53 equal temperament

53 equal temperament, also known as 53-tone equal temperament or 53-EDO, is a musical tuning system that divides the octave into 53 equally spaced semitones, with each step measuring approximately 22.6415 cents (1200/53 cents).^[1] This fine-grained division allows for highly accurate approximations of just intonation intervals within the 5-limit, particularly the perfect fifth (3:2 ratio) at 701.89 cents, deviating from the just 701.96 cents by -0.07 cents.^[2] The major third (5:4 ratio) is approximated at 384.91 cents, -1.40 cents from the just 386.31 cents, outperforming the 12-tone equal temperament's 400 cents (+13.69 cents deviation).^[1] Historically, the properties of 53 equal temperament were first noted in ancient China by Jing Fang (78–37 BCE), who observed that 53 just perfect fifths nearly equal 31 octaves, differing by the small Mercator's comma of about 3.615 cents.^[2] This insight was rediscovered in Europe by Nicholas Mercator in the 17th century, who calculated the comma and noted the tuning's precision for fifths, with each tempered fifth flattened by roughly 0.0682 cents from purity.^[2] William Holder in 1694 highlighted its close approximation to just major thirds (within 1.4 cents), and Isaac Newton explored similar ideas in unpublished work around 1664–1665.^[2] Mathematically, 53 ET is nearly equivalent to an extended Pythagorean tuning, as 53 steps of (3/2)^{1/53} approximate the octave with minimal error, making it a bridge between rational just intonation and equal division.^[2][3] In modern music theory and practice, 53 equal temperament is valued in microtonal composition for its ability to resolve small commas like the Holdrian comma while supporting complex harmonies without the distortions of coarser tunings like 12 ET.^[1] It has been implemented in software such as Ableton Live for experimental music, where it enables precise control over intervals like the syntonic comma (81:80), though major thirds may still exhibit slight tempering compared to pure just intonation.^[1][4] The system also relates to historical proposals, such as Georg Philipp Telemann's 18th-century "new musical system," which shares structural elements like dividing the whole tone into 9 parts and the diatonic semitone into 4, forming a subset approximating 53 ET's pure fifths.^[4] Overall, 53 ET stands out among equal temperaments for its balance of accuracy and usability in both theoretical analysis and creative applications.^[3]

Fundamentals

Definition and characteristics

53 equal temperament, also known as 53-TET, is a musical tuning system that divides the octave—defined by a frequency ratio of 2:1—into 53 equal steps. Each step represents a frequency ratio of $2^{1/53}$21/53, corresponding to approximately 22.641 cents on the cent scale, where the octave spans 1200 cents.^[2][3] This temperament exhibits high accuracy in approximating intervals of just intonation, particularly those within the 5-limit, which include ratios formed by the primes 2, 3, and 5, such as the perfect fifth (3/2) and major third (5/4). Such precise approximations enable detailed exploration of microtonal harmonies and subtle interval variations beyond standard Western tuning systems.^[2][5] In contrast to 12 equal temperament (12-TET), which introduces noticeable compromises in interval purity to facilitate modulation across all keys, 53-TET supports near-perfect extensions of Pythagorean tuning, where intervals are generated primarily from stacked perfect fifths. This results in a generator interval that aligns closely with the pure fifth, allowing for extended chains of fifths with minimal cumulative error.^[2][3] The basic generator of 53-TET is the approximated perfect fifth, spanned by 31 steps, or $31/53$31/53 of the octave, which deviates from the just fifth by only about 0.068 cents.^[2]

