|
|
| Line 1: |
Line 1: |
| | | Greetings! I am Myrtle Shroyer. Years ago we moved to North Dakota. Doing ceramics is what her family members and her appreciate. Bookkeeping is what I do.<br><br>my website; [http://www.animecontent.com/user/D2456 over the counter std test] |
| [[Image:Spectrogram of violin.png|thumb|250px|A [[spectrogram]] of a violin waveform, with linear frequency on the vertical axis and time on the horizontal axis. The bright lines show how the spectral components change over time. The intensity coloring is logarithmic (black is −120 dBFS).]]
| |
| [[Music]] theorists sometimes use [[mathematics]] to understand music, and although music has no axiomatic foundation in modern mathematics, mathematics is "the basis of sound" and sound itself "in its musical aspects... exhibits a remarkable array of number properties", simply because nature itself "is amazingly mathematical".<ref>Reginald Smith Brindle, ''The New Music'', Oxford University Press, 1987, pp 42-3</ref> Though ancient Chinese, Egyptians and Mesopotamians are known to have studied the mathematical principles of sound,<ref>Reginald Smith Brindle, ''The New Music'', Oxford University Press, 1987, p 42</ref> the [[Pythagoreanism|Pythagoreans]] of ancient Greece are the first researchers known to have investigated the expression of [[musical scale]]s in terms of numerical [[ratio]]s,<ref>Plato, (Trans. Desmond Lee) ''The Republic'', Harmondsworth Penguin 1974, page 340, note.</ref> particularly the ratios of small integers. Their central doctrine was that "all nature consists of [[harmony]] arising out of numbers".<ref>Sir James Jeans, ''Science and Music'', Dover 1968, p. 154.</ref>
| |
| | |
| From the time of [[Plato]], harmony was considered a fundamental branch of [[physics]], now known as [[musical acoustics]]. Early [[Indian music scale|Indian]] and [[Chinese musicology|Chinese]] theorists show similar approaches: all sought to show that the mathematical laws of [[harmonic]]s and [[rhythm]]s were fundamental not only to our understanding of the world but to human well-being.<ref>Alain Danielou, ''Introduction to the Study of Musical Scales'', Mushiram Manoharlal 1999, Chapter 1 ''passim''.</ref> [[Confucius]], like Pythagoras, regarded the small numbers 1,2,3,4 as the source of all perfection.<ref>Sir James Jeans, ''Science and Music'', Dover 1968, p. 155.</ref>
| |
| | |
| To this day mathematics has more to do with acoustics than with composition, and the use of mathematics in composition is historically limited to the simplest operations of counting and measuring.{{Citation needed|date=May 2012}} The attempt to structure and communicate new ways of composing and hearing music has led to musical applications of [[set theory]], [[abstract algebra]] and [[number theory]]. Some composers have incorporated the [[golden ratio]] and [[Fibonacci numbers]] into their work.<ref>Reginald Smith Brindle, ''The New Music'', Oxford University Press, 1987, Chapter 6 ''passim''</ref><ref>{{cite web |title=Eric - Math and Music: Harmonious Connections |url=http://www.eric.ed.gov/ERICWebPortal/recordDetail?accno=ED388615}}</ref>
| |
| | |
| ==Time, rhythm and meter==
| |
| {{Main|Meter (music)}}
| |
| Without the boundaries of rhythmic structure – a fundamental equal and regular arrangement of [[Pulse (music)|pulse]] [[Repetition (music)|repetition]], [[Beat (music)|accent]], [[Phrase (music)|phrase]] and duration – music would be impossible.<ref>Arnold Whittall, in ''The Oxford Companion to Music'', OUP, 2002, Article: ''Rhythm''</ref> In Old English the word "rhyme", derived to "rhythm", became associated and confused with ''rim'' – "number"<ref>''Chambers' Twentieth Century Dictionary'', 1977, p. 1100</ref> – and modern musical use of terms like [[Meter (music)|meter]] and [[bar (music)|measure]] also reflects the historical importance of music, along with astronomy, in the development of counting, arithmetic and the exact measurement of time and [[Frequency|periodicity]] that is fundamental to physics.
