Psycho-Acoustic Temperament Hierarchy (PATH) is a tuning theory and method for pianos and other fixed keyboard instruments with natural variations in inharmonicity. It can be used for any keyboard instrument with a fixed pitch, even with no inharmonicity. It is easy enough for instrumentalists to implement, test, and therefore maintain, due to its design leveraging scientific reproducibility.
It is based on new tuning and contrapuntal theories, also presented, providing a new foundation for future tuning and orchestration techniques.
A theory is presented describing the mathematic and scientific effects of tuning for psychoacoustic consonance, and a technique is developed from the theory. Instead of focusing on mitigating dissonance, the focus is on distributing consonance.
A newly discovered mathematical relationship between Pythagorean Tempermant 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 recursive-descent optimization algorithm that solves for contrapuntal consonance and "shaped" dissonance under the theory, noting that consonance can be held fixed for entire classes of intervals relating to counterpoint intervals.
No beat counting is used. Only consonant-sounding 8ths and 5ths are used, tuning most 8ths, 5ths, and 4ths with deliberate psychoacoustic properties, while letting the logarithmic inharmonicity of the instrument evenly distribute the remaining intervals. A self-similar, hierarchical temperament pattern evenly distributes the contrapuntal psychoacoustic consonant properties, rather than focusing on "tempering" the dissonances within a single, small temperament range. 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.
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 uses 5ths (modulo 7). A second order interval to intersect, and thereby connect, the non-factoring chains logically expresses the comma-spreads between each chain.
PATH is based on chains of 5ths instead of octaves. 5ths are 7 notes apart, meaning that 7 independent chains exist. The same is possible with 4ths and 5 independent chains. Each has different properties and side-effects to be considered.
PATH uses 8ths (modulo 12) to connect the 5ths (modulo 7).
One way to spread the Pythagorean Comma is to use a 1st-order chain of 8ths, and a 2nd-order chain of 5ths, beat counting a circle of 5ths against a tempo that happens to line up with a starting note, and the inharmonicity of the instrument. But PATH shows that you can swap the orders, and connect 8ths purely, in a way that distributes the comma in a self-similar leaf pattern.
PATH connects by 8ths. A note of N % 7 connects 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]. 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 |
| 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 so far may be thought of as a functional medium with input variables and output variables.
The input variables are:
The output variables are:
Due to the gradient effects of non-ideal inharmonicity, an implementation should use short chains. The shortest chain-connection pattern possible is 2 5ths up, and 1 8th down. So a temperament "leaf" can be created from that. Then a "trunk" of 8ths can connect several leaf temperaments in a self-similar pattern.
Self-similar patterns are known for being mathematically ideal for distributing effects evenly. For example:
First, create the smallest connected series of chains possible. Move ascending in 5ths against the 1st harmonic, and descend in octaves against the second harmonic, only to not-yet-tuned modulo-7 (perfect 5th) intervals. This creates a temperament "leaf node".
| From | To | Chain | Segment | Memo |
|---|---|---|---|---|
| Ref | C4 | 0 | 3 | |
| C4 | F3 | Down a 5th | ||
| C4 | G4 | |||
| G4 | G3 | 2 | 3 | Down an 8th |
| G3 | D4 | |||
| D4 | A4 | |||
| A4 | A3 | 4 | 3 | Down an 8th |
| A3 | E4 | |||
| E4 | B4 | |||
| B4 | B3 | 6 | 3 | Down an 8th |
| B3 | F♯4 | |||
| F♯4 | F♯3 | 1 | 3 | Down an 8th |
| F♯3 | C♯4 | |||
| C♯4 | G♯4 | |||
| G♯4 | G♯3 | 3 | 3 | Down an 8th |
| G♯3 | D♯4 | |||
| D♯4 | A♯4 | |||
| A♯4 | A♯3 | 5 | 3 | Down an 8th |
| A♯3 | F4 |
F3-B4 are now tuned, in 3 contiguous groups of notes:
The 5th chains are:
| Chain | Nº | Note |
|---|---|---|
| 0 | 0 | F3 |
| 7 | C4 | |
| 14 | G4 | |
| 2 | 2 | G3 |
| 9 | D4 | |
| 16 | A4 | |
| 4 | 4 | A3 |
| 11 | E4 | |
| 18 | B4 | |
| 6 | 6 | B3 |
| 13 | F♯4 | |
| 1 | 1 | F♯3 |
| 8 | C♯4 | |
| 15 | G♯4 | |
| 3 | 3 | G♯3 |
| