This article gives the practical tuning procedure for a Baroque-biased Clavitone temperament. It is designed to favor Baroque contrapuntal interval priorities, especially stable 8ths, 5ths, 4ths, and selected bridge intervals, while leaving the remaining irregularity where it is less disruptive musically.
For the theory, model, and rationale, see Psycho-Acoustic Temperament Hierarchy. This article is the tuner-facing guide: what to tune first, how the keyboard is divided into phases, and what the current hybrid traversal is doing in practice.
The tables below give the order of operations and the interval type to be checked at each step. They are not universal fixed-frequency targets. For octave rows, tune the octave as a consonant local stretch on the instrument in front of you. In grouped rows, move through the listed notes in sequence while continuing to sound back through the chain as it grows.
Use the same two checks while building each chain, and again before moving on to the next phase. This running recheck is what improves convergence.
Begin with a compact rational spine around C4. This gives a stable central region before any wider octave propagation begins.
| From | To | Interval | Goal |
|---|---|---|---|
| Reference | C4 | Set the central reference note. | |
| C4 | F3 | Down a 5th | Tune a rational 2/3 5th. |
| F3 | A♯2 | Down a 5th | Continue the lower branch of the spine. |
| C4 | G4 | Up a 5th | Tune a rational 3/2 5th. |
| G4 | D5 | Up a 5th | Continue the upper branch of the spine. |
| D5 | A5 | Up a 5th | Complete the first rational region. |
Before continuing, recheck the primary 5th spine, then run the harmonic-series sweep through the central region.
Once the first 5th spine is in place, use stretched octaves to stabilize the surrounding region before crossing to the second rational chain.
| Anchor | Expand To | Goal |
|---|---|---|
| C4 | C5, C3 | Set the central octave frame. |
| F3 | F4, F5 | Stabilize the lower-left branch. |
| A♯2 | A♯3, A♯4 | Carry the lower branch upward. |
| G4 | G3, G5 | Stabilize the upper-right branch around the center. |
| D5 | D4, D3 | Carry the D branch back through the middle. |
| A5 | A4, A3 | Complete the first octave-defined region. |
Before continuing, recheck the octave families just established around the first rational region, then repeat the harmonic-series sweep across that region.
This is the distinctive crossover. Use one major 3rd bridge and one 4th bridge to enter the second rational 5th spine.
| From | To | Interval | Goal |
|---|---|---|---|
| C4 | E4 | Major 3rd | Set the deliberate upper bridge at 5/4. |
| E4 | E3 | Down a stretched octave | Carry the bridge into the lower register. |
| E3 | B2 | Down a 4th | Set the deliberate lower bridge at 3/4. |
| B2 | F♯3 | Up a 5th | Begin the second rational spine. |
| F♯3 | C♯4 | Up a 5th | Continue the second spine. |
| C♯4 | G♯4 | Up a 5th | Continue the second spine. |
| G♯4 | D♯5 | Up a 5th | Complete the second rational region. |
Before continuing, recheck the bridge chain C4-E4-E3-B2 and the new 5th spine from B2 through D♯5, then shift the harmonic-series sweep chromatically across both regions.
After the second spine is stable, propagate stretched octaves outward from the new anchors. Work one chain at a time, keeping each local region consonant before moving on.
| Anchor | Continue By Octaves |
|---|---|
| E4 | E5, then E2 and E1 from E3 |
| B2 | B3, B4, B1, B0 |
| F♯3 | F♯4, F♯5, F♯2, F♯1 |
| C♯4 | C♯5, C♯3, C♯2, C♯1 |
| G♯4 | G♯5, G♯3, G♯2, G♯1 |
| D♯5 | D♯4, D♯3, D♯2, D♯1 |
| Anchor | Continue By Octaves |
|---|---|
| C3 | C2, C1 |
| D3 | D2, D1 |
| F3 | F2, F1 |
| G3 | G2, G1 |
| A3 | A2, A1, A0 |
| A♯2 | A♯1, A♯0 |
Before continuing, recheck each octave chain completed in the bass and middle registers, then use the harmonic-series sweep chromatically to confirm that the broader field still holds together.
Once the middle and bass are stable, continue the same octave process upward from the established upper anchors.
| Anchor | Continue Upward |
|---|---|
| A♯4 | A♯5, A♯6, A♯7 |
| B4 | B5, B6, B7 |
| C5 | C6, C7, C8 |
| C♯5 | C♯6, C♯7 |
| D5 | D6, D7 |
| D♯5 | D♯6, D♯7 |
| E5 | E6, E7 |
| F5 | F6, F7 |
| F♯5 | F♯6, F♯7 |
| G5 | G6, G7 |
| G♯5 | G♯6, G♯7 |
| A5 | A6, A7 |
Before continuing, recheck the upper octave chains you have just extended, then run the harmonic-series sweep chromatically through the finished temperament.
When the full traversal is complete, make one final pass using the same two checks.
For an audio demonstration of this temperament, see Clavitone Tuning demonstrated by BWV 846, Prelude in C Major, by J.S. Bach, performed by Ben Woolley.
For the theory, inharmonicity model, and repository outputs behind this guide, see Psycho-Acoustic Temperament Hierarchy. The public modeling code and generated sequence artifacts are in the PATH repository.
Psycho-Acoustic Temperament Hierarchy (PATH) is a tuning theory and reference framework for pianos and other fixed keyboard instruments with natural variations in inharmonicity. It can also be used for fixed-pitch instruments with little or no inharmonicity. It is intended to be reproducible enough that instrumentalists can implement, test, and maintain practical methods derived from it.
For the tuner-facing procedure derived from this theory, see Baroque Clavitone Temperament.
The theory treats tuning as a distribution of consonance rather than as a local mitigation of dissonance. The current PATH method therefore treats temperament as a graph-traversal problem: decide which rational interval chains are built first, which note regions are stabilized by stretched octaves, and where the traversal crosses between regions.
A newly discovered mathematical relationship between Pythagorean Temperament and Equal Temperament is described, where stretched octaves determine the relationship between the two systems. This allows Pythagorean rational tuning methods to be used, but where the Pythagorean Comma is distributed across 7 octaves instead of 1, and further mitigated by following the natural inharmonicity of the instrument.
The theory describes an inharmonicity coefficient that stretches octaves across the Pythagorean Comma, and the resulting psychoacoustic properties that vary across arbitrary inharmonicity coefficients in between that wide value and 0. The technique is developed as a hybrid graph traversal that groups rational 5th chains and stretched octaves so contrapuntal consonance stays flatter across longer runs, while the remaining irrationality is left where it is less harmful musically.
No beat counting is used. The current method uses consonant-sounding stretched octaves, rational 5ths, a selected rational 4th, and a selected rational major 3rd. Rather than rapidly switching between rational 5ths and irrational octave stretches, PATH stabilizes local note regions before crossing to the next rational bridge. Rather than subtracting dissonance evenly, the method adds consonance evenly, and is able to be performed by an ear without special practice outside of normal musical training. That is, piano players can be piano tuners, fixing tuning problems iteratively over time, as was done for most of the history of the piano.
It will be shown mathematically that a higher level of inharmonicity is closer to ideal, and it is easier to tune by ear through metrics of consonance rather than through beat counting.
It will be shown that the preference for beat counting has design problems, both aesthetic and scientific.
It will be shown that the psychoacoustic definition of counterpoint suits a particular design for its tuning system, and that the controlling design variable is the traversal of the interval graph. If several consonant relationships can be tested together within one region, the error does not compound as quickly as when tuning repeatedly alternates between rational 5ths and irrational stretched octaves.
The typical method for tuning a piano is based on a 12 note reference temperament of a single octave to be extrapolated across the remaining octaves. This is a general tuning method that makes sense when thinking in terms of interval ratios first, as expected when tuning is presented only as a mathematical problem. It is optimal for instruments of low inharmonicity, and low amounts of change in inharmonicity across pitch, like the middle of a Steinway. However, most pianos are not Steinways, most pianos have arbitrary levels of inharmonicity, and most concert music spans many octaves.
Smaller ratios are now favored when piano tuning by ear because the beats can more easily be "felt", because the beats are faster. That is, the tuning errors are more significant as beats when tuning smaller intervals, so it is easier to mitigate beat errors of the smaller intervals. However, faster beats are also harder to converge on.
Beat counting itself has theoretical flaws which defy the common sense notion that consonance is much easier felt than the specifics of a dissonance. It is possible to reformulate tuning as a distribution of consonance with known dissonant effects, allowing any ear to keep a piano tuning maintained with nominal musical training.
Even counterpoint only concerns itself with the property of consonance, discarding any consideration for classes of dissonance. It is, after all, the masking effect of consonance that counterpoint addresses. The first actual compositional problem in music was not dissonance, but the interference that harmonic consonance had on polyphonic melodic lines. Instead of trying to control dissonance, counterpoint controls consonance.
