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In music, hemiola (also hemiolia) is the ratio 3:2. The equivalent Latin term is sesquialtera. In rhythm, hemiola refers to three beats of equal value in the time normally occupied by two beats. In pitch, hemiola refers to the interval of a perfect fifth.

Etymology

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The word hemiola comes from the Greek adjective ἡμιόλιος, hemiolios, meaning "containing one and a half," "half as much again," "in the ratio of one and a half to one (3:2), as in musical sounds."[1] The words "hemiola" and "sesquialtera" both signify the ratio 3:2, and in music were first used to describe relations of pitch. Dividing the string of a monochord in this ratio produces the interval of a perfect fifth. Beginning in the 15th century, both words were also used to describe rhythmic relationships, specifically the substitution (usually through the use of coloration—red notes in place of black ones, or black in place of "white", hollow noteheads) of three imperfect notes (divided into two parts) for two perfect ones (divided into three parts) in tempus perfectum or in prolatio maior.[2][3]

Rhythm

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In rhythm, hemiola refers to three beats of equal value in the time normally occupied by two beats.[4]

Vertical hemiola: sesquialtera

[edit]

The Oxford Dictionary of Music illustrates hemiola with a superimposition of three notes in the time of two and vice versa.[5]

\new Staff << \new voice \relative c' { \clef percussion \time 6/8 \set Score.tempoHideNote = ##t \tempo 4. = 80 \stemDown \repeat volta 2 { g4. g } } \new voice \relative c' { \stemUp \repeat volta 2 { f4 f f } } >>

One textbook states that, although the word "hemiola" is commonly used for both simultaneous and successive durational values, describing a simultaneous combination of three against two is less accurate than for successive values and the "preferred term for a vertical two against three … is sesquialtera."[6] The New Harvard Dictionary of Music states that in some contexts, a sesquialtera is equivalent to a hemiola.[7] Grove's Dictionary, on the other hand, has maintained from the first edition of 1880 down to the most recent edition of 2001 that the Greek and Latin terms are equivalent and interchangeable, both in the realms of pitch and rhythm,[8][3] although David Hiley, E. Thomas Stanford, and Paul R. Laird hold that, though similar in effect, hemiola properly applies to a momentary occurrence of three duple values in place of two triple ones, whereas sesquialtera represents a proportional metric change between successive sections.[9]

Sub-Saharan African music

[edit]

A repeating vertical hemiola is known as polyrhythm, or more specifically, cross-rhythm. The most basic rhythmic cell of sub-Saharan Africa is the 3:2 cross-rhythm. Novotney observes: "The 3:2 relationship (and [its] permutations) is the foundation of most typical polyrhythmic textures found in West African musics."[10] Agawu states: "[The] resultant [3:2] rhythm holds the key to understanding ... there is no independence here, because 2 and 3 belong to a single Gestalt."[11]

Ghanaian gyil

In the following example, a Ghanaian gyil plays a hemiola as the basis of an ostinato melody. The left hand (lower notes) sounds the two main beats, while the right hand (upper notes) sounds the three cross-beats.[12]

\new Staff << \new voice \relative c' { \clef treble \time 6/8 \set Score.tempoHideNote = ##t \tempo 4. = 80 \stemDown \repeat volta 2 { b4. d } } \new voice \relative c' { \stemUp \repeat volta 2 { fis8[ r fis] r[ a r] } } >>

European music

[edit]

In compound time (6
8 or 6
4), where a regular pattern of two beats to a measure is established at the start of a phrase, this changes to a pattern of three beats at the end of the phrase.

Archaic hemiola

The minuet from J. S. Bach's keyboard Partita No. 5 in G major articulates groups of 2 times 3 quavers that are really in 6
8 time, despite the 3
4 metre stated in the initial time-signature.[13] The latter time is restored only at the cadences (bars 4 and 11–12):

Bach: Minuet from Partita 5 in G bars 1–12
Bach: Minuet from Partita 5 in G bars 1–12

Later in the same piece, Bach creates a conflict between the two metres (6
8 against 3
4):

Bach Minuet from Partita 5 in G bars 37–52
Bach: Minuet from Partita 5 in G, bars 37–52

