Generalizing Musical Intervals - Dmitri Tymoczko
... twentieth-century mathematics. My aims are threefold. First, I want to explore the fascinating intellectual-historical resonance between these two very different, and yet in some sense very similar intellectual projects—letting our understanding of each deepen and enrich our understanding of the oth ...
... twentieth-century mathematics. My aims are threefold. First, I want to explore the fascinating intellectual-historical resonance between these two very different, and yet in some sense very similar intellectual projects—letting our understanding of each deepen and enrich our understanding of the oth ...
Appendix I: Microtonality in 20 Century Western Art Music: A
... had written his first major quarter-tone composition in Vienna in 1920, i.e. his String Quartet No.2 op.7 in which the quarter-tones are used as part of a highly chromatic lateromantic tonal pitch language (the piece was premiered in Berlin in 1922). Hába continued to write quarter-tone music (thoug ...
... had written his first major quarter-tone composition in Vienna in 1920, i.e. his String Quartet No.2 op.7 in which the quarter-tones are used as part of a highly chromatic lateromantic tonal pitch language (the piece was premiered in Berlin in 1922). Hába continued to write quarter-tone music (thoug ...
Chapter 1: Introduction to Pairwise Well-Formed Scales
... The notion of a pairwise well-formed scale is my own extension of the theory to a class that includes (but is not limited to) some scales that incorporate the step interval of the augmented second. These scales are prominent in world musics, and have been appropriated by composers of Western art mus ...
... The notion of a pairwise well-formed scale is my own extension of the theory to a class that includes (but is not limited to) some scales that incorporate the step interval of the augmented second. These scales are prominent in world musics, and have been appropriated by composers of Western art mus ...
Pythagoras and Music
... in contemporary times, by scientists working with quantum relationships. The theories set forth by Pythagoras are complex to those uninitiated into mathematical and musical analysis, but certain concepts are important to set forth here. Nicomachus of Gerasa, a theorist in the first or second century ...
... in contemporary times, by scientists working with quantum relationships. The theories set forth by Pythagoras are complex to those uninitiated into mathematical and musical analysis, but certain concepts are important to set forth here. Nicomachus of Gerasa, a theorist in the first or second century ...
DevStat8e_15_03
... By considering other values of 0 and the decision resulting from each one, the following general fact emerges: Every number inside the interval (15.7) specifies a value of 0 for which t of (15.8) leads to nonrejection of H0,whereas every number outside the interval (15.7) corresponds to a t for wh ...
... By considering other values of 0 and the decision resulting from each one, the following general fact emerges: Every number inside the interval (15.7) specifies a value of 0 for which t of (15.8) leads to nonrejection of H0,whereas every number outside the interval (15.7) corresponds to a t for wh ...
Pythagorean whole tone - Jacobs University Mathematics
... (Here a scale is a set of tones with mathematically defined frequency relationships. This is to distinguish from the various scale modes, defined by the particular order in which whole tones and semitones succeed each other.) A.Diederich – International University Bremen – USC – MMM – Spring 2005 ...
... (Here a scale is a set of tones with mathematically defined frequency relationships. This is to distinguish from the various scale modes, defined by the particular order in which whole tones and semitones succeed each other.) A.Diederich – International University Bremen – USC – MMM – Spring 2005 ...
Scales - SmartSite
... The sharps and flats of the key signatures occur, as you just learned, in a predetermined order. Each of the twelve key signatures serves for two keys: a major key and its relative minor. Thus C major and A minor (the relative minor of C major) share the same key signature, i.e., no sharps or flats. ...
... The sharps and flats of the key signatures occur, as you just learned, in a predetermined order. Each of the twelve key signatures serves for two keys: a major key and its relative minor. Thus C major and A minor (the relative minor of C major) share the same key signature, i.e., no sharps or flats. ...
Estimation of harpsichord inharmonicity
... In the Western musical tradition at least, this gives rise to the need for temperament, because it is not possible to accommodate all pure intervals within the small set of pitch classes available. The two most consonant intervals are the octave (frequency ratio 2:1) and perfect fifth (frequency rat ...
... In the Western musical tradition at least, this gives rise to the need for temperament, because it is not possible to accommodate all pure intervals within the small set of pitch classes available. The two most consonant intervals are the octave (frequency ratio 2:1) and perfect fifth (frequency rat ...