This may be unintuitive for an instrumentalist struggling to maintain the pitch of the instrument, but for a composer it is the unexpected harmonic alignments that interfere with the control of melodic lines. Counterpoint allows the composer to move compounded melodic lines across consonance in expected, controlled patterns that become useful compositional devices. The instrumentalist is focusing intervals on controlled pitch, while the composer is focusing intervals on producing controlled harmonic patterns. The composer is rarely bothered about a particular pitch, but an instrumentalist might be obsessed. The different tasks have very different mindsets of approach.
During the development of counterpoint in the Baroque period, tuning systems were theoretically designed around rational intervals, but tunings could only be implemented by ear, so tunings could only be implemented along psychoacoustic alignments which aren't truly rational. Even though ratios were prescribed, measurement was done by ear, so the inharmonicity of octaves, and the lack of understanding of inharmonicity, limited tuning system design to a single octave of 5ths, because octaves could not reproduce the effects of the ratios. That prevented the formal distribution of the Pythagorean Comma across multiple octaves by ear, which is now possible.
At that time, tuning beats wide to spread the Pythagorean Comma was understood, but there was considerable debate, one side arguing to spread the Pythagorean Comma completely evenly, the other side arguing to leave select intervals beatless. The argument to spread evenly is to have effects be reproducible across key changes, but it is actually easier to converge on a tuning when fewer intervals are to be tempered.
On the side of even temperament, the composer is asking for reproducibility, and on the side of consonant temperament, the tuner is asking for reproducibility. The debate effectively ends with the tuner giving up attempting to converge on an optimal tuning, the composer expecting convergence, and the instrumentalist working with whatever has converged. But there should be no reason why both cannot demand reproducibility through a proper foundational development of the field of tuning theory, a theory enabling the development of technique to optimize with reproducible psychoacoustic effects.
Piano tuners often analyze interval harmonic alignments by showing how all harmonics align against all others, as tables of Cartesian products in spreadsheets. However, there is usually only one harmonic that is significantly affecting the resulting beat envelope, the harmonic directly aligned. The harmonic partials of all tones have a quickly diminishing slope of harmonic amplitudes, for even a perfect square wave has a quickly diminishing slope.
The amplitude envelope has significant psychoacoustics because of the way that aural nerves are triggered by pressure potential. That is, small variations in pressure are lost in the sense organ nerve paths. One way to model the psychoacoustics mathematically is by convolving a gamma distribution function over the pressure levels. This is known as a gammatone filter, often used to model the physiology of nerves in the aural sense, which should also apply to beat amplitude envelopes.
For each Pythagorean ratio interval, it can be theorized intuitively which harmonic would be the main driver of the beat envelope, even without analysis with a gammatone filter.
Only the 8th is significantly relevant to inharmonicity. A perfect 4th against a stretched octave and a perfect 5th is (half a cent) near a stretched even-temperament. That is, the difference of error measurement is insignificant despite the application of consonance in 5ths and 8ths.
The smaller a Pythagorean ratio, the more chord-centric its tuning. That is, the harmonic relationships of a minor 3rd have far more to do with how well the top note lines up with the 5th harmonic of a chord than with the harmonics of the bottom note of the minor 3rd. One of the main reasons for using stretch tuning is to maintain alignment with a chord of notes that spans the harmonic series.
This actually leads to a definition of consonant intervals in counterpoint. The Pythagorean ratios n+1/n — 2/1, 3/2, 4/3, 5/4, etc.) with denominators that are powers of 2 (2n+1/2n — 2/1, 3/2, 5/4, 9/8, etc.) are more consonant than Pythagorean Ratios which are not (4/3, 6/5, 7/6, 8/7, etc.). The reason is because the bottom note of the ratio is phase aligned with the fundamental, and the top note phase aligns itself with the bottom note over the denominated period — the fundamental. The result is a consonant overall phase alignment with the fundamental.
Indeed, any interval with a denominator that is a power of 2, and a numerator above the denominator, satisfies this definition, including 5/2 (C4, E5), or 9/2 (C3, D5). But try 5/3 (G4, E5) and notice the difference. That extends the definition beyond special Pythagorean ratios, into a definition based purely on the harmonic series.
This theory is demonstrated in many ways:
This leads to a new mathematical and psychoacoustic descriptive model for the contrapuntal descriptions of intervals. And the tuning system can be optimized to satisfy these definitions of counterpoint.
A tuning system could be designed to favor the development of a temperament for solving 8ths, 5ths, and 4ths, leaving the rest to have irrational relationships. If the 4ths and 5ths in the second octave of the harmonic series are in tune, the major and minor 3rds of the 3rd octave are less significant. This aligns with the expectations of counterpoint. Such a system is Equal Pythagorean Temperament.
Piano strings have the property of inharmonicity. Inharmonicity is where each harmonic is not a perfect multiple of the fundamental frequency, but rather each harmonic is slightly sharp a frequency offset coefficient scaled by the harmonic number. The frequency offset coefficient is known as the inharmonicity coefficient.
Is there a reasonable inharmonicity coefficient that aligns well with Pythgarean ratios? Do typical inharmonicities align well with Pythagorean ratios? It turns out that there is, and they can.
The inharmonicity coefficient of a tone is a parameter to a function that defines the harmonic frequencies of a tone that stretch (are sharp in pitch, or raised in frequency) in proportion to the inharmonicity coefficient. A coefficient of 0 means that there is no inharmonicity, where harmonics are perfect integral multiples of the fundamental frequency. A coefficient of greater than 0 means that each harmonic is sharp proportionally by the coefficient.
Given a stretch factor, there is an inharmonicity coefficient that causes the harmonic to align with the stretched harmonic. The 2nd and 3rd harmonic functions are:
The function for the 2nd harmonic is for "open piped" tones, tones which do not skip harmonics. The function for the 3rd harmonic is for "closed piped" tones, tones which skip even harmonics.
There is an inharmonicity coefficient for the 2nd harmonic of Pythagorean Equal Temperament. Just provide the Pythagorean Equal Comma as an argument to the 2nd Harmonic Stretch Coefficient function:
Then provide that coefficient to the Harmonic Coefficient Ratio function to return all of the harmonics of an Ideal Equal Pythagorean Tone, where the 2nd harmonic lines up exactly with the stretched octave of Equal Pythagorean, lining up both octaves and 5ths.
| Harmonic h | Ratio h(1 + 1/2c(h2 - 1)) | Volume -6dB/octave |
|---|---|---|
| 1 | 1.0 | -0.0 |
| 2 | 2.0038754738031512… | -6.0 |
| 3 | 3.015501895212605 | -9.509775004326938… |
| 4 | 4.038754738031512… | -12.0 |
| 5 | 5.077509476063025… | -13.931568569324174… |
| 6 | 6.135641583110294… | -15.509775004326936… |
| 7 | 7.217026532976469… | -16.844129532345626… |
Physical inharmonicity naturally stretches the octave toward the Pythagorean Comma. There is a sort of "Goldilocks Zone" between the Inharmonicity Coefficient of 0, and the Inharmonicity Coefficient that aligns with the Equal Pythagorean Comma, where 0 means that octaves are exactly 2/1, and where an Equal Pythagorean Inharmonicity means that 5ths are exactly 3/2.
The following table shows the boundaries of values within this "Goldilocks Zone" where the s octave stretch can freely be adjusted to align with the local effective inharmonicity of a range of notes. This represents an ideal tempering for inharmonic octaves. Due to its mathematical expression as a form of equal temperament, with fractional exponents, an implementation requires high mathematical precision, which could be very difficult to perform as a tuning sequence without digital tools.
| Comma | Equal | Equal Pythagorean |
|---|---|---|
| IC | 0 | 0.0012918246010504102… |
| Stretch | 1 |
s
(312/219)1/7 |
| 1.0 | 1.0019377369015756… | |
| Octave | 2 |
2s
2(312/219)1/7 |
| 2.0 | 2.0038754738031512… | |
| 5th | 27/12 | 3/2 |
| 1.4983070768766815… | 1.5 | |
| 4th | 25/12 |
(2s)5/12
(2(312/219)1/7)5/12 |
| 1.3348398541700344… | 1.3359169825354331… | |
| Major 3rd |
24/12 |
(2s)4/12
(2(312/219)1/7)4/12 |
| 1.2599210498948732… | 1.2607343233213617… | |
| Minor 3rd |
23/12 |
(2s)3/12
(2(312/219)1/7)3/12 |
| 1.189207115002721… | 1.1897827894843853… |
It is already shown that a Pythagorean method can equate a particular equal method. Can a modified Pythagorean method approximate the mathematical approach of equal tempering? How would a particular modification affect the temperament bias? The following table re-expresses the Equal Inharmonicity axis as a Pythagorean Inharmonicity Axis.
| Comma | Pythagorean | Equal Pythagorean |
|---|---|---|
| IC | 0 | 0.0012918246010504102… |
| Stretch | 1 |
s
(312/219)1/7 |
| 1.0 | 1.0019377369015756… | |
| Octave | 2 |
2s
2(312/219)1/7 |
| 2.0 | 2.0038754738031512… | |
| 5th | 3/2 | 3/2 |
| 1.5 | 1.5 | |
| 4th |
311/21126
311/217 |
311/211(2s)6
311/211(2(312/219)1/7)6 |
| 1.35152435302734375 | 1.3359169825354331… | |
| Major 3rd |
34/2422
34/26 |
34/24(2s)2
34/24(2(312/219)1/7)2 |
| 1.265625 | 1.2607343233213617… | |
| Minor 3rd |
39/2925
39/214 |
39/29(2s)5
39/29(2(312/219)1/7)5 |
| 1.20135498046875 | 1.1897827894843853… |
However, there are three problems with this:
What are the effects of assuming that the piano has the desired coefficient across a small tempered region without the desired coefficient? The coefficient is only involved in the 8ths lining up with the 2nd harmonic because the 5ths line up with the 1st harmonic. This means that 5ths are independent of the coefficient, leaving only 8ths (and therefore resulting 4ths) dependent on the coefficient.
