Pythagorean tuning
Pythagorean tuning is a system of musical tuning based on the ratios of small whole numbers, primarily the octave (2:1) and the perfect fifth (3:2), which generates a diatonic scale through successive stacking of these intervals adjusted by octaves.[1] Attributed to the ancient Greek philosopher Pythagoras in the 6th century BCE, it forms the foundation of early Western music theory by aligning tones with simple harmonic relationships derived from string lengths or frequencies.[2] This tuning system constructs the major scale using seven distinct notes, with interval ratios such as the major second (9/8), major third (81/64), perfect fourth (4/3), perfect fifth (3/2), major sixth (27/16), and major seventh (243/128).[3] By repeatedly applying the 3:2 ratio for fifths—starting from a base tone and reducing by octaves when exceeding the next octave—it produces consonant perfect fifths and fourths that closely match the natural overtones of vibrating strings.[1] However, stacking twelve perfect fifths results in a slight discrepancy known as the Pythagorean comma (approximately 23.46 cents), where the total span exceeds seven octaves by a small interval (ratio 531441:524288), preventing the circle of fifths from closing perfectly.[2] Historically, Pythagorean tuning influenced medieval and Renaissance music, emphasizing melodic purity in monophonic and early polyphonic contexts, though its wide major thirds (81/64, about 407.8 cents) were considered dissonant compared to the just intonation major third (5/4, 386 cents).[4] Its advantages include mathematical simplicity and strong fifths that support modal music, but limitations such as the comma and uneven semitones (e.g., 256/243 for the diatonic semitone) led to the development of alternative systems like meantone temperament and equal temperament for greater harmonic flexibility in later Western music.[3] Today, it remains relevant in historical performance practice, microtonal exploration, and as a pedagogical tool for understanding the mathematical basis of consonance.[1]Fundamentals
Definition and principles
Pythagorean tuning is a system of musical intonation in which intervals are derived from simple integer frequency ratios, primarily using the perfect fifth (3:2) and the octave (2:1) to generate a diatonic scale with pure, consonant sounds.[5] In this approach, musical intervals represent ratios of the frequencies of vibrating sounds, where the octave serves as the foundational interval with a ratio of 2:1, corresponding to a pitch doubling that the human ear perceives as the same note at a higher register, measured as exactly 1200 cents in logarithmic terms.[6] The perfect fifth, with its 3:2 ratio, approximates 702 cents and forms the basis for stacking intervals to approximate the octave, yielding consonant harmonies rooted in natural acoustic principles.[7] The key principles of Pythagorean tuning emphasize reliance on small prime integers—chiefly 2 and 3—to create intervals that align with the harmonic series, promoting stability in monophonic and early polyphonic music without equal temperament's compromises.[8] This method divides the octave through successive approximations via the 3:2 fifth, resulting in a scale where most intervals, such as the fourth (4:3), are pure and free of beats when performed on instruments like the monochord.[5] Unlike later systems, it prioritizes fifths over thirds, reflecting an acoustic focus on vertical sonorities that sound naturally harmonious due to their low-integer ratios.[6] Historically, Pythagorean tuning is named after the ancient Greek philosopher Pythagoras (c. 570–495 BCE), who discovered the correspondence between simple integer ratios and musical concords through experiments with vibrating strings on a monochord, identifying the octave (2:1), fifth (3:2), and fourth (4:3) as fundamental.[8] Although Pythagoras laid the groundwork by linking mathematics to acoustics, the complete tuning system as a diatonic scale emerged later among his followers and in subsequent Greek theory, influencing Western music for centuries.[5]Ratio-based construction
Pythagorean tuning builds the scale through a systematic process of stacking perfect fifths, each with a frequency ratio of 3:2, starting from a base note and adjusting for octaves by dividing by powers of 2 to confine the pitches within a single octave range.[1][9] This method generates the notes sequentially via the circle of fifths, where each step ascends by a fifth, forming a spiral approximation rather than a closed circle due to the incommensurability of the ratios.[10] The construction begins with the base note, conventionally assigned the ratio 1:1 (for example, C). To derive the next note (G), multiply by 3:2, yielding 3:2. For the following note (D), multiply the current ratio by 3:2 again (9:4), then divide by 2 to reduce the octave, resulting in 9:8. This process continues: A is obtained by 9:8 × 3:2 = 27:16; E by 27:16 × 3:2 = 81:32, then ÷2 = 81:64; B by 81:64 × 3:2 = 243:128. The seventh note, F, is typically derived as the pure fourth from the base (inverse of the fifth, adjusted by an octave: 1 ÷ 3:2 × 2 = 4:3), completing the diatonic scale within one octave.[1][9] The ratios for the notes in the C major scale, generated by stacking perfect fifths with F derived as a perfect fourth from C, are as follows:| Note | Ratio |
