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71-521: The interval 3:1 (often called by a new name, tritave ) serves as the fundamental harmonic ratio, replacing the diatonic scale's 2:1 (the octave) with a perfect twelfth (an octave higher than a perfect fifth). For any pitch that is part of the BP scale, all pitches one or more tritaves higher or lower are part of the system as well, and are considered equivalent. The BP scale divides the tritave into 13 steps, either equal tempered (the most popular form), or in

142-443: A are the labels assigned to the desired pitch ( n ) and the reference pitch ( a ). These two numbers are from a list of consecutive integers assigned to consecutive semitones. For example, A 4 (the reference pitch) is the 49th key from the left end of a piano (tuned to 440 Hz ), and C 4 ( middle C ), and F ♯ 4 are the 40th and 46th keys, respectively. These numbers can be used to find

213-455: A dominant seventh chord , but occasionally in minor as a minor seventh chord v with passing function : As defined by the 19th century musicologist Joseph Fétis , the dominante was a seventh chord over the first note of a descending perfect fifth in the basse fondamentale or root progression, the common practice period dominant seventh he named the dominante tonique . Dominant chords are important to cadential progressions . In

284-592: A justly tuned version. Compared with octave-repeating scales, the BP scale's intervals are more consonant with certain types of acoustic spectra . The scale was independently described by Heinz Bohlen , Kees van Prooijen and John R. Pierce . Pierce, who, with Max Mathews and others, published his discovery in 1984, renamed the Pierce 3579b scale and its chromatic variant the Bohlen–Pierce scale after learning of Bohlen's earlier publication. Bohlen had proposed

355-531: A ratio of ≈ 517:258 or ≈ 2.00388:1 rather than the usual 2:1, because 12 perfect fifths do not equal seven octaves. During actual play, however, violinists choose pitches by ear, and only the four unstopped pitches of the strings are guaranteed to exhibit this 3:2 ratio. Five- and seven-tone equal temperament ( 5 TET Play and {{7 TET }} Play ), with 240 cent Play and 171 cent Play steps, respectively, are fairly common. 5 TET and 7 TET mark

426-494: A standard pitch of 440 Hz, called A 440 , meaning one note, A , is tuned to 440  hertz and all other notes are defined as some multiple of semitones away from it, either higher or lower in frequency. The standard pitch has not always been 440 Hz; it has varied considerably and generally risen over the past few hundred years. Other equal temperaments divide the octave differently. For example, some music has been written in 19 TET and 31 TET , while

497-544: A tritave ( play ), and split into 13 equal parts. This provides a very close match to justly tuned ratios consisting only of odd numbers. Each step is 146.3 cents ( play ), or 3 13 {\textstyle {\sqrt[{13}]{3}}} . Wendy Carlos created three unusual equal temperaments after a thorough study of the properties of possible temperaments with step size between 30 and 120 cents. These were called alpha , beta , and gamma . They can be considered equal divisions of

568-469: A 5th removed), and especially the first two of these. The scheme I-x-V-I symbolizes, though naturally in a very summarizing way, the harmonic course of any composition of the Classical period . This x , usually appearing as a progression of chords , as a whole series, constitutes, as it were, the actual "music" within the scheme, which through the annexed formula V-I, is made into a unit, a group, or even

639-427: A different choice of perfect fifth. Dominant (music) In music , the dominant is the fifth scale degree ( [REDACTED] ) of the diatonic scale . It is called the dominant because it is second in importance to the first scale degree, the tonic . In the movable do solfège system, the dominant note is sung as "So(l)". The triad built on the dominant note is called the dominant chord . This chord

710-484: A full octave (1200 cents), divided by 146.3… cents per step, gives 8.202087 steps per octave. Dividing the tritave into 13 equal steps tempers out, or reduces to a unison, both of the intervals 245:243 (about 14 cents, sometimes called the minor Bohlen–Pierce diesis ) and 3125:3087 (about 21 cents, sometimes called the major Bohlen–Pierce diesis) in the same way that dividing the octave into 12 equal steps reduces both 81:80 ( syntonic comma ) and 128:125 ( 5-limit limma ) to