This doesn't mean that it is not possible to abuse a 5th to help an 8th line up with a particular coefficient. The point is: why formulate the temperament for beat offsets at all? Instead of spreading out mitigations that are difficult to implement, why not distribute evenly the desired properties that are much easier to feel? The proximity of the resultant 4ths to even-temperament shows that optimized solutions can have the power of mitigation, but better converge to a solution.
That is, the psychoacoustic value of a targeted approach is logically equal to a mitigated approach, so optimize the problem toward maximum consonance value, rather than minimum dissonance error, because the "cost function" toward a maximum is far better at converging on a solution.
Consonance converges better because the beat amplitude envelope is a mathematical effect that is directly observed as a percept by the ear, but the timing of the beat amplitude envelope itself is not directly observed as a percept. The requirement of a fine-grained, absolute sense of time is a tall order for humans. As far as I know, that sense is not part of human physiology, and has to be developed 100% cognitively.
When an optimization solution has trouble converging on a solution, it indicates a problem with an input variable. The beat rate is a bad input variable for humans without an electronic sensor to measure it properly.
Dissonance itself is not technically an error, but an harmonic property that is used in composition and orchestration. Tuning systems typically treat it as an error, with a design to mitigate the error.
There are mainly two ways to deal with dissonance as an error:
The beat-tempo required to implement a rational beat spread of a synchronous system is dependent on the inharmonicity coefficient, so it is very difficult to converge on a solution quickly. Interestingly, the inability to rationally converge on a beat would classify the tuning as an irrational beat spread, and therefore have the psychoacoustic properties of lowering the amplitude of dissonance, the properties of an asynchronous system. Synchronous tuning is more suited to implementation on virtual instruments.
This might explain why modern tuning practice prefers to use the error mitigation approach to achieve a stretched even-temperament, because an approximate asynchronous solution is aesthetically equal to a perfect asynchronous solution. However, the difficulty of the practice limits amateur adoption, creating an "ivory tower" mysticism of skill around the practice, but it should be possible to tune an instrument as long as there is some sense of unison. A musician should not need to hire a professional tuner.
There is also a scientific problem with the inability to converge on a tuning. The lack of convergence violates the principle of reproducibility, lowering the descriptiveness of the medium. The RPT exam requires a master tuning as a reference, but this tuning is not reproducible, so the exam is not technically scientific.
If the RPT exam were instead a measure of ear-tuning skill, it would be an exam of self-consistency of tuning, where each examinee were to set their own master tuning, where the master tuning system were evaluated independently of tuning skill. The lack of a formal tuning theory makes it difficult to distinguish hand and ear skill from the tuning system because the theory helps define the boundaries of the tuning system, and define the precise techniques that comprise the skill.
Regular effects are important for compositional design. A composition may only be designed for a particular temperament when its effects are repeatable. As a composer, I have abused awkward regularities in virtual instruments that are not real properties of the physical instrument. It is extremely important for the musician to have precise control over the instrument.
Tuning systems have 3 demands, that they be:
A theorem could be stated that only 2 out of the 3 of the demands may be satisfied at any time. But as in any problem of entropy, the optimal solution involves directing the flow of thermodynamic energy through a system. The system is to be broken down into problem domains, and the demand pressures are to be guided through system components.
The PATH tuning system proposes a hybrid tuning system where 8ths, 5ths, and 4ths are controlled rationally, but remaining intervals, and stretch-gaps are left to logarithmic effects. This takes advantage of the properties of both even tempering and rational tuning, and most importantly, incorporates arbitrary inharmonicity as a design condition.
In PATH, a rational component is separated from an irrational component. The rational component satisfies demands 1 and 2, and the irrational component satisfies demands 1 and 3. In each case, demand 1 is satisfied, the aesthetic demand of the listener, leaving the tuner to balance the demands across 2 and 3, but this balance is defined by PATH in a formal way, where demand 3 is implied by the way that demand 2 is implemented.
No matter what the inharmonicity, tuning 5ths against the 1st harmonic has a constant ideal aesthetic effect. However, maintaining that ideal assumes that 8ths will be able to stretch back across the 5ths spanning the ideal inharmonicity coefficient.
In theory, all notes can be tuned with just 5ths in a single direction, creating 7 parallel chains of 5ths that are only orthogonal (independent) along the 5ths. These chains can be identified by a modulo 7 against the note number, 0 through 6 (N % 7). For example, chains of 3 5ths:
| Chain Modulo | 1st Link | 2nd Link | 3rd Link | |||
|---|---|---|---|---|---|---|
| Nº | Note | Nº | Note | Nº | Note | |
| 0 | 0 | C4 | 7 | G4 | 14 | D5 |
| 1 | 1 | C♯4 | 8 | G♯4 | 15 | D♯5 |
| 2 | 2 | D4 | 9 | A4 | 16 | E5 |
| 3 | 3 | D♯4 | 10 | A♯4 | 17 | F5 |
| 4 | 4 | E4 | 11 | B4 | 18 | F♯5 |
| 5 | 5 | F4 | 12 | C5 | 19 | G5 |
| 6 | 6 | F♯4 | 13 | C♯5 | 20 | G♯5 |
That orthogonality can be seen by unrolling the table columns:
| Chain Modulo | Nº | Note | Link |
|---|---|---|---|
| 0 | 0 | C4 | 1st |
| 1 | 1 | C♯4 | |
| 2 | 2 | D4 | |
| 3 | 3 | D♯4 | |
| 4 | 4 | E4 | |
| 5 | 5 | F4 | |
| 6 | 6 | F♯4 | |
| 0 | 7 | G4 | 2nd |
| 1 | 8 | G♯4 | |
| 2 | 9 | A4 | |
| 3 | 10 | A♯4 | |
| 4 | 11 | B4 | |
| 5 | 12 | C5 | |
| 6 | 13 | C♯5 | |
| 0 | 14 | D5 | 3rd |
| 1 | 15 | D♯5 | |
| 2 | 16 | E5 | |
| 3 | 17 | F5 | |
| 4 | 18 | F♯5 | |
| 5 | 19 | G5 | |
| 6 | 20 | G♯5 |
A tuning system may be described simply as a function that takes a single parameter of a note, and returns a frequency. That is not necessarily even the case, since other parameters may be used, like musical context, or instrument properties. More generally, a tuning system is a system for tuning the pitches of notes.
The possible scope of theory for tuning systems is vast. There are a lot of observations about it, and a variety of techniques, but not a lot of descriptive theory. Even the mathematics of tuning have trouble relating to specific aesthetic effects.
Parallel orthogonal chains of any interval may be created. Most tuning systems are based on allocating intervals within an octave. If all octaves are constrained to the same ratio, that leaves 12 "chains" of octaves. For example, all C notes are one chain, and all C♯ notes are a second chain, and so on, until reaching another C.
PATH organizes notes as 7 parallel chains of 5ths (N % 7). These chains are the rational reference structures of the temperament, not the whole temperament by themselves.
Parallel chains of other intervals are also possible, but 5ths are especially useful because they preserve strong rational alignment while spanning the keyboard efficiently.
The key tempering choice is not a beat target inside one interval. It is the traversal of the interval graph: which rational 5ths are established first, which note regions are stabilized by stretched octaves, and where a smaller number of rational 4ths or major 3rds bridge between regions.
Longer runs of compatible consonances flatten the local error surface. If several intervals can be checked together within one connected region, the residual error does not compound as quickly as when the tuning alternates immediately between rational 5ths and irrational stretched octaves.
The current PATH method therefore uses grouped rational 5th spines and grouped octave propagation. A note of N % 7 connects by 8th to a note (N - 8) % 7, equal to (N + 2) % 7. A circle of chains connected by 8ths is therefore [0, 2, 4, 6, 1, 3, 5]. This connection structure remains the basis of the graph. For example:
| Note | N % 7 |
(N + 2) % 7 |
||
|---|---|---|---|---|
| Chain | Octave | Chain | Octave | |
| D | 0 | 5 | 2 | 4 |
| D♯ | 1 | 5 | 3 | 4 |
| E | 2 | 5 | 4 | 4 |
| F | 3 | 5 | 5 | 4 |
| F♯ | 4 | 5 | 6 | 4 |
| G | 5 | 5 | 0 | 4 |
| G♯ | 6 | 5 | 1 | 4 |
In circle sequence, moving 5ths to keep within the table range, as an example:
| Note | N % 7 |
(N + 2) % 7 |
||
|---|---|---|---|---|
| Chain | Octave | Chain | Octave | |
| D | 0 | 5 | 2 | 4 |
| E | 2 | 5 | 4 | 4 |
| F♯ | 4 | 5 | 6 | 4 |
| G♯ | 6 | 5 | 1 | 4 |
| D♯ | 1 | 5 | 3 | 4 |
| F | 3 | 5 | 5 | 4 |
| G | 5 | 5 | 0 | 4 |
In fact, the number of 5ths moved is arbitrary. For example, not moving 5ths at all, and moving from D4 to D3 also moves from chain 2 to chain 4.
| Note | N % 7 |
(N + 2) % 7 |
||
|---|---|---|---|---|
| Chain | Octave | Chain | Octave | |
| D | 0 | 5 | 2 | 4 |
| 2 | 4 | 4 | 3 | |
This means that any 7 chains of 5ths can have 7 random (arbitrarily tuned) 8ths connecting them, as long as each 8th connects to a distinct chain connected to no other 8th. However, the ratio between each chain would vary depending on the distance away from the ideal inharmonicity coefficient of the stretch of each 8th. That is, if the ideal coefficient is used, the ratios between each respective chain will be equal, and all ratios will be per-interval constant, a perfect even-stretch temperament.
For inharmonicity coefficients below the ideal coefficient, the speed of the beats between the 8ths and the 4ths are inverse-proportional to each other. That is, the ideal coefficient has perfect 8ths, and a coefficient of 0 has perfect 4ths, and there is a gradient between them. Most pianos have very small coefficients in the midrange, but 8ths tend to be orchestrated in the outer ranges. This also means that the chains of 5ths will drift away from even pitch distribution, greatly affecting the intervals not yet defined, favoring short chains, lowering the total comma-drift across the edges of the chains.
The same situation happens in typical Pythagorean Comma spreads of 1st-order 8ths and 2nd-order 5ths, but the 4ths and 5ths have the swapping gradients, instead of 4ths and 8ths. The same thing can be done with 8ths and Major 3rds, but a 3rd order of chains would be required, because the 2nd-order modulo 4 is a factor of the 1st-order modulo 12, so it would not fully connect to the 1st order. Also, using a 2nd-order modulo that factors into the 1st-order modulo creates a fixed period of effects of the 2nd-order within the 1st-order. For example, a Major 3rd pattern within an 8th repeats the Major 3rd pattern (modulo 4) 3 times within the 8th (modulos 12/4 = 3).
It is possible to use a 1st-order of 4ths, and a 2nd-order of 8ths, but that lowers the number of 8ths over which to spread the comma, requiring a much higher inharmonicity coefficient. However, using 4ths would mean that only 5 parallel chains would need to optimized, rather than 7 for 5ths. That is, 5ths have less drift than 4ths, since the 5th circle spans a greater distance.
So far in the definition of PATH, there are definitions for 8ths, 5ths, and the resulting 4ths. All other intervals are undefined at this point. Defining 8ths and 5ths also implicitly defines 4ths. Theoretically, any modulo A against modulo B also implicitly defines modulo A-B, wherever the modulo B connects.
In non-ideal inharmonicity conditions, an interesting property emerges from the non-equality of the smaller intervals: the logarithmic effects of differences in inharmonicity create logarithmic, irrational distributions of the smaller intervals, gaining the benefit of shaping the dissonant noise, lowering its average amplitude. That is, the imperfections are moved to where they benefit the solution. This may seem similar to "noise shaping", but it is even better, because the noise is part of the solution.
The system may be thought of as a functional medium with input variables and output variables.
The input variables are:
The output variables are:
The practical consequence is that PATH should not alternate blindly between one rational 5th and one stretched octave. When a region is stabilized by several compatible intervals at once, the error is distributed more evenly, and harmonically aligned chords are easier to preserve.
The preferred current traversal therefore begins with a rational 5th spine around the reference note, expands that region by stretched octaves, bridges to a second rational 5th spine by a rational major 3rd and rational 4th, and then again propagates by stretched octaves. The hierarchy is still present, but it is expressed directly by graph traversal.
The current practical implementation of this theory is Baroque Clavitone Temperament, a Baroque-biased practical tuning guide derived from the hybrid PATH traversal.
In practical terms, the theory is realized by building a rational 5th spine around C4, stabilizing that region by stretched octaves, bridging by a rational major 3rd and rational 4th into a second rational region, and then propagating stretched octaves outward. The full tuner-facing sequence, listening priorities, and demonstration video are kept in the separate practical article so this article can remain focused on theory and modeling.
The current PATH traversal is modeled in the public PATH repository. The generator in path.py constructs candidate tuning graphs and emits a serial tuning sequence, a markdown table, and an interval-offset report for each candidate. Running python3 path.py writes the current hybrid artifacts as hybrid_sequence.txt, hybrid_markdown.md, and hybrid_report.txt.
The public repository also emits the full modeled sequence table for the current Baroque Clavitone Temperament. In that output, the Ratio column is the realized multiplier from the parent note to the child note, that is, child frequency / parent frequency. Rational bridge intervals therefore appear as exact ratios such as 1.50000, 0.66667, 1.25000, and 0.75000, while the octave rows show the modeled stretched-octave ratio, or its inverse when descending, at that pitch.
Those octave ratios are generated from steinway_inharmonicity_coefficient_func() in path.py, a smooth empirical Steinway B inharmonicity model. The underlying piano-string inharmonicity equation and example Steinway B fitted values were documented on the original Verituner site, which is now offline; an archived copy is available at web.archive.org. PATH then converts that local inharmonicity model into the octave ratios used in the traversal by way of the second-harmonic coefficient relation defined earlier in this article.
The interval-graph alternatives are sketched in graph.md, and the current public modeling and bridge-generation logic is in path.py. Readers who want to model their own sequences may fork the repository, define a different traversal in path.py, choose different rational bridges or octave-propagation orderings, and compare the emitted reports before publishing a new serial tuning sequence.
This article presents a new mathematical discovery that will revolutionize musical tuning. The discovery reveals:
Let us start with two constants:
An octave represents a doubling of frequency, an interval ratio of 2/1. The series of octaves therefore forms an exponential curve where each octave is the next power of 2. With 12 notes in between each octave, each note is 1/12th the next power of 2. Therefore, the note ratio function is Æ’(N) = 2N/12, and the ratio note function is the inverse Æ’(R) = 12(log2(R)).
For awareness, the modern tuning unit is known as a cent, 1/100th of a note or 1/1200th of an octave, defined as Æ’(R) = 1200(log2(R)).
| Note | N | Æ’(N) | Ratio | Hz (A4=440) |
|---|---|---|---|---|
| A4 | 0 | 20/12 | 1.0 | 440 |
| A♯4, Bâ™4 | 1 | 21/12 | 1.0594630943592953… | 466.1637615180899… |
| B4 | 2 | 22/12 | 1.122462048309373… | 493.8833012561241… |
| C5 | 3 | 23/12 | 1.189207115002721… | 523.2511306011972… |
| C♯5, Dâ™5 | 4 | 24/12 | 1.2599210498948732… | 554.3652619537442… |
| D5 | 5 | 25/12 | 1.3348398541700344… | 587.3295358348151… |
| D♯5, Eâ™5 | 6 | 26/12 | 1.4142135623730951… | 622.2539674441618… |
| E5 | 7 | 27/12 | 1.4983070768766815… | 659.2551138257398… |
| F5 | 8 | 28/12 | 1.5874010519681994… | 698.4564628660078… |
| F♯5, Gâ™5 | 9 | 29/12 | 1.6817928305074292… | 739.9888454232689… |
| G5 | 10 | 210/12 | 1.7817974362806788… | 783.9908719634986… |
| G♯5, Aâ™5 | 11 | 211/12 | 1.887748625363387… | 830.6093951598903… |
| A5 | 12 | 212/12 | 2.0 | 880 |
The advantage of using an exponential curve is that each note is the same ratio from the next. Each difference of some number of notes is the same ratio as any other difference of that number of notes. That is a reflection of the exponent law xmxn = xm+n. For example, the 4 note interval between C and E is the same ratio as the interval between A and C♯, about 1.2599210498948732. Transposing a series of intervals starting from one note to starting from another note results in the same sequence of ratios. That is, all intervals are equally tempered, so the method is known as Equal Temperament.