|---|---|
| C | 1:1 |
| D | 9:8 |
| E | 81:64 |
| F | 4:3 |
| G | 3:2 |
| A | 27:16 |
| B | 243:128 |
| C | 2:1 |
Interval Structure
Perfect fifths and octaves
In Pythagorean tuning, the perfect fifth is the foundational interval, defined by a frequency ratio of 3:2, where the higher note's frequency is one-and-a-half times that of the lower note.[11] This ratio yields an acoustically pure interval that is highly consonant, producing a smooth, stable sound without perceptible beats when performed on instruments like strings or voices.[12] In the diatonic scale, the perfect fifth spans seven semitones, serving as the primary generator for constructing the tuning system's pitches.[13] The octave, with a frequency ratio of 2:1, acts as the tuning's bounding interval, representing the perceptual equivalence of notes differing by a doubling or halving of frequency.[11] It normalizes all generated pitches to a single octave range, ensuring the scale remains within a practical perceptual span while maintaining the purity of the underlying ratios.[11] Pythagorean tuning builds its scale through the stacking of pure perfect fifths, where six such intervals—each at 3:2—produce all seven distinct pitches of the diatonic scale.[11] The octave provides closure by reducing these stacked fifths modulo powers of 2.[11] Acoustically, the perfect fifth's consonance arises from its early appearance in the harmonic series, as the third harmonic of the fundamental aligns closely with the second harmonic of the fifth, minimizing dissonance and enabling beat-free intervals in performance.[14] This alignment, rooted in the physics of vibrating strings and air columns, underpins the tuning's emphasis on natural resonance.[14]Derived major and minor intervals
In Pythagorean tuning, major and minor intervals beyond the perfect fifth and octave are derived by stacking multiple perfect fifths (ratio 3:2) and adjusting by octaves (ratio 2:1) to reduce the result within a single octave, ensuring all ratios are powers of 2 and 3. This process generates the composite intervals of the diatonic scale through successive multiplications and divisions, prioritizing the purity of fifths while accepting approximations for other intervals.[11][15] The major second is obtained by two stacked fifths reduced by one octave: \frac{(3/2) \times (3/2)}{2} = \frac{9}{8} The major third arises from four fifths reduced by two octaves: \frac{(3/2)^4}{2^2} = \frac{81}{64} The major sixth comes from three fifths reduced by one octave: \frac{(3/2)^3}{2} = \frac{27}{16} The major seventh is derived from five fifths reduced by two octaves: \frac{(3/2)^5}{2^2} = \frac{243}{128} These ratios reflect the "wide" whole tones and thirds characteristic of Pythagorean intonation.[2][16] Minor intervals in Pythagorean tuning are typically obtained as complements to major intervals within the octave or through the circle of fifths, yielding narrower approximations compared to other tunings. The minor second, or limma, is the complement of the major seventh: \frac{2}{243/128} = \frac{256}{243} The minor third is the complement of the major sixth: \frac{2}{27/16} = \frac{32}{27} The minor sixth is the complement of the major third: \frac{2}{81/64} = \frac{128}{81} The minor seventh is the complement of the major second: \frac{2}{9/8} = \frac{16}{9} These derivations ensure the intervals fit the 3-limit constraint of Pythagorean tuning.[11][15] The following table summarizes the key derived intervals with their ratios:| Interval | Ratio |
|---|---|
| Major second | 9:8 |
| Major third | 81:64 |
| Major sixth | 27:16 |
| Major seventh | 243:128 |
| Minor second | 256:243 |
| Minor third | 32:27 |
| Minor sixth | 128:81 |
| Minor seventh | 16:9 |
Mathematical Properties
Interval sizes in cents
In Pythagorean tuning, intervals are quantified using cents, a logarithmic unit that divides the octave into 1200 equal parts, allowing precise comparison across tuning systems. The size of any interval in cents is calculated as $1200 \times \log_2(r), where r is the frequency ratio of the interval. This formula applies uniformly to all Pythagorean intervals, which are derived from stacking perfect fifths (3:2 ratio) and adjusting for octaves, resulting in a characteristic "wolf" fifth in the full chromatic scale due to the Pythagorean comma.[11] The following table lists the approximate sizes in cents for the 12 chromatic intervals in Pythagorean tuning, starting from unison, based on standard ratios. These values highlight deviations from equal temperament (where all semitones are 100 cents), such as the diatonic semitone at about 90 cents and the chromatic semitone at about 114 cents. Notably, the Pythagorean major third measures approximately 408 cents, which is wider than the just intonation major third of 386 cents, while the minor third is narrower at 294 cents compared to just intonation's 316 cents.[11][17]| Interval | Cents |
|---|---|
| Unison | 0.00 |
| Minor second | 90.22 |
| Major second | 203.91 |
| Minor third | 294.13 |
| Major third | 407.82 |
| Perfect fourth | 498.04 |
| Augmented fourth (tritone) | 611.73 |
| Perfect fifth | 701.96 |
| Minor sixth | 792.18 |
| Major sixth | 905.87 |
| Minor seventh | 996.09 |
| Major seventh | 1109.78 |
| Octave | 1200.00 |