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Limit (music)
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Limit (music)
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The first 16 harmonics, with frequencies and log frequencies (not drawn to scale).

In music theory, limits or harmonic limits are a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term limit was introduced by Harry Partch,[1] who used it to give an upper bound on the complexity of harmony, hence the name.

The harmonic series and the evolution of music

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Overtone series, partials 1-5 numbered Playⓘ.

Harry Partch, Ivor Darreg, and Ralph David Hill are among the many microtonalists to suggest that music has been slowly evolving to employ higher and higher harmonics in its constructs (see emancipation of the dissonance).[citation needed] In medieval music, only chords made of octaves and perfect fifths (involving relationships among the first three harmonics) were considered consonant. In the West, triadic harmony arose (contenance angloise) around the time of the Renaissance, and triads quickly became the fundamental building blocks of Western music. The major and minor thirds of these triads invoke relationships among the first five harmonics.

Around the turn of the 20th century, tetrads debuted as fundamental building blocks in African-American music.[citation needed] In conventional music theory pedagogy, these seventh chords are usually explained as chains of major and minor thirds. However, they can also be explained as coming directly from harmonics greater than 5. For example, the dominant seventh chord in 12-ET approximates 4:5:6:7 (albeit very poorly), while the major seventh chord approximates 8:10:12:15.

Odd-limit and prime-limit

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In just intonation, intervals between pitches are drawn from the rational numbers. Since Partch, two distinct formulations of the limit concept have emerged: odd limit and prime limit. Odd limit and prime limit n do not include the same intervals even when n is an odd prime.

Odd limit

[edit]

For a positive odd number n, the n-odd-limit contains all rational numbers such that the largest odd number that divides either the numerator or denominator is not greater than n.

In Genesis of a Music, Harry Partch considered just intonation rationals according to the size of their numerators and denominators, modulo octaves.[2] Since octaves correspond to factors of 2, the complexity of any interval may be measured simply by the largest odd factor in its ratio. Partch's theoretical prediction of the sensory dissonance of intervals (his "One-Footed Bride") are very similar to those of theorists including Hermann von Helmholtz, William Sethares, and Paul Erlich.[3]

See § Examples, below.

Identity

[edit]
For other uses, see Identity (music).

An identity is each of the odd numbers below and including the (odd) limit in a tuning. For example, the identities included in 5-limit tuning are 1, 3, and 5. Each odd number represents a new pitch in the harmonic series and may thus be considered an identity: 

C  C  G  C  E  G  B  C  D  E  F  G  ...
1  2  3  4  5  6  7  8  9  10 11 12 ...

According to Partch: "The number 9, though not a prime, is nevertheless an identity in music, simply because it is an odd number."[4] Partch defines "identity" as "one of the correlatives, 'major' or 'minor', in a tonality; one of the odd-number ingredients, one or several or all of which act as a pole of tonality".[5]

Odentity and udentity are short for over-identity and under-identity, respectively.[6] According to music software producer Tonalsoft: "An udentity is an identity of an utonality".[7]

Prime limit

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First 32 harmonics, with the harmonics unique to each limit sharing the same color.

For a prime number n, the n-prime-limit contains all rational numbers that can be factored using primes no greater than n. In other words, it is the set of rationals with numerator and denominator both n-smooth.

p-Limit Tuning. Given a prime number p, the subset of consisting of those rational numbers x whose prime factorization has the form with forms a subgroup of (). ... We say that a scale or system of tuning uses p-limit tuning if all interval ratios between pitches lie in this subgroup.[8]

In the late 1970s, a new genre of music began to take shape on the West coast of the United States, known as the American gamelan school. Inspired by Indonesian gamelan, musicians in California and elsewhere began to build their own gamelan instruments, often tuning them in just intonation. The central figure of this movement was the American composer Lou Harrison[citation needed]. Unlike Partch, who often took scales directly from the harmonic series, the composers of the American Gamelan movement tended to draw scales from the just intonation lattice, in a manner like that used to construct Fokker periodicity blocks. Such scales often contain ratios with very large numbers, that are nevertheless related by simple intervals to other notes in the scale.

Prime-limit tuning and intervals are often referred to using the term for the numeral system based on the limit. For example, 7-limit tuning and intervals are called septimal, 11-limit is called undecimal, and so on.