The disadvantage of using an exponential curve is that intervals other than an octave are non-integer powers, resulting in irrational ratios. To mitigate, 12 is selected as the number of notes per octave since 12 is the lowest number of notes where the Pythagorean ratios closely align, and the Pythagorean ratio of next complexity from an octave, 3/2, forms a Circle of 5ths of 12 steps that line up near the 12 equally tempered steps. The existence of this second level of complexity from an octave provided a second degree of harmonic modulation.
Observe the closeness of the Pythagorean ratios in the table rows where N is 7 (3/2), 5 (4/3), 4 (5/4), and 3 (6/5). For further analysis of other numbers of steps, and interval circles, see a demonstration program.
| System | Pythagorean | Equal | ||||
|---|---|---|---|---|---|---|
| Interval | N | N+1/N | Pythagorean Ratio | t | 2t/12 | Equal Ratio |
| 8th (Octave) | 1 | 2/1 | 2.0 | 12 | 212/12, 21/1 | 2.0 |
| Perfect 5th | 2 | 3/2 | 1.5 | 7 | 27/12 | 1.4983070768766815… |
| Perfect 4th | 3 | 4/3 | 1.333… | 5 | 25/12 | 1.3348398541700344… |
| Major 3rd | 4 | 5/4 | 1.25 | 4 | 24/12, 21/3 | 1.2599210498948732… |
| Minor 3rd | 5 | 6/5 | 1.2 | 3 | 23/12, 21/4 | 1.189207115002721… |
For example, the Perfect 5th is the 7th note of a 12 note octave (7 12ths of an octave). That is, 27/12, or about 1.4983070768766815, which is very close to 1.5, or 3/2, the Pythagorean ratio for a Perfect 5th. When the ratio 27/12 is multiplied against iself 12 times, the /12 is canceled out, becoming 27, or the interval of 7 octaves. That is, ascending 12 5ths is the same as ascending 7 octaves.
The Circle of 5ths is the observation that ascending 5ths, descending an octave any time a 5th ascends outside an octave, cycles through all 12 notes of an octave. That mathematical property exists because the numbers 7 and 12 are coprime integers. Pythagorean tuning is simply cycling through the 12 notes in an octave, assuming that a 5th is the Pythagorean ratio of a Perfect 5th, 3/2.
| Note | Sequence |
12-tet |
Octave |
Pythagorean | Ratio | Equal |
|---|---|---|---|---|---|---|
| C | 0 | 0 | 0 | 30/2020 | 1.0 | 2(7)(0)-(12)(0)/12, 20/12 |
| G | 1 | 7 | 0 | 31/2120 | 1.5 | 2(7)(1)-(12)(0)/12, 27/12 |
| D | 2 | 2 | 1 | 32/2221 | 1.125 | 2(7)(2)-(12)(1)/12, 22/12 |
| A | 3 | 9 | 1 | 33/2321 | 1.6875 | 2(7)(3)-(12)(1)/12, 29/12 |
| E | 4 | 4 | 2 | 34/2422 | 1.265625 | 2(7)(4)-(12)(2)/12, 24/12 |
| B | 5 | 11 | 2 | 35/2522 | 1.8984375 | 2(7)(5)-(12)(2)/12, 211/12 |
| F♯ | 6 | 6 | 3 | 36/2623 | 1.423828125 | 2(7)(6)-(12)(3)/12, 26/12 |
| C♯ | 7 | 1 | 4 | 37/2724 | 1.06787109375 | 2(7)(7)-(12)(4)/12, 21/12 |
| G♯ | 8 | 8 | 4 | 38/2824 | 1.601806640625 | 2(7)(8)-(12)(4)/12, 28/12 |
| D♯ | 9 | 3 | 5 | 39/2925 | 1.20135498046875 | 2(7)(9)-(12)(5)/12, 23/12 |
| A♯ | 10 | 10 | 5 | 310/21025 | 1.802032470703125 | 2(7)(10)-(12)(5)/12, 210/12 |
| E♯, F | 11 | 5 | 6 | 311/21126 | 1.35152435302734375 | 2(7)(11)-(12)(6)/12, 25/12 |
| B♯, C | 12 | 0 | 7 | 312/21227 | 1.0136432647705078125 | 2(7)(12)-(12)(7)/12, 20/12 |
A clear pattern emerges, leading to a generalization of cycles for many intervals. However, not all cycles cover all 12 notes in an octave. The Circle of 5ths and Circle of 4ths do, noting that 4ths can be seen as 5ths inverted from an octave. The other circles do not hit 12 notes because the denominator of 12 in each interval formula reduces to a lowest common denominator, since each interval numerator is a factor of 12. For example, 24/12 reduces to 21/3, indicating that the cycle only actually covers 3 notes. Therefore, a simple lowest-common-denominator algorithm determines whether the cycle covers the full range of notes.
| Ratio | Interval | 12-tet, i=Sequence | Octave | Pythagorean | Equal, 2t/12 | Coverage |
|---|---|---|---|---|---|---|
| 2/1 | 212/12, 21/1 | t=12i-12o, 12i % 12 | o=[12i/12] | 2i/1i2o | 212i-12o/12 | 1/12 |
| 3/2 | 27/12 | t=7i-12o, 7i % 12 | o=[7i/12] | 3i/2i2o | 27i-12o/12 | 12/12 |
| 4/3 | 25/12 | t=5i-12o, 5i % 12 | o=[5i/12] | 4i/3i2o | 25i-12o/12 | 12/12 |
| 5/4 | 24/12, 21/3 | t=4i-12o, 4i % 12 | o=[4i/12] | 5i/4i2o | 24i-12o/12 | 3/12 |
| 6/5 | 23/12, 21/4 | t=3i-12o, 3i % 12 | o=[3i/12] | 6i/5i2o | 23i-12o/12 | 4/12 |
| n/d | 2n/12 | t=ni-12o, ni % 12 | o=[ni/12] | ni/di2o | 2ni-12o/12 | LCD(n, 12)/12 |
The Circle Coverage, LCD(n, 12)/12 shows how many tones in an octave are reached by the circle. Therefore the truth value of whether the circle covers all notes in an octave is LCD(n, 12) = 12, also expressed as GCD(n, 12) = 1, both of which express whether n and 12 are mutually prime, or Coprime. That is, if the numerator and denominator of the fraction of the steps of an octave are coprime, then the circle covers all steps of an octave.
A comma is a ratio that represents the difference between two different tunings of the same note. Each circle sequence has a posiiton 12 that represents the same note as position 0, completing a cycle. However, by using rational ratios, the value specified at position 12 is different from position 0. That difference is expressed as a ratio and denoted as a Pythagorean Comma, specifically the Circle of 5ths. The Circle of 5ths based on the Pythagorean ratio of 3/2 is used because it is the closest comma to 1/1 from a circle whose octave steps are coprime. Only the circles of 5ths (7/12), 4ths (5/12), and Minor 2nds (1/12) are coprime, and therefore span all 12 octave steps.
| Interval | Beginning | End | ||||
|---|---|---|---|---|---|---|
| PR | OS | CP | Symbol | Ratio | Symbol | Ratio |
| 2/1 | 12/12 | No | 20/1020 | 1/1 | 212/112212 | 4096/4096 1.0 |
| 3/2 | 7/12 | Yes | 30/2020 | 1/1 | 312/21227 | 531441/524288 1.0136432647705078125 |
| 4/3 | 5/12 | Yes | 40/3020 | 1/1 | 412/31225 | 16777216/17006112 0.986540368545144239\ 906217247069… |
| 5/4 | 4/12 | No | 50/4020 | 1/1 | 512/41224 | 244140625/268435456 0.909494701772928237\ 9150390625 |
| 6/5 | 3/12 | No | 60/5020 | 1/1 | 612/51223 | 2176782336/1953125000 1.114512556032 |
| 9/8 | 2/12 | No | 90/8020 | 1/1 | 912/81222 | 282429536481/274877906944 1.027472668214613804\ 59368228912353515625 |
| 18/17 | 1/12 | Yes | 180/17020 | 1/1 | 1812/171221 | 1156831381426176/1165244474459522 0.992779976032713423\ 409459963891… |
Therefore the Pythagorean Comma is:
At this point, two constants remain:
With respect to changing the number of notes with a constant octave, there are multiple methods. The Pythagorean ratios can be compared against Equal note positions, a modern method made possible by logarithm tables. Simply replace the constant 12 in the Equal functions with a variable S.