Interval approximations

53 equal temperament excels in approximating 5-limit just intonation intervals, with errors generally below 1.5 cents for major consonances, far surpassing the approximations in smaller equal divisions like 12-TET (where the major third deviates by +13.69 cents and the perfect fifth by -1.96 cents). This accuracy stems from 53-TET's ability to closely match ratios involving primes 2, 3, and 5, tempering out tiny commas like the schisma and Holdrian comma to align tempered steps with harmonic series overtones.^[6][5] Key 5-limit approximations are detailed in the table below, showing the just interval cents, best 53-TET step count, tempered cents, and deviation: These values demonstrate 53-TET's strengths: the perfect fifth and major whole tone serve as near-perfect matches (errors under 0.2 cents), while the major and minor thirds remain perceptually pure, with deviations below the just noticeable difference for trained listeners (around 5-6 cents for thirds). The system's overall maximum error for 5-limit intervals stays under 2 cents, positioning it as one of the finest equal temperaments for 5-limit harmony among equal divisions up to 72-TET.^[7][6][5] Extensions to 7-limit just intonation are reasonable but less precise; for example, the harmonic seventh (7/4) at 968.83 cents is best approximated by 43 steps yielding 973.58 cents, with an error of +4.76 cents. This allows exploration of septimal harmonies, though with noticeable sharpening compared to 5-limit purity.^[7]

Mathematical properties

Step size and calculations

In 53 equal temperament (53-TET), the octave is divided into 53 equal steps, each with a frequency ratio of $2^{1/53}$21/53.^[6] This step size corresponds to exactly $1200/53$1200/53 cents, which approximates 22.6415 cents.^[6] By definition, 53 such steps span the full octave of 1200 cents precisely, establishing octave equivalence in the tuning system.^[8] To generate an interval with frequency ratio $r$r in 53-TET, the number of steps $n$n is determined by rounding to the nearest integer: $n = \round(53 \cdot \log_2 r)$n=\round(53⋅log2​r).^[8] The corresponding cent value of this approximation is then $n \times (1200/53)$n×(1200/53). For comparison, the just intonation cent value of the interval is given by $1200 \cdot \log_2 r$1200⋅log2​r.^[6] A key generator in 53-TET is the perfect fifth, approximated by 31 steps, yielding $31 \times (1200/53) \approx 701.89$31×(1200/53)≈701.89 cents, compared to the just perfect fifth of approximately 701.96 cents.^[8] These step-based calculations provide the foundation for approximating musical intervals in 53-TET, as explored further in the interval approximations section.

Holdrian and Mercator's commas

The Holdrian comma, named after the 17th-century mathematician William Holder, is a small musical interval equivalent to a single step in 53 equal temperament, with a size of approximately 22.6415 cents and a frequency ratio of $2^{1/53}$21/53. This interval arises from theoretical divisions of the octave into 53 equal parts, approximating the structure of diatonic scales by treating the tone as composed of nine such commas, leading to an octave of five tones and two diatonic semitones (5 × 9 + 2 × 4 = 53 commas). In this context, the Holdrian comma serves as the fundamental unit for 53-TET's close approximation of Pythagorean tuning, where stacking intervals reveals discrepancies tempered out by the equal division.^[9] Mercator's comma, named after Nicholas Mercator who first proposed the 53-division in 1672 using logarithmic calculations, is a much smaller interval measuring about 3.615 cents. It represents the discrepancy between 53 Pythagorean perfect fifths (each with ratio 3/2) and 31 octaves, calculated as $1200 \times (53 \log_2 (3/2) - 31)$1200×(53log2​(3/2)−31) cents, yielding a frequency ratio of approximately $3^{53}/2^{84}$353/284. This comma highlights 53-TET's precision in Pythagorean approximation, as the tuning flattens each fifth by roughly 0.0682 cents ($3.615 / 53$3.615/53), thereby setting Mercator's comma to zero and ensuring that 53 tempered fifths exactly equal 31 octaves.^[10][2] The two commas are interconnected in 5-limit tuning theory: the Holdrian comma provides the granular step size for interval approximations, while Mercator's comma quantifies the residual error in the extended chain of fifths that 53-TET eliminates, enabling near-Pythagorean equivalence without the accumulation of larger deviations like the 23.46-cent Pythagorean comma. In modern interpretations, these commas facilitate extended just intonation systems, where 53-TET's steps allow composers to explore 7-limit and higher harmonics (e.g., approximations to 7/4 at 43 steps or 9/7 at 19 steps) with errors under 10 cents, extending beyond traditional diatonic frameworks.^[11]

