below the staff. An octave is the interval between one musical pitch and another with double or half its frequency . For example, if one note has a frequency of 440  Hz ,

1020-454: Is the same in every octave (the Bohlen–Pierce scale is one exception). An octave-repeating scale can be represented as a circular arrangement of pitch classes, ordered by increasing (or decreasing) pitch class. For instance, the increasing C major scale is C–D–E–F–G–A–B–[C], with the bracket indicating that the last note is an octave higher than the first note, and the decreasing C major scale

1071-404: The diapason ) is a series of eight notes occupying the interval between (and including) two notes, one having twice the frequency of vibration of the other. 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

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1122-556: The harmonic overtones series. Many musical scales in the world are based on this system, except most of the musical scales from Indonesia and the Indochina Peninsulae, which are based on inharmonic resonance of the dominant metalophone and xylophone instruments. Some scales use a different number of pitches. A common scale in Eastern music is the pentatonic scale, which consists of five notes that span an octave. For example, in

1173-478: The scientific , Helmholtz , organ pipe, and MIDI note systems. In scientific pitch notation, a specific octave is indicated by a numerical subscript number after note name. In this notation, middle C is C 4 , because of the note's position as the fourth C key on a standard 88-key piano keyboard, while the C an octave higher is C 5 . The notation 8 or 8 is sometimes seen in sheet music , meaning "play this an octave higher than written" ( all' ottava : "at

1224-615: The Chinese culture, the pentatonic scale is usually used for folk music and consists of C, D, E, G and A, commonly known as gong, shang, jue, chi and yu. Some scales span part of an octave; several such short scales are typically combined to form a scale spanning a full octave or more, and usually called with a third name of its own. The Turkish and Middle Eastern music has around a dozen such basic short scales that are combined to form hundreds of full-octave spanning scales. Among these scales Hejaz scale has one scale step spanning 14 intervals (of

1275-483: The Latin scala , which literally means " ladder ". Therefore, any scale is distinguishable by its "step-pattern", or how its intervals interact with each other. Often, especially in the context of the common practice period , most or all of the melody and harmony of a musical work is built using the notes of a single scale, which can be conveniently represented on a staff with a standard key signature . Due to

1326-591: The Sonido 13 Symphonic Orchestra that performed in different parts of the world, playing microtonal music composed by Carrillo in different intervals. In 1933, Ahualulco, the town where Carrillo was born, was renamed to Ahualulco del Sonido 13 in honor of Carrillo's work. Carrillo was, "closely associated with the Díaz regime ," and preferred neo-classicism to nationalism . Octave In music , an octave ( Latin : octavus : eighth) or perfect octave (sometimes called

1377-519: The beginning of the end and the point of departure of a new musical generation which will transform everything." Carrillo attended the National Conservatory of Music in Mexico City, where he studied violin, composition, physics , acoustics , and mathematics . The laws that define music intervals instantly amazed Carrillo, which led him to conduct experiments on his violin. He began analyzing

1428-449: The chord. The word is also used to describe melodies played in parallel one or more octaves apart (see example under Equivalence, below). While octaves commonly refer to the perfect octave (P8), the interval of an octave in music theory encompasses chromatic alterations within the pitch class, meaning that G ♮ to G ♯ (13 semitones higher) is an Augmented octave (A8), and G ♮ to G ♭ (11 semitones higher)

1479-409: The constituent intervals of a scale have a large role in the cognitive perception of its sonority, or tonal character. "The number of the notes that make up a scale as well as the quality of the intervals between successive notes of the scale help to give the music of a culture area its peculiar sound quality." "The pitch distances or intervals among the notes of a scale tell us more about the sound of

1530-426: The convention "that scales are uniquely defined by specifying the intervals within an octave". The conceptualization of pitch as having two dimensions, pitch height (absolute frequency) and pitch class (relative position within the octave), inherently include octave circularity. Thus all C ♯ s (or all 1s, if C = 0), any number of octaves apart, are part of the same pitch class . Octave equivalence