Examples

[edit]
ratio interval odd-limit prime-limit audio
3/2 perfect fifth 3 3 Playⓘ
4/3 perfect fourth 3 3 Playⓘ
5/4 major third 5 5 Playⓘ
5/2 major tenth 5 5 Playⓘ
5/3 major sixth 5 5 Playⓘ
7/5 lesser septimal tritone 7 7 Playⓘ
10/7 greater septimal tritone 7 7 Playⓘ
9/8 major second 9 3 Playⓘ
27/16 Pythagorean major sixth 27 3 Playⓘ
81/64 ditone 81 3 Playⓘ
243/128 Pythagorean major seventh 243 3 Playⓘ

Beyond just intonation

[edit]

In musical temperament, the simple ratios of just intonation are mapped to nearby irrational approximations. This operation, if successful, does not change the relative harmonic complexity of the different intervals, but it can complicate the use of the harmonic limit concept. Since some chords (such as the diminished seventh chord in 12-ET) have several valid tunings in just intonation, their harmonic limit may be ambiguous.

See also

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References

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[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Limit (music)

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In music theory, particularly within the frameworks of and microtonal tuning systems, a limit refers to the highest allowed in the of the rational frequency ratios that define the intervals between pitches. This numerical bound quantifies the complexity of a tuning, where lower limits yield simpler, more harmonies based on fundamental overtones, while higher limits introduce greater variety and potential dissonance. The most common formulation is the prime limit, which restricts ratios to those whose prime factors do not exceed the specified number n. For instance, a 3-prime-limit system, known as , employs only the primes 2 and 3, generating intervals such as the (3/2) and whole tone (9/8). Expanding to a 5-prime-limit incorporates the prime 5, enabling consonant major and minor thirds (5/4 and 6/5, respectively), which form the basis of classic triads like 4:5:6. Higher prime limits, such as 7-limit (adding ratios like 7/4) or 11-limit, allow for more intricate scales and chords, often explored in contemporary and to approximate subtler harmonic nuances. An alternative is the odd limit, which considers only the largest odd number in the ratio rather than primes, emphasizing intervallic concordance within chords; for example, a 5-odd-limit includes ratios up to 5 but excludes even factors beyond powers of 2 for certain analyses. Composer notably advocated for odd limits in his theoretical work, using them to construct scales and instruments that prioritize sensory consonance over traditional . In practice, limits guide the generation of scale lattices or tonality diamonds, mapping all possible intervals within the constraint to facilitate composition and performance in non-standard tunings.

Introduction

Definition and Origins

In music theory, particularly within , a limit denotes the highest odd (in the case of odd-limit tunings) or (in prime-limit tunings) used in the prime factorization of rational frequency ratios that define musical intervals, thereby restricting harmonic complexity to simpler from the series. This classification helps quantify the consonance and structural simplicity of a tuning system by excluding higher partials that introduce greater dissonance or microtonal subtlety. The concept of the limit was introduced by American composer and theorist in his 1949 book Genesis of a Music, where it served as a foundational tool for exploring extended scales beyond traditional . Partch drew inspiration from ancient Greek ideas of harmonic ratios—such as Pythagorean intervals based on simple whole-number proportions—but formalized the limit as a precise metric for modern microtonal composition, emphasizing its role in creating coherent sonic worlds. A basic illustration of the limit appears in the 3-limit, which incorporates only the first three harmonics (1, 2, and 3), yielding the octave (2:1) and perfect fifth (3:2) as primary intervals. The 5-limit extends this by including the fifth harmonic, adding the major third (5:4) and enabling richer triadic harmonies while still maintaining relative simplicity. These foundational examples underscore how limits progressively unlock new expressive possibilities in just intonation.