For example, see how 19 Equal Temperament approximates Pythagorean ratios:
| System | Pythagorean | Equal | ||||
|---|---|---|---|---|---|---|
| Interval | N | N+1/N | Pythagorean Ratio | t | 2t/19 | Equal Ratio |
| 8th (Octave) | 1 | 2/1 | 2.0 | 19 | 219/19, 21/1 | 2.0 |
| Perfect 5th | 2 | 3/2 | 1.5 | 11 | 211/19 | 1.4937589616544857… |
| Perfect 4th | 3 | 4/3 | 1.333… | 8 | 28/19 | 1.3389041012244722… |
| Major 3rd | 4 | 5/4 | 1.25 | 6 | 26/19 | 1.2446925894640233… |
| Minor 3rd | 5 | 6/5 | 1.2 | 5 | 25/19 | 1.200102719578103… |
Since 19 is a prime number, every interval is able to form a circle that covers every note in the octave.
However, in ancient times, logarithm tables did not exist. Therefore, circles of each Pythagorean ratio were found in continuing the orbital spiraling of the ratio within a single constant octave. The formulas above demonstrate that the number of octaves cycled equals the step number of an octave.
For example:
| Interval | Beginning | End | ||||
|---|---|---|---|---|---|---|
| PR | OS | CP | Symbol | Ratio | Symbol | Ratio |
| 3/2 | 7/12 | Yes | 30/2020 | 1/1 | 312/21227 | 531441/524288 1.0136432647705078125 |
| 3/2 | 11/19 | Yes | 30/2020 | 1/1 | 319/219211 | 1162261467/1073741824 1.082440341822803020477294921875 |
| 3/2 | 31/53 | Yes | 30/2020 | 1/1 | 353/253231 | 1938324566768\ 0019896796723 /1934281311383\ 4066795298816 1.0020903140410861725914298316\ 173442351619980381083330200908\ 67648832499980926513671875 |
Notice that the comma resuting from the 19th 5th is wider than the comma of the 12th 5th. Even though powers of 1/19ths align more closely to Pythagorean ratios, the wider comma ultimately leads larger variations from equal when implemented with Pythagorean ratios rather than calculating fractional powers. Therefore, the use of 19 note octaves is a modern device.
In ancient times, it was observed that cycling 3/2 53 times is very close to the 31st octave, representing the 31st note of a 53 note octave. The first known documentation of this comma comes from Jing Fang, so this will be called Fang's Comma.
Instead of thinking of fractional powers aligning with Pythagorean ratios, it is possible to work in reverse, to stack Pythagorean ratios to produce a set of fractional powers. Indeed, that is the effect of the 12 notes of the circle of 5ths, or any Pythagorean circle. The proximity of Pythagorean Tuning to Equal Tuning is the proximity of the circle of 5ths of 12 notes to fractional powers of 1/12th.
An intuitive way to understanding this fact is to first see that stacking any interval N number of times is represented mathematically by simply raising the interval ratio by the Nth power, which outlines an exponential curve perfectly. The mechanism that subdivides that exponential curve into a fraction is simply the process of wrapping the values of each degree of power around a single octave. Since the prime ratio of the wrapped interval will never meet the octave ratio, the subdivision is never perfect—however, the subdivision will iteratively approach perfection as more iterations pass more closely by the octave.
Although this is not the preferred method for calculating a single arbitrary logarithm, it is an efficient way to generate an entire structured logarithm table using a few ratios. An approximate logarithm table between 1 and 2, one octave, is produced by simply iterating a ratio until a reasonably close comma is found, and sorting by the evaluated ratios.
For example, The Circle of 53 5th:
Sequence |
Octave |
Pythagorean | Ratio |
|---|---|---|---|
| 0 | 0 | 30/2020 | 1.0 |
| 1 | 0 | 31/2120 | 1.5 |
| 2 | 1 | 32/2221 | 1.125 |
| 3 | 1 | 33/2321 | 1.6875 |
| 4 | 2 | 34/2422 | 1.265625 |
| 5 | 2 | 35/2522 | 1.8984375 |
| 6 | 3 | 36/2623 | 1.423828125 |
| 7 | 4 | 37/2724 | 1.06787109375 |
| 8 | 4 | 38/2824 | 1.601806640625 |
| 9 | 5 | 39/2925 | 1.20135498046875 |
| 10 | 5 | 310/21025 | 1.802032470703125 |
| 11 | 6 | 311/21126 | 1.35152435302734375 |
| 12 | 7 | 312/21227 | 1.0136432647705078125 |
| 13 | 7 | 313/21327 | 1.52046489715576171875 |
| 14 | 8 | 314/21428 | 1.1403486728668212890625 |
| 15 | 8 | 315/21528 | 1.71052300930023193359375 |
| 16 | 9 | 316/21629 | 1.2828922569751739501953125 |
| 17 | 9 | 317/21729 | 1.92433838546276092529296875 |
| 18 | 10 | 318/218210 | 1.4432537890970706939697265625 |
| 19 | 11 | 319/219211 | 1.082440341822803020477294921875 |
| 20 | 11 | 320/220211 | 1.6236605127342045307159423828125 |
| 21 | 12 | 321/221212 | 1.217745384550653398036956787109375 |
| 22 | 12 | 322/222212 | 1.8266180768259800970554351806640625 |
| 23 | 13 | 323/223213 | 1.369963557619485072791576385498046875 |
| 24 | 14 | 324/224214 | 1.02747266821461380459368228912353515625 |
| 25 | 14 | 325/225214 | 1.541209002321920706890523433685302734375 |
| 26 | 15 | 326/226215 | 1.15590675174144053016789257526397705078125 |
| 27 | 15 | 327/227215 | 1.733860127612160795251838862895965576171875 |
| 28 | 16 | 328/228216 | 1.30039509570912059643887914717197418212890625 |
| 29 | 16 | 329/229216 | 1.950592643563680894658318720757961273193359375 |
| 30 | 17 | 330/230217 | 1.46294448267276067099373904056847095489501953125 |
| 31 | 18 | 331/231218 | 1.0972083620045705032453042804263532161712646484375 |
| 32 | 18 | 332/232218 | 1.64581254300685575486795642063952982425689697265625 |
| 33 | 19 | 333/233219 | 1.2343594072551418161509673154796473681926727294921875 |
| 34 | 19 | 334/234219 | 1.85153911088271272422645097321947105228900909423828125 |
| 35 | 20 | 335/235220 | 1.3886543331620345431698382299146032892167568206787109375 |
| 36 | 21 | 336/236221 | 1.041490749871525907377378672435952466912567615509033203125 |
| 37 | 21 | 337/237221 | 1.5622361248072888610660680086539287003688514232635498046875 |
| 38 | 22 | 338/238222 | 1.171677093605466645799551006490446525276638567447662353515625 |
| 39 | 22 | 339/239222 | 1.7575156404081999686993265097356697879149578511714935302734375 |
| 40 | 23 | 340/240223 | 1.318136730306149976524494882301752340936218388378620147705078125 |
| 41 | 23 | 341/241223 | 1.9772050954592249647867423234526285114043275825679302215576171875 |
| 42 | 24 | 342/242224 | 1.482903821594418723590056742589471383553245686925947666168212890625 |
| 43 | 25 | 343/243225 | 1.11217786619581404269254255694210353766493426519446074962615966796875 |
| 44 | 25 | 344/244225 | 1.668266799293721064038813835413155306497401397791691124439239501953125 |
| 45 | 26 | 345/245226 | 1.25120009947029079802911037655986647987305104834376834332942962646484375 |
| 46 | 26 | 346/246226 | 1.876800149205436197043665564839799719809576572515652514994144439697265625 |
| 47 | 27 | 347/247227 | 1.40760011190407714778274917362984978985718242938673938624560832977294921875 |
| 48 | 28 | 348/248228 | 1.0557000839280578608370618802223873423928868220400545396842062473297119140625 |
| 49 | 28 | 349/249228 | 1.58355012589208679125559282033358101358933023306008180952630937099456787109375 |
| 50 | 29 | 350/250229 | 1.1876625944190650934416946152501857601919976747950613571447320282459259033203125 |
| 51 | 29 | 351/251229 | 1.78149389162859764016254192287527864028799651219259203571709804236888885498046875 |
| 52 | 30 | 352/252230 | 1.3361204187214482301219064421564589802159973841444440267878235317766666412353515625 |
| 53 | 31 | 353/253231 | 1.002090314041086172591429831617344235161998038108333020090867648832499980926513671875 |
