Tuning is not a staircase leading toward one perfect system. It is the way a community makes a continuous world of frequencies usable: for a season, a ritual, a drone, an ensemble, a material, a repertoire, or a global exchange. This unit develops the mathematics of pitch organization while placing the major systems in their physical and cultural contexts. You will learn to:
A tuning system is a coordinate system for pitch. The octave folds logarithmic frequency into a circle; each culture chooses where to place landmarks on that circle and how finely to divide it. Equal steps become regular polygons, just ratios become constellations, key colour becomes regional hue bias, and beats become visible shimmer. Chromatone can make these choices inspectable rather than hiding them behind a single default grid.
Throughout the unit, ask:
What must sound well together, and with what? The answer may be the year, the hour, a drone, a bronze ensemble, a choir, a keyboard, a repertoire, or an international standard. A system is not "in tune" in the abstract. It is a negotiated relationship among bodies, instruments, environments, and expectations.
This unit closes the Sound module. It uses:
The ear responds to ratios more consistently than to frequency differences. The jump from 100 Hz to 200 Hz is an octave, and so is the jump from 1000 Hz to 2000 Hz. A linear ruler cannot represent those equal musical distances. The cent scale can: $$ \text = 1200 \times \log_2\left(\frac\right) $$
Simple ratios produce stable periodic relationships:
| Ratio | Interval | Cents | Frequency relation |
|---|---|---|---|
| 2:1 | octave | 1200 | the upper tone completes two cycles for one |
| 3:2 | perfect fifth | 701.96 | three cycles for two |
| 4:3 | perfect fourth | 498.04 | four cycles for three |
| 5:4 | just major third | 386.31 | five cycles for four |
| 6:5 | just minor third | 315.64 | six cycles for five |
| The octave lets us identify frequencies separated by powers of two. But twelve pure fifths do not equal seven octaves: | |||
| $$ | |||
| \left(\frac\right)^ \div 2^7 = \frac | |||
| $$ | |||
| The excess is the Pythagorean comma, about 23.46 cents. On a pitch circle, the fifth spiral misses its starting point. On a calendar, a 360-day ideal year misses the solar year by about 5.24 days. These are not the same number, but they have the same structural lesson: a repeating human map does not perfectly contain a physical cycle. |
The Pythagorean comma and the syntonic comma must remain distinct:
| Comma | Ratio | Cents | Problem |
|---|---|---|---|
| Pythagorean | $531441:524288$ | 23.46 | twelve pure fifths fail to close at seven octaves |
| Syntonic | $81:80$ | 21.51 | a fifth-generated major third is sharper than a harmonic major third |
| The syntonic comma is the difference between: | |||
| $$ | |||
| \left(\frac\right)^4 \div 2^2 = \frac \approx 407.82\text | |||
| $$ | |||
| and the just major third: | |||
| $$ | |||
| \frac \approx 386.31\text. | |||
| $$ | |||
| One comma concerns closing a fifth cycle. The other concerns the conflict between fifth-generated and harmonic-series thirds. Temperaments are different policies for distributing these discrepancies. |
The history of tuning does not begin in one place or serve one musical purpose. In Han-dynasty China, the twelve lu pitch standards were associated with the twelve months. The sanfen sunyi method alternated fifth-like and fourth-like operations to generate the pipes. Pitch was part of calendrical and state order: the year was cast in bamboo and bronze. Accounts of flying ashes describe pitch pipes placed in a sealed chamber and filled with reed ash. At a seasonal moment, the ash was believed to leave the pipe whose pitch corresponded to the earth's qi. Whether or not the apparatus worked acoustically, the image shows what tuning could do: connect an institution to a cycle larger than performance. The return problem appeared here as a calendrical problem. Jing Fang extended the system to sixty lu and noticed the near-return after fifty-three steps. He Chengtian proposed distributing the error so the cycle would close. Qian Lezhi imagined 360 standards, one for each degree of a cosmological circle. In 1584, Zhu Zaiyu published a precise mathematical account of equal temperament, including the twelfth root of two, in an explicitly calendrical and metrological context. This publication priority belongs to Zhu; Simon Stevin developed related work independently in Europe, but his manuscript remained unpublished until 1884. Other cyclic traditions offer parallel models. The Byzantine oktoechos organizes eight modes through a weekly cycle, while the Gregorian temporale joins modes and melodies to the liturgical year. These are not primitive attempts at a universal scale. They are different ways of making pitch participate in time. Teaching image: show a circle of months, a set of pitch pipes, and a fifth spiral that misses its return. The visual question is not "which one is correct?" but "which cycle is this system keeping?"