Purpose in Music Theory

In music theory, limits play a crucial role in classifying intervals and tuning systems within by restricting the prime factors of their ratios or the odd s involved, thereby providing a framework for comparing harmonic structures across different scales. The prime-limit approach designates the highest permitted in the of any , such as in 3-limit tuning (also known as ), which exclusively uses the primes 2 and 3 to generate intervals like the (3/2). Similarly, the odd-limit method, pioneered by composer , limits intervals to those derived from the first n odd-numbered partials of the series, emphasizing consonance through simpler odd s beyond the fundamental (2:1). This dual classification system allows theorists to delineate the scope of harmonic resources in a given tuning, facilitating precise analysis of consonance and complexity. By imposing these caps on primes or odd numbers, limits simplify the vast array of possible rational intervals, avoiding overly complex ratios that could introduce dissonance or impracticality in performance. For instance, extending beyond 5-limit introduces higher primes like 7, which yield more intricate intervals such as seventh (7/4), but lower limits ensure greater consonance and feasibility for fixed-pitch instruments by prioritizing simpler, more stable harmonies. This reduction in complexity not only streamlines scale construction but also enhances perceptual clarity, as lower-limit intervals align more closely with the human ear's preference for pure, beat-free sounds derived from small-integer ratios. Theoretically, limits enable a systematic of xenharmonic scales—those departing from the standard 12-tone —by providing bounded parameters for generating novel tunings that balance innovation with acoustic purity. This methodical bounding supports compositional experimentation, as seen in software tools like Scala, which leverages limit-based algorithms to produce and archive thousands of scales for practical application in digital synthesis and notation. Such benefits have made limits indispensable for advancing microtonal , allowing composers to navigate possibilities without unbounded proliferation.

Theoretical Foundations

The Harmonic Series

The harmonic series in acoustics and consists of an infinite sequence of frequencies that are successive multiples of a f0f_0, denoted as the first or fundamental tone. The frequency of the nnth is given by fn=n⋅f0,f_n = n \cdot f_0, where n=1,2,[3,… ](/page/3Dots)n = 1, 2, [3, \dots](/page/3_Dots). This series emerges naturally from the of ideal strings, pipes, or membranes under simple physical conditions, producing a spectrum of above the fundamental. In music, the lower harmonics—typically the first through eighth partials—hold particular relevance, as their simple integer ratios yield consonant intervals when two or more are combined. For example, the second harmonic (octave above the fundamental) and third harmonic form a perfect fifth with a frequency ratio of 3/23/2, while the fourth and fifth produce a major third at 5/45/4. These ratios approximate just intonation intervals, where consonance arises from the alignment of partials with minimal interference or beating; higher partials introduce greater dissonance due to closer frequency proximities. Hermann von Helmholtz's analysis of partial tones demonstrated that consonance is tied to the low-order harmonics' smooth fusion in perception, influencing the physiological basis of musical harmony. More generally, the interval between the mmth and nnth harmonics (with m>nm > n) has a of m/nm/n, providing a simplified acoustic model for pitch relationships. These series-derived ratios underpin principles, where tuning systems prioritize simple fractions for stability. In musical practice, early frameworks emphasized low harmonics for core consonances, whereas contemporary theory extends to higher orders to explore increased timbral and intervallic complexity.

Just Intonation Principles

Just intonation is a tuning system in which musical intervals are tuned to exact rational frequency ratios derived from small whole numbers, such as 3/2 for the and 5/4 for the , contrasting with the approximate intervals of . This approach aims to produce pure, consonant sounds by aligning the harmonics of simultaneously sounding notes, fostering a sense of harmonic fusion. The principles of are rooted in the harmonic series, where the frequencies of are multiples of a fundamental pitch, yielding simple that define intervals. Consonance increases as involve smaller prime numbers, such as 2, 3, and 5 in the five-limit system, because these produce periodic waveforms with minimal beating between partials. Within this framework, the , with a of 81/80 (approximately 21.5 cents), emerges as the interval between the Pythagorean (81/64) and the just (5/4), representing a key adjustment in to prioritize purer thirds over stacked fifths. A primary limitation of arises from its reliance on fixed rational pitches, which can lead to discrepancies known as commas during modulation between keys, as accumulating small intervals like the disrupt overall coherence. To address this, musical limits—such as odd-limit and prime-limit systems—constrain the allowable ratios to a , helping to manage complexity while preserving consonance.