| |
| | |
| ==Musical form==
| |
| {{Main|Musical form}}
| |
| Musical form is the plan by which a short piece of music is extended. The term "plan" is also used in architecture, to which musical form is often compared. Like the architect, the composer must take into account the function for which the work is intended and the means available, practicing economy and making use of repetition and order.<ref>Imogen Holst, ''The ABC of Music'', Oxford 1963, p.100</ref> The common types of form known as [[Binary form|binary]] and [[Ternary form|ternary]] ("twofold" and "threefold") once again demonstrate the importance of small integral values to the intelligibility and appeal of music.
| |
| | |
| The word "rhyme" was not derived from "rhythm"{{contradiction-inline |reason=Contradicts (referenced) rhyme/rhythm statement in previous section. |date=July 2013}} (see Oxford and Collins dictionaries) but from old English "rime". The spelling of "rime" was later affected by the spelling of "rhythm", although the two are totally different.
| |
| | |
| ==Frequency and harmony==
| |
| [[Image:Chladini.Diagrams.for.Quadratic.Plates.svg|thumb|220px|Chladni figures produced by sound vibrations in fine powder on a square plate. ([[Ernst Chladni]], ''Acoustics'', 1802)]]
| |
| A [[musical scale]] is a discrete set of [[pitch (music)|pitch]]es used in making or describing music. The most important scale in the Western tradition is the [[diatonic scale]] but many others have been used and proposed in various historical eras and parts of the world. Each pitch corresponds to a particular frequency, expressed in hertz (Hz), sometimes referred to as cycles per second (c.p.s.). A scale has an interval of repetition, normally the [[octave]]. The [[octave]] of any pitch refers to a frequency exactly twice that of the given pitch. Succeeding superoctaves are pitches found at frequencies four, eight, sixteen times, and so on, of the fundamental frequency. Pitches at frequencies of half, a quarter, an eighth and so on of the fundamental are called suboctaves. There is no case in musical harmony where, if a given pitch be considered accordant, that its octaves are considered otherwise. Therefore any note and its octaves will generally be found similarly named in musical systems (e.g. all will be called '''doh''' or '''A''' or '''Sa''', as the case may be). When expressed as a frequency bandwidth an octave '''A<sub>2</sub>–A<sub>3</sub>''' spans from 110 Hz to 220 Hz (span=110 Hz). The next octave will span from 220 Hz to 440 Hz (span=220 Hz). The third octave spans from 440 Hz to 880 Hz (span=440 Hz) and so on. Each successive octave spans twice the frequency range of the previous octave.
| |
| | |
| Because we are often interested in the relations or [[ratio]]s between the pitches (known as [[Interval (music)|intervals]]) rather than the precise pitches themselves in describing a scale, it is usual to refer to all the scale pitches in terms of their ratio from a particular pitch, which is given the value of one (often written '''1/1'''), generally a note which functions as the [[tonic (music)|tonic]] of the scale. For interval size comparison [[cent (music)|cent]]s are often used.
| |
| | |
| [[Image:4Octaves.and.Frequencies.svg|thumb|right|200px|The exponential nature of octaves when measured on a linear frequency scale.]]
| |
| [[Image:4Octaves.and.Frequencies.Ears.svg|right|thumb|200px|This diagrams presents octaves as they appear in the sense of musical intervals, equally spaced.]]
