Semitone
A semitone, also known as a half step or minor second, is the smallest musical interval commonly used in Western tonal music, representing the pitch distance between two adjacent notes on a piano keyboard, such as from C to C-sharp or E to F.[1][2][3] In the standard equal-tempered tuning system, which divides the octave into 12 equal parts, each semitone has a frequency ratio of $2^{1/12} and measures exactly 100 cents, where an octave spans 1200 cents.[4][5] Semitones form the foundational building blocks of musical scales and harmonies, with two semitones combining to create a whole tone (or major second).[6] For instance, the C major scale ascends through the pattern of whole, whole, half, whole, whole, whole, half steps—corresponding to the intervals C–D (tone), D–E (tone), E–F (semitone), F–G (tone), G–A (tone), A–B (tone), and B–C (semitone)—demonstrating how semitones define key structural points like the leading tone.[6] This intervallic framework applies across major and minor keys, enabling modulation and chromaticism in compositions.[7] In acoustic terms, the semitone's perceptual size can vary slightly with frequency due to just noticeable differences in pitch (approximately 5–8 cents), but equal temperament ensures consistent spacing for practical performance on instruments like the piano and guitar.[4] Historically rooted in Western music theory since the development of diatonic scales, the semitone facilitates expressive techniques such as dissonance resolution and is essential in genres from classical to contemporary, where it underpins chord progressions and melodic contours.[8][9]Definition and Fundamentals
Core Definition
In Western music theory, a semitone is defined as the smallest interval commonly used, representing the distance between two adjacent notes in the chromatic scale.[2] This interval corresponds to a minor second, spanning one half step between pitches.[10] In the diatonic scale, semitones occur as the smaller steps between certain natural notes, such as from E to F or B to C.[11] The prefix "semi-" in semitone indicates "half," signifying that it is half the size of a whole tone, which is a major second consisting of two semitones.[12] For example, the interval from C to C♯ exemplifies a semitone, as these notes are neighboring keys on a piano keyboard.[1] In the standard equal temperament tuning system of Western music, an octave encompasses exactly 12 semitones, providing the foundational division for pitch organization across instruments and compositions.[13]Acoustic and Frequency Basis
A semitone, as an interval, corresponds to specific frequency ratios that define its acoustic properties. In just intonation, the diatonic semitone—often encountered in scalar contexts like the major scale between the third and fourth degrees or seventh and eighth—has a frequency ratio of 16:15, approximately 1.0667.[14] This ratio yields a perceptual size of about 111.7 cents, calculated using the standard formula for musical intervals. Another common just intonation semitone, the chromatic variety (such as between C and C♯), uses a ratio of 25:24, approximately 1.0417, measuring roughly 70.7 cents.[15] These ratios derive from simple integer proportions that promote consonance when combined in harmonies, reflecting the harmonic series' influence on tone relationships.[16] In equal temperament, widely used in modern Western music, the semitone is standardized to a frequency ratio of $2^{1/12}, approximately 1.05946, ensuring the octave divides evenly into 12 equal steps.[17] This geometric progression arises because an octave spans a 2:1 frequency ratio, and raising the twelfth root evenly distributes the logarithmic scale of pitch perception. The interval measures exactly 100 cents in this system, providing a consistent approximation that tempers the variable just ratios for instrumental versatility across keys. Human perception of pitch is logarithmic, meaning equal intervals are sensed as proportional changes in frequency rather than arithmetic differences; a note twice as high sounds an octave higher, regardless of absolute frequency.[18] This psychophysical principle underpins the cent as a unit of measure, defined by the formula c = 1200 \times \log_2 \left( \frac{f_2}{f_1} \right), where f_2 and f_1 are the higher and lower frequencies, respectively.[19] The factor of 1200 assigns the full octave (ratio 2:1) exactly 1200 cents, so each equal-tempered semitone receives $1200 / 12 = 100 cents, as \log_2 (2^{1/12}) = 1/12. For just intonation examples, applying the formula to 16/15 gives $1200 \times \log_2 (16/15) \approx 111.7 cents, while 25/24 yields $1200 \times \log_2 (25/24) \approx 70.7 cents, illustrating how cents quantify deviations from equal temperament for precise tuning analysis.[15]Notation and Interval Equivalents