781-787: A highly precise, simple and ingenious method for arithmetic calculation of equal temperament mono-chords in 1584" and that Stevin "offered a mathematical definition of equal temperament plus a somewhat less precise computation of the corresponding numerical values in 1585 or later." The developments occurred independently. Kenneth Robinson credits the invention of equal temperament to Zhu and provides textual quotations as evidence. In 1584 Zhu wrote: Kuttner disagrees and remarks that his claim "cannot be considered correct without major qualifications". Kuttner proposes that neither Zhu nor Stevin achieved equal temperament and that neither should be considered its inventor. Chinese theorists had previously come up with approximations for 12 TET , but Zhu

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852-457: A proper fraction in the relationship q t = s also defines a unique family of one equal temperament and its multiples that fulfil this relationship. For example, where k is an integer, 12 k EDO sets q = ⁠ 1 / 2 ⁠ , 19 k EDO sets q = ⁠ 1 / 3 ⁠ , and 31 k EDO sets q = ⁠ 2  / 5 ⁠ . The smallest multiples in these families (e.g. 12, 19 and 31 above) has

923-418: A ratio equal to the 12th root of 2 , ( 2 12 {\textstyle {\sqrt[{12}]{2}}} ≈ 1.05946). That resulting smallest interval, ⁠ 1 / 12 ⁠ the width of an octave, is called a semitone or half step. In Western countries the term equal temperament , without qualification, generally means 12 TET . In modern times, 12 TET is usually tuned relative to

994-513: A ratio slightly larger than an octave, so each of the 39 equal steps is slightly smaller than half of one of the 12 equal steps of the standard scale. Dividing each step of the Bohlen-Pierce scale into fifths (so that the tritave is divided into 65 steps) results in a very accurate octave (41 steps) and perfect fifth (24 steps), as well as approximations for other just intervals. The scale is practically identical to 41-tone equal division of

1065-402: A semitone and a tone are in this equal temperament, one can find the number of steps it has in the octave. An equal temperament with the above properties (including having no notes outside the circle of fifths) divides the octave into 7 t − 2 s steps and the perfect fifth into 4 t − s steps. If there are notes outside the circle of fifths, one must then multiply these results by n ,

1136-521: A similar scale: Bohlen originally expressed the BP scale in both just intonation and equal temperament . The tempered form, which divides the tritave into thirteen equal steps, has become the most popular form. Each step is √ 3 = 3 = 1.08818… above the next, or 1200 log 2  (3) = 146.3… cents per step. The octave is divided into a fractional number of steps. Twelve equally tempered steps per octave are used in 12-tet . The Bohlen–Pierce scale could be described as 8.202087-tet, because

1207-416: A tempered perfect twelfth (1902.4 cents, about a half cent larger than a just twelfth) is divided into 65 equal steps, resulting in a seeming paradox: Taking every fifth degree of this octave-based scale yields an excellent approximation to the non-octave-based equally tempered BP scale. Furthermore, an interval of five such steps generates (octave-based) MOSes (moments of symmetry) with 8, 9, or 17 notes, and

1278-403: A unison. A 7-limit linear temperament tempers out both of these intervals; the resulting Bohlen–Pierce temperament no longer has anything to do with tritave equivalences or non-octave scales, beyond the fact that it is well adapted to using them. A tuning of 41 equal steps to the octave ( 1200 ⁄ 41 = 29.27 cents per step) would be quite logical for this temperament. In such a tuning,

1349-502: A whole piece. In music theory , the dominant triad is a major chord , symbolized by the Roman numeral "V" in the major scale . In the natural minor scale , the triad is a minor chord , denoted by "v". However, in a minor key , the seventh scale degree is often raised by a half step ( ♭ [REDACTED] to ♮ [REDACTED] ), creating a major chord . These chords may also appear as seventh chords : typically as

1420-452: Is based on step sizes and functions not used in the BP scale, it is often called by a new name, tritave ( play ), in BP contexts, referring to its role as a pseudooctave , and using the prefix "tri-" (three) to distinguish it from the octave. In conventional scales, if a given pitch is part of the system, then all pitches one or more octaves higher or lower also are part of the system and, furthermore, are considered equivalent . In

1491-535: Is based on the Fibonacci sequence , although it was created from combination tones , and contains a complex network of harmonic relations due to the inclusion of coinciding harmonics of stacked 833 cent intervals. For example, "step 10 turns out to be identical with the octave (1200 cents) to the base tone, at the same time featuring the Golden Ratio to step 3". Alternate scales may be specified by indicating