| |
| :{|class="wikitable"
| |
| !Common name
| |
| !Example<br>name {{right|Hz}}
| |
| !Multiple<br>of fundamental
| |
| !Ratio<br>within octave
| |
| !Cents<br>within octave
| |
| |-
| |
| |Fundamental
| |
| |A<sub>2</sub>, {{right|110}}
| |
| |{{center|1''x''}}
| |
| |1/1 = 1''x''
| |
| |{{right|0}}
| |
| |-
| |
| |rowspan=2 | Octave
| |
| |rowspan=2 | A<sub>3</sub> {{right|220}}
| |
| |rowspan=2 | {{center|2''x''}}
| |
| |2/1 = 2''x''
| |
| |{{right|1200}}
| |
| |-
| |
| |2/2 = 1''x''
| |
| |{{right|0}}
| |
| |-
| |
| |Perfect Fifth
| |
| |E<sub>4</sub> {{right|330}}
| |
| |{{center|3''x''}}
| |
| |3/2 = 1.5''x''
| |
| |{{right|702}}
| |
| |-
| |
| |rowspan=2 | Octave
| |
| |rowspan=2 | A<sub>4</sub> {{right|440}}
| |
| |rowspan=2 | {{center|4''x''}}
| |
| |4/2 = 2''x''
| |
| |{{right|1200}}
| |
| |-
| |
| |4/4 = 1''x''
| |
| |{{right|0}}
| |
| |-
| |
| |Major Third
| |
| |C{{music|#}}<sub>5</sub> {{right|550}}
| |
| |{{center|5''x''}}
| |
| |5/4 = 1.25''x''
| |
| |{{right|386}}
| |
| |-
| |
| |Perfect Fifth
| |
| |E<sub>5</sub> {{right|660}}
| |
| |{{center|6''x''}}
| |
| |6/4 = 1.5''x''
| |
| |{{right|702}}
| |
| |-
| |
| |[[Harmonic seventh]]
| |
| |G<sub>5</sub> {{right|770}}
| |
| |{{center|7''x''}}
| |
| |7/4 = 1.75''x''
| |
| |{{right|969}}
| |
| |-
| |
| |rowspan=2 | Octave
| |
| |rowspan=2 | A<sub>5</sub> {{right|880}}
| |
| |rowspan=2 | {{center|8''x''}}
| |
| |8/4 = 2''x''
| |
| |{{right|1200}}
| |
| |-
| |
| |8/8 = 1''x''
| |
| |{{right|0}}
| |
| |}
| |
| | |
| ==Tuning systems==
| |
| {{Main|Musical tuning|Musical temperament}}
| |
| [[5-limit tuning]], the most common form of [[just intonation]], is a system of tuning using tones that are [[regular number]] [[harmonic]]s of a single [[fundamental frequency]]. This was one of the scales [[Johannes Kepler]] presented in his [[Harmonices Mundi]] (1619) in connection with planetary motion. The same scale was given in transposed form by [[Alexander Malcolm]] in 1721 and by theorist [[Jose Wuerschmidt]] in the 20th century. A form of it is used in the music of northern India. American composer [[Terry Riley]] also made use of the inverted form of it in his "Harp of New Albion". Just intonation gives superior results when there is little or no [[chord progression]]: voices and other instruments gravitate to just intonation whenever possible. However, as it gives two different whole tone intervals (9:8 and 10:9) a keyboard instrument so tuned cannot change key.<ref>Jeremy Montagu, in ''The Oxford Companion to Music'', OUP 2002, Article: ''just intonation''.</ref> To calculate the frequency of a note in a scale given in terms of ratios, the frequency ratio is multiplied by the tonic frequency. For instance, with a tonic of [[A440 (pitch standard)|A4]] (A natural above middle C), the frequency is 440 [[hertz|Hz]], and a justly tuned fifth above it (E5) is simply 440×(3:2) = 660 Hz.
| |
| | |
| [[Image:HarmonicIdentities.Names.Frequencies.svg|thumb|right|200px|frame|The first 16 harmonics, their names and frequencies, showing the exponential nature of the octave and the simple fractional nature of non-octave harmonics.]]
| |
| [[Image:Normalized.HarmonicIdentities.Names.Frequencies.svg|frame|right|The first 16 harmonics, with frequencies and log frequencies.]]