History

Early developments

The conceptual origins of 53 equal temperament trace back to ancient musical theories, with the earliest known reference in ancient China. Jing Fang (78–37 BCE), a music theorist, observed that 53 just perfect fifths (3:2) nearly equal 31 octaves, differing by the small Mercator's comma of about 3.615 cents.^[2] This insight highlighted the close approximation achievable with 53 divisions. In the Ottoman theoretical tradition, which evolved from Islamic foundations, the octave was systematically divided into 53 equal commas (koma) to notate the microtonal perdes (pitches) of makam music, providing a high-resolution framework for traditional scales. This approach originated in the 13th-century work of Safi al-Din al-Urmawi, who extended the 17-tone Pythagorean scale using abjad notation, and was further refined in Ottoman texts to embrace intervals with errors under 1 cent when mapped to 53 equal steps.^[12] Such non-Western precursors emphasized practical microtonal hierarchies over equal division but highlighted the utility of 53 as a theoretical benchmark for fine-tuning.

Key contributors and publications

Isaac Newton explored ideas similar to 53 equal temperament in unpublished work around 1664–1665.^[2] The development of 53 equal temperament was formalized in the 17th century by Nicholas Mercator, a German mathematician and astronomer, who provided the first precise mathematical account of dividing the octave into 53 equal parts to achieve a closer approximation to the pure fifth than previous systems. In his 1668 treatise Logarithmotechnia, Mercator employed logarithmic calculations to determine the size of the "artificial comma" required for this temperament, expressing it as the ratio 353/284, which highlighted its utility for theoretical tuning precision. This work laid the groundwork for understanding 53-TET as a high-resolution system capable of tempering out the Pythagorean comma with minimal error in fifths. Building on Mercator's calculations, William Holder, an English natural philosopher and music theorist, further advocated for 53 equal temperament in his 1694 publication A Treatise of the Natural Grounds, and Principles of Harmony. Holder confirmed the system's Pythagorean closeness by noting that 53 steps approximate 31 octaves and 36 just fifths exceptionally well, while also pointing out its superior rendering of the just major third (5/4) compared to 12-TET. He favored the Holdrian comma (approximately 22.64 cents) as the unit for this scale, emphasizing its practical theoretical advantages over slightly larger divisions like 55-TET. In the 18th and 19th centuries, 53-TET received mentions in broader discussions of temperament theory by figures such as Leonhard Euler, whose 1739 Tentamen novae theoriae musicae explored comma tempering and interval approximations, influencing subsequent analyses of tuning accuracy. The 20th-century revival of microtonal music brought renewed attention to 53-TET through Russian composer Leonid Sabaneyev, who proposed it in the early 1900s as a viable system for modern orchestration.^[13] Post-2000, digital tools have amplified its accessibility; for instance, the Scala software, developed by Manu Nagel and hosted by the Huygens-Fokker Foundation, supports 53-TET tuning files for synthesis and composition, enabling widespread experimentation in electronic and contemporary music.^[14]