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1562-447: Is correspondingly 6 semitones on top and 4 semitones on bottom (0, 4, 10; play ). 5:7:9 is the first inversion of the major triad (0, 4, 7; play ). A study of chromatic triads formed from arbitrary combinations of the 13 tones of the chromatic scale among twelve musicians and twelve untrained listeners found semitones 0, 1, 2 to be the most dissonant chord ( play ), but 0, 11, 13 ( play )

1633-437: Is highly unequal; however, in 1972 Surjodiningrat, Sudarjana and Susanto analyze pelog as equivalent to 9 TET (133-cent steps Play ). A Thai xylophone measured by Morton in 1974 "varied only plus or minus 5 cents" from 7 TET . According to Morton, A South American Indian scale from a pre-instrumental culture measured by Boiles in 1969 featured 175 cent seven-tone equal temperament, which stretches

1704-508: Is said to have dominant function , which means that it creates an instability that requires the tonic for resolution . Dominant triads, seventh chords , and ninth chords typically have dominant function. Leading-tone triads and leading-tone seventh chords may also have dominant function. In very much conventionally tonal music , harmonic analysis will reveal a broad prevalence of the primary (often triadic) harmonies: tonic, dominant, and subdominant (i.e., I and its chief auxiliaries

1775-477: Is similar to 4:5:6 s (the just major chord), more similar than that of the minor chord. This similarity suggests that our ears will also perceive 3:5:7 as consonant. The 3:5:7 chord may thus be considered the major triad of the BP scale. It is approximated by an interval of 6 equal-tempered BP semitones ( play one semitone ) on bottom and an interval of 4 equal-tempered semitones on top (semitones 0, 6, 10; play ). A minor triad

1846-425: Is slightly higher than in conventional 12 tone equal temperament. Because a perfect fifth is in 3:2 relation with its base tone, and this interval comprises seven steps, each tone is in the ratio of 3 / 2 7 {\textstyle {\sqrt[{7}]{3/2}}} to the next (100.28 cents), which provides for a perfect fifth with ratio of 3:2, but a slightly widened octave with

1917-572: Is that 12 EDO is the smallest equal temperament to closely approximate 5 limit harmony, the next-smallest being 19 EDO.) Each choice of fraction q for the relationship results in exactly one equal temperament family, but the converse is not true: 47  EDO has two different semitones, where one is ⁠ 1  / 7 ⁠ tone and the other is ⁠ 8  / 9 ⁠ , which are not complements of each other like in 19  EDO ( ⁠ 1  / 3 ⁠ and ⁠ 2  / 3 ⁠ ). Taking each semitone results in

1988-411: Is the dominant scale degree in the main key. If, for example, a piece is written in the key of C major , then the tonic key is C major and the dominant key is G major since G is the dominant note in C major. In sonata form in major keys, the second subject group is usually in the dominant key. The movement to the dominant was part of musical grammar, not an element of form. Almost all music in

2059-500: Is the ratio: where the ratio r divides the ratio p (typically the octave, which is 2:1) into n equal parts. ( See Twelve-tone equal temperament below. ) Scales are often measured in cents , which divide the octave into 1200 equal intervals (each called a cent). This logarithmic scale makes comparison of different tuning systems easier than comparing ratios, and has considerable use in ethnomusicology . The basic step in cents for any equal temperament can be found by taking

2130-502: The Arab tone system uses 24 TET . Instead of dividing an octave, an equal temperament can also divide a different interval, like the equal-tempered version of the Bohlen–Pierce scale , which divides the just interval of an octave and a fifth (ratio 3:1), called a "tritave" or a " pseudo-octave " in that system, into 13 equal parts. For tuning systems that divide the octave equally, but are not approximations of just intervals,

2201-474: The OEIS ) is the sequence of divisions of octave that provides better and better approximations of the perfect fifth. Related sequences containing divisions approximating other just intervals are listed in a footnote. The equal-tempered version of the Bohlen–Pierce scale consists of the ratio 3:1 (1902 cents) conventionally a perfect fifth plus an octave (that is, a perfect twelfth), called in this theory