Hemiola is found in many Renaissance pieces in triple rhythm. One composer who exploited this characteristic was the 16th-century French composer Claude Le Jeune, a leading exponent of musique mesurée à l'antique. One of his best-known chansons is "Revoici venir du printemps", where the alternation of compound-duple and simple-triple metres with a common counting unit for the beat subdivisions can be clearly heard:

Claude LeJeune, Revoici venir du printemps
Claude LeJeune, "Revoici venir du printemps", bars 1–4 of the upper vocal line. Listen on YouTube

The hemiola was commonly used in baroque music, particularly in dances, such as the courante and minuet. Other composers who have used the device extensively include Corelli, Handel, Weber and Beethoven. A spectacular example from Beethoven comes in the scherzo from his String Quartet No. 6. As Philip Radcliffe puts it, "The constant cross-rhythms shifting between 3
4 and 6
8, more common at certain earlier and later periods, were far from usual in 1800, and here they are made to sound especially eccentric owing to frequent sforzandi on the last quaver of the bar... it looks ahead to later works and must have sounded very disconcerting to contemporary audiences."[14]

Beethoven Scherzo from Op 18 No 6
Beethoven Scherzo from Op. 18, No. 6, violin and cello only Listen on YouTube

Later in the nineteenth century, Tchaikovsky frequently used hemiolas in his waltzes, as did Richard Strauss in the waltzes from Der Rosenkavalier, and the third movement of Robert Schumann's Piano Concerto is noted for the ambiguity of its rhythm. John Daverio says that the movement's "fanciful hemiolas... serve to legitimize the dance-like material as a vehicle for symphonic elaboration."[15]

Schumann Piano Concerto Finale bars 120–127
Schumann Piano Concerto Finale bars 120–127

Johannes Brahms was particularly famous for exploiting the hemiola's potential for large-scale thematic development. Writing about the rhythm and meter of Brahms's Symphony No. 3, Frisch says "Perhaps in no other first movement by Brahms does the development of these elements play so critical a role. The first movement of the third is cast in 6
4 meter that is also open, through internal recasting as 3
2 (a so-called hemiola). Metrical ambiguity arises in the very first appearance of the motto [opening theme]."[16]

Brahms Symphony No. 3, opening bars
Brahms, Symphony No. 3, opening bars

At the beginning of the second movement, Assez vif – très rythmé, of his String Quartet (1903), Ravel "uses the pizzicato as a vehicle for rhythmic interplay between 6
8 and 3
4."[17]

Ravel Quartet, second movement
Second movement of Ravel Quartet

Horizontal hemiola

[edit]

Peter Manuel, in the context of an analysis of the flamenco soleá song form, refers to the following figure as a horizontal hemiola or "sesquialtera" (which mistranslates as: "six that alters"). It is "a cliché of various Spanish and Latin American musics ... well established in Spain since the sixteenth century", a twelve-beat scheme with internal accents, consisting of a 6
8 bar followed by one in 3
4, for a 3 + 3 + 2 + 2 + 2 pattern.[18]

Horizontal hemiola

This figure is a common African bell pattern, used by the Hausa people of Nigeria, in Haitian Vodou drumming, Cuban palo, and many other drumming systems. The horizontal hemiola suggests metric modulation (6
8 changing to 3
4). This interpretational switch has been exploited, for example, by Leonard Bernstein, in the song "America" from West Side Story, as can be heard in the prominent motif (suggesting a duple beat scheme, followed by a triple beat scheme):

Horizontal hemiola in Bernstein's, "America" from West Side Story

Pitch

[edit]

The perfect fifth

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Hemiola can be used to describe the ratio of the lengths of two strings as three-to-two (3:2), that together sound a perfect fifth.[2] The early Pythagoreans, such as Hippasus and Philolaus, used this term in a music-theoretic context to mean a perfect fifth.[19]

Just perfect fifth on C

The justly tuned pitch ratio of a perfect fifth means that the upper note makes three vibrations in the same amount of time that the lower note makes two. In the cent system of pitch measurement, the 3:2 ratio corresponds to approximately 702 cents, or 2% of a semitone wider than seven semitones. The just perfect fifth can be heard when a violin is tuned: if adjacent strings are adjusted to the exact ratio of 3:2, the result is a smooth and consonant sound, and the violin sounds in tune. Just perfect fifths are the basis of Pythagorean tuning, and are employed together with other just intervals in just intonation. The 3:2 just perfect fifth arises in the justly tuned C major scale between C and G.[20]