| |
| {|class="wikitable"
| |
| ! Semitone || Ratio || [[Interval (music)|Interval]] || [[Harmonic series (music)|Natural]] || Half Step
| |
| |-
| |
| ! 0
| |
| | 1:1 || [[unison]] || 480 || 0
| |
| |-
| |
| ! 1
| |
| | 16:15 || minor [[semitone]] || 512 || 16:15
| |
| |-
| |
| ! 2
| |
| | 9:8 || [[major second]] || 540 || 135:128
| |
| |-
| |
| ! 3
| |
| | [[sesquiquintum|6:5]] || [[minor third]] || 576 || 16:15
| |
| |-
| |
| ! 4
| |
| | [[sesquiquartum|5:4]] || [[major third]] || 600 || 25:24
| |
| |-
| |
| ! 5
| |
| | [[sesquitertium|4:3]] || [[perfect fourth]] || 640 || 16:15
| |
| |-
| |
| ! 6
| |
| | 45:32 || diatonic [[tritone]] || 675 || 135:128
| |
| |-
| |
| ! 7
| |
| | [[sesquialterum|3:2]] || [[perfect fifth]] || 720 || 16:15
| |
| |-
| |
| ! 8
| |
| | 8:5 || [[minor sixth]] || 768 || 16:15
| |
| |-
| |
| ! 9
| |
| | 5:3 || [[major sixth]] || 800 || 25:24
| |
| |-
| |
| ! 10
| |
| | 9:5 || [[minor seventh]] || 864 || 27:25
| |
| |-
| |
| ! 11
| |
| | 15:8 || [[major seventh]] || 900 || 25:24
| |
| |-
| |
| ! 12
| |
| | 2:1 || [[octave]] || 960 || 16:15
| |
| |}
| |
| | |
| [[Pythagorean tuning]] is tuning based only on the perfect consonances, the (perfect) octave, perfect fifth, and perfect fourth. Thus the major third is considered not a third but a ditone, literally "two tones", and is (9:8)<sup>2</sup> = 81:64, rather than the independent and harmonic just 5:4 = 80:64 directly below. A whole tone is a secondary interval, being derived from two perfect fifths, (3:2)<sup>2</sup> = 9:8.
| |
| | |
| The just major third, 5:4 and minor third, 6:5, are a [[syntonic comma]], 81:80, apart from their Pythagorean equivalents 81:64 and 32:27 respectively. According to [[Carl Dahlhaus]] (1990, p. 187), "the dependent third conforms to the Pythagorean, the independent third to the harmonic tuning of intervals."
| |
| | |
| Western [[common practice period|common practice music]] usually cannot be played in just intonation but requires a systematically tempered scale. The tempering can involve either the irregularities of [[well temperament]] or be constructed as a [[regular temperament]], either some form of [[equal temperament]] or some other regular meantone, but in all cases will involve the fundamental features of [[meantone temperament]]. For example, the root of chord '''ii''', if tuned to a fifth above the dominant, would be a major whole tone (9:8) above the tonic. If tuned a just minor third (6:5) below a just subdominant degree of 4:3, however, the interval from the tonic would equal a minor whole tone (10:9). Meantone temperament reduces the difference between 9:8 and 10:9. Their ratio, (9:8)/(10:9) = 81:80, is treated as a unison. The interval 81:80, called the [[syntonic comma]] or comma of Didymus, is the key comma of meantone temperament.
| |
| | |
| In [[equal temperament]], the octave is divided into twelve equal parts, each semitone (half-step) is an interval of the [[twelfth root of two]] so that twelve of these equal half steps add up to exactly an octave. With fretted instruments it is very useful to use equal temperament so that the frets align evenly across the strings. In the European music tradition, equal temperament was used for lute and guitar music far earlier than for other instruments, such as [[musical keyboard]]s. Because of this historical force, twelve-tone equal temperament is now the dominant intonation system in the Western, and much of the non-Western, world.