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2272-654: The University of Toronto , directed the development of a "Stredici", a string instrument tuned to the Bohlen–Pierce scale. The five-meter long instrument was used in concerts in Boston in 2012. A first Bohlen–Pierce symposium took place in Boston on March 7 to 9, 2010, produced by composer Georg Hajdu ( Hochschule für Musik und Theater Hamburg ) and the Boston Microtonal Society . Co-organizers were

2343-430: The frequency ratio of the interval between two adjacent notes, is the twelfth root of two : This interval is divided into 100 cents. To find the frequency, P n , of a note in 12  TET , the following formula may be used: In this formula P n represents the pitch, or frequency (usually in hertz ), you are trying to find. P a is the frequency of a reference pitch. The indes numbers n and

2414-484: The logarithm of a multiplication reduces it to addition. Furthermore, by applying the modular arithmetic where the modulus is the number of divisions of the octave (usually 12), these integers can be reduced to pitch classes , which removes the distinction (or acknowledges the similarity) between pitches of the same name, e.g., c is 0 regardless of octave register. The MIDI encoding standard uses integer note designations. 12 tone equal temperament, which divides

2485-412: The logarithmic changes in pitch frequency. In classical music and Western music in general, the most common tuning system since the 18th century has been 12 equal temperament (also known as 12 tone equal temperament , 12 TET or 12 ET , informally abbreviated as 12 equal ), which divides the octave into 12 parts, all of which are equal on a logarithmic scale , with

2556-574: The 8-note scale (comprising degrees 0, 5, 10, 15, 20, 25, 30, and 35 of the 41-equal scale) could be considered the octave-equivalent version of the Bohlen–Pierce scale. The following are the thirteen notes in the scale (cents rounded to nearest whole number): Justly tuned Equal-tempered play equal tempered Bohlen–Pierce scale What does music using a Bohlen–Pierce scale sound like, aesthetically ? Dave Benson suggests it helps to use only sounds with only odd harmonics, including clarinets or synthesized tones, but argues that because "some of

2627-414: The BP scale, if a given pitch is present, then none of the pitches one or more octaves higher or lower are present, but all pitches one or more tritaves higher or lower are part of the system and are considered equivalent. The BP scale's use of odd integer ratios is appropriate for timbres containing only odd harmonics. Because the clarinet 's spectrum (in the chalumeau register) consists of primarily

2698-452: The Bohlen–Pierce scale, performed more than 40 compositions in the novel system and introduced several new musical instruments. Performers included German musicians Nora-Louise Müller and Ákos Hoffman on Bohlen–Pierce clarinets and Arturo Grolimund on Bohlen–Pierce pan flute as well as Canadian ensemble tranSpectra, and US American xenharmonic band ZIA, led by Elaine Walker. Other non-octave tunings investigated by Bohlen include twelve steps in

2769-870: The Boston Goethe Institute , the Berklee College of Music , the Northeastern University and the New England Conservatory of Music. The symposium participants, which included Heinz Bohlen, Max Mathews, Clarence Barlow , Curtis Roads , David Wessel, Psyche Loui, Richard Boulanger, Georg Hajdu, Paul Erlich , Ron Sword, Julia Werntz, Larry Polansky , Manfred Stahnke, Stephen Fox, Elaine Walker, Todd Harrop, Gayle Young, Johannes Kretz, Arturo Grolimund, Kevin Foster, presented 20 papers on history and properties of

2840-430: The additional property of having no notes outside the circle of fifths . (This is not true in general; in 24  EDO , the half-sharps and half-flats are not in the circle of fifths generated starting from C .) The extreme cases are 5 k EDO , where q = 0 and the semitone becomes a unison, and 7 k EDO , where q = 1 and the semitone and tone are the same interval. Once one knows how many steps

2911-410: The distance between two adjacent steps of the scale is the same interval . Because the perceived identity of an interval depends on its ratio , this scale in even steps is a geometric sequence of multiplications. (An arithmetic sequence of intervals would not sound evenly spaced and would not permit transposition to different keys .) Specifically, the smallest interval in an equal-tempered scale

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2982-416: The dominant key. Modulation to the dominant often creates a sense of increased tension; as opposed to modulation to the subdominant (fourth note of the scale), which creates a sense of musical relaxation. The vast majority of harmonies designated as "essential" in the basic frame of structure must be I and V–the latter, when tonal music is viewed in broadest terms , an auxiliary support and embellishment of

3053-434: The eighteenth century went to the dominant: before 1750 it was not something to be emphasized; afterward, it was something that the composer could take advantage of. This means that every eighteenth-century listener expected the movement to the dominant in the sense that [one] would have been puzzled if [one] did not get it; it was a necessary condition of intelligibility. Music which modulates (changes key) often modulates to

3124-407: The end, 12-tone equal temperament won out. This allowed enharmonic modulation , new styles of symmetrical tonality and polytonality , atonal music such as that written with the 12-tone technique or serialism , and jazz (at least its piano component) to develop and flourish. In 12 tone equal temperament, which divides the octave into 12 equal parts, the width of a semitone , i.e.

3195-549: The endpoints of the syntonic temperament 's valid tuning range, as shown in Figure ;1 . According to Kunst (1949), Indonesian gamelans are tuned to 5 TET , but according to Hood (1966) and McPhee (1966) their tuning varies widely, and according to Tenzer (2000) they contain stretched octaves . It is now accepted that of the two primary tuning systems in gamelan music, slendro and pelog , only slendro somewhat resembles five-tone equal temperament, while pelog

3266-786: The five chords rated most consonant by trained musicians are approximately diatonic intervals, suggesting that their training influenced their selection and that similar experience with the BP scale would similarly influence their choices. Compositions using the Bohlen–Pierce scale include "Purity", the first movement of Curtis Roads ' Clang-Tint . Other computer composers to use the BP scale include Jon Appleton , Richard Boulanger ( Solemn Song for Evening (1990)), Georg Hajdu , Juan Reyes' ppP (1999-2000), Ami Radunskaya 's "A Wild and Reckless Place" (1990), Charles Carpenter ( Frog à la Pêche (1994) & Splat ), and Elaine Walker ( Stick Men (1991), Love Song , and Greater Good (2011)). David Lieberman, an associate professor of architecture at

3337-565: The following frequencies, respectively: The intervals of 12  TET closely approximate some intervals in just intonation . The fifths and fourths are almost indistinguishably close to just intervals, while thirds and sixths are further away. In the following table, the sizes of various just intervals are compared to their equal-tempered counterparts, given as a ratio as well as cents. Violins, violas, and cellos are tuned in perfect fifths ( G D A E for violins and C G D A for violas and cellos), which suggests that their semitone ratio

3408-638: The former, for which it is the principal medium of tonicization . The dominant is an important concept in Middle Eastern music . In the Persian Dastgah , Arabic maqam and the Turkish makam , scales are made up of trichords , tetrachords , and pentachords (each called a jins in Arabic ) with the tonic of a maqam being the lowest note of the lower jins and the dominant being that of

3479-452: The frequency of C 4 and F ♯ 4 : To convert a frequency (in Hz) to its equal 12  TET counterpart, the following formula can be used: E n is the frequency of a pitch in equal temperament, and E a is the frequency of a reference pitch. For example, if we let the reference pitch equal 440 Hz, we can see that E 5 and C ♯ 5 have

3550-554: The intervals sound a bit like intervals in [the more familiar] twelve-tone scale , but badly out of tune ", the average listener will continually feel "that something isn't quite right", due to social conditioning . Mathews and Pierce conclude that clear and memorable melodies may be composed in the BP scale, that "counterpoint sounds all right", and that "chordal passages sound like harmony", presumably meaning progression , "but without any great tension or sense of resolution". In their 1989 study of consonance judgment, both intervals of

3621-615: The length was halved. Zhu created several instruments tuned to his system, including bamboo pipes. Some of the first Europeans to advocate equal temperament were lutenists Vincenzo Galilei , Giacomo Gorzanis , and Francesco Spinacino , all of whom wrote music in it. Simon Stevin was the first to develop 12  TET based on the twelfth root of two , which he described in van de Spiegheling der singconst ( c.  1605 ), published posthumously in 1884. Plucked instrument players (lutenists and guitarists) generally favored equal temperament, while others were more divided. In