Types of Limits

Odd-Limit

The odd-limit in music theory refers to a measure of complexity for intervals in just intonation, defined as the highest odd integer appearing in the numerator or denominator of a reduced ratio after removing factors of 2 (which represent octaves). This approach ignores even harmonics beyond the octave and focuses solely on odd components to characterize the tuning system. For example, the interval 5/4 (a major third) has an odd-limit of 5, as its reduced form has odd parts 5 and 1, while 7/4 (a harmonic seventh) has an odd-limit of 7 and is thus excluded from a 5-odd-limit tuning. To calculate the odd-limit of a p/qp/q in lowest terms, first extract the odd part of pp by dividing out all factors of 2, and do the same for qq; the odd-limit is then the maximum of these two odd parts. This is formally known as the Kees height (KH) of the interval, where KH(p:qp:q) = \max(p', q') with p′p' and q′q' as the odd cores. For instance, the 28:30 reduces to 14:15, with odd parts 7 and 15, yielding an odd-limit of 15; in a 15-odd-limit system, this interval is permitted, but it would be excluded in lower limits like 11. Such systems generate scales by considering all ratios whose odd parts are odd integers up to the limit value, providing a structured way to extend beyond basic harmonic series approximations. One key advantage of odd-limit tunings is their emphasis on the "color" contributed by higher odd harmonics, which add timbral richness and dissonance without relying on even that merely reinforce octaves. This selective inclusion promotes a sense of corporeal in acoustic instruments, as odd harmonics dominate the spectra of many non-sine wave sounds. extensively employed 11-odd-limit tunings in his compositions, deriving his influential 43-tone scale from ratios involving odd integers up to 11 (such as 1, 3, 5, 7, 9, and 11), which allowed for novel intervals like 11/10 while maintaining perceptual coherence. In odd-limit systems, each odd integer up to the limit corresponds to a unique pitch class identity, often visualized in structures like the tonality diamond, where rows or columns represent otonal (overtone-based) or utonal (subharmonic-based) series anchored by that odd number. This conceptual framework underscores the individuality of each generator—e.g., 5 evokes a distinct "fifthness" separate from 3's "thirdness"—fostering intuitive navigation of extended scales. As an alternative, prime-limit tunings restrict to prime factors rather than all odd integers.

Prime-Limit

In music theory, particularly within , the prime-limit specifies the highest allowed in the of any used to represent musical intervals. This constraint limits the harmonic complexity by permitting only products of primes up to that number, including powers thereof, while always incorporating 2 for equivalence. For instance, a 7-limit system allows primes 2, 3, 5, and 7, thus including intervals like the septimal 7/5 but excluding the undecimal semi-augmented fourth 11/8, as its factorization involves the prime 11. To determine the prime-limit of a given , one factorizes its numerator and denominator into prime factors and identifies the largest such prime across the ratio in lowest terms. Powers of 2 are inherently included but do not raise the limit beyond the highest odd prime; this method contrasts with the odd-limit, a looser variant that measures the maximum odd integer remaining after extracting factors of 2 from numerator and denominator separately, thereby excluding some composite odds unless the limit value accommodates their full odd component. The prime-limit approach promotes harmonic purity and minimal complexity by prioritizing lower primes, which yield more intervals derived from early , and facilitates systematic exploration in xenharmonic theory through bounded generation of scales and chord structures. Notably, the 5-limit and 5-prime-limit coincide in their allowable intervals, but the paradigms diverge at points like the interval 9/8, which the odd-limit permits only at 9 (due to its odd numerator 9) while the prime-limit includes it as early as 3 (since 9 = 3²).