| |
| | |
| Equally-tempered scales have been used and instruments built using various other numbers of equal intervals. The [[19 equal temperament]], first proposed and used by [[Guillaume Costeley]] in the 16th century, uses 19 equally spaced tones, offering better major thirds and far better minor thirds than normal 12-semitone equal temperament at the cost of a flatter fifth. The overall effect is one of greater consonance. [[24 equal temperament]], with 24 equally spaced tones, is widespread in the pedagogy and [[Arabic_maqam#Notation|notation]] of [[Arabic music]]. However, in theory and practice, the intonation of Arabic music conforms to [[Rational number|rational ratios]], as opposed to the [[Irrational number|irrational ratios]] of equally-tempered systems. While any analog to the equally-tempered [[quarter tone]] is entirely absent from Arabic intonation systems, analogs to a three-quarter tone, or [[neutral second]], frequently occur. These neutral seconds, however, vary slightly in their ratios dependent on [[Arabic maqam|maqam]], as well as geography. Indeed, Arabic music historian [[Habib Hassan Touma]] has written that "the breadth of deviation of this musical step is a crucial ingredient in the peculiar flavor of Arabian music. To temper the scale by dividing the octave into twenty-four quarter-tones of equal size would be to surrender one of the most characteristic elements of this musical culture."<ref>{{cite book|last=Touma|first=Habib Hassan|title=The Music of the Arabs|year=1996|publisher=Amadeus Press|location=Portland, OR|isbn=0-931340-88-8|pages=22–24}}</ref>
| |
| | |
| The following graph reveals how accurately various equal-tempered scales approximate three important harmonic identities: the major third (5th harmonic), the perfect fifth (3rd harmonic), and the "[[harmonic seventh]]" (7th harmonic). [Note: the numbers above the bars designate the equal-tempered scale (i.e., "12" designates the 12-tone equal-tempered scale, etc.)]
| |
| :{|class="wikitable"
| |
| !Note
| |
| !Frequency (Hz)
| |
| !Frequency<br>Distance from<br>previous note
| |
| !Log frequency<br>log<sub>2</sub> ''f''
| |
| !Log frequency<br>Distance from<br>previous note
| |
| |-
| |
| |A<sub>2</sub>
| |
| |110.00
| |
| |N/A
| |
| |6.781
| |
| |N/A
| |
| |-
| |
| |A{{music|#}}<sub>2</sub>
| |
| |116.54
| |
| |6.54
| |
| |6.864
| |
| |0.0833 (or 1/12)
| |
| |-
| |
| |B<sub>2</sub>
| |
| |123.47
| |
| |6.93
| |
| |6.948
| |
| |0.0833
| |
| |-
| |
| |C<sub>3</sub>
| |
| |130.81
| |
| |7.34
| |
| |7.031
| |
| |0.0833
| |
| |-
| |
| |C{{music|#}}<sub>3</sub>
| |
| |138.59
| |
| |7.78
| |
| |7.115
| |
| |0.0833
| |
| |-
| |
| |D<sub>3</sub>
| |
| |146.83
| |
| |8.24
| |
| |7.198
| |
| |0.0833
| |
| |-
| |
| |D{{music|#}}<sub>3</sub>
| |
| |155.56
| |
| |8.73
| |
| |7.281
| |
| |0.0833
| |
| |-
| |
| |E<sub>3</sub>
| |
| |164.81
| |
| |9.25
| |
| |7.365
| |
| |0.0833
| |
| |-
| |
| |F<sub>3</sub>
| |
| |174.61
| |
| |9.80
| |
| |7.448
| |
| |0.0833
| |
| |-
| |
| |F{{music|#}}<sub>3</sub>
| |
| |185.00
| |
| |10.39
| |
| |7.531
| |
| |0.0833
| |
| |-
| |
| |G<sub>3</sub>
| |
| |196.00
| |
| |11.00
| |
| |7.615
| |
| |0.0833
| |
| |-
| |
| |G{{music|#}}<sub>3</sub>
| |
| |207.65
| |
| |11.65
| |
| |7.698
| |
| |0.0833
| |
| |-
| |
| |A<sub>3</sub>
| |
| |220.00
| |
| |12.35
| |
| |7.781
| |
| |0.0833
| |
| |}
| |
| | |
| Below are [[Ogg Vorbis]] files demonstrating the difference between just intonation and equal temperament. You may need to play the samples several times before you can pick the difference.
| |
| *[[media:Comparison 550Hz - 554Hz.ogg|Two sine waves played consecutively]] – this sample has half-step at 550 Hz (C{{music|#}} in the just intonation scale), followed by a half-step at 554.37 Hz (C{{music|#}} in the equal temperament scale).
| |
| *[[media:Major third comparison.ogg|Same two notes, set against an A440 pedal]] – this sample consists of a "[[Dyad (music)|dyad]]". The lower note is a constant A (440 Hz in either scale), the upper note is a C{{music|#}} in the equal-tempered scale for the first 1", and a C{{music|#}} in the just intonation scale for the last 1". [[Beat (acoustics)|Phase]] differences make it easier to pick the transition than in the previous sample.
| |
| | |
| ==Connections to set theory==
| |
| {{Main|Set theory (music)}}
| |
| Musical set theory uses some of the concepts from mathematical [[set theory]] to organize musical objects and describe their relationships. To analyze the structure of a piece of (typically atonal) music using musical set theory, one usually starts with a set of tones, which could form motives or chords. By applying simple operations such as [[transposition (music)|transposition]] and [[inversion (music)|inversion]], one can discover deep structures in the music. Operations such as transposition and inversion are called [[isometries]] because they preserve the intervals between tones in a set.
| |
| | |
| ==Connections to abstract algebra==
| |
| {{Main|Abstract algebra}}
| |
| Expanding on the methods of musical set theory, some theorists have used abstract algebra to analyze music. For example, the notes in an equal temperament octave form an [[abelian group]] with 12 elements. It is possible to describe [[just intonation]] in terms of a [[free abelian group]].<ref>{{cite web |title=Algebra of Tonal Functions. |url=http://sonantometry.blogspot.com/2007_05_01_archive.html}}</ref>
| |
| | |
| [[Transformational theory]] is a branch of music theory developed by [[David Lewin]]. The theory allows for great generality because it emphasizes transformations between musical objects, rather than the musical objects themselves.
| |
| | |
| Theorists have also proposed musical applications of more sophisticated algebraic concepts. Mathematician [[Guerino Mazzola]] has applied [[topos theory]] to music,{{Citation needed|date=January 2010}} though the result has been controversial.{{Citation needed|date=January 2010}}
| |
| | |
| The chromatic scale has a free and transitive action of the [[cyclic group]] <math>\mathbb{Z}/12\mathbb{Z}</math>, with the action being defined via [[transposition (music)|transposition]] of notes. So the chromatic scale can be thought of as a [[torsor]] for the group <math>\mathbb{Z}/12\mathbb{Z}</math>.
| |
| | |
| ==The golden ratio and Fibonacci numbers==
| |
| {{unreferenced section|date=December 2011}}
| |
| [[James Tenney]] reconceived his piece "For Ann (Rising)", which consists of up to twelve computer-generated tones that [[glissando]] upwards (see [[Shepard tone]]), as having each tone start so each is the golden ratio (in between an [[equal-tempered]] [[minor sixth|minor]] and [[major sixth]]) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced.{{citation needed|date=August 2012}}
| |
| | |
| [[Ernő Lendvaï]] analyzes [[Béla Bartók]]'s works as being based on two opposing systems: those of the golden ratio and the [[acoustic scale]]. In Bartók's ''[[Music for Strings, Percussion, and Celesta]]'', the [[xylophone]] progression at the beginning of the 3rd movement occurs at the intervals 1:2:3:5:8:5:3:2:1. French composer [[Erik Satie]] used the golden ratio in several of his pieces, including ''Sonneries de la Rose Croix''.{{citation needed|date=August 2012}}
| |
| | |
| The golden ratio is also apparent in the organization of the sections in the music of [[Claude Debussy|Debussy]]'s ''Image, "Reflections in Water"'', in which the sequence of keys is marked out by the intervals 34, 21, 13, and 8 (a descending Fibonacci sequence), and the main climax sits at the [[golden ratio|φ]] position. "Prelude to the Afternoon of a Faun" also reaches a climax point, marked by the entrance of the antique cymbal, at the φ position.{{citation needed|date=August 2012}}
| |
| | |
| Many of the important musical events in [[Krzysztof Penderecki]]'s "Threnody for the Victims of Hiroshima" occur at φ positions.{{citation needed|date=August 2012}}
| |
| | |