Notation and scales

Notation systems

Sagittal notation provides a comprehensive system for representing microtonal intervals in 53 equal temperament (53-TET), using arrow-based accidentals to denote steps from the 12-tone equal temperament baseline. Each Sagittal accidental corresponds to a specific number of 53-TET steps, with the basic up-arrow (↑) symbolizing a single step of approximately 22.64 cents (1/53 of an octave), and down-arrow (↓) for the inverse. For larger alterations, combinations of symbols are employed, such as the up-arrow with a slash (⩨) for two steps or more complex glyphs for finer distinctions, enabling precise notation of intervals like the Holdrian comma as a single step. This system, developed by George Secor and Dave Keenan, is particularly adapted for 53-TET in contexts like Turkish makam music, where it distinguishes microtonal nuances beyond standard sharps and flats.^[15][16] Fractional step notation offers a mathematical approach to labeling pitches in 53-TET, expressing them as fractions of the octave relative to a base note, such as C + 31/53 to approximate the perfect fifth (G) at roughly 701.96 cents. In this system, any pitch is denoted as base + n/53, where n ranges from 0 to 52, reflecting the equal division of the octave into 53 parts, each with a frequency ratio of $2^{1/53}$21/53. This notation emphasizes the temperament's precision in approximating just intervals, such as the major third at 17/53 of the octave. It is commonly used in computational music applications and theoretical analyses to map scale degrees without relying on visual symbols.^[6][5] Keyboard layouts for 53-TET instruments extend traditional designs to accommodate the full 53 steps per octave, often building on Adriaan Fokker's 31-note keyboard principles by adding keys for the remaining divisions. These layouts typically arrange keys in a linear or isomorphic grid to facilitate playing microtonal scales, with each key corresponding to one 53-TET step, allowing direct access to intervals like the 31-step fifth. Modern implementations, such as those on the Lumatone isomorphic keyboard, map 53-TET across multiple octaves using consistent interval patterns for intuitive navigation. Such designs are essential for performing 53-TET music on physical instruments, though they require expanded key counts compared to standard 12-key keyboards.^[17][18] Modern digital notation tools, including MuseScore plugins, support 53-TET through customizable accidentals and tuning parameters. The Microtonal EDO plugin, for instance, enables retuning of notes to 53-TET steps using arrow symbols (e.g., ^ for one step up, ^^ for two), integrated with key signatures and transposition functions that adjust pitches by fractional octaves. This allows composers to notate and playback 53-TET scores accurately, often incorporating Kite Giedraitis' ups-and-downs convention for microtonal alterations. These plugins bridge traditional staff notation with high-resolution temperaments, facilitating composition and analysis in software environments.^[19]

Scale diagrams and chord structures

In 53 equal temperament (53-TET), scale diagrams typically illustrate the division of the octave into 53 equal steps, each approximately 22.6415 cents, providing high-fidelity approximations to just intonation intervals. A linear diagram positions the notes sequentially from 0 to 53, with the root at step 0 (unison, 0 cents) and returning to the octave at step 53 (1200 cents). Key approximate just notes are labeled based on their proximity to 5-limit ratios, such as the perfect fifth at 31 steps (701.887 cents, approximating 3/2 at 701.955 cents, error -0.068 cents) and the major third at 17 steps (384.906 cents, approximating 5/4 at 386.314 cents, error -1.408 cents).^[7] For visual clarity, the following table summarizes selected steps in a C-based 53-TET scale, highlighting approximations to common just intervals: This tabular representation aids accessibility by enumerating positions without requiring graphical tools, though circular diagrams can also depict the scale as a clock-like wheel to emphasize enharmonic equivalences.^[20] Chord structures in 53-TET leverage its precise interval approximations for consonant harmony. The major triad is constructed at steps 0-17-31, yielding intervals of a near-just major third (17 steps) and perfect fifth (31 steps), with the upper minor third (14 steps) completing the stack; this configuration tempers the schisma (1.953 cents) for smooth voice leading in 5-limit contexts. The minor triad uses 0-14-31, approximating a just minor third (14 steps to 6/5) below the fifth, offering a subtly warmer tonality than in 12-TET due to reduced beating. Quartal harmony examples include stacking perfect fourths (22 steps each, approximating 4/3 at +0.068 cents error), such as the quartal triad at 0-22-44, which evokes suspended, open sonorities suitable for modal interchange. The Pythagorean chain in 53-TET forms a closed circle of fifths, where each fifth spans 31 steps, and 53 such fifths total 1643 steps—exactly 31 octaves (31 × 53 steps)—returning precisely to the unison without residual comma, as the Mercator comma (≈3.615 cents) is fully tempered out. This equivalence to an extended 3-limit Pythagorean tuning enables seamless chains of pure-sounding fifths across the full 53-note set.^[20][2] For 7-limit extensions, 53-TET supports chords incorporating the harmonic seventh (7/4 ≈968.826 cents, best at 43 steps with +4.759 cents error), such as the otonal tetrad at 0-17-31-43, blending 5-limit triads with septimal color for richer harmonic series approximations; this structure tempers the septimal kleisma while maintaining consonance in modal progressions.

