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{ \override Score.TimeSignature#'stencil = ##f \relative c' { \clef treble \time 4/4 \key c \major <c c'>1 } }
A perfect octave between two Cs
Perfect octave
Inverseunison
Name
Other names-
AbbreviationP8
Size
Semitones12
Interval class0
Just interval2:1[1]
Cents
12-Tone equal temperament1200[1]
Just intonation1200[1]

In music, an octave (Latin: octavus: eighth) or perfect octave (sometimes called the diapason)[2] is an interval between 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".[3] The interval between the first and second harmonics of the harmonic series 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 is sometimes abbreviated 8a or 8va (Italian: all'ottava), 8va bassa (Italian: all'ottava bassa, sometimes also 8vb), or simply 8 for the octave in the direction indicated by placing this mark above or below the staff.

Explanation and definition

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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, 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 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 ) and 4 (or ) times the frequency, respectively.

The number of octaves between two frequencies is given by the formula:

  • Oscillogram of middle C (261.62 Hz) (scale: 1 square is equal to 1 millisecond)
    Oscillogram of middle C (261.62 Hz) (scale: 1 square is equal to 1 millisecond)
  • C5, an octave above middle C. The frequency is twice that of middle C (523.25 Hz).
    C5, an octave above middle C. The frequency is twice that of middle C (523.25 Hz).
  • C3, an octave below middle C. The frequency is half that of middle C (130.81 Hz).
    C3, an octave below middle C. The frequency is half that of middle C (130.81 Hz).

Music theory

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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.

{ \override Score.TimeSignature #'stencil = ##f \relative c' { \clef treble \key c \major \time 8/4 \once \override NoteHead.color = #red c4 d e f g a b \once \override NoteHead.color = #red c } }

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 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) 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.

Notation

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Octave of a pitch

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Octaves are identified with various naming systems. Among the most common are 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 C4, 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 C5.

Piano Keyboard
An 88-key piano, with the octaves numbered and Middle C (turquoise) and A440 (yellow) highlighted
{ \override Score.SpacingSpanner.strict-note-spacing = ##t \set Score.proportionalNotationDuration = #(ly:make-moment 1/32) \override Score.TimeSignature #'stencil = ##f \relative c,,,, { \clef bass \time 11/1 \key c \major c1 c' c' c' c' \clef treble c' c' c' c' c' c' } }
Scientific C−1 C0 C1 C2 C3 C4 C5 C6 C7 C8 C9
Helmholtz C,,, C,, C, C c c' c'' c''' c'''' c''''' c''''''
Organ 64 foot 32 foot 16 foot 8 foot 4 foot 2 foot 1 foot 3 line 4 line 5 line 6 line
Name Dbl contra Sub contra Contra Great Small 1 line 2 line 3 line 4 line 5 line 6 line
MIDI note 0 12 24 36 48 60 72 84 96 108 120

Ottava alta and bassa

[edit]
{ \relative c' { \clef treble \time 4/4 \key c \major c4 e g2 \ottava #-1 c,4 e g2 \ottava #-2 c,4 e g2 } }
Similar example with 8vb and 15mb
{ \relative c''' { \clef treble \time 4/4 \key c \major c4 e g2 \ottava #1 c,4 e g2 \ottava #2 c,4 e g2 } }
Example of the same three notes expressed in three ways: (1) regularly, (2) in an 8va bracket, and (3) in a 15ma bracket

The notation 8a or 8va is sometimes seen in sheet music, meaning "play this an octave higher than written" (all' ottava: "at the octave" or all' 8va). 8a or 8va stands for ottava, the Italian word for octave (or "eighth"); the octave above may be specified as ottava alta or ottava sopra). Sometimes 8va 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 8vb (ottava bassa or ottava sotto) is also used. Similarly, 15ma (quindicesima) means "play two octaves higher than written" and 15mb (quindicesima bassa) means "play two octaves lower than written."

The abbreviations col 8, coll' 8, and c. 8va stand for coll'ottava, meaning "with 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.[4]

Equivalence

[edit]

{ \relative c' { \clef treble \time 4/4 \key c \major <c c'>4^\markup { "(1) Parallel octaves (doubled)" } <c c'> <g' g'> <g g'> <a a'> <a a'> <g g'>2 <f f'>4 <f f'> <e e'> <e e'> <d d'> <d d'> <c c'>2 } }
{ \relative c' { \clef treble \time 4/4 \key c \major <c g'>4^\markup { "(2) Parallel fifths" } <c g'> <g' d'> <g d'> <a e'> <a e'> <g d'>2 <f c'>4 <f c'> <e b'> <e b'> <d a'> <d a'> <c g'>2 } }
{ \relative c' { \clef treble \time 4/4 \key c \major <c d>4^\markup { "(3) Parallel seconds" } <c d> <g' a> <g a> <a b> <a b> <g a>2 <f g>4 <f g> <e f> <e f> <d e> <d e> <c d>2 } }
Demonstration of octave equivalence. The melody to "Twinkle, Twinkle, Little Star" with parallel harmony. The melody is paralleled in three ways: (1) in octaves (consonant and equivalent); (2) in fifths (fairly consonant but not equivalent); and (3) in seconds (neither consonant nor equivalent).

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 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.[5]

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 the convention "that scales are uniquely defined by specifying the intervals within an octave".[6] The conceptualization of pitch as having two dimensions, pitch height (absolute frequency) and pitch class (relative position within the octave), inherently include octave circularity.[6] Thus all C♯s (or all 1s, if C = 0), any number of octaves apart, are part of the same pitch class.

Octave equivalence is a part of most musical cultures, but is far from universal in "primitive" and early music.[7] 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 the strings, with indications to tune the remaining two strings an octave from two of the seven tuned strings.[8] 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."[9]

Monkeys experience octave equivalence, and its biological basis apparently is an octave mapping of neurons in the auditory thalamus of the mammalian brain.[10] Studies have also shown the perception of octave equivalence in rats,[11] human infants,[12] and musicians[13] but not starlings,[14] 4–9-year-old children,[15] or non-musicians.[13][6][clarification needed]

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Octave

View on Grokipedia
from Grokipedia
In music, an octave is the interval between two pitches in which the higher pitch has a exactly double that of the lower pitch, creating a sensation of tonal similarity and consonance. This interval encompasses eight degrees of the —from a starting note to the note bearing the same name an eighth higher—and forms the foundational building block of musical scales and across many traditions. The octave's perceptual equivalence arises from the series, where the second of a coincides with the first of the octave above, producing a unified auditory impression. The recognition of the octave as a fundamental musical unit traces back to ancient civilizations, with systematic exploration attributed to the Greek philosopher around the 6th century BCE. identified the octave's 2:1 through experiments involving vibrating strings of equal tension but varying lengths, observing that a 1:2 length produced the octave interval. This discovery integrated music with , influencing subsequent theorists like and , who incorporated the octave into frameworks for tuning systems and cosmology, viewing it as a symbol of unity and perfection in the "music of the spheres." In Western music theory and practice, the octave is typically divided into 12 equal semitones using equal temperament, a tuning system that approximates the pure 2:1 ratio while enabling seamless key changes. Adopted widely since the 18th century for keyboard instruments like the piano, equal temperament divides the octave logarithmically, with each semitone corresponding to a frequency multiplication factor of 21/122^{1/12}, approximately 1.0595. This standardization facilitates composition and performance in all 12 major and minor keys without accumulating intonation errors, though it slightly compromises the purity of certain intervals like the fifth. Beyond Western contexts, the octave appears universally in musical cultures, often structuring scales and instruments, underscoring its perceptual primacy in human audition.