Historical Development

Evolution in Western Music

In music and the subsequent Pythagorean tradition, tuning practices were confined to the 3-limit , relying primarily on the ratios of octaves (2:1) and perfect fifths (3:2) to generate scales. This approach, attributed to around the 6th century BCE, emphasized pure consonances suitable for monophonic melodies and early theoretical frameworks, avoiding higher primes for simplicity and mathematical purity. In the 2nd century CE, Claudius Ptolemy proposed a tuning resembling 5-limit , incorporating major thirds (5:4) alongside fifths and octaves. By the medieval period, this 3-limit Pythagorean tuning persisted in Western , underpinning —a monophonic vocal tradition developed in the 9th-10th centuries CE—which favored unaccompanied lines with intervals derived from stacked fifths and octaves, ensuring harmonic stability without complex vertical sonorities. The transition from to in the late medieval era, particularly with the rise of and motets in the 12th-13th centuries, began to challenge the limitations of 3-limit tuning, as multiple voices demanded more vertical intervals beyond mere fifths and octaves. Although early retained much of the Pythagorean framework for its melodic incisiveness, the increasing complexity of interwoven lines gradually necessitated broader harmonic resources, setting the stage for later expansions. While non-Western traditions, such as those in ancient Indian or , independently explored analogous interval ratios, the Western shift toward uniquely drove the incorporation of higher limits to support richer textures. During the and into the period, musical practices evolved to embrace 5-limit , incorporating the 5 to enable major (5:4) and minor (6:5) thirds, which formed the basis of stable triads essential for emerging harmonic progressions. Theorists like Bartolomeus Ramis de Pareia in 1482 advocated this system to resolve dissonances in polyphonic compositions, while Gioseffo Zarlino's 1558 formalized 5-limit ratios as ideal for . To approximate these pure intervals on fixed-pitch instruments like organs and harpsichords, meantone temperaments—such as quarter-comma meantone introduced around 1523—flattened fifths slightly to prioritize sweeter thirds, facilitating the triad-centered music of composers like and . In the 19th century, equal temperament became the dominant tuning system in Western music, enabling the chromaticism and modulations central to Romantic harmony, as seen in the works of composers like Richard Wagner. Theoretical interest in just intonation persisted, and by the early 20th century, explorations of 7-limit intervals—such as the natural seventh (7:4)—emerged in efforts to integrate higher harmonics from the overtone series, supporting experimental and microtonal innovations distinct from traditional equal temperament practices.

Contributions of Key Theorists

, an influential American composer and theorist active from the 1940s through the 1970s, advanced the concept of musical limits by developing an 11-limit system that incorporated intervals derived from the seventh and eleventh harmonics of the natural series. In his seminal work Genesis of a Music (first published in 1949 and revised in 1974), Partch defined a "limit" as the highest odd integer factor in interval ratios, prioritizing sensory directness and corporeal experience over abstract mathematical precision in . He derived his renowned 43-tone scale from an 11-limit diamond, which allowed for a richer palette of intervals while rejecting equal temperament's compromises. Partch's innovations extended beyond theory into practice, as he built custom instruments to realize his scales and composed works that integrated , , and visual elements, emphasizing music's ritualistic and humanistic dimensions. His approach influenced subsequent explorations in microtonality, including the creation of American gamelan ensembles by composers like , who drew from Partch's principles to blend Western and Southeast Asian traditions. Additionally, Partch's reinterpretations of American vernacular forms, such as hobo speeches and folk narratives, echoed elements of African-American in their rhythmic and intonational freedom, inspiring later fusions of microtonal techniques with . Earlier in the 20th century, Max F. Meyer laid foundational ideas for prime-limit systems through his 1929 book The Musician's Arithmetic: Drill Problems for an Introduction to the Scientific Study of Musical Composition. Meyer introduced a 7-limit tonality diamond, mapping intervals using prime factors up to 7, which provided a structured approach to just intonation ratios and influenced later theorists like Partch in constructing higher-limit frameworks. His work emphasized mathematical rigor in musical education, treating tuning as an arithmetic exercise to generate harmonious scales without equal temperament. In more recent decades, the xenharmonics community has extended limit theory into higher realms, with figures like Erv Wilson contributing innovative scale designs up to the 31-limit through combinatorial methods and diagrammatic notations. Wilson's systems, often realized via software tools for tuning exploration, have enabled composers to access complex harmonies beyond traditional odd- or prime-limits, fostering that integrates computational analysis with auditory intuition.