Indian theory speaks of 22 shruti, literally something heard. The number has been interpreted in several ways: as a theoretical partition, a perceptual vocabulary, or a family of interval positions. Fixed reconstructions such as those associated with Alain Daniélou remain useful models but are not a complete description of practice. N. A. Jairazbhoy and later performance research emphasize flexible intonation, ornament, context, and the relationship to the drone. The raga is not simply a scale. Ragas can be associated with times of day and seasons: Bhairav with dawn, Yaman with evening, and Megh or Malhar with the monsoon. The tanpura's sustained sa acts as an axis; other tones can lean, glide, and return around it. Pitch organization here is temporal and relational rather than a fixed keyboard grid.
Safi al-Din al-Urmawi's Kitab al-Adwar, the Book of Cycles, described a 17-step octave and included neutral interval categories associated with Zalzal. Later Ottoman theory, including the work of Dimitrie Cantemir, recorded makam through fine interval distinctions. The Arel-Ezgi-Uzdilek tradition formalized an octave as 53 commas, with a whole tone of nine commas. These grids are important theoretical tools, but practice exceeds them. Maqam and Persian dastgah use inflection, regional style, and expressive movement. A neutral third may be a range of gestures rather than a single fixed frequency. The 1932 Cairo Congress recommended quarter-tone standardization for teaching and notation, but living maqam practice did not simply become 24-TET.
In the seasonal practices described by Thomas Turino, tarka music is associated with the wet season and carnival, while siku belongs to the dry season. A siku melody is split between the ira and arka rows: two players interlock to produce one line. The system organizes not only pitch but also weather, repertoire, and social cooperation.
The phrase "in tune" can mean "in tune with this instrument" or "in tune with these people."
Indonesian slendro is often described as five roughly equal steps, but measured sets vary and are not identical 5-TET. Pelog generally uses seven unequal steps, with regional and set-specific practice. A gamelan is tuned to itself: its bronze, resonators, room, repertoire, and hereditary tuner form one acoustic ecology. Paired instruments may be deliberately detuned to create ombak, a slow beating or wave. The beating is not an error to remove. It is a valued animation of the ensemble. Embat names the individual intervallic character of a set. Here the Sethares principle becomes concrete: consonance depends on the relationship between interval and timbre, and bronze instruments can support a different set of preferred relationships from harmonic strings.
Mbira tunings can be maker's signatures rather than standard copies. Gyil xylophones use local scale practices, resonators, and buzzing membranes whose timbre belongs to the instrument's identity. In Central African polyphony, the agreement is made through coordinated performance, not necessarily by matching an external frequency chart. Drone traditions provide another solution. Indian tanpura, Byzantine ison, Scottish bagpipes, and other drone-based practices establish a stable reference while permitting flexible intonation above it. One voice supplies continuity; other voices can negotiate detail. Europe also had plural pitch worlds. Baroque Chorton and Kammerton, organ pitch and chamber pitch, coexisted across towns and institutions. A440 did not replace an ancient universal. It replaced many local treaties with a modern international one.