Fundamentals

Definition

An octave is the interval between two musical pitches in which the higher pitch has exactly double the of the lower pitch, corresponding to a ratio of 2:1. This interval serves as a foundational element in the organization of musical sound, establishing a sense of repetition and equivalence that underpins scales, harmonies, and melodic structures across musical traditions. The term "octave" derives from the Latin octavus, meaning "eighth," reflecting its position as the eighth note in a , where the final note returns to the starting pitch at double the . The term entered English by the mid-17th century, though the concept was previously referred to as the "eighth" from the mid-15th century, emphasizing the interval's role in completing the cycle of eight notes before the pattern repeats. A primary example of an octave is the span from one C to the next higher C on a piano keyboard, which encompasses 12 semitones and represents the largest interval in the diatonic scale before the sequence of pitches recurs. Unlike smaller intervals such as seconds or fifths, the octave provides a perceptual boundary that reinforces tonal identity and facilitates the hierarchical arrangement of musical elements.

Physical Properties

The octave is defined acoustically by a precise frequency ratio of 2:1, meaning that the higher note has exactly twice the of the lower note. If the lower note has a ff in hertz (Hz), the upper note has a of 2f2f Hz. This relationship holds regardless of the absolute frequencies involved, as long as the ratio is maintained. This doubling of results in an inverse relationship with , since the vv in air is constant and given by v=fλv = f \lambda, where λ\lambda is the . Consequently, doubling the halves the , from λ\lambda to λ/2\lambda / 2. This change influences perception, as shorter wavelengths interact differently with musical instruments' resonators and the , contributing to the distinct yet related tonal qualities of notes an octave apart. In the harmonic series produced by a vibrating source, such as a or air column, the octave corresponds to the first , which is the second . The ff is the first , followed by the second at 2f2f, representing the octave above the fundamental; subsequent harmonics are integer multiples, with the octave interval recurring at each doubling. This positioning in the harmonic series underscores the octave's foundational role in the content of musical tones. The octave is quantified in cents using a to measure intervals precisely, where one octave spans exactly cents in tuning. The formula for the interval in cents between two frequencies f1f_1 and f2f_2 (with f2>f1f_2 > f_1) is given by cents=1200×log⁡2(f2f1).\text{cents} = 1200 \times \log_2 \left( \frac{f_2}{f_1} \right). For an octave, where f2/f1=2f_2 / f_1 = 2, this yields 1200×log⁡2(2)=[1200](/page/1200)1200 \times \log_2(2) = [1200](/page/1200) cents, providing a standardized metric for acoustic analysis.

Music Theory

Interval Characteristics

In Western , the octave spans 12 semitones, with the perfect at 0 cents and the octave at 1200 cents. The octave exhibits the strongest consonance among musical intervals due to its simple frequency ratio of 2:1, which is perceived as more unified than the perfect fifth's ratio of 3:2. In interval inversion, the octave is the inverse of the perfect unison, and vice versa, such that transposing one note by an octave reverses their roles while preserving the perfect quality. Compound intervals exceed one octave; for instance, a compound octave, or double octave, encompasses 24 semitones across two octaves and functions equivalently to a simple octave when reduced by an octave. In , the octave adheres precisely to the 2:1 ratio, emphasizing pure relationships through rational proportions. In contrast, maintains the octave at exactly 2:1 while dividing it into 12 logarithmically equal semitones (each approximately 100 cents) to facilitate modulation across keys, though this introduces approximations in other just intervals for greater practical versatility.

Equivalence in Scales and Harmony

Octave equivalence forms a foundational principle in Western music theory, where pitches separated by one or more octaves are regarded as representatives of the same note class, such as C4 and C5 both denoting "C". This concept allows for the cyclical nature of scales, where ascending through a series of intervals returns to the starting pitch class after completing the octave, facilitating endless repetition without altering the perceived note identity. In the , the octave is divided into seven distinct steps—five whole tones and two semitones—before returning to the tonic, creating the structural basis for modes such as the scales. This division, exemplified by the scale (C-D-E-F-G-A-B-C), establishes tonal hierarchies and enables modal transpositions within the octave framework, where the pattern of intervals repeats invariantly across octaves due to equivalence. Harmonically, the octave plays a key role in chord construction and voicing; a root-position triad, the basic building block of , spans a from root to fifth, which is less than a full octave, ensuring compact sonorities that define chord quality without octave displacement. Octave doubling, wherein chord tones are replicated at the octave above or below, enhances the and fullness of the without altering its intervallic structure or perceived identity, a practice long established in Western composition to reinforce sonic presence. Various tuning systems incorporate the octave as a pure interval of 2:1 frequency ratio, while adjusting internal divisions to optimize consonance. In , stacking twelve perfect fifths (each 3:2) approximates the octave after adjustments, prioritizing fifth purity at the expense of thirds. Meantone temperaments, such as quarter-comma meantone, narrow the fifths slightly to achieve purer major thirds (), with the overall scale resolving to a just octave. Well-tempered systems, including those approaching , distribute inconsistencies across all keys while maintaining the pure octave as the bounding interval, enabling modulation without dissonant accumulations.