Alternative Names
In music theory, the semitone is primarily synonymous with the minor second, particularly in diatonic contexts where it represents the smallest interval between scale degrees.[20] The minor second is understood as the smaller half of a whole tone, distinguishing it from the larger major second, which spans two semitones.[21] From the perspective of the unison interval, the semitone is equivalently termed an augmented unison, serving as its enharmonic equivalent by raising the upper note by one semitone.[21] Common English-language alternatives include "half step" and "half tone," both denoting the minimal pitch distance in Western music.[1] The term "demi-ton," originating from French music theory, similarly translates to half tone and has influenced historical nomenclature for this interval.[22] In harmonic analysis, the minor second is frequently used to describe melodic or vertical intervals within scales and chords, emphasizing its role in tension and resolution.[20] Conversely, the augmented unison appears in chord naming conventions, such as labeling the interval from C to C♯ in an augmented unison chord.[23] In unequal temperaments, semitones are distinguished as major or minor based on their relative sizes within the whole tone, with the minor semitone being the smaller variant.[23]Musical Notation
In musical notation, semitones are primarily indicated through chromatic accidentals, which alter the pitch of a note by one semitone from its position in the prevailing key signature. The sharp symbol (♯) raises a note by a semitone, the flat symbol (♭) lowers it by a semitone, and the natural symbol (♮) cancels any previous sharp or flat, restoring the note to its original pitch class in the key.[24][25] These accidentals are placed before the notehead on the staff and apply to all subsequent notes on the same line or space within the same measure unless canceled.[26] An ascending semitone can be notated as the interval from C to C♯ (using a sharp) or from B to C (a diatonic step without alteration), while a descending semitone appears as C to B or C♯ to C (using a natural after a sharp).[27] These notations ensure precise representation of the smallest interval in the chromatic scale, allowing performers to execute half steps accurately across the staff.[28] Enharmonic equivalents further illustrate semitone notation, where different symbols denote the same pitch but serve contextual purposes in harmony or key. For instance, from C, the note D♭ (a flat) and C♯ (a sharp) are enharmonically equivalent, both one semitone above C, yet chosen based on the musical context to simplify reading or fit the key signature.[29][30] In staff notation, semitones maintain consistent representation across clefs, such as treble, bass, or alto, though their vertical positions shift; for example, the semitone from E to F occupies adjacent lines in the treble clef without a space between. For transposing instruments, written notation adjusts to account for the instrument's pitch displacement: a B♭ clarinet, which sounds a major second (two semitones) lower than written, requires the player to read a semitone-altered part to produce concert pitch, such as notating D to sound C.[31][32] On the piano keyboard, semitones correspond to the layout of white and black keys, where adjacent keys—regardless of color—form a half step; the black keys fill gaps between most white keys (e.g., C to C♯/D♭), except between E-F and B-C, which are natural white-key semitones. This alternating pattern visually reinforces the diatonic whole steps (two semitones, skipping a key) and half steps essential for reading and playing chromatic passages.[33][34]Historical Development
Ancient and Medieval Origins