Practical Examples

Intervals and Scales by Limit

In , the 3-limit system, known as , generates intervals using only the prime factors 2 and 3, producing a scale based on stacked perfect fifths of 3/2 (approximately 701.96 cents) and perfect fourths of 4/3 (approximately 498.05 cents). This system lacks consonant thirds, as the major third requires the prime 5, resulting in a dissonant Pythagorean of 81/64 (approximately 407.82 cents) that is avoided in harmonic contexts. Extending to the 5-limit incorporates the prime 5, enabling the syntonic diatonic scale with added major thirds of (approximately 386.31 cents) and minor thirds of 6/5 (approximately 315.64 cents). This scale resolves the through the of 81/80 (approximately 21.51 cents), which measures the discrepancy between the Pythagorean and just major thirds, allowing for more triads like the major chord. The 7-limit further enriches the palette by including the prime 7, introducing septimal intervals such as the natural seventh of 7/4 (approximately 968.83 cents), the septimal of 7/5 (approximately 582.51 cents), and the septimal of 7/6 (approximately 266.87 cents). An example 7-limit scale is a 7-note progression incorporating the 7/6 , as seen in certain folk traditions, which provides subtler shadings than 5-limit equivalents. Higher limits enable microtonal extensions; for instance, the 7-limit Bohlen-Pierce scale divides the 3:1 tritave into 13 just intervals using primes up to 7, offering non-octave-based structures for . Similarly, 13-limit tunings incorporate the prime 13 for even finer divisions, facilitating complex microtonal scales beyond traditional diatonic frameworks. The following table summarizes select representative intervals across these limits, with cent values calculated relative to the (1200 cents):
Interval NameRatioCentsLimit
Perfect Fifth3/2701.963
Perfect Fourth4/3498.053
Major Third5/4386.315
Minor Third6/5315.645
Septimal Minor Third7/6266.877
Septimal Tritone7/5582.517
Natural Seventh7/4968.837

Audio and Sensory Aspects

Lower-limit intervals in , such as the (3/2), are perceived as particularly pure and because their simple integer ratios result in the alignment of low-order partials from the harmonic series, minimizing auditory beating and roughness between . This purity arises from the partials of the two tones coinciding or being sufficiently separated to avoid interference within critical bandwidths, as modeled in psychoacoustic studies of sensory consonance. In contrast, intervals derived from higher prime limits, while still when purely tuned, introduce greater complexity in partial interactions; for instance, the septimal major seventh (7/4) exhibits a subtler form of roughness compared to the (5/4) due to the involvement of higher harmonics that create less intense but more nuanced beating patterns. Human auditory perception prioritizes lower harmonics in the processing of complex tones, with the fundamental frequency and initial overtones dominating the sense of pitch and timbre before higher partials contribute to finer details. This evolutionary adaptation likely stems from the acoustic properties of natural sounds, where low-frequency components carry primary structural information, influencing the preference for simple ratios in musical intervals. Composer Harry Partch emphasized this bodily connection in his theory of corporeal music, proposing a "monophony of natural size" where the fundamental tone has a wavelength of approximately one foot to resonate with the human body's scale, enhancing sensory immersion and physical response to just intonation. A representative example is the 5-limit major triad, tuned to the ratios 4:5:6 (e.g., C-E-G), which produces a smooth, stable sonority with minimal dissonance due to the close alignment of its partials, often described as warm and foundational in audio demonstrations. In comparison, a 7-limit extension incorporating the or (7/4), as heard in septimal dominant seventh chords (e.g., G-B-D-F, with the seventh tuned to 7/4 from the root), introduces a richer, more resonant texture with subtle harmonic color from the seventh harmonic, though it can evoke a slightly edgier tension from increased partial density. Perceptual dissonance in these intervals can be approximated through models like Sethares' dissonance curve, which descriptively captures roughness as peaking when nearby partials beat rapidly (around 20-30 Hz) and dipping to minima at simple just ratios, reflecting how lower-limit feel effortlessly blended while higher-limit ones add layered sensory depth without overt clash.

Extensions and Modern Applications

Beyond Just Intonation

In tempered tuning systems, the concept of limits from is adapted through approximations, where equal divisions of the octave provide close but imperfect representations of rational intervals. For instance, 12-tone equal temperament (12-TET) effectively supports 5-limit by closely approximating key intervals such as the , measured at 700 cents compared to the just value of approximately 702 cents. This near-match enables 12-TET to function as a practical compromise for 5-limit triadic , tempering the to allow modulation across all keys without drastic dissonance. However, higher-limit intervals, like the septimal major seventh of 7/47/4 at 969 cents, deviate more significantly in 12-TET, as its equal steps yield irrational ratios that only coarsely approximate these pure tones, limiting the system's capacity for extended harmonic complexity. To extend beyond basic 5-limit approximations, theorists introduce the notion of an "effective limit" in tempered systems, defined by the best-fitting intervals within the equal division. For example, 19-TET enhances support for 7-limit intervals over 12-TET, with its fifth at about 695 cents tempering it downward to better approximate septimal ratios like 7/67/6 and 7/47/4, though still tempering commas such as the septimal to maintain uniformity. This adaptive approach evaluates a temperament's quality by how well it realizes higher-limit consonances without requiring pure ratios, allowing 19-TET to facilitate 7-limit scales and tetrads in a . Similarly, 31-TET excels in approximating 11-limit intervals, offering deviations around 1 cent for ratios involving 11, such as the neutral third 11/911/9 at 347 cents. Challenges in these adaptations include dissonant "wolf" intervals that arise from comma tempering, particularly in meantone systems where flattening fifths to pure major thirds creates out-of-tune fifths elsewhere. In quarter-comma meantone, the wolf fifth—approximated poorly compared to the 5-limit 3/23/2—disrupts chromatic modulation, but limits provide a framework for assessment: higher divisions like 31-TET minimize such wolves by closing the circle of fifths while supporting up to 11-limit harmony with minimal error. This evaluation underscores how limits guide the selection of temperaments for specific harmonic goals, balancing purity against practicality.

