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Topic: Mathematics of the Western music scale

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The Mathematics of Western music

Minor scale

Hexatonic

Meantone temperament

Equal temperament

Interval (music)

Regular temperament

Chord (music)

Major triad

Well temperament

Pentatonic

Physics of music

Pitch (music)

Music theory

World music

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	Mathematics of the Western music scale: Definition and Links by Encyclopedian.com
		When the original sound sources are periodic, as musical instruments approximately are, the note consists of several frequencies: A fundamental and overtones, that are integer multiples of the fundamental.
		To obtain a scale of 12 notes the major tone 9:8 is equated with the minor tone 10:9 and to two semitones 256:225.
		Any scale in which the ratio of any note to the tonic is an integer ratio is called a scale of just intonation.
	www.encyclopedian.com /th/The-Mathematics-of-Western-music.html (1455 words)

	Mathematics of musical scales - Wikipedia, the free encyclopedia
		The most important scale in the Western tradition is the diatonic scale, but the scales used and proposed in various historical eras and parts of the world have been many and varied.
		Below is a typical example of a 5-limit justly tuned scale, one of the scales Johannes Kepler presents in his Harmonice Mundi or Harmonics of the World of 1619, in connection with planetary motion.
		The same scale was given in transposed form by Alexander Malcolm in 1721 and theorist Jose Wuerschmidt in the last century and is used in an inverted form in the music of northern India.
	en.wikipedia.org /wiki/Mathematics_of_musical_scales (1167 words)

	Mathematics of the Western music scale
		A scale is in just intonation if the ratios between the frequencies for all degrees of the scale are either ratios of small integers, or obtained by a succession of such ratios.
		To calculate the frequency of a note in a scale given in terms of ratios, the frequency ratio is multiplied by the frequency we associate to the unison, which will often be the tonic frequency.
		The scale of the Western tradition is by its very nature neither one of just intonation nor one defined only in practice, but is a systematically tempered scale.
	www.mp3.fm /Mathematics_of_the_Western_music_scale.htm (949 words)

	Learn more about Chord (music) in the online encyclopedia. (Site not responding. Last check: 2007-10-13)
		The most commonly used chords in western music, triads are the basis of diatonic harmony, and are composed of three notes: a root note, a note which is an interval of a third above the root, and a note which is an interval of a fifth above the root.
		Using the same scale (and thus, implicitly, the key of C major) a chord may be constructed using the D as the root note.
		In a diatonic context, the third of the chord is the leading-tone of the scale, which has a strong tendency to pull towards the tonal center, or root note, of the key.
	www.onlineencyclopedia.org /c/ch/chord__music_.html (2645 words)

	Music (Site not responding. Last check: 2007-10-13)
		The musical tunes create the melodic pattern in the spirit of ascents and descents maintaining harmonic relations in the context of the spirit of alamkaras, murchanas, tanas and ten essentials which tend to render music lively and vibrating with the new spirit.
		Theories on music were accordingly the guiding principles on the pattern of the demonstration of music in different periods of our cultural history and had also functioned as a grammar of principles on the practical demonstration of music and these principles had obviously been followed by posterity.
		So music is not just a media for the joyful soothing of the tired nerves of the body or the peaceful satisfaction of the human senses; music is truly an aspect ofSadhana in which the Yoga system enables the human being to realise the truth in oneself, to realise the self in you.
	www.indiangyan.com /books/otherbooks/finearts/music.shtml (11107 words)

	Chord (music)
		In music and music theory, a chord (from the middle English cord, short for accord) is three or more different notes or pitches sounding simultaneously, or nearly simultaneously, over a period of time.
		Music is said to be chord-based when the melody is determined by the chords and not by melodic concerns such as modal frames.
		Chords are also distinguished and notated by the scale degree, pitch, or note of their root and bass, although there are many different conventions for indicating the quality of the chord, and the inversion of the chord (determined by which note of the chord serves as the bass note); see Inverted Triads below).
	www.mp3.fm /Chord_(music).htm (1189 words)

	Math and Music
		Mathematics is the study of mathematics using mathematics.
		Thus, there is a natural connection between mathematics and music: Both are experienced as pure objects of the brain, and both have meaning outside of the brain only by artificial connections.
		The second constant value in music is the 12th root of 2 (1.0594630943593...) which is the ratio of the frequencies between half tones.
	members.cox.net /mathmistakes/music.htm (1842 words)

	Music
		The scale was rediscovered in the late 15th century, and eventually became the basic scale used in western music.
		The scale used on piano and fretted instruments is actually an approximation to the exact ratios of the diatonic scale, because modern western music uses an equal temperament scale, allowing music to be transposed while slightly sacrificing the euphony of chords.
		For instance, since a major scale is defined to have exact ratios of frequencies 9:8, 5:4, 4:3, 3:2, 5:3, 2:1, etc., changing the tonic from a C to a D (using a C with a frequency of 264 Hertz, i.e.
	www.halexandria.org /dward113.htm (1843 words)