Musica Ficta and
... Several theorists added that these rules should be effected where possible without resort to musica ficta – if the interval or progression can be corrected within the system of musica recta, this should be done. (Whereas Jehan de Murs’ examples invariably sharpen a leading note in the upper part, us ...
... Several theorists added that these rules should be effected where possible without resort to musica ficta – if the interval or progression can be corrected within the system of musica recta, this should be done. (Whereas Jehan de Murs’ examples invariably sharpen a leading note in the upper part, us ...
Inner Octaves and Eastern Music
... material vibrating at a definite frequency inducing waves of acoustical energy in a medium of transmission. The essence of sound, as in all other natural phenomena, is vibration. A vibration is a continuously and smoothly alternating pulse of energy. Vibrations are scaled by their frequencies (rates ...
... material vibrating at a definite frequency inducing waves of acoustical energy in a medium of transmission. The essence of sound, as in all other natural phenomena, is vibration. A vibration is a continuously and smoothly alternating pulse of energy. Vibrations are scaled by their frequencies (rates ...
Notes - Stanford University
... same pitch at the same volume level, or that an opera singer sings the same note on an “oo” and an “ee” vowel. It is easy to hear the differences between these sounds, but how are they manifested in the physical sound waves? The answer lies in that, even though our brain perceives the sinusoidal wav ...
... same pitch at the same volume level, or that an opera singer sings the same note on an “oo” and an “ee” vowel. It is easy to hear the differences between these sounds, but how are they manifested in the physical sound waves? The answer lies in that, even though our brain perceives the sinusoidal wav ...
Hybrid Thinking in Meloharmony
... Renaissance (Pietro Aron is a prominent name in this context) the major third became a just 5/4 within meantone keyboard tunings, while the fifth was heavily tempered (down by 5.4 cents against just 3/2). The unequal size of the intervallic steps (from 76 to 117 cents, with the famous "wolf" between ...
... Renaissance (Pietro Aron is a prominent name in this context) the major third became a just 5/4 within meantone keyboard tunings, while the fifth was heavily tempered (down by 5.4 cents against just 3/2). The unequal size of the intervallic steps (from 76 to 117 cents, with the famous "wolf" between ...
The History of Tuning
... Euclid’s comments are from the Math Department: Pythagoras studied in Babylonia when it was part of Chaldean Empire and returned home to Samos, a Greek Island and started a school. This first school was not successful; it attracted only one student, which Pythagoras had to pay to attend. Next, Pytha ...
... Euclid’s comments are from the Math Department: Pythagoras studied in Babylonia when it was part of Chaldean Empire and returned home to Samos, a Greek Island and started a school. This first school was not successful; it attracted only one student, which Pythagoras had to pay to attend. Next, Pytha ...
EKU Music Theory Study Guide with PAGE NUMBERS
... Interval quality—describes the quality of the interval with the following descriptions: perfect (P), major (M), minor (m), diminished (o), and augmented (+). Interval size—describes the size of the interval which is measured by how far apart the notes are on the staff with the following numbers: uni ...
... Interval quality—describes the quality of the interval with the following descriptions: perfect (P), major (M), minor (m), diminished (o), and augmented (+). Interval size—describes the size of the interval which is measured by how far apart the notes are on the staff with the following numbers: uni ...
EKU Music Theory Study Guide - Music Theory And Composition
... Interval quality—describes the quality of the interval with the following descriptions: perfect (P), major (M), minor (m), diminished (o), and augmented (+). Interval size—describes the size of the interval which is measured by how far apart the notes are on the staff with the following numbers: uni ...
... Interval quality—describes the quality of the interval with the following descriptions: perfect (P), major (M), minor (m), diminished (o), and augmented (+). Interval size—describes the size of the interval which is measured by how far apart the notes are on the staff with the following numbers: uni ...
Music Theory And Composition - Eastern Kentucky University
... Interval quality—describes the quality of the interval with the following descriptions: perfect (P), major (M), minor (m), diminished (o), and augmented (+). Interval size—describes the size of the interval which is measured by how far apart the notes are on the staff with the following numbers: uni ...
... Interval quality—describes the quality of the interval with the following descriptions: perfect (P), major (M), minor (m), diminished (o), and augmented (+). Interval size—describes the size of the interval which is measured by how far apart the notes are on the staff with the following numbers: uni ...