Other intervals

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Later Greek authors such as Aristoxenus and Ptolemy use the word to describe smaller intervals as well, such as the hemiolic chromatic pyknon, which is one-and-a-half times the size of the semitone comprising the enharmonic pyknon.[21]

See also

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References

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Hemiola

View on Grokipedia
from Grokipedia
Hemiola is a rhythmic device in music theory characterized by a 3:2 , where three evenly spaced pulses are superimposed over the time typically occupied by two, creating a temporary shift in perceived meter without altering the notated . This technique, derived from the Greek term hemiolia meaning "one and a half," originally referred to pitch intervals like the but evolved primarily into a rhythmic concept involving cross-rhythms or . There are two primary types of hemiola: horizontal, which occurs within a single melodic line by grouping notes to emphasize a duple feel against a triple meter (or vice versa), and vertical, which involves simultaneous contrasting rhythms between multiple parts, such as one line in groups of three against another in groups of two. Horizontal hemiolas often use ties or accents to alter perception, while vertical ones produce polyrhythmic textures that add tension and complexity to harmonic progressions. These ratios can extend to broader forms like or 12:8, enhancing and driving musical phrases toward resolution. Originating in 15th-century European polyphony, as seen in works by composers like Guillaume Dufay, hemiola became a staple in dances such as the and by composers including Handel, and later in Classical and Romantic repertoire by figures including , Beethoven, and Brahms. It also appears prominently in non-Western traditions, including African, Cuban, and , and in modern compositions like Leonard Bernstein's "America" from , where a 3:2 pattern evokes Latin rhythms. In contemporary genres such as and , hemiola interacts with to create dynamic call-and-response patterns and intricate textures.

Definition and Etymology

Core Concept

Hemiola refers to a fundamental musical ratio of 3:2, in which three equal units are perceived or structured within the temporal or spatial framework typically occupied by two equal units. This principle creates a perceptual overlap or superposition, allowing for rhythmic ambiguity or harmonic consonance depending on its application. The term originates from hēmiólios, meaning "one and a half," reflecting its proportional essence. In rhythmic hemiola, the 3:2 ratio manifests as a temporal grouping where three notes or pulses of equal duration are superimposed over the space of two beats, often producing a sense of metric shift or accentuation. This can be mathematically represented as a 32\frac{3}{2} superposition in time, where the durations align periodically every six units (the lowest common multiple of 2 and 3). In contrast, pitch hemiola applies the same ratio to frequency relationships, defining intervals such as the perfect fifth, where the higher frequency f2f_2 relates to the lower f1f_1 by f2/f1=32f_2 / f_1 = \frac{3}{2}. For string instruments, this corresponds inversely to length division, with the shorter string vibrating at 1.5 times the frequency of the longer one. Rhythmic hemiola appears in notation through proportional substitutions, such as aligning a measure in 34\frac{3}{4} with one in 68\frac{6}{8}, where in the former fit the duple grouping of the latter. In historical , it was indicated by techniques like coloration, altering note values to achieve the 3:2 proportion without changing the overall mensuration. These representations highlight hemiola's role in enhancing structural tension or resolution across musical dimensions.

Linguistic Origins

The term "hemiola" originates from the adjective hēmiólion (ἡμιόλιον), literally meaning "half as much again" or "one and a half times," denoting the mathematical proportion of 3:2. This concept entered Western intellectual tradition through the writings of the Roman philosopher and mathematician Anicius Manlius Severinus (c. 480–524 CE), who in his De institutione arithmetica libri duo (c. 500–510 CE) discussed superparticular ratios, including the hemiolios as the simplest such proportion where the larger number exceeds the smaller by half of it (3 = 2 + 1). Boethius extended this arithmetic application to musical theory in De institutione musica libri quinque (c. 500–510 CE), using sesquialter (the Latin equivalent of the Greek hemiolios) to describe harmonic intervals derived from dividing the monochord string in the 3:2 ratio, yielding the (diapente), thus linking numerical proportion to auditory consonance in Pythagorean tradition. During the Renaissance, the term reemerged in Western music theory, adapted into Latin as sesquialtera (meaning "one-and-a-half times"), a direct equivalent to the Greek hemiolios. Johannes Tinctoris (c. 1435–1511), a prominent theorist at the Aragonese court in Naples, systematically incorporated sesquialtera into his Proportionale musices (c. 1474), a treatise dedicated to musical proportions in mensural notation and polyphony. There, Tinctoris applied it primarily to pitch ratios for monochord divisions and interval construction, while also addressing its role in rhythmic proportions within contrapuntal compositions, emphasizing audible consonance over speculative arithmetic. This marked a shift from Boethius's philosophical framework to practical applications in contemporary polyphonic music, where sesquialtera signified a proportional relationship ensuring coherent voice leading. The earliest documented uses of these terms in musical treatises appear in mid-15th-century works on proportions in polyphony, predating Tinctoris slightly with references in Prosdocimo de' Beldomandi's Brevis summula proportionum quantum ad musicam pertinet (1409) and Ugolino of Orvieto’s Declaratio musice discipline (c. 1430), where sesquialtera denotes the 3:2 ratio for both intervallic tuning and mensural adjustments in sacred and secular compositions. Over subsequent centuries, the terminology evolved: sesquialtera dominated 15th- and 16th-century mensural notation for rhythmic subdivisions (often notated with a ♩3 or similar sign), while the Greek-derived "hemiola" (or Italian emiolia) gradually resurfaced in theoretical writings by the late 16th century, as seen in Gioseffo Zarlino's Le istitutioni harmoniche (1558), bridging pitch and rhythm. By the 19th century, "hemiola" became the standard English and modern term, encompassing both applications without distinction, reflecting a unified conceptual framework in post-Renaissance theory.

Rhythmic Hemiola

Vertical Form

Vertical hemiola, also known as sesquialtera, involves the simultaneous layering of three notes or beats against two of equal duration, forming a 3:2 polyrhythmic that introduces rhythmic tension through conflicting streams. This vertical alignment contrasts with linear rhythmic patterns, emphasizing polyrhythmic interplay across voices or instruments rather than sequential shifts. In from the period, vertical hemiola was realized through sesquialtera proportion, where three colored notes of the same value (such as semibreves or minims) were performed in the temporal space of two uncolored notes of that value, often indicated by coloration or a "3/2" sign to denote the temporary metric adjustment. Modern equivalents typically superimpose triplet patterns in 3/2 meter over duple divisions, such as three quarter notes against two dotted quarter notes, allowing composers to evoke similar polyrhythmic effects without archaic notation. The acoustic result of this 3:2 layering produces as accents from one stream offset those of the other, while perceptually it fosters metric ambiguity, where listeners may momentarily align with either the triple or duple interpretation, heightening dramatic contrast until resolution. In practice, sesquialtera served to temporarily alter duple mensurations to triple for expressive variety, applying the proportion selectively to for polyrhythmic depth without disrupting the overall structure.

Horizontal Form

Horizontal hemiola refers to a rhythmic device in which a sequence of notes is reinterpreted through successive changes in metric grouping, typically alternating between duple and triple divisions of the same temporal span. This creates a linear progression that shifts the perceived meter without altering the underlying or , often manifesting as a 3:2 over time. For instance, a span of six eighth notes can first be grouped as two sets of three (evoking a 6/8 feel) and then rearticulated as three sets of two (suggesting a 3/4 feel), emphasizing different beats to drive the reinterpretation. In notation, horizontal hemiola is commonly illustrated by alternating between time signatures such as 6/8 and 3/4, where both encompass the same six eighth notes per measure but differ in accentuation and beaming. In 6/8, the beats are grouped as two dotted quarter notes (1-2-3, 4-5-6), projecting a duple subdivision, while a switch to 3/4 rephrases them as three quarter notes (1-2, 3-4, 5-6), imposing a triple feel through accents on the new downbeats. This technique can also appear within a single using ties or phrasing slurs to override the written meter, such as tying two quarter notes in 3/4 to simulate a duple . The perceptual effect of horizontal hemiola is a of forward and metric ambiguity, where the listener experiences a temporary or intensification as the grouping shifts, often heightening tension toward a . This reinterpretation engages the ear in resolving the conflicting pulses, fostering rhythmic vitality in monophonic or homorhythmic textures. Unlike s, which involve simultaneous layering of contrasting rhythms across multiple voices, horizontal hemiola operates sequentially within a unified rhythmic stream, avoiding vertical superposition and focusing instead on temporal regrouping.

Cultural and Historical Applications

African Traditions

In Sub-Saharan African music, particularly in West African traditions, vertical hemiola manifests as a foundational , most commonly in the 3:2 ratio where three pulses overlay two, creating patterns that drive communal performances. This rhythmic device is integral to genres such as and jùjú, where drumming ensembles layer three-against-two patterns to produce a syncopated, propulsive groove that emphasizes collective interplay over individual lines. For instance, music often features initial syncopated rhythms like the 3:3:2 (tresillo) pattern, which derives from traditional West African polyrhythms and contributes to the genre's danceable energy. Specific instruments exemplify this integration, such as the gyil, a used in Ghanaian music among the Dagara and Lobi peoples, where performers execute 3:2 ostinatos by alternating two main beats in the left hand with three cross-beats in the right, forming the basis of melodic and ic structures. In Ewe drumming ensembles from southeastern , hemiola appears in layered patterns during call-and-response sequences, as seen in the style; the support drum () often plays a C3/2 cycle—dividing two bell (gankogui) pulses into three—against the steady 4-pulse meter of the ensemble, fostering rhythmic density and improvisation. These examples highlight vertical hemiola's role in creating polyrhythmic textures without reliance on Western notation, as the music relies on oral transmission and embodied learning predating colonial influences. Culturally, hemiola in these traditions serves as a for social harmony, symbolizing the interdependence of community members navigating life's complexities together, much like musicians maintain purpose amid cross-beats to achieve unity. Ewe master drummer C.K. Ladzekpo describes the 3:2 relationship as the "foundation of our ," underscoring its embodiment of relational homogeneity in Niger-Congo musical practices. This rhythmic philosophy extends to communal performances, where interlocking patterns dissolve individual egos into a collective "sound of togetherness," reflecting broader social values of cooperation and shared narrative. In the , hemiola influenced modern genres like , pioneered by , who blended traditional 3:2 polyrhythms with elements to create extended grooves emphasizing layered percussion and call-and-response. Tracks such as "Water No Get Enemy" showcase polyrhythmic complexity through interlocking drum patterns and guitar lines that retain the 3:2 timeline from roots, adapting indigenous rhythms for political expression and urban dance contexts. Kuti's approach, informed by Yoruba traditions and his 1969 U.S. exposures, positioned hemiola as a bridge between ancestral practices and global fusion, amplifying Afrobeat's communal and resistive ethos.

European and Western Traditions

In the period, hemiola emerged as a key rhythmic device in European polyphony, particularly through the use of sesquialtera, a 3:2 proportional mensuration that created textural contrast in motets. Composers like employed sesquialtera innovatively to shift rhythmic layers, appearing in approximately 17% of his works, often to heighten expressive depth and differentiate sections within sacred vocal pieces. For instance, in the motet Tu pauperum refugium, attributed to Josquin, proportio sesquialtera is indicated by a change in mensuration sign, accelerating the tempus while maintaining the tactus, thereby producing a layered rhythmic interplay that underscores the text's emotional weight. Theoretical foundations for such rhythmic proportions were articulated by Gioseffo Zarlino in his 1558 treatise Le Istitutioni harmoniche, where he explored mensural notations derived from simple ratios like 3:2, influencing the systematic notation of hemiola in polyphonic music. Zarlino distinguished between perfect (triple) and imperfect (duple) tempus, advocating for proportional changes to enhance musical flow without disrupting the underlying tactus, a concept that shaped compositional practices across Europe. These ideas provided a framework for later developments, emphasizing hemiola's role in balancing rhythmic complexity with structural clarity. During the Baroque era, vertical hemiola—where simultaneous voices emphasize conflicting duple and triple subdivisions—gained prominence in contrapuntal works, as seen in Johann Sebastian Bach's fugues from . In the B-minor fugue (BWV 869), Bach employs hemiola through beaming patterns that suggest rhythmic ambivalence, creating tension in cadential passages by overlaying 3:2 patterns across voices. This technique drove contrapuntal momentum, a practice echoed in the Classical period by , who used vertical hemiola in symphonies like the Eroica (Symphony No. 3) to propel rhythmic energy, particularly in the first movement's development section where syncopated hemiolas disrupt the metric flow for dramatic intensity. In the 19th-century Romantic era, horizontal hemiola—grouping notes across bar lines to imply a metric shift—became integral to character pieces evoking , notably in Frédéric Chopin's mazurkas. These works feature horizontal hemiola to introduce that mimics the lilting asymmetry of Polish mazur, as in the in , Op. 17 No. 3, where measures 14–16 overlay triple meter with duple phrasing, enhancing the dance-like propulsion and emotional nuance. Such applications extended hemiola's Western legacy, transforming it from a polyphonic tool into a vehicle for personal expression and nationalistic rhythm.

Modern and Global Usage

In , horizontal hemiola was employed to create metric ambiguity and rhythmic tension. Leonard Bernstein's "America" from (1957) exemplifies this through its alternation between 6/8 and 3/4 groupings, where phrases in 3/4 overlay the underlying 6/8 pulse, producing a playful yet disorienting shift that enhances the song's energetic dance sequence. Similarly, Maurice Ravel's (1928) incorporates subtle hemiola at phrase endings, where chord changes and rhythmic accents briefly disrupt the steady 3/4 , releasing built-up tension and contributing to the work's hypnotic escalation. In popular and global fusions, vertical hemiola has become prominent in genres, layering 3:2 ratios to generate syncopated grooves. In , particularly the rhythm, vertical hemiola manifests as a "3 against 2" in taconeo (footwork) solos and compás sequences, where the dancer's triple subdivisions clash against the duple pulse of palmas (handclaps) and guitar, fostering an improvisational intensity central to the form. similarly utilizes vertical hemiola through the tresillo pattern—a 3:2 rhythmic motif—evident in works by artists like , where it underpins clave rhythms to blend Afro-Cuban elements with swing, creating a propulsive yet off-kilter feel. In , the 3:2 hemiola appears within tala cycles such as (a 16-beat structure), often in Carnatic compositions where korvais (rhythmic cadences) employ two-measure hemiolas to resolve phrases against the cyclic pulse, heightening dramatic closure in percussion solos. Contemporary digital applications have integrated hemiola into electronic music production and multimedia scoring for enhanced textural complexity. Software like facilitates vertical hemiola via plugins and session automation, allowing producers to layer 3:2 patterns across drum racks and melodic tracks, as seen in genres like IDM and where such devices disrupt steady grids to evoke unease or euphoria. In film scores, composers such as employ hemiola to build , superimposing triple rhythms over duple meters in cues for action sequences, a technique that amplifies perceptual tension by challenging listeners' metric expectations. Recent 21st-century scholarship has linked hemiola to cognitive music , exploring its neural impacts through empirical studies. Research using EEG has shown that intentional switches from ternary to binary rhythms in hemiola passages elicit enhanced positive peak responses in the compared to simple metrical patterns, suggesting involvement of motor and auditory areas in processing these ambiguities. Post-2000 analyses further indicate that hemiola activates and pre-supplementary motor areas during and , reducing activation relative to isochronous rhythms and highlighting its role in entrainment and temporal prediction in listeners.

Pitch Hemiola

Perfect Fifth

In pitch hemiola, the arises from the 3:2 frequency ratio, where the higher note's frequency is three-halves that of the lower note. Acoustically, this can be demonstrated using a monochord, a single-string instrument where the determines pitch; dividing the into segments with a 3:2 ratio produces frequencies in the inverse 2:3 ratio, resulting in the sound of a , which spans approximately 702 cents. The size of this interval is mathematically derived using the formula for cents in the equal-tempered scale: interval size (cents)=1200×log⁡2(32)≈701.96\text{interval size (cents)} = 1200 \times \log_2\left(\frac{3}{2}\right) \approx 701.96 This perfect fifth slightly exceeds the 700 cents of the fifth, creating a subtly wider and purer sound in contexts emphasizing natural harmonics. Historically, the 3:2 formed the foundation of , a system attributed to the philosopher around the 6th century BCE, which generated scales through successive stacking of these intervals. This tuning influenced ancient Greek musical scales, such as the diatonic genus, and persisted into the , where it underpinned parallel in early polyphonic music, prioritizing fifth-based consonances over thirds. Perceptually, the perfect fifth's consonance stems from its simple 3:2 ratio, which minimizes beating and produces a stable, harmonious blend due to low dissonance in their partials. It also aligns closely with the harmonic series, appearing as the ratio between the fundamental and the second (third harmonic), reinforcing its role as a primary building block of tonal harmony.

Other Intervals

In , stacking multiple perfect fifths based on the hemiola of 3:2 generates secondary intervals such as the whole tone with a of 9:89:8 (approximately 204 cents), obtained by two fifths minus one , and the ditone with a of 81:6481:64 (approximately 408 cents), derived from four fifths minus two s. These intervals form the building blocks of the Pythagorean scale, where successive applications of the 3:2 approximate diatonic steps without . Ancient Greek music theory recognized the hemiolic interval of 3:2 as the , a foundational interval used in constructing scales from in various , including the enharmonic genus with its pyknon of two small intervals below a larger tone, the overall tetrachord spanning a 4:3 fourth. In modern microtonal contexts, hemiola-derived ratios from 3:2 inform explorations of non-octave intervals, particularly 3:2-based septimal intervals in 7-limit tuning, such as the septimal of 9:7 (approximately 435 cents), which extends Pythagorean principles into richer harmonic palettes. A key example of such stacking is the , arising from twelve 3:2 fifths exceeding seven octaves by the ratio 531441524288\frac{531441}{524288}, equivalent to 23.46 cents, highlighting the tempering needed for closed scales.

Role in Tuning Systems

In , the is constructed through a chain of pure perfect fifths with the frequency ratio of 3:2, starting from a fundamental tone and stacking up to seven notes, which approximates the but accumulates a discrepancy known as the upon completing the circle of twelve fifths, resulting in one or more intervals that disrupt smooth modulation across all keys. This system prioritizes the consonance of the 3:2 ratio for melodic and harmonic purity in monophonic and early polyphonic contexts, influencing Western music theory from through the . Just intonation extends this principle by deriving all intervals from simple whole-number ratios, including pure 3:2 fifths, to achieve maximal consonance without tempering, though practical implementations on fixed-pitch instruments often incorporate meantone temperaments that narrow the fifth slightly from 3:2 to render major thirds () nearly pure, thereby enhancing chordal . Early string instruments like viols were commonly tuned in such meantone systems during the , allowing performers to adjust frets or strings for approximate just intervals in ensemble playing. In non-Western traditions, the 3:2 ratio integrates into indigenous tuning frameworks; for instance, Indian classical music's sruti system divides the into 22 microtonal intervals, with the core swaras (notes) structured around just ratios like 3:2 for the , providing a flexible basis for elaboration. Similarly, tunings frequently employ Pythagorean-derived intervals, centering the 3:2 fifth as a foundational element for modal scales that incorporate quarter tones for expressive nuance. Contemporary microtonal extensions adapt the hemiola for synthesizers and electronic music by equal-tempering the 3:2 fifth into finer divisions beyond the 12-tone system; ' alpha scale, for example, divides it into 9 equal steps (approximately 77.78 cents each), while her beta scale uses 11 steps (approximately 63.8 cents each), enabling non-octave-repeating harmonies in works like those on Beauty in the Beast. These approaches, implemented in digital tuning software, bridge traditional ratios with modern equal divisions to explore expanded tonal palettes.

References

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  2. https://soundformationmusic.com/what-is-a-hemiola/
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  4. https://fiveable.me/key-terms/ap-music-theory/hemiola
  5. https://www.classical-music.com/articles/brahms-style-guide
  6. https://online.berklee.edu/takenote/cuban-music-fusion-pianists-to-put-in-heavy-rotation/
  7. https://www.bandsman.co.uk/downloads/eastern.pdf
  8. https://lilypond.org/doc/v2.23/Documentation/music-glossary/hemiola
  9. https://dictionary.onmusic.org/terms/1697-hemiola_36
  10. https://music.arts.uci.edu/dobrian/polytemporal/polytemporaltechniques.htm
  11. https://microtonal.miraheze.org/wiki/Perfect_fifth
  12. http://www.medieval.org/emfaq/anaigeon/e_coloration.html
  13. https://classicalliberalarts.com/wp-content/uploads/BOETHIUS-Bower-1989-Fundamentals_of_Music.pdf
  14. https://ora.ox.ac.uk/objects/uuid:e01e0aba-11dd-4910-bde6-09b4edf75134
  15. https://www.open-access.bcu.ac.uk/handle/2436/3780
  16. https://core.ac.uk/download/477653566.pdf
  17. https://www.chrysalis-foundation.org/musical-mathematics-pages/ibn-sina-stifel-and-zarlino/
  18. https://acda-publications.s3.us-east-2.amazonaws.com/choral_journals/April_2002_Kingsbury_S.pdf
  19. https://read.dukeupress.edu/journal-of-music-theory/article/57/1/87/14455/Metrical-Displacement-and-Metrically-Dissonant
  20. https://simssa.ca/assets/files/thomae_2019_dlfm_mensural_paper.pdf
  21. https://douglasniedt.com/hemiola.html
  22. https://www.skoove.com/blog/what-is-hemiola-in-music/
  23. https://online.ucpress.edu/jams/article/78/2/517/212248/Whose-Decolonization-Prospects-for-Decolonizing
  24. https://chromatone.center/theory/rhythm/system/crossbeat/
  25. http://d-scholarship.pitt.edu/10261/1/OADosunmuDissertation_ETD_1.pdf
  26. https://www.academia.edu/45436527/Taking_the_Measure_of_Josquin
  27. https://acda-publications.s3.us-east-2.amazonaws.com/choral_journals/March_1982_Hawthorne_W.pdf
  28. https://www.cambridge.org/core/books/tactus-mensuration-and-rhythm-in-renaissance-music/theory/4D0D1FCC9FDE9D9CFD8144DC6DFCA2CA
  29. https://lirias.kuleuven.be/retrieve/105957
  30. https://pure.qub.ac.uk/files/95442104/Deciphering_Tomita_19Jul2015.pdf
  31. https://openscholar.huji.ac.il/sites/default/files/gileadbar-elli/files/bach-wtc1.pdf
  32. https://orchestrasounds.com/2015/11/01/59-rhythmic-irregularities-1/
  33. https://symposium.music.org/60-1/item/11481-tracing-as-an-analytical-approach-in-the-music-theory-classroom-exploring-chopin-s-mazurka-in-a-flat-major-op-17-no-3.html
  34. https://read.dukeupress.edu/journal-of-music-theory/article/53/1/57/98284/Metrical-Dissonance-and-Directed-Motion-in
  35. https://www.fachords.com/hemiola/
  36. https://aithor.com/essay-examples/maurice-ravel-bolero-analysis-of-music-piece
  37. https://estudioflamenco.com/wp-content/uploads/2016/05/flamenco.pdf
  38. https://www.bradmehldau.com/hemiola/
  39. https://iftawm.org/journal/oldsite/articles/2020a/Wells_AAWM_Vol_8_1.pdf
  40. https://splice.com/blog/polyrhythms-music-producer/
  41. https://mtosmt.org/issues/mto.16.22.3/mto.16.22.3.murphy.html
  42. https://www.frontiersin.org/journals/psychology/articles/10.3389/fpsyg.2014.01257/full
  43. https://academic.oup.com/book/5956/chapter/149294756
  44. https://www.whipplemuseum.cam.ac.uk/explore-whipple-collections/acoustics/monochord
  45. https://hubguitar.com/music-theory/the-perfect-fifth
  46. https://sengpielaudio.com/calculator-centsratio.htm
  47. http://www.medieval.org/emfaq/harmony/pyth4.html
  48. https://www.britannica.com/art/tuning-and-temperament/Classic-tuning-systems
  49. https://digitalcommons.cedarville.edu/cgi/viewcontent.cgi?article=1005&context=music_and_worship_student_presentations
  50. https://pmc.ncbi.nlm.nih.gov/articles/PMC4568680/
  51. https://www.audiolabs-erlangen.de/resources/MIR/FMP/C5/C5S1_Intervals.html
  52. https://www.math.drexel.edu/~dp399/textbooks/mathmusic/pythagorus.html
  53. https://microtonal.miraheze.org/wiki/7-limit_tuning
  54. http://www.medieval.org/emfaq/harmony/pyth5.html
  55. https://www.cfmta.org/docs/essays/2023/2023-A-Ketchum.pdf
  56. https://archive.org/download/tuningtemperamen00barb/tuningtemperamen00barb.pdf
  57. https://iupress.org/9780253021236/meantone-temperaments-on-lutes-and-viols/
  58. https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1041&context=univstudiespapers
  59. https://www.academia.edu/8447879/Arabian_and_Turkish_tuning_systems
  60. https://www.wendycarlos.com/resources/pitch.html
  61. https://microtonal.miraheze.org/wiki/Beta_scale
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