Applications

In Turkish makam theory

In Turkish makam theory, 53 equal temperament serves as a foundational model for approximating the microtonal intervals of the traditional 24-tone Arel-Ezgi-Uzdilek (AEU) system, dividing the octave into 53 equal steps known as komas (each approximately 22.64 cents, corresponding to the Holdrian comma).^[21] This structure refines the quarter-tone framework by incorporating additional commas, allowing for precise representation of maqam perdes (notes) with errors under 1 cent compared to historical tunings like the Abjad scale.^[12] For instance, the bakiya, a small whole tone of roughly 90 cents, is closely approximated by 4/53 of the octave (90.57 cents), enabling nuanced distinctions between small (4-koma) and large (5-koma) tones within the 9-koma whole tone.^[22] Specific makams benefit from these approximations, as seen in Hüseyni, where key intervals align well with 53-TET steps. The major second (9/8, approximately 203.91 cents) corresponds to 9 steps (203.77 cents), while the augmented second known as asiran (81/64, 407.82 cents) is rendered at 18 steps (407.55 cents), facilitating accurate scale construction.^[12] Ottoman composers and theorists, such as Dimitri Kantemir in the 18th century, theoretically employed comma-based divisions akin to 53-TET for notating makam melodies, though practical implementation remained limited to melodic traditions rather than fixed-pitch instruments.^[21] One advantage of 53-TET over 12-TET in makam performance lies in its superior handling of commas, which supports more authentic simulation of meend—the glissando-like ornamentations central to expressive phrasing—through finer step sizes that better approximate just intonation microintervals (e.g., a perfect fifth at 31/53 ≈ 701.89 cents versus 702 cents just).^[23] This granularity reduces the distortion of subtle pitch inflections inherent in 12-TET's coarser 100-cent steps, preserving the melodic fluidity of makam.^[22] Contemporary Turkish musicians have integrated 53-TET into digital tools for electronic maqam compositions, such as the Mus2okur software, which generates AEU-based scales and enables modal explorations via microtonal harmonies.^[22] Similarly, computer applications like those developed for 53-TET modal interchange allow for algorithmic generation of maqam progressions, bridging traditional theory with modern production techniques in electronic contexts.^[6]

In Western and modern music

In Western music, 53 equal temperament (53-TET) has been employed in experimental and microtonal contexts to extend traditional harmonic practices, leveraging its precise approximations to 5-limit just intonation intervals, such as a perfect fifth of 701.89 cents—nearly identical to the Pythagorean interval of 702 cents. This makes it suitable for meantone-derived works, where it tempers out small commas like the schisma (1.95 cents) and kleisma (1.97 cents), allowing composers to explore nuanced dissonances and resolutions beyond 12-TET while maintaining compatibility with classical chord structures.^[6][2] Twentieth-century microtonal pioneers, including Harry Partch and James Tenney, advanced the use of high-division equal temperaments and just intonation in Western experimental music. Partch developed the 43-tone scale for his corporeal music, while Tenney explored spectral pieces with microtonal elements, such as in Glissando (1982), primarily favoring just intonation. 53-TET has emerged as a practical system for approximating just and spectral tunings in later microtonal compositions.^[24][6] In contemporary settings, 53-TET has gained traction in electronic and popular music production. Multi-instrumentalist Jacob Collier employs microtonal tunings in his harmonic experiments, enhancing chord voicings in albums like Djesse Vol. 1 (2018) to achieve subtle beats and extended tertian stacks.^[6] Recent technological advancements have further popularized 53-TET in the 2020s. Ableton Live's tuning framework includes native 53-EDO presets, facilitating its use in electronic composition and live performance for genres like IDM and ambient. The Lumatone isomorphic controller, introduced in 2022, supports 53-EDO mappings on its hexagonal grid, enabling intuitive playability for microtonal producers and performers in software environments like DAWs.^[1][25]