Suppose an ensemble ascends through four pure fifths from C to E. The result is the Pythagorean third, about 407.82 cents. If the singers then lower E to the just third at 386.31 cents, the system has dropped by one syntonic comma. Repeat the operation and the pitch drifts. This is the comma pump. A cappella groups can drift because every local interval can be pure while the larger cycle loses its starting point. A fixed-pitch instrument must decide in advance which relationships to adjust. Tuning is therefore a kind of ensemble governance: local agreement and global stability cannot both be maximized everywhere.
The following systems are essential, but they are one regional lineage among many. Their sequence is not a universal history of musical intelligence.
Pythagorean tuning builds pitches by stacking $3:2$ fifths and reducing by octaves. Fifths and fourths are exceptionally close to their simple ratios. The major third, however, is 81:64, about 407.82 cents, roughly 21.5 cents sharper than 5:4. Best suited to: music that privileges fifths, fourths, and modal relationships. Cost: thirds generated by the fifth cycle are noticeably sharp; the circle does not close.
Five-limit just intonation builds a key or region from ratios such as 5:4, 6:5, 5:3, and 15:8. A C-major example is:
| Note | Ratio to C | Cents |
|---|---|---|
| C | 1:1 | 0 |
| D | 9:8 | 203.91 |
| E | 5:4 | 386.31 |
| F | 4:3 | 498.04 |
| G | 3:2 | 701.96 |
| A | 5:3 | 884.36 |
| B | 15:8 | 1088.27 |
| C | 2:1 | 1200 |
| The advantage is beat-free alignment for the chosen ratios and timbre. The cost is that a ratio pure in one key may not be pure after modulation. JI is not one fixed scale: changing the harmonic limit, reference, comma choices, or melodic context changes the available ratios. Harry Partch's 43-tone, 11-limit system and La Monte Young's 7-limit The Well-Tuned Piano show that just intonation can be a large compositional world rather than a return to a small ancient scale. |
Quarter-comma meantone tempers each fifth by roughly one quarter of the syntonic comma, about 5.38 cents. Four narrowed fifths then produce a pure 5:4 major third. The result is a family of especially sweet thirds in nearby keys. Advantages: excellent major thirds; a luminous Renaissance and early Baroque sound. Cost: limited modulation and a wolf fifth. On a twelve-note keyboard, the accumulated adjustments are forced into one interval, often heard as a harsh, widened fifth of roughly 737.6 cents. The wolf is not a failure of arithmetic. It is the visible remainder of a decision to preserve other intervals.
Well-temperaments such as Werckmeister III, Kirnberger III, Vallotti, and Young distribute the comma unequally across several fifths. All major keys become usable, but they do not sound identical. Some keys have purer thirds and fifths; others carry more beating and tension. This creates key colour: a distinct intervallic and affective character for each key. Bach's Well-Tempered Clavier demonstrates a world in which every key is available, not necessarily a world of modern equal temperament. The precise temperament Bach used remains debated; the broader historical point is secure. Playing the work in 12-TET keeps the architecture while flattening some of its acoustic lighting.
12-TET divides the octave into twelve equal logarithmic steps: $$ +r = \sqrt12 \approx 1.059463094 $$ The frequency of step $n$ above $f_0$ is: $$ +f_n = f_0 \times 2^ $$ Every fifth is 700 cents, about 1.96 cents narrower than a pure fifth. The major third is 400 cents, about 13.69 cents wider than a just third. No interval except the octave has a simple integer ratio. Advantages: unrestricted transposition, enharmonic equivalence, standardized instruments, and easy international exchange. Costs: no pure thirds, flattened key colour, and a bias toward harmonic timbres and modulating keyboard traditions. 12-TET is best described as dominant and useful in Western practice, not physically inevitable or universally optimal. It is an honest middle ground: small enough to navigate, regular enough to build, and close enough to a pure fifth for many listeners. It is a treaty, not a law of nature.