Contemporary Uses in Composition and Technology

In composition, the concept of limits continues to influence microtonal and experimental works, particularly through frameworks that prioritize harmonic purity. Composer incorporated 7-limit intervals in pieces such as Harmonium No. 7 (2000), where ratios like 8/7 and 7/6 form extended tonal structures derived from low-prime harmonics, creating subtle dissonances that evolve over time. Similarly, in spectralism, Gérard Grisey extended harmonic explorations to 13-limit approximations in works like Les Espaces Acoustiques (1976–1985), using spectral analysis of acoustic instruments to reconstruct pitches from partials up to the 13th, blending natural overtones with composed microintervals for timbral depth. These approaches highlight limits as tools for bridging acoustic analysis and perceptual innovation in post-tonal music. Technological advancements have democratized limit-based composition through specialized software and hardware. The Scala program facilitates the creation and export of odd-limit scales, allowing users to generate tunings restricted to primes up to a specified limit, such as 7-limit or 13-limit, for integration into workstations. Tune Smithy extends this by enabling fractal-like generation of microtonal scales within odd-limits, supporting real-time experimentation with patterns in algorithmic compositions. tools like incorporate microtonal tuners that support xenharmonic scales approximating high limits, up to 23-limit intervals, via retuning and MPE-compatible synthesizers, enabling precise control over interval ratios in electronic productions. In modern electronic and cross-cultural contexts, limits inform diverse applications, often approximating traditional microtonal practices. has utilized custom microtonal tunings in albums like Drukqs (2001) to craft dissonant, evolving synth textures that challenge norms. Influences from African-American traditions, including the "blue notes" of and —rooted in microtonal inflections from West African scales—are reinterpreted in contemporary works using limit-restricted tunings to evoke cultural resonance. Gamelan-inspired compositions similarly adapt Indonesian slendro and pelog tunings, which feature microtonal intervals akin to 5-limit or 7-limit , into Western experimental pieces for layered polyrhythms and . Mathematical models further refine these applications by quantifying perceptual effects. William Sethares' critical bandwidth dissonance , which models roughness as a function of frequency separation within auditory critical bands, provides a basis for evaluating consonance in odd-limit scales, helping composers select intervals that minimize sensory dissonance while maximizing timbral . The is given by: d(f1,f2)=A1(αx(e−x−e−b)1+αx(e−x−e−b))+A2((x−γ)2e−β(x−γ)21+(x−γ)2e−β(x−γ)2)d(f_1, f_2) = A_1 \left( \frac{\alpha x (e^{-x} - e^{-b})}{1 + \alpha x (e^{-x} - e^{-b})} \right) + A_2 \left( \frac{(x- \gamma)^2 e^{-\beta (x- \gamma)^2}}{1 + (x- \gamma)^2 e^{-\beta (x- \gamma)^2}} \right) where x=2(f2−f1)ERB(f1)x = \frac{2(f_2 - f_1)}{\text{ERB}(f_1)}, ERB denotes the equivalent rectangular bandwidth, and parameters A1,A2,α,b,β,γA_1, A_2, \alpha, b, \beta, \gamma are empirically derived constants. This approach integrates psychoacoustics into limit-based design, influencing both software algorithms and compositional decisions. As of 2025, emerging AI tools for generation, such as those using models trained on spectral datasets, are increasingly supporting microtonal and features, including higher prime limits, to create novel harmonic structures.