The concept of the semitone emerged in ancient Greek music theory through Pythagoras' division of the tetrachord, a foundational four-note sequence spanning a perfect fourth (ratio 4:3), into two whole tones (each with ratio 9:8) followed by a smaller interval known as the leimma, with ratio 256:243, approximating 90 cents.[35] This leimma represented the earliest formalized semitone-like interval, distinguishing it from larger tones as the remainder after subtracting two tones from the fourth, and it became integral to constructing scales and modes.[36] In Greek modal systems, such as the Dorian mode, the semitone (leimma) occupied a specific position within the tetrachord structure, typically between the third and fourth notes, creating the pattern of two tones followed by a semitone in the lower tetrachord (e.g., intervals yielding E-D-C-B in descending form, with the leimma between C and B).[37] This placement contributed to the mode's characteristic sound, emphasizing stability and ethos, as the semitone provided melodic tension and resolution within the diatonic framework of ancient Greek harmoniai.[38] Medieval music theory inherited and adapted these ideas, with Boethius in his 6th-century treatise De institutione musica describing the semitonium as an interval smaller than the whole tone (tonus), named not because it was exactly half but because it was incomplete relative to the tone, often quantified as the Pythagorean limma (256:243).[39] Boethius positioned the semitonium as a perceptual and mathematical unit within larger intervals like the diatessaron (two tones plus one semitonium), influencing subsequent quadrivial studies of music as a branch of mathematics.[39] By the 11th century, Guido d'Arezzo advanced practical applications through his hexachord system, a six-note solmization framework (ut-re-mi-fa-sol-la) that incorporated a fixed semitone between mi and fa, visualized on the Guidonian hand as a mnemonic diagram for sight-singing and scale navigation.[40] This system allowed mutation between hexachords to traverse the gamut, with semitones enabling modal flexibility, while musica ficta introduced chromatic alterations (e.g., b mollis or b durum) to adjust semitones for melodic and harmonic propriety in polyphony.[41] Early medieval approximations of the diatonic semitone also explored ratios beyond strict Pythagoreanism, such as 16:15 (approximately 112 cents), which represented a just intonation variant for the interval between scale degrees like E-F, offering a purer sonic quality in theoretical discussions of tonal divisions.[42]Renaissance to Modern Evolution
During the Renaissance period, the semitone gained prominence through the incorporation of chromatic semitones in polyphonic compositions, allowing for greater expressive depth and harmonic complexity. Composers such as Josquin des Prez exemplified this evolution in works like his motets and masses, where chromatic alterations introduced semitones that deviated from diatonic frameworks to enhance emotional tension and resolution.[43] This shift was supported by the development of meantone tuning systems around the early 16th century, which prioritized pure major thirds—approximating a 5:4 ratio—over the Pythagorean tuning's emphasis on perfect fifths, thereby reducing dissonant "wolf" intervals and making chromatic semitones more consonant in practice.[44] Meantone temperaments, such as quarter-comma meantone, tempered the major thirds to about 386 cents while enlarging diatonic semitones to roughly 117 cents and shrinking chromatic ones to 76 cents, facilitating the polyphonic textures of the era.[45] In the Baroque era, the semitone's role expanded with the advent of well temperaments, which distributed irregularities across the octave to enable modulation between all keys without extreme dissonance. Andreas Werckmeister's temperaments, detailed in his 1681 treatise Orgel-Probe, represented a key innovation by tempering fifths unequally to create a "well-tempered" system usable in every key, with semitones varying slightly but approaching the 100-cent equal division seen in later standards.[46] These systems built on meantone by allowing composers like Johann Sebastian Bach to explore chromatic progressions freely, as the semitone served as a bridge for key changes in intricate fugues and suites. By the early 18th century, such temperaments had become precursors to equal temperament, standardizing the semitone's perceptual uniformity while preserving some tonal color.[47] The 18th and 19th centuries marked the semitone's standardization through Johann Sebastian Bach's The Well-Tempered Clavier (1722), a collection of 48 preludes and fugues that demonstrated the semitone's functionality across all 24 major and minor keys in a well-tempered framework, influencing keyboard composition and pedagogy profoundly.[48] This work underscored the semitone's versatility in harmonic exploration, paving the way for equal temperament's widespread adoption by the mid-19th century, particularly with the rise of chromaticism in Romantic music, where the semitone was fixed at exactly 100 cents (a frequency ratio of $2^{1/12}) to support unrestricted modulation.[49] Hermann von Helmholtz's On the Sensations of Tone (1863) provided a scientific foundation for this evolution, analyzing the semitone's acoustic properties through resonance and beat frequencies to explain its perceptual role in consonance and dissonance.[50] The 20th century saw challenges to the semitone's dominance as the smallest melodic unit, with microtonal experiments subdividing it further for novel expressive possibilities. Czech composer Alois Hába pioneered quarter-tone systems in works like his Suite for Quarter-Tone Piano (op. 1a, 1918) and subsequent string quartets from 1919 onward, dividing the semitone into two 50-cent intervals to create atonal and non-Western-inspired scales that expanded beyond equal temperament's constraints.[51] These innovations, influencing modernist composers, highlighted the semitone's historical contingency while reinforcing its foundational status in Western music theory.Semitones Across Tuning Systems
Pythagorean and Just Intonation
In Pythagorean tuning, the primary semitone is the limma, with a frequency ratio of \frac{256}{243}, equivalent to approximately 90.225 cents. This interval arises from stacking perfect fifths of ratio \frac{3}{2} (approximately 701.955 cents each) and reducing by octaves, specifically calculated as \left( \frac{3}{2} \right)^{-5} \times 2^{3} = \frac{2^{8}}{3^{5}} = \frac{256}{243}, representing the remainder after seven octaves and twelve fifths in the circle of fifths. The limma forms the diatonic semitones in the Pythagorean diatonic scale, such as between B and C or E and F, emphasizing the system's reliance on powers of 2 and 3 for interval purity, particularly in fifths and octaves.[52][53] In 5-limit just intonation, which incorporates the prime 5 alongside 2 and 3, semitones exhibit greater variety to achieve purer consonant intervals like major and minor thirds. The diatonic semitone has a ratio of \frac{16}{15}, approximately 111.731 cents, derived as the difference between a perfect fourth (\frac{4}{3}) and a major third (\frac{5}{4}), or \frac{4/3}{5/4} = \frac{16}{15}; an example occurs from E (\frac{5}{4}) to F (\frac{4}{3}). The chromatic semitone, smaller at \frac{25}{24} or about 70.672 cents, appears in enharmonic adjustments, such as from E to F♭, calculated as the difference between a minor third (\frac{6}{5}) and a minor second derived from overtones, or \frac{5/4}{6/5} = \frac{25}{24}. These ratios prioritize harmonic consonance from the overtone series, with the diatonic semitone larger than the Pythagorean limma to accommodate the just major third of 386.314 cents.[54][55] Extended just intonations, such as 7-limit systems incorporating the prime 7, introduce further semitone varieties for enhanced harmonic flexibility, often using smaller prime factors to refine intervals beyond 5-limit purity. A representative 7-limit semitone is \frac{21}{20}, approximately 85.420 cents, used for chromatic steps like C to C♯ in certain scales, derived from ratios involving the prime 7 such as (3×7)/(4×5). Smaller intervals like the septimal diesis \frac{36}{35}, about 48.770 cents, emerge as differences such as between \frac{7}{6} and \frac{6}{5}, enabling microtonal distinctions in extended harmonies. These additions allow for ratios involving 7 to approximate or refine semitones closer to perceptual neutrality.[56][57] Within these systems, comparisons highlight trade-offs in interval purity: Pythagorean tuning yields eleven pure fifths but culminates in a wolf interval—a narrow fifth of approximately 678.49 cents (flat by the Pythagorean comma of 23.463 cents from a pure fifth), between notes like G♯ and E♭ due to the circle of fifths mismatch. Just intonation, by contrast, achieves purer major thirds (e.g., \frac{5}{4} \approx 386.314 cents vs. Pythagorean's \frac{81}{64} \approx 407.820 cents) and avoids localized wolves through flexible ratio selection from the harmonic series, though it may introduce enharmonic distinctions or comma adjustments for octave equivalence.[44][58]| System | Semitone Type | Ratio | Cents (approx.) |
|---|---|---|---|
| Pythagorean | Limma (diatonic) | 256/243 | 90.225 |
| 5-Limit Just | Diatonic | 16/15 | 111.731 |
| 5-Limit Just | Chromatic | 25/24 | 70.672 |
| 7-Limit Just | Example (chromatic) | 21/20 | 85.420 |
| 7-Limit Just | Septimal diesis | 36/35 | 48.770 |