| [[Boards of Canada]] have discussed using the golden ratio and Fibonacci numbers in their work.<ref>http://bocpages.org/wiki/Big_Country</ref> Song titles such as "Music is Math", and numerous songs featuring samples of people counting or discussing numbers, also illustrate the influence of mathematics on their music.{{citation needed|date=August 2012}}
| |
| | |
| Australian composer Scott Sanders, in his piece "Sweets from Dr Phil" uses [[modular arithmetic]] to limit the Fibonacci series to a set of digits modulo 2 to modulo 16, performed by 15 instances of the same timbre to reveal a pseudo-melody apparent in the upper notes of the combination of these instruments. A system of directly mapping digits to pitch classes and durations determines the notes each instrument plays.<ref>http://www.thehouseofandersen.com/music/sweets%20from%20dr%20phil.mp3</ref><ref>http://www.thehouseofandersen.com/scores/sweets%20from%20dr%20phil.pdf</ref><ref>http://www.thehouseofandersen.com</ref>
| |
| | |
| The song, "[[Lateralus (song)|Lateralus]]", by [[Tool (band)|Tool]] incorporates the [[Fibonacci number|Fibonacci sequence]].<ref>{{cite web| url=http://www.upvenue.com/music-news/blog-headline/1142/fibonacci-in-tool-s-lateralus.html |title=Fibonacci in Tool's Lateralus|publisher=UpVenue|accessdate=9 August 2011}}</ref> The theme of the song describes the desire of humans to explore and to expand for more knowledge and a deeper understanding of everything. The lyrics "spiral out", refers to this desire and also to the [[Fibonacci number|Fibonacci spiral]], which is formed by creating and arranging squares for each number in the sequence's 1,1,2,3,5,8,... pattern, and drawing a curve that connects to two corners of each square. This would, allowed to continue onwards, theoretically create a never-ending and infinitely-expanding spiral. Related to this, the song's main theme features successive time signatures 9/8, 8/8, and 7/8.<ref>{{cite web|url=http://www.guitaretab.com/t/tool/21818.html|title=Tool - Lateralus tab|publisher=GuitareTab!|accessdate=9 August 2011}}
| |
| </ref> The number 987 is the sixteenth integer of the Fibonacci sequence.<ref>{{cite web|url=http://indigo.ie/~peter/Fib1.htm|title=Fibonacci and extensions|publisher=indigo.ie|accessdate=9 August 2011}}</ref>
| |
| | |
| ==See also==
| |
| {{Portal|Music|Mathematics}}
| |
| *[[Equal temperament]]
| |
| *[[Interval (music)]]
| |
| *[[Musical tuning]]
| |
| *[[Piano key frequencies]]
| |
| *[[3rd bridge]] (harmonic resonance based on equal string divisions)
| |
| *''[[The Glass Bead Game]]''
| |
| *[[Non-Pythagorean scale]]
| |
| *[[Tonality Diamond]]
| |
| *[[Utonality and otonality]]
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| ==External links==
| |
| *[http://www.harmonics.com/scales Database of all the possible 2048 musical scales in 12 note equal temperament and other alternatives in meantone tunings]
| |
| *[http://www-personal.umd.umich.edu/~tmfiore/1/musictotal.pdf ''Music and Math'' by Thomas E. Fiore]
| |
| *[http://thinkzone.wlonk.com/Music/12Tone.htm Twelve-Tone Musical Scale.]
| |
| *[http://sonantometry.blogspot.com Sonantometry or music as math discipline.]
| |
| *[http://www.maths.abdn.ac.uk/~bensondj/html/music.pdf Music: A Mathematical Offering by Dave Benson].
| |
| *[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=1313&bodyId=1470 Nicolaus Mercator use of Ratio Theory in Music] at [http://mathdl.maa.org/convergence/1/ Convergence]
| |
| *[http://sites.google.com/site/abimepublications/home/maths-and-music ''The Glass Bead Game''] Hermann Hesse gave music and mathematics a crucial role in the development of his Glass Bead Game.
| |
| *[http://www.aboutscotland.com/harmony/prop.html Harmony and Proportion. Pythagoras, Music and Space].
| |
| | |
| {{Music theory}}
| |
| {{Mathematics-footer|state=collapsed}}
| |
| | |
| {{DEFAULTSORT:Music And Mathematics}}
| |
| [[Category:Mathematics and culture]]
| |
| [[Category:Mathematics of music| ]]
| |