1581-439: The first degree is, obviously, 0 semitones from the tonic (and therefore coincides with it), the second is 2 semitones from the tonic, the third is 4 semitones from the tonic, and so on. Again, this implies that the notes are drawn from a chromatic scale tuned with 12-tone equal temperament. For some fretted string instruments, such as the guitar and the bass guitar , scales can be notated in tabulature , an approach which indicates

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1632-510: The frequency, respectively. The number of octaves between two frequencies is given by the formula: Most musical scales are written so that they begin and end on notes that are an octave apart. For example, the C major scale is typically written C D E F G A B C (shown below), the initial and final Cs being an octave apart. Because of octave equivalence, notes in a chord that are one or more octaves apart are said to be doubled (even if there are more than two notes in different octaves) in

1683-438: The fret number and string upon which each scale degree is played. Composers transform musical patterns by moving every note in the pattern by a constant number of scale steps: thus, in the C major scale, the pattern C–D–E might be shifted up, or transposed , a single scale step to become D–E–F. This process is called "scalar transposition" or "shifting to a new key" and can often be found in musical sequences and patterns. (It

1734-449: The higher tone has an oscillation frequency of a fixed ratio (by a factor equal to the twelfth root of two , or approximately 1.059463) higher than the frequency of the lower one. A scale uses a subset consisting typically of 7 of these 12 as scale steps. Many other musical traditions use scales that include other intervals. These scales originate within the derivation of the harmonic series . Musical intervals are complementary values of

1785-475: The key of G major (which uses an F ♯ ). Composers also often modulate to other related keys. In some Romantic music era pieces and contemporary music, composers modulate to "remote keys" that are not related to or close to the tonic. An example of a remote modulation would be taking a song that begins in C major and modulating (changing keys) to F ♯ major. Through the introduction of blue notes , jazz and blues employ scale intervals smaller than

1836-453: The mammalian brain . Studies have also shown the perception of octave equivalence in rats, human infants, and musicians but not starlings, 4–9-year-old children, or non-musicians. Sources Scale (music) In music theory , a scale is "any consecutive series of notes that form a progression between one note and its octave ", typically by order of pitch or fundamental frequency . The word "scale" originates from

1887-477: The middle eastern type found 53 in an octave) roughly similar to 3 semitones (of the western type found 12 in an octave), while Saba scale , another of these middle eastern scales, has 3 consecutive scale steps within 14 commas, i.e. separated by roughly one western semitone either side of the middle tone. Gamelan music uses a small variety of scales including Pélog and Sléndro , none including equally tempered nor harmonic intervals. Indian classical music uses

1938-498: The music than does the mere number of tones." Scales may also be described by their symmetry, such as being palindromic , chiral , or having rotational symmetry as in Messiaen's modes of limited transposition . The notes of a scale form intervals with each of the other notes of the chord in combination . A 5-note scale has 10 of these harmonic intervals, a 6-note scale has 15, a 7-note scale has 21, an 8-note scale has 28. Though

1989-878: The nature of intervals. Carrillo became an excellent musician at the Conservatory and received a scholarship to study at the Leipzig Royal Conservatory. After Carrillo returned to Mexico in 1918, he became conductor of the National Symphony Orchestra and in 1920 he also became principal of the National Conservatory of Music. It was during this time that he began to invest a significant amount of time on Sonido 13. His achievements in this area were extensive and consisted of writing over 20 books, making more than 40 compositions, patenting fifteen pianos capable of producing small intervals, and organizing

2040-668: The note one octave above is at 880 Hz, and the note one octave below is at 220 Hz. The ratio of frequencies of two notes an octave apart is therefore 2:1. Further octaves of a note occur at 2 n {\displaystyle 2^{n}} times the frequency of that note (where n is an integer), such as 2, 4, 8, 16, etc. and the reciprocal of that series. For example, 55 Hz and 440 Hz are one and two octaves away from 110 Hz because they are + 1 ⁄ 2 (or 2 − 1 {\displaystyle 2^{-1}} ) and 4 (or 2 2 {\displaystyle 2^{2}} ) times