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3692-414: The number of nonoverlapping circles of fifths required to generate all the notes (e.g., two in 24  EDO , six in 72  EDO ). (One must take the small semitone for this purpose: 19  EDO has two semitones, one being ⁠ 1  / 3 ⁠ tone and the other being ⁠ 2  / 3 ⁠ . Similarly, 31  EDO has two semitones, one being ⁠ 2  / 5 ⁠ tone and

3763-427: The number of steps in a semitone be s , and the number of steps in a tone be t . There is exactly one family of equal temperaments that fixes the semitone to any proper fraction of a whole tone, while keeping the notes in the right order (meaning that, for example, C , D , E , F , and F ♯ are in ascending order if they preserve their usual relationships to C ). That is, fixing q to

3834-436: The octave except that each step is slightly smaller (less than a hundredth of a cent per step). Equal temperament An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave (or other interval) 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

3905-473: The octave into 12 intervals of equal size, is the musical system most widely used today, especially in Western music. The two figures frequently credited with the achievement of exact calculation of equal temperament are Zhu Zaiyu (also romanized as Chu-Tsaiyu. Chinese: 朱載堉 ) in 1584 and Simon Stevin in 1585. According to F.A. Kuttner, a critic of giving credit to Zhu, it is known that Zhu "presented

3976-703: The octave slightly, as with instrumental gamelan music. Chinese music has traditionally used 7 TET . Other equal divisions of the octave that have found occasional use include 13 EDO , 15 EDO , 17 EDO , and 55 EDO. 2, 5, 12, 41, 53, 306, 665 and 15601 are denominators of first convergents of log 2 (3), so 2, 5, 12, 41, 53, 306, 665 and 15601 twelfths (and fifths), being in correspondent equal temperaments equal to an integer number of octaves, are better approximations of 2, 5, 12, 41, 53, 306, 665 and 15601 just twelfths/fifths than in any equal temperament with fewer tones. 1, 2, 3, 5, 7, 12, 29, 41, 53, 200, ... (sequence A060528 in

4047-413: The odd harmonics 3:1; 5:3, 7:3; 7:5, 9:5; 9:7, and 15:7; while the 39-step scale includes all of those and many more (11:5, 13:5; 11:7, 13:7; 11:9, 13:9; 13:11, 15:11, 21:11, 25:11, 27:11; 15:13, 21:13, 25:13, 27:13, 33:13, and 35:13), while still missing almost all of the even harmonics (including 2:1; 3:2, 5:2; 4:3, 8:3; 6:5, 8:5; 9:8, 11:8, 13:8, and 15:8). The size of this scale is about 25 equal steps to

4118-436: The odd harmonics, and the instrument overblows at the twelfth (or tritave) rather than the octave as most other woodwind instruments do, there is a natural affinity between it and the Bohlen–Pierce scale. At the suggestion of composer Georg Hajdu , clarinet maker Stephen Fox developed the first Bohlen–Pierce soprano clarinets and began offering them for sale in early 2006. He produced the first BP tenor clarinet (six steps below

4189-409: The other being ⁠ 3  / 5 ⁠ ). The smallest of these families is 12 k EDO , and in particular, 12  EDO is the smallest equal temperament with the above properties. Additionally, it makes the semitone exactly half a whole tone, the simplest possible relationship. These are some of the reasons 12  EDO has become the most commonly used equal temperament. (Another reason

4260-477: The perfect fifth. Each of them provides a very good approximation of several just intervals. Their step sizes: Alpha and beta may be heard on the title track of Carlos's 1986 album Beauty in the Beast . In this section, semitone and whole tone may not have their usual 12 EDO meanings, as it discusses how they may be tempered in different ways from their just versions to produce desired relationships. Let

4331-502: The same scale based on consideration of the influence of combination tones on the Gestalt impression of intervals and chords. The intervals between BP scale pitch classes are based on odd integer frequency ratios, in contrast with the intervals in diatonic scales, which employ both odd and even ratios found in the harmonic series . Specifically, the BP scale steps are based on ratios of integers whose factors are 3, 5, and 7. Thus