Notation

Pitch Designation

Scientific pitch notation is a widely used system for designating specific pitches by combining the note's letter name with a subscript octave number, where middle C is labeled C₄ and serves as the reference point for the fourth octave. Octaves are numbered sequentially starting from C₀ in the sub-contra range, ascending through higher registers such as the contra (C₁ to B₁), great (C₂ to B₂), and so on, with each octave encompassing the 12 semitones from C to B. This notation aligns with the international concert pitch standard, where A₄ is defined as exactly 440 Hz, providing a consistent frequency reference for tuning across instruments and ensembles. Helmholtz pitch notation, developed by the physicist in the 19th century, employs a system of letter cases, primes (apostrophes), and commas to indicate octave ranges relative to middle C, denoted as c′ (lowercase c with a single prime). Lowercase letters without modifiers represent the octave below middle C (e.g., c for the C below middle C), while uppercase letters without modifiers represent the great octave, two octaves below middle C (e.g., C for the C two octaves below middle C); additional apostrophes raise the pitch by further octaves (e.g., c″ for one octave above middle C, c′″ for two octaves above), and commas lower it (e.g., C,, for two octaves below the great octave). This relative system facilitates quick identification of pitch height in theoretical and analytical contexts, particularly in acoustics and . On a standard keyboard, which comprises spanning approximately seven octaves and a minor third, the range extends from A₀ (the lowest note) to C₈ (the highest), with octave numbering following such that the white keys from C to B fall within the same octave designation. For instance, the lowest C is C₁, middle C is C₄ (located near the center of the keyboard), and the highest C is C₈, allowing performers and composers to map pitches systematically across the instrument's full extent. In electronic music and digital interfaces, the Musical Instrument Digital Interface (MIDI) standard assigns numerical values to pitches from 0 to 127, with middle C fixed as note number 60 regardless of octave labeling conventions, and each successive octave increasing the note number by 12 to reflect the 12 semitones per octave in equal temperament. This numerical progression enables precise control and transposition in synthesizers, sequencers, and software, where, for example, the C one octave below middle C is 48 and one octave above is 72.

Markings in Scores

In musical notation, octave displacements are indicated primarily through specialized symbols known as ottava markings, which transpose passages by one or more octaves to reduce the need for numerous lines and improve readability. The ottava alta (8va), derived from the Italian term for "at the high octave," consists of the abbreviation "8va" placed above the staff, often accompanied by a curved bracket or dashed line extending over the affected notes or measures. This instructs the performer to play the indicated passage one octave higher than written, commonly applied to melodic lines in the treble register during the Classical and Romantic eras to simplify high-range notation. The counterpart, ottava bassa (8vb or 8ba), meaning "at the low octave," appears below the staff with a similar or line, directing the notes to be performed one octave lower than notated. This marking is frequently used in bass lines or left-hand parts to avoid excessive lines below the staff, particularly in orchestral scores or keyboard music where low registers predominate. For instance, in piano reductions of symphonic works, 8vb allows the or lines to be written in a more comfortable range without altering the visual layout. The extent of the marking is delineated by the bracket's endpoints, ensuring precise application to specific passages. For displacements beyond a single octave, double octave markings employ terms like quindicesima (15ma for two octaves higher or 15mb for lower), placed analogously above or below the staff with brackets. These are less common but appear in virtuosic passages, such as rapid scalar runs in or concertos, to denote extreme registers efficiently. Rarely, markings for three octaves, such as 22ma (ventiduesima), are used in contemporary or experimental scores for even wider transpositions, though they are exceptional due to the practical limits of most instruments' ranges. Historically, Baroque composers relied on notation in continuo parts, where numerical figures above the bass line implied harmonic intervals, including octave doublings realized by keyboardists or other instruments to reinforce the fundamental pitch. This system, prevalent in works by composers like J.S. Bach, allowed improvisational octave additions without explicit symbols, thickening the texture while maintaining harmonic flexibility. In modern practice, particularly for , markings—such as "Ped." followed by an asterisk for release—enable sustained octave effects by allowing low bass octaves to resonate beneath melodic lines, creating a fuller sonority without continuous manual holding of notes. These pedal indications, often detailed in Romantic-era scores like those of Chopin, enhance the illusion of orchestral depth in solo writing.