	Music - Crystalinks
		The history of music in relation to human beings predates the written word and is tied to the development and unique expression of various human cultures.
		The definition of music as sound with particular characteristics is taken as a given by psychoacoustics, and is a common one in musicology and performance.
		Music theory is the study of music, generally in a highly technical manner outside of other disciplines.
	www.crystalinks.com /music.html (1820 words)

	Music and the Western Mysteries
		The music was believed to be poetic and rhythmic and played in modes that had specific purposes in rituals or healing.
		MUSIC: This selection by Dufay entitled Nunca rosarium flores was written to commemorate the completion of the dome of the church of Santa Maria de Fiore.
		MUSIC: The musical selection is from Act Two of the opera where we hear the flute played by Tamino and his passage with Pamina through the trials of fire and water.
	sric-canada.org /Music.htm (6384 words)

	Anilyong Apóy / Hapò - Fatima Lasay (Site not responding. Last check: 2007-10-13)
		Another definition of algorithm in relation to the science of mathematics is that it is "a process, or set of rules, usually one expressed in algebraic notation, now used especially in computing, machine translation and linguistics." When applied to the study of mathematics, the term has appeared in print since roughly 1938.
		In "A Mountain Burial" it is the narrative structure in the music that is important, not the algorithm.
		The resulting music is whimsical - at least I imagine the earthworms, a few and short at first then rising in number and getting longer and eventually re-enter the cycle - the structure of the music, the sound of the music resembles the nature of the algorithm.
	www.upd.edu.ph /~cfa/digitalmedia/digiteer/ring/index.html (4070 words)

	Music
		While some might "note" that there are only 12 "notes" in the scale, if you don't have a root and octave, a start and an end, you have no means of calculating the gradations in between, so this 13th note as the octave is essential to computing the frequencies of the other notes.
		, referring to the eight whole tones of the complete musical scale, which in the key of C are C-D-E-F-G-A-B-C. In a scale, the dominant note is the 5th note of the major scale, which is also the 8th note of all 13 notes that comprise the octave.
		As an example, the climax of songs is often found at roughly the phi point (61.8%) of the song, as opposed to the middle or end of the song.
	goldennumber.net /music.htm (554 words)

	Chord (music) (Site not responding. Last check: 2007-10-13)
		In music and music theory a chord (from the middle English cord short accord) is three or more notes sounding simultaneously or near simultaneously over period of time.
		Broadly any combination of or more notes is a chord although the common practice period in western music and most popular some combinations were given more prominence than Thus in common usage a chord is only those groups of three which are tonal or have diatonic functionality.
		Using the same scale (and thus implicitly key of C major) a chord may constructed using the D as the root This would be D (root) F (third) (fifth).
	www.freeglossary.com /Chord_(music) (2060 words)

	Music and Mathematics
		In western music an octave is devided into 12 equal steps.
		Euler's music theory assigns to a frequency ratio p/q a number G(p,q)=G(p/q) called "gradus suavitatis" which could be translated as "degree of pleasure".
		The "antique doric tune" is obtained by taking from c-d and d-e the frequency relation 9/8 and from e-f the relation 128/117, (c-f has then a relation 4/3) then again 9/8 from f-g, g-a and a-h and again 128/117 from h to c.
	www.mathematik.com /Piano/index.html (228 words)

	ABC Mathematics - Words to think about numbers, space and algebra
		Morris Kline's Mathematics in Western Culture, 1953 (Pelican 1972) tells the history of mathematics and relates it to the general history of human thought and to philosophy, art, and science.
		House numbers are usually on an ordinal scale because 1 is usually at the beginning of the road and the higher the number the further down the road it is.
		Catchup mathematics is an opportunity for people who may have missed out on the chance to appreciate and enjoy mathematics, the first time around.
	www.mdx.ac.uk /www/study/glonum.htm (2494 words)

	Music + Mathematics
		There is much talk about the relationship between mathematics and music, which mostly consists of speculation by those on the outside, concerning some of the obvious things they have in common.
		The important thing to realize is that numbers and math are not cold and lifeless, and that music, which is a tangible incarnation of numbers, reflects in its beauty and emotion some of the beauty and emotion in the world of mathematics.
		Our Western musical scale paralleled the evolution of the keyboard, and finally reached its modern form at the time of J.S. Bach, who was one of its champions.
	www.woodpecker.com /writing/essays/math+music.html (3165 words)

	Mathematics (Rome Reborn: The Vatican Library & Renaissance Culture)
		The mathematics and astronomy of the Greeks had been known in medieval western Europe only through often imperfect translations, some of them made from Arabic intermediary texts rather than the Greek originals.
		Shown here is Proposition 13, with many scholia, concerned with the ratio to the diameters of the moon and sun of the line subtending the arc dividing the light and dark portions of the moon in a lunar eclipse.
		One of the most powerful creations of Greek science was the mathematical astronomy created by Hipparchus in the second century B.C. and given final form by Ptolemy in the second century A.D. Ptolemy's work was known in the Middle Ages through imperfect Latin versions.
	www.loc.gov /exhibits/vatican/math.html (2972 words)