References

  1. http://www.tonalsoft.com/enc/l/limit.aspx
  2. http://www.patmissin.com/tunings/tun1.html
  3. https://masa.plainsound.org/pdfs/JI.pdf
  4. http://www.tonalsoft.com/sonic-arts/mclaren/partch/layers.htm
  5. https://skemman.is/bitstream/1946/8545/1/Lokaritgerd.pdf
  6. https://www.huygens-fokker.org/scala/
  7. https://www.feynmanlectures.caltech.edu/I_50.html
  8. https://www.mpi.nl/world/materials/publications/levelt/Plomp_Levelt_Tonal_1965.pdf
  9. https://www.mtosmt.org/issues/mto.96.2.6/mto.96.2.6.walker.html
  10. https://music.arts.uci.edu/abauer/3.1/notes/Hasegawa_Just_Intonation.pdf
  11. https://arxiv.org/pdf/1603.08904
  12. https://soundamerican.org/issues/just-intonation/why-how-and-wherefore
  13. https://sethares.engr.wisc.edu/paperspdf/Erlich-MiddlePath.pdf
  14. http://lumma.org/tuning/faq/
  15. https://www.dbdoty.com/Words/primersample.pdf
  16. https://www.cfmta.org/docs/essays/2023/2023-A-Ketchum.pdf
  17. https://hasegawa.research.mcgill.ca/pdf/Hasegawa-L%27intonation_juste_2013_EN.pdf
  18. http://www.medieval.org/emfaq/harmony/pyth5.html
  19. https://mtosmt.org/issues/mto.19.25.1/mto.19.25.1.willis.html
  20. https://www.kylegann.com/PartchKeynote.html
  21. https://www.sandiegouniontribune.com/2016/03/19/qa-jon-szanto-on-unique-legacy-of-harry-partch/
  22. https://ojs.uph.edu/index.php/JSM/article/download/4799/2257
  23. https://www.cambridge.org/core/journals/tempo/article/relative-intonation-nonsymmetrical-implications-of-linear-and-logarithmic-intervallic-measurement/1D3C37CDC46FF8F494254997C666491F
  24. https://www.chrysalis-foundation.org/musical-mathematics-pages/meyer-diamond/
  25. https://www.xenharmonikon.org/2018/10/06/forward-excerpt-to-microtonality-and-the-tuning-systems-of-erv-wilson/
  26. https://cmloegcmluin.wordpress.com/2013/01/18/ned-talk-3-0-an-introduction-to-xenharmonics/
  27. https://www.patmissin.com/tunings/tun1.html
  28. http://www.microtonal-synthesis.com/scale_just_intonation.html
  29. https://chromatone.center/theory/notes/temperaments/just/
  30. https://www.lpthe.jussieu.fr/~talon/MUSIC6.PDF
  31. http://www.tonalsoft.com/enc/b/bohlen-pierce.aspx
  32. https://www.huygens-fokker.org/bpsite/intervals.html
  33. http://www.r-5.org/files/books/rx-music/tuning/William_A_Sethares-Tuning_Timbre_Spectrum_Scale-EN.pdf
  34. https://pmc.ncbi.nlm.nih.gov/articles/PMC5819010/
  35. https://digitalcommons.du.edu/cgi/viewcontent.cgi?article=1044&context=sbs
  36. https://www.patmissin.com/tunings/audio.html
  37. https://www.youtube.com/watch?v=DZPBAbZWJog
  38. https://sethares.engr.wisc.edu/paperspdf/Erlich-22.pdf
  39. http://www.tonalsoft.com/enc/h/hindemith-12et-5limit-ji.aspx
  40. https://www.chrisvaisvil.com/microtonal-theory-pages/about-19-edo/
  41. https://www.huygens-fokker.org/docs/terp31.html
  42. https://www.huygens-fokker.org/docs/rap31.html
  43. https://alex-vaughan.com/pdfs/variation_transformation_and_development_in_gerard_grisey%27s_les_espaces_acoustiques.pdf
  44. http://www.tunesmithy.co.uk/
  45. https://www.ableton.com/en/manual/live-instrument-reference/
  46. https://microtonal.miraheze.org/wiki/Microtonal_music
  47. https://emusicology.org/article/id/4566/
  48. https://huygens-fokker.org/en/microtonality/
  49. https://www.ableton.com/en/blog/microtonal-music-production/
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