There is a reasonable comma at the 12th iteration, the Pythagorean Comma. Sort the first 12 values to find 12 1/12th powers:
log2(arg) |
Sequence |
Octave |
Pythagorean | Ratio |
|---|---|---|---|---|
| 0/12 | 0 | 0 | 30/2020 | 1.0 |
| 1/12 | 7 | 4 | 37/2724 | 1.06787109375 |
| 2/12 | 2 | 1 | 32/2221 | 1.125 |
| 3/12 | 9 | 5 | 39/2925 | 1.20135498046875 |
| 4/12 | 4 | 2 | 34/2422 | 1.265625 |
| 5/12 | 11 | 6 | 311/21126 | 1.35152435302734375 |
| 6/12 | 6 | 3 | 36/2623 | 1.423828125 |
| 7/12 | 1 | 0 | 31/2120 | 1.5 |
| 8/12 | 8 | 4 | 38/2824 | 1.601806640625 |
| 9/12 | 3 | 1 | 33/2321 | 1.6875 |
| 10/12 | 10 | 5 | 310/21025 | 1.802032470703125 |
| 11/12 | 5 | 2 | 35/2522 | 1.8984375 |
| 0/12 | 12 | 7 | 312/21227 | 1.0136432647705078125 |
The next-smallest comma is at the 53rd iteration, Fang's Comma. Sort the first 53 values to find 53 1/53rd powers:
log2(arg) |
Sequence |
Octave |
Pythagorean | Ratio |
|---|---|---|---|---|
| 0/53 | 0 | 0 | 30/2020 | 1.0 |
| 1/53 | 12 | 7 | 312/21227 | 1.0136432647705078125 |
| 2/53 | 24 | 14 | 324/224214 | 1.02747266821461380459368228912353515625 |
| 3/53 | 36 | 21 | 336/236221 | 1.041490749871525907377378672435952466912567615509033203125 |
| 4/53 | 48 | 28 | 348/248228 | 1.0557000839280578608370618802223873423928868220400545396842062473297119140625 |
| 5/53 | 7 | 4 | 37/2724 | 1.06787109375 |
| 6/53 | 19 | 11 | 319/219211 | 1.082440341822803020477294921875 |
| 7/53 | 31 | 18 | 331/231218 | 1.0972083620045705032453042804263532161712646484375 |
| 8/53 | 43 | 25 | 343/243225 | 1.11217786619581404269254255694210353766493426519446074962615966796875 |
| 9/53 | 2 | 1 | 32/2221 | 1.125 |
| 10/53 | 14 | 8 | 314/21428 | 1.1403486728668212890625 |
| 11/53 | 26 | 15 | 326/226215 | 1.15590675174144053016789257526397705078125 |
| 12/53 | 38 | 22 | 338/238222 | 1.171677093605466645799551006490446525276638567447662353515625 |
| 13/53 | 50 | 29 | 350/250229 | 1.1876625944190650934416946152501857601919976747950613571447320282459259033203125 |
| 14/53 | 9 | 5 | 39/2925 | 1.20135498046875 |
| 15/53 | 21 | 12 | 321/221212 | 1.217745384550653398036956787109375 |
| 16/53 | 33 | 19 | 333/233219 | 1.2343594072551418161509673154796473681926727294921875 |
| 17/53 | 45 | 26 | 345/245226 | 1.25120009947029079802911037655986647987305104834376834332942962646484375 |
| 18/53 | 4 | 2 | 34/2422 | 1.265625 |
| 19/53 | 16 | 9 | 316/21629 | 1.2828922569751739501953125 |
| 20/53 | 28 | 16 | 328/228216 | 1.30039509570912059643887914717197418212890625 |
| 21/53 | 40 | 23 | 340/240223 | 1.318136730306149976524494882301752340936218388378620147705078125 |
| 22/53 | 52 | 30 | 352/252230 | 1.3361204187214482301219064421564589802159973841444440267878235317766666412353515625 |
| 23/53 | 11 | 6 | 311/21126 | 1.35152435302734375 |
| 24/53 | 23 | 13 | 323/223213 | 1.369963557619485072791576385498046875 |
| 25/53 | 35 | 20 | 335/235220 | 1.3886543331620345431698382299146032892167568206787109375 |
| 26/53 | 47 | 27 | 347/247227 | 1.40760011190407714778274917362984978985718242938673938624560832977294921875 |
| 27/53 | 6 | 3 | 36/2623 | 1.423828125 |
| 28/53 | 18 | 10 | 318/218210 | 1.4432537890970706939697265625 |
| 29/53 | 30 | 17 | 330/230217 | 1.46294448267276067099373904056847095489501953125 |
| 30/53 | 42 | 24 | 342/242224 | 1.482903821594418723590056742589471383553245686925947666168212890625 |
| 31/53 | 1 | 0 | 31/2120 | 1.5 |
| 32/53 | 13 | 7 | 313/21327 | 1.52046489715576171875 |
| 33/53 | 25 | 14 | 325/225214 | 1.541209002321920706890523433685302734375 |
| 34/53 | 37 | 21 | 337/237221 | 1.5622361248072888610660680086539287003688514232635498046875 |
| 35/53 | 49 | 28 | 349/249228 | 1.58355012589208679125559282033358101358933023306008180952630937099456787109375 |
| 36/53 | 8 | 4 | 38/2824 | 1.601806640625 |
| 37/53 | 20 | 11 | 320/220211 | 1.6236605127342045307159423828125 |
| 38/53 | 32 | 18 | 332/232218 | 1.64581254300685575486795642063952982425689697265625 |
| 39/53 | 44 | 25 | 344/244225 | 1.668266799293721064038813835413155306497401397791691124439239501953125 |
| 40/53 | 3 | 1 | 33/2321 | 1.6875 |
| 41/53 | 15 | 8 | 315/21528 | 1.71052300930023193359375 |
| 42/53 | 27 | 15 | 327/227215 | 1.733860127612160795251838862895965576171875 |
| 43/53 | 39 | 22 | 339/239222 | 1.7575156404081999686993265097356697879149578511714935302734375 |
| 44/53 | 51 | 29 | 351/251229 | 1.78149389162859764016254192287527864028799651219259203571709804236888885498046875 |
| 45/53 | 10 | 5 | 310/21025 | 1.802032470703125 |
| 46/53 | 22 | 12 | 322/222212 | 1.8266180768259800970554351806640625 |
| 47/53 | 34 | 19 | 334/234219 | 1.85153911088271272422645097321947105228900909423828125 |
| 48/53 | 46 | 26 | 346/246226 | 1.876800149205436197043665564839799719809576572515652514994144439697265625 |
| 49/53 | 5 | 2 | 35/2522 | 1.8984375 |
| 50/53 | 17 | 9 | 317/21729 | 1.92433838546276092529296875 |
| 51/53 | 29 | 16 | 329/229216 | 1.950592643563680894658318720757961273193359375 |
| 52/53 | 41 | 23 | 341/241223 | 1.9772050954592249647867423234526285114043275825679302215576171875 |
| 0/53 | 53 | 31 | 353/253231 | 1.002090314041086172591429831617344235161998038108333020090867648832499980926513671875 |
Notice some patterns:
The sequence and octave values for the sorted 53 5ths are the sequence and octave values for 12 5ths (12 sequence, 7 octave), incremented, and modulo the sequence and octave values for the comma of 53 5ths. That is, the sequence values increment 12 modulo 53 (12, 24, 36, 48, [wrapping around 53] 7 …), and the octave values increment 7 modulo 31 (7, 14, 21, 28, [wrapping around 31] 4 …). This is also true of the next-smallest comma, with sequence of 359 (incrementing 53 modulo 359), octave 210 (incrementing 31 modulo 210).
The previous pattern can be stated precisely with the continued-fraction convergents of the base 2 logarithm of the Pythagorean 5th.
For each convergent, stack 3/2 ak times and subtract octaves with ⌊nbk/ak⌋ so that every ratio stays within the range [1, 2). Sorting that set by pitch is equivalent to sorting the fractional parts of nbk/ak.
Proposition. Consecutive convergents satisfy the unimodular identity akbk-1 - ak-1bk = (-1)k-1. Therefore ak-1 is the modular inverse of bk, up to sign, modulo ak. This is why the sorted circle of size ak is traversed by stepping the sequence by the previous denominator ak-1 modulo ak.
Corollary. The musical examples in this article are exactly the first two non-trivial instances of that rule:
Theorem. Let dS and dL be the small and large adjacent step ratios in the sorted ak-note circle. Then the gap between the two step sizes is a fractional power of the previous comma:
In exact multiplicative terms, the previous comma becomes the next circle's first difference between small and large steps. The step pattern itself is balanced: in combinatorics it is a Christoffel, or Sturmian, word built recursively from the previous convergents.
This formalizes the earlier observation that a comma becomes the next-smaller circle's first step, and that the previous comma also determines the ordering rule for the next circle.
To traverse the set of all possible commas, start with each Pythagorean ratio in one dimension, and cycle a spiraling orbit through octaves in a second dimension of commas, determined as they pass as near as possible to each octave. Enumerating the sorted set of produced notes approximates fractional powers, which can be mapped inversely to form a logarithm table.