	Music in Egypt - Seven Millennia of Performance
		The Arab musical tradition as it is known today developed between the AD 7th and 13th Centuries in the courts of Islam.
		An intriguing side-effect of improvisational music forms is the use of notes not actually present in the formal musical scale used by the artist.
		The music of the Middle East is now coming full-circle as the modern musicians assimilate elements from many of the musical forms that had their beginnings in Arabic music.
	www.touregypt.net /music.htm (2089 words)

	Mathematics of Music
		The musical scale used in western music originated with the ancient Greeks.
		Your challenge is to measure the frequencies of all the notes of a chromatic scale and then to determine a mathematical pattern.
		Determine the frequencies of the notes of a musical scale.
	www.narragansett.k12.ri.us /NHS/scienceweb/quansweb/sound3.htm (912 words)

	Jeff's Music Theory Page Psychoacoustics
		Still, we have marked in blue the ones that occur in the major scale, and in red the additional ones that occur in the minor scale.
		The generally accepted solution is to build the chromatic scale in intervals determined by iteratively multiplying the base frequency by a factor c, where c is the twelvth root of 2 (approximately 1.059463).
		The diatonic major scale is made up of the {1, 3, 5, 6, 8, 10, 12} notes of the chromatic scale, or equivalently, the intervals (2, 2, 1, 2, 2, 2, 1) of chromatic steps.
	home.austin.rr.com /jmjensen/musicTheory.html (2924 words)

	Awesome Library - Arts
		Describes early musical instruments, such as the Flute, Recorder, Whistle, Tabor Pipe, Lute, Viols and Violins, Guitar and Vihuela, Cittern and Bandora, Gittern and Citole, Bagpipe, Shawm and Curtal, Crumhorn and Rackett, Psaltery and Dulcimer, and Cornett.
		Provides an introduction to reading music, musical instruments, the sounds of different instruments, musical genres, This Day in Music and more.
		The music is slow to load, but provides recorded sound, not imitation.
	www.awesomelibrary.org /Classroom/Arts/Music/Music.html (865 words)

	SPAMS Email Archive - Lauren Williams - The Mathematics of 12-Tone Music (Site not responding. Last check: 2007-10-13)
		The notes of a 12-tone piece are based on a "tone row," (a particular ordering of the 12 pitches of the chromatic scale), as well as related tone rows that are the result of applying various operations to the original.
		Since a tone-row is an ordering of 12 elements, it can be thought of as an element of the symmetric group; furthermore, each operation on a tone row is the result of multiplying the tone row by a particular element of the symmetric group.
		We will examine the question of how to define equivalence classes of tone rows, with a cute graphic notation to represent them, and then use some group theory to calculate how rare symmetric tone rows are.
	www-math.mit.edu /spams/spring2003/email10.htm (304 words)

	Music, classical, ethnomusicology books from the University of Chicago Press
		… Rothstein is a wonderful guide to the architecture of musical space, its tensions and relations, its resonances and proportions.
		Gerber, Rebecca L.: Sacred Music from the Cathedral at Trent: Trent, Museo Provinciale d'Arte, Codex 1375 (olim 88)
		Charry, Eric: Mande Music: Traditional and Modern Music of the Maninka and Mandinka of Western Africa
	www.press.uchicago.edu /Subjects/virtual_music.html (1381 words)

	MP3.FM - MP3 Downloads, Song Lyrics, Guitar Tabs, Musical Instruments, FREE CDs, Ringtones, Videos
		Interactive music – Interval (music) – Interval class –
		List of major chord shapes for guitar – List of popular music terms – Legato
		Octave –; Octave illusion – Opera – Operetta – Oratorio – Orchestra – Orchestration – Organ – Organology – Ornament (music) – Ostinato – Overtone
	www.mp3.fm (139 words)

	Chocolate...iHygenic.com
		Chocolate also releases endorphins in the brain, which makes it nice regardless of taste (like a drug).
		For notation of inverted chord chord symbols see: figured bass.
		Seventh chords may be thought of as the next natural step in composing tertian chords is to add the note a third above the fifth of the chord, or the seventh of the chord.
	www.ihygenic.com /chocolate.htm (2362 words)

	Musical Info (Site not responding. Last check: 2007-10-13)
		The Acoustical Society of America have some interesting sounds (in.au format), including Shepard's Discrete Scale and Risset's Continuous Scale used to show Circularity in Pitch Judgment
		Two in a Bar Rule - UK Government plans to outlaw live music :-(
		circonium : music - the sound of berlin
	www.canopus22.demon.co.uk /links/musinfo.htm (205 words)

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