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4402-500: The scale contains consonant harmonies based on the odd harmonic overtones 3:5:7:9 ( play ). The chord formed by the ratio 3:5:7 ( play ) serves much the same role as the 4:5:6 chord (a major triad play ) does in diatonic scales (3:5:7 = 1: ⁠1 + 2 / 3 ⁠ : ⁠2 + 1 / 3 ⁠ and 4:5:6 = 2: ⁠2 + 1 / 2 ⁠ :3 = 1: ⁠1 + 1 / 4 ⁠ : ⁠1 + 1 / 2 ⁠ ). 3:5:7 s intonation sensitivity pattern

4473-588: The size of equal tempered steps, for example Wendy Carlos ' 78-cent alpha scale and 63.8-cent beta scale , and Gary Morrison's 88-cent scale (13.64 steps per octave or 14 per 1232-cent stretched octave). This gives the alpha scale 15.39 steps per octave and the beta scale 18.75 steps per octave. Paul Erlich proposed dividing each step of the Bohlen–Pierce into thirds so that the tritave is divided into 39 equal steps instead of 13 equal steps. The scale, which can be viewed as three evenly staggered Bohlen-Pierce scales, gives additional odd harmonics. The 13-step scale hits

4544-575: The soprano) in 2010 and the first epsilon clarinet (four steps above the soprano) in 2011. A contra clarinet (one tritave lower than the soprano) is now (2020) played by Nora Mueller, Luebeck, Germany. A diatonic Bohlen–Pierce scale may be constructed with the following just ratios (chart shows the "Lambda" (λ) scale): play just Bohlen–Pierce "Lambda" scale contrast with just major diatonic scale A just BP scale may be constructed from four overlapping 3:5:7 chords, for example, V, II, VI, and IV, though different chords may be chosen to produce

4615-406: The strongest cadence, the authentic cadence (example shown below), the dominant chord is followed by the tonic chord. A cadence that ends with a dominant chord is called a half cadence or an "imperfect cadence". The dominant key is the key whose tonic is a perfect fifth above (or a perfect fourth below) the tonic of the main key of the piece. Put another way, it is the key whose tonic

4686-660: The term equal division of the octave , or EDO can be used. Unfretted string ensembles , which can adjust the tuning of all notes except for open strings , and vocal groups, who have no mechanical tuning limitations, sometimes use a tuning much closer to just intonation for acoustic reasons. Other instruments, such as some wind , keyboard , and fretted instruments, often only approximate equal temperament, where technical limitations prevent exact tunings. Some wind instruments that can easily and spontaneously bend their tone, most notably trombones , use tuning similar to string ensembles and vocal groups. In an equal temperament,

4757-434: The tonic to rise to what was note III (semitone 3), which therefore may be considered the dominant . One may consider VIII (semitone 10) the analogue of the subdominant . 3:1 serves as the fundamental harmonic ratio, replacing the diatonic scale's 2:1 (the octave ). ( play ) This interval is a perfect twelfth in diatonic nomenclature ( perfect fifth when reduced by an octave), but as this terminology

4828-410: The tritave, named A12 by Enrique Moreno and based on the 4:7:10 chord Play , seven steps in the octave ( 7-tet ) or similar 11 steps in the tritave, and eight steps in the octave, based on 5:7:9 Play and of which only the just version would be used. Additionally, the pentave can be divided into eight steps which approximates chords of the form 5:9:13:17:21:25. The Bohlen 833 cents scale

4899-408: The width of p above in cents (usually the octave, which is 1200 cents wide), called below w , and dividing it into n parts: In musical analysis, material belonging to an equal temperament is often given an integer notation , meaning a single integer is used to represent each pitch. This simplifies and generalizes discussion of pitch material within the temperament in the same way that taking

4970-488: Was considered the most consonant by the trained subjects (because it sounds like an octave-dropped major triad) and 0, 7, 10 ( play ) was judged most consonant by the untrained subjects. Every tone of the Pierce 3579b scale is in a major and minor triad except for tone II of the scale. There are thirteen possible keys. Modulation is possible through changing a single note. Moving note II up one semitone causes

5041-518: Was the first person to mathematically solve 12 tone equal temperament, which he described in two books, published in 1580 and 1584. Needham also gives an extended account. Zhu obtained his result by dividing the length of string and pipe successively by 2 12 {\textstyle {\sqrt[{12}]{2}}} ≈ 1.059463 , and for pipe length by 2 24 {\displaystyle {\sqrt[{24}]{2}}} ≈ 1.029302 , such that after 12 divisions (an octave),

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