2091-486: The notes of a scale, it is customary that each scale degree be assigned its own letter name: for example, the A major scale is written A–B–C ♯ –D–E–F ♯ –G ♯ rather than A–B–D ♭ –D–E–E [REDACTED] –G ♯ . However, it is impossible to do this in scales that contain more than seven notes, at least in the English-language nomenclature system. Scales may also be identified by using

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2142-403: The notes of the C major scale using A = 1, B = 2, C = 3, and so on. When we do so, we create a new scale called the A minor scale . See the musical note article for how the notes are customarily named in different countries. The scale degrees of a heptatonic (7-note) scale can also be named using the terms tonic , supertonic , mediant , subdominant , dominant , submediant , subtonic . If

2193-685: The octave" or all' 8 ). 8 or 8 stands for ottava , the Italian word for octave (or "eighth"); the octave above may be specified as ottava alta or ottava sopra ). Sometimes 8 is used to tell the musician to play a passage an octave lower (when placed under rather than over the staff), though the similar notation 8 ( ottava bassa or ottava sotto ) is also used. Similarly, 15 ( quindicesima ) means "play two octaves higher than written" and 15 ( quindicesima bassa ) means "play two octaves lower than written." The abbreviations col 8 , coll' 8 , and c. 8 stand for coll'ottava , meaning "with

2244-471: The octave", i.e. to play the notes in the passage together with the notes in the notated octaves. Any of these directions can be cancelled with the word loco , but often a dashed line or bracket indicates the extent of the music affected. After the unison , the octave is the simplest interval in music. The human ear tends to hear both notes as being essentially "the same", due to closely related harmonics. Notes separated by an octave "ring" together, adding

2295-444: The primary or original scale. See: modulation (music) and Auxiliary diminished scale . In many musical circumstances, a specific note of the scale is chosen as the tonic —the central and most stable note of the scale. In Western tonal music, simple songs or pieces typically start and end on the tonic note. Relative to a choice of a certain tonic, the notes of a scale are often labeled with numbers recording how many scale steps above

2346-489: The principle of octave equivalence, scales are generally considered to span a single octave, with higher or lower octaves simply repeating the pattern. A musical scale represents a division of the octave space into a certain number of scale steps, a scale step being the recognizable distance (or interval ) between two successive notes of the scale. However, there is no need for scale steps to be equal within any scale and, particularly as demonstrated by microtonal music , there

2397-483: The scale is not a chord , and might never be heard more than one note at a time, still the absence, presence, and placement of certain key intervals plays a large part in the sound of the scale, the natural movement of melody within the scale, and the selection of chords taken naturally from the scale. A musical scale that contains tritones is called tritonic (though the expression is also used for any scale with just three notes per octave, whether or not it includes

2448-425: The strings, with indications to tune the remaining two strings an octave from two of the seven tuned strings. Leon Crickmore recently proposed that "The octave may not have been thought of as a unit in its own right, but rather by analogy like the first day of a new seven-day week". Monkeys experience octave equivalence, and its biological basis apparently is an octave mapping of neurons in the auditory thalamus of

2499-405: The subtonic is a semitone away from the tonic, then it is usually called the leading-tone (or leading-note); otherwise the leading-tone refers to the raised subtonic. Also commonly used is the (movable do) solfège naming convention in which each scale degree is denoted by a syllable. In the major scale, the solfège syllables are: do, re, mi, fa, so (or sol), la, ti (or si), do (or ut). In naming

2550-425: The tonic they are. For example, the notes of the C major scale (C, D, E, F, G, A, B) can be labeled {1, 2, 3, 4, 5, 6, 7}, reflecting the choice of C as tonic. The expression scale degree refers to these numerical labels. Such labeling requires the choice of a "first" note; hence scale-degree labels are not intrinsic to the scale itself, but rather to its modes. For example, if we choose A as tonic, then we can label

2601-428: The way the pitch of a string changed depending on the finger position, concluding that there had to be a way to split the string into an infinite number of parts. One day, Carrillo was able to divide the fourth string of his violin with a razor into 16 parts in the interval between the notes G and A, thus creating 16 unique sounds. This event was the beginning of Sonido 13 that led Carrillo to study more about physics and

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