Perception and Applications

Psychoacoustic Effects

The human auditory system perceives octave intervals as highly consonant due to the harmonic alignment of their fundamental frequencies, where the higher tone's frequency is exactly double that of the lower, leading the brain to fuse the two into a single perceptual pitch class. This fusion arises from the shared harmonic series structure, minimizing sensory dissonance and promoting a unified tonal identity, as demonstrated in psychoacoustic studies on interval perception. Octaves distinguish between pitch height, which varies linearly with on a , and pitch chroma, the note identity that repeats every octave regardless of absolute height. This separation allows listeners to recognize the same across octaves while perceiving differences in or register; notably, the critical bandwidth of human hearing, the range within which tones interact strongly, approximates one octave at lower frequencies, facilitating this perceptual grouping. Neurologically, the organizes pitch representations logarithmically along the tonotopic axis, compressing frequency doublings (octaves) into equivalent perceptual units despite their linear physical separation, which underpins octave equivalence. This mapping reflects the cochlea's place-code mechanism, where neural responses to octave-related tones overlap significantly, enhancing chroma-based processing over height. A striking demonstration of these mechanisms is the octave illusion, exemplified by the , where overlapping sine waves separated by octaves create an ambiguous auditory continuum that perceptually ascends or descends indefinitely without resolution. This illusion exploits the brain's logarithmic pitch processing and chroma equivalence, as the fading in and out of octave components tricks the into perceiving continuous motion along the .

Historical and Cultural Uses

In Greek music theory, the tetrachord served as a foundational unit, spanning four notes over a perfect fourth (ratio 4:3), effectively half an octave, and combining two such tetrachords with an intervening whole tone to form the complete octave. During the medieval period, the octave evolved within European solmization practices, particularly through Guido d'Arezzo's innovations around the 11th century, which introduced the Guidonian hand as a mnemonic diagram mapping pitches across the hexachord system. The hexachord, comprising six notes with intervals of two whole tones, a semitone, and two more whole tones, allowed singers to navigate the octave by overlapping these units—starting on C (naturalis), F (mollis), or G (durum)—using syllables ut, re, mi, fa, sol, and la to facilitate sight-singing and modal transposition without fixed notation. This system persisted into the Renaissance, dividing the octave into manageable segments for polyphonic composition and vocal training in monastic and courtly settings. In non-Western traditions, the octave has been subdivided in diverse ways reflecting cultural tunings. , as formalized in Bharata Muni's (circa 200 BCE–200 CE), divides the octave (saptaka) into 22 shrutis, microtonal intervals that underpin the nuanced pitch inflections of ragas, allowing for expressive variations beyond the seven swaras. Chinese music historically employs pentatonic scales within the octave, with the five core tones—gong, shang, jue, zhi, and yu—derived from ancient pitch standards like the sanfen sunyi method, structuring melodies in instruments such as the and emphasizing cyclical harmony over . Among African traditions, the Shona mbira's tuning uses empirical adjustments of reed frequencies to produce intervals that approximate but vary from those of , as measured in cents, enabling idiomatic polyrhythms and overtones in Zimbabwean gourd-resonated performances. The 20th century saw expansions of the octave in microtonal and electronic contexts, challenging Western . Composer developed a 43-tone scale per octave based on an 11-limit diamond, using custom instruments like the Chromelodeon to realize compositions such as Delusion of the Fury, which explored corporeal and ritualistic timbres beyond 12-tone constraints. In electronic music, techniques from the 1990s onward incorporated synthesized octave anomalies—such as abrupt pitch doublings or halvings from digital buffer errors—as cultural artifacts, transforming technological malfunctions into aesthetic elements in works by artists like Yasunao Tone, who manipulated CD skips to evoke impermanence and noise in experimental .