The previous formulas show that the note in an octave of a Pythagorean ratio is also the number of octaves that the ratio will span in a cycle. That is effectively using the Pythagorean circle as a logarithm table. Therefore, a logarithm can be used to calculate commas in reverse. That is, for any N number of notes, the number of octaves spanned to form a cycle comma will be the N number of notes times the base 2 logarithm of the cycled Pythagorean ratio, rounding to the nearest integer: the rounded integer of the Ratio Note Function.
For example, for a 12 note circle of 3/2 5ths, the number of octaves spanned will be 7, because 12 times the base 2 logarithm of 3/2 is 7.019550008653875…, nearest to 7, with a proximity in proportion to the resulting octave comma. The resulting Pythagorean comma formula can be computed from these values, 312/21227. And since the 27 can be 27.019550008653875… to remove the comma, it is clear that the smaller the comma, the nearer the rounded logarithm would be to the unrounded logarithm.
This leads to a fast way to find small commas without needing to evaluate large powers until the comma needs to be fully expressed. Generally:
The full space of commas can also be explored computationally. One direct method is to iterate a Pythagorean ratio through octaves and record each new closest return toward 1/1. For the 5th, this recovers the familiar record commas 12/7, 53/31, 359/210, 665/389, and so on.
A more direct search begins from the logarithm itself. Let α = log2(3/2). For each candidate note count N, evaluate the nearest octave counts ⌊Nα⌋ and ⌈Nα⌉, compute the corresponding comma, and retain the best results. This recovers not only the main convergents, but also the intermediate best approximants, also known as semiconvergents.
Therefore comma space is not merely a short list of historically famous commas. It is a structured search space around the line D = N(log2(3/2)), and can be explored efficiently by integer arithmetic around that line. The convergent structure described above explains why the best circles appear in such a regular sequence.
This leads to a specific computational result. For a chosen comma pair of N notes and D octaves, the resulting octave-domain logarithm table can be generated as a stream. Each entry is derived from the previous one by a constant update rule: advance the sequence, apply the next octave carry, and emit the next rational point in the table. No transcendental function needs to be evaluated for each entry once the generating logarithm has been fixed.
Any method that emits N table entries must still write N outputs, so no exact table-building method can asymptotically do better than linear time in the size of the produced table. In that precise sense, a streamed comma-space construction is asymptotically optimal for building large rational logarithm tables on a regular octave lattice.
This should not be confused with the more general problem of evaluating log(x) for arbitrary scattered inputs. For generic logarithm evaluation, modern range-reduction methods with polynomial or rational approximations remain superior. The advantage here is different: exact indexing, reproducible ordering, bulk generation, and immediate compatibility with rational tuning structures.
There are far superior methods for calculating powers and logarithms today, so what is the significance of Pythagorean methods in tuning today? Although pitch is approximately sensed logarithmically, a general principle true of many senses, the problem of harmony relates directly to the phase-level interference (interferometry) between different intervals. The use of ratios, the rational means, provides for a level of interferometric control over harmonic effects, the ends, and therefore better harmony.
Changing the number of notes in an octave has been considered. Now consider equalities for changing the ratios of:
To temper the Pythagorean ratio in the numerator of a comma, solve for a reduction of the ratio that meets the octave count of the Ratio Octave Span, the denominator of the comma. Equalities of that comma denominator:
g is the Nth root of the Ratio Comma.
g = (Ratio Comma)1/N
g is the ratio between the Pythagorean Ratio and its Equal Ratio.
g = (Pythagorean Ratio)/(Equal Ratio)
To temper the octave in the denominator of a comma, solve for an octave stretch that meets the Pythagorean numerator of the comma. Equalities of that comma numerator:
With respect to changing the ratio of an octave, it is already effectively common due to physical inharmonicity in instruments, like pianos, but the ratio of an octave can also be changed to temper a circle comma.
When tuning, the ratios used are to be selected by the tuner. To this day, the standard practice of the tuner is to bind the distribution of the Pythagorean Comma to within a single reference octave. That provides 12 opportunities to spread the comma across the reference octave, distributing parts of the comma ratio to different parts of the octave.
There is actually no need to constrain the error to a single octave. This is already the case when piano tuners use a stretch tuning, where the octave is already stretched toward the Pythagorean Comma, not to deal with the comma, but to deal with the octave that is slightly sharp due to the physical property of the inharmonicity of piano strings.
There is a single global solution to an inharmonicity that stretches octaves to reach the Pythagorean Comma across 12 5ths of 7 stretched 8ths. This provides for an effective comma of 1, a form of equal tempering, but solving for different intervals. This is the Equal Pythagorean Comma. Stacking 12 3/2 5ths, never wrapping down octaves to stay within a single octave, the Pythagorean Comma ends up placed at the end of the 7th octave. Each octave is uniform, and effectively is spread the 7th root of the Pythagorean Comma, approximately 7 times closer to 1.
Instead of thinking in terms of 2/1 octaves, add a stretch factor variable next to the octave ratio, and solve for it within the Pythagorean equations. It is then seen that merely stretching the octaves to meet the Pythagorean Comma fits neatly into the equation for Equal Temperament, forming an Equal Pythagorean Temperament where the octaves are tempered instead of the 5ths.
The root R of a number N will be expressed as a reciprocated power, the denominator of a fractional power, N1/R, according to the law of exponents. This makes it easier to visualize the algebra. For example, the square root of 5, the second root of 5, would be expressed as 51/2.
Equal Pythagorean has no comma, expressed in Pythagorean form:
Equal Pythagorean has no comma, merely by the fact that it can be expressed in Equal form. Therefore, solving any temperament for an Equal form shows how to remove the temperament comma:
As demonstrated, the comma of 12 3/2 intervals across 7 octaves has a much smaller comma, the Equal Pythagorean Comma. This comma is so small that it is below the Just-Noticeable Difference, and smaller than Fang's Comma. When octaves are stretched by this comma, the comma of the temperament is completely cancelled out to 1/1, with Equal intervals that are perceptually identical to the intervals of Equal Temperament.
Therefore, an Equal Pythagorean tuning can have rational 5ths and octaves across the entire range of the keyboard, and can be expressed with both the Pythagorean Temperament formula, and the Equal Temperament formula, unifying the two domains.
| Note | Sequence |
12-tet |
Octave |
Pythagorean | Equal | Ratio |
|---|---|---|---|---|---|---|
| C | 0 | 0 | 0 | 30/20(2s)0 | (2s)0/12 | 1.0 |
| G | 1 | 7 | 0 | 31/21(2s)0 | (2s)7/12 | 1.5 |
| D | 2 | 2 | 1 | 32/22(2s)1 | (2s)2/12 | 1.12282426199… |
| A | 3 | 9 | 1 | 33/23(2s)1 | (2s)9/12 | 1.68423639299… |
| E | 4 | 4 | 2 | 34/24(2s)2 | (2s)4/12 | 1.26073432332… |
| B | 5 | 11 | 2 | 35/25(2s)2 | (2s)11/12 | 1.89110148498… |
| F♯ | 6 | 6 | 3 | 36/26(2s)3 | (2s)6/12 | 1.41558308615… |
| C♯ | 7 | 1 | 4 | 37/27(2s)4 | (2s)1/12 | 1.05963402267… |
| G♯ | 8 | 8 | 4 | 38/28(2s)4 | (2s)8/12 | 1.589451034… |
| D♯ | 9 | 3 | 5 | 39/29(2s)5 | (2s)3/12 | 1.18978278948… |
| A♯ | 10 | 10 | 5 | 310/210(2s)5 | (2s)10/12 | 1.78467418423… |
| E♯, F | 11 | 5 | 6 | 311/211(2s)6 | (2s)5/12 | 1.33591698254… |
| B♯, C | 12 | 0 | 7 | 312/212(2s)7 | (2s)0/12 | 1.0 |
Now, all 5ths are Pythagorean 3/2, and octaves are 2 times the Equal Pythagorean Comma, which is a valid inharmonicity property, and still very close to 2/1. All intervals are equally tempered. The 12th position lines up with the 0th position, so there is no unbalanced comma to be spread.
| System | Pythagorean | Equal Pythagorean | |||||
|---|---|---|---|---|---|---|---|
| Interval | N | N+1/N | Ratio | t | 3s/2s(2s)o | (2s)t/12 | Ratio |
| 8th (Octave) | 1 | 2/1 | 2.0 | 12 | 312/212(2s)7 | (2s)12/12 | 2.0038754738… |
| Perfect 5th | 2 | 3/2 | 1.5 | 7 | 31/21(2s)0 | (2s)7/12 | 1.5 |
| Perfect 4th | 3 | 4/3 | 1.333… | 5 | 311/211(2s)6 | (2s)5/12 | 1.33591698254… |
| Major 3rd | 4 | 5/4 | 1.25 | 4 | 34/24(2s)2 | (2s)4/12 | 1.26073432332… |
| Minor 3rd | 5 | 6/5 | 1.2 | 3 | 39/29(2s)5 | (2s)3/12 | 1.18978278948… |