Musical Mathematics: The Mathematics of Equal Temperament
... Many people seem to know that there is a connection between math and music, but few seem to be able to explain this connection in any detail. As a middle school math teacher, I have had students, parents and friends encourage me to use this connection in my teaching to make math more meaningful, but ...
... Many people seem to know that there is a connection between math and music, but few seem to be able to explain this connection in any detail. As a middle school math teacher, I have had students, parents and friends encourage me to use this connection in my teaching to make math more meaningful, but ...
Music Theory Review Guide - Guitar
... Interval quality—describes the quality of the interval with the following descriptions: perfect (P), major (M), minor (m), diminished (o), and augmented (+). Interval size—describes the size of the interval which is measured by how far apart the notes are on the staff with the following numbers: uni ...
... Interval quality—describes the quality of the interval with the following descriptions: perfect (P), major (M), minor (m), diminished (o), and augmented (+). Interval size—describes the size of the interval which is measured by how far apart the notes are on the staff with the following numbers: uni ...
Adding New Pitches
... partials of two tones, should also explain-again, in part-why equaltempered major thirds could sometimes seem dissonant, as well as various other traditionally consonant intervals that have been microtonally altered. An equal-tempered C-E could be perceived as dissonant because of the beats the E cr ...
... partials of two tones, should also explain-again, in part-why equaltempered major thirds could sometimes seem dissonant, as well as various other traditionally consonant intervals that have been microtonally altered. An equal-tempered C-E could be perceived as dissonant because of the beats the E cr ...
Tonal Function in Harmonic Scales
... Melodic intervals similar in size to the major third and the minor sixth approximate simple frequency ratios (4:5×2(n-1) and 5:2(n+2), respectively), but they are also only a semitone from the prototypical p4 and p5 ratios. For this reason, these intervals are inherently ambiguous because they can b ...
... Melodic intervals similar in size to the major third and the minor sixth approximate simple frequency ratios (4:5×2(n-1) and 5:2(n+2), respectively), but they are also only a semitone from the prototypical p4 and p5 ratios. For this reason, these intervals are inherently ambiguous because they can b ...
(pdf)
... are a half step apart. Definition 1.2. A diatonic scale is a seven-note musical scale with 5 whole steps and 2 half steps, where the half steps have the maximum separation usually 2 or 3 notes apart. In this paper we will simply consider the major scale, which is one of the diatonic scales. Definiti ...
... are a half step apart. Definition 1.2. A diatonic scale is a seven-note musical scale with 5 whole steps and 2 half steps, where the half steps have the maximum separation usually 2 or 3 notes apart. In this paper we will simply consider the major scale, which is one of the diatonic scales. Definiti ...
The Rosegarden Codicil: Rehearsing Music in Nineteen
... though there have been several subsequent studies (for example: Blackwood 1985 and 1991) which are more sophisticated mathematically and methodologically. The following quote from Mandelbaum’s introduction remains largely true forty-five years later: ‘In spite of the tremendous changes in musical ta ...
... though there have been several subsequent studies (for example: Blackwood 1985 and 1991) which are more sophisticated mathematically and methodologically. The following quote from Mandelbaum’s introduction remains largely true forty-five years later: ‘In spite of the tremendous changes in musical ta ...
Music Theory: Exploring The 432Hz Tuning Debate
... The music and audio industry currently uses the A = 440 Hz universal standard pitch tuning around the world. It wasn’t always this way. In fact, 440 Hz has been the standard for less than a hundred years, a drop in the bucket in terms of music history. The tuning of A = 432 Hz, also used throughout ...
... The music and audio industry currently uses the A = 440 Hz universal standard pitch tuning around the world. It wasn’t always this way. In fact, 440 Hz has been the standard for less than a hundred years, a drop in the bucket in terms of music history. The tuning of A = 432 Hz, also used throughout ...
Tuning Systems
... The rst thing musicians must do before they can play together is "tune". For musicians in the standard Western music1 tradition, this means agreeing on exactly what pitch2 (what frequency3 ) is an "A", what is a "B at" and so on. Other cultures not only have dierent note names and dierent scales ...
... The rst thing musicians must do before they can play together is "tune". For musicians in the standard Western music1 tradition, this means agreeing on exactly what pitch2 (what frequency3 ) is an "A", what is a "B at" and so on. Other cultures not only have dierent note names and dierent scales ...