References

  1. http://hyperphysics.phy-astr.gsu.edu/hbase/Music/mussca.html
  2. https://www.phys.uconn.edu/~gibson/Notes/Section3_2/Sec3_2.htm
  3. https://isle.hanover.edu/Ch13Music/Ch13OctaveToneSimilarity.html
  4. https://www.mtsu.edu/faculty/wroberts/teaching/scales.php
  5. http://hyperphysics.phy-astr.gsu.edu/hbase/Music/et.html
  6. https://openbooks.library.umass.edu/physicsofmusic/chapter/lab-activity-7/
  7. https://math.libretexts.org/Courses/College_of_the_Canyons/Math_100%3A_Liberal_Arts_Mathematics_(Saburo_Matsumoto)/07%3A_Mathematics_and_the_Arts/7.03%3A_Musical_Scales
  8. https://www.etymonline.com/word/octave
  9. https://cmtext.indiana.edu/acoustics/chapter1_pitch.php
  10. https://www.open.edu/openlearn/science-maths-technology/engineering-technology/sound-music-technology-an-introduction/content-section-8.1
  11. https://www.phys.uconn.edu/~gibson/Notes/Section4_2/Sec4_2.htm
  12. http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cents.html
  13. https://www.oberton.org/en/overtone-singing/harmonic-series/
  14. https://www.cs.cmu.edu/~music/cmp/archives/cmsip/readings/music-theory.htm
  15. https://musictheorymaterials.utk.edu/wp-content/uploads/2018/09/Intervals.pdf
  16. https://opencurriculum.org/5570/tuning-systems/
  17. https://pressbooks.nebraska.edu/openmusictheory/chapter/pitch-and-pitch-class/
  18. https://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Koss.pdf
  19. https://music.arts.uci.edu/abauer/4.1/readings/Laitz_Complete_Musician_Oxford_29_Symmetrical_Harmony.pdf
  20. https://www.math.drexel.edu/~dp399/textbooks/mathmusic/pythagorus.html
  21. https://crypto.stanford.edu/~blynn/sound/tuning.html
  22. https://milnepublishing.geneseo.edu/fundamentals-function-form/chapter/5-pitch/
  23. https://www.iso.org/standard/3601.html
  24. https://www.flutopedia.com/octave_notation.htm
  25. https://ultimatemusictheory.com/piano-key-numbers/
  26. https://midi.org/community/midi-specifications/midi-octave-and-note-numbering-standard
  27. https://newt.phys.unsw.edu.au/jw/notes.html
  28. https://dictionary.onmusic.org/terms/19-8vb
  29. https://dictionary.onmusic.org/terms/5-15ma
  30. https://musictheory.pugetsound.edu/mt21c/FiguredBassHistoricalContext.html
  31. https://academic.oup.com/cercor/article/23/11/2601/299137
  32. https://pubs.aip.org/asa/jasa/article-pdf/36/12/2346/12060753/2346_1_online.pdf
  33. http://www.tonalsoft.com/monzo/aristoxenus/tetrachord-tutorial.aspx
  34. http://assets.cambridge.org/97805218/84150/excerpt/9780521884150_excerpt.pdf
  35. http://www.medieval.org/emfaq/harmony/hex1.html
  36. https://www.ece.lsu.edu/kak/shruti.pdf
  37. https://digitalcommons.oberlin.edu/cgi/viewcontent.cgi?article=1029&context=ost
  38. https://pubs.aip.org/asa/jasa/article/123/2/1169/992979/Vibrational-frequencies-and-tuning-of-the-African
  39. http://www.tonalsoft.com/monzo/partch/scale/partch43-lattice.aspx
  40. https://ethw.org/w/images/d/d0/Davies.pdf
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