An $N$-tone equal temperament divides the octave into $N$ equal steps: $$ \text = \frac\text, \qquad \text = \sqrtN $$
| System | Step size | What it can offer |
|---|---|---|
| 5-TET | 240 cents | a compact pentatonic approximation; useful comparison with slendro, though no slendro set is thereby explained |
| 7-TET | 171.43 cents | a distinct heptatonic grid associated with some comparisons to Southeast Asian and African practices, without replacing local practice |
| 19-TET | 63.16 cents | useful approximations of major and minor thirds; a compact alternative keyboard |
| 24-TET | 50 cents | quarter-tone notation and a rough grid for some Arabic, Persian, and twentieth-century Western practices |
| 31-TET | 38.71 cents | strong approximations of many 5-limit intervals; associated with Huygens and Fokker |
| 53-TET | 22.64 cents | excellent approximations of both fifths and thirds; related to Mercator, Holder, and some Ottoman theory |
| These are maps, not cultures. Calling a maqam "24-TET" or slendro "5-TET" can be useful for a synthesis experiment, but it must not erase gesture, tuning variation, instrument construction, or social meaning. |
Fifty-three equal steps approximate the octave with unusually fine resolution. A fifth is about 31 steps, or 701.89 cents, and a just major third is about 17 steps, or 384.91 cents. The system is historically interesting because several traditions approached 53 divisions independently or for different reasons: Jing Fang's near-return, Mercator's mathematical observation, Holder's proposal, and the 53-comma language of Ottoman theory. Similar numbers do not prove cultural identity. They show that some mathematical approximations are attractive when people seek a closed circle with many near-pure landmarks.
The 22 shruti of Indian theory should not be translated into twenty-two equal steps. The term can refer to heard distinctions, theoretical positions, and intonational resources within raga practice. The drone, melodic motion, ornament, and raga grammar determine how a pitch is realized. A 22-TET synthesizer can be an interesting instrument, but it is not a complete model of shruti.
Slendro is often represented as five roughly spaced notes; pelog as seven unequal notes. Actual gamelan sets vary in interval sizes and tuning curves. Paired detuning, bronze inharmonicity, resonator design, embat, repertoire, and ritual all matter. A polygon drawn over the pitch circle is a visual abstraction of a set, not the set itself.
Reference pitch has never been eternal:
Piano strings are stiff and inharmonic. Their partials follow an approximation such as: $$ +f_n = n f_0 \sqrt $$ If a tuner set every octave by the fundamentals alone, upper partials would not align. Tuners therefore stretch octaves: low notes are tuned relatively low and high notes relatively high so the partials fit the instrument's body. The Railsback curve, measured by O. L. Railsback in 1938, can reach roughly 30-35 cents sharp in the highest register of a concert grand, with the bass comparably displaced. Each piano receives its own negotiated version of the nominal grid. Electronic tuners estimate inharmonicity and calculate a custom stretch curve. The supposedly universal system is always bent by matter.
William Sethares formalized a powerful extension of the unit's central principle: find the dissonance curve of a timbre, then choose intervals that fit it. This helps explain why a harmonic-spectrum instrument, a bronze gamelan, a bell, and an inharmonic digital sound may prefer different pitch relationships. The modern frontier includes:
Log frequency modulo the octave gives a circular coordinate. With a reference frequency REF, the smooth hue mapping is:
const hue = (((Math.log2(f / REF) % 1) + 1) % 1) * 360;REF is a chosen cultural zero, such as A4 = 440 Hz. The modulo operation removes whole octaves and keeps the position within one octave. The result is not a claim that pitch and colour are naturally identical; it is a useful visual instrument. Any tuning can become a constellation on the same circle:
const stars = ratios.map((ratio) =>
(((Math.log2(ratio) % 1) + 1) % 1) * 360
);
const nTet = (steps) =>
Array.from({ length: steps }, (_, index) => index * 360 / steps);Examples:
REF in the hue mapping. Observe that changing the reference rotates the entire visual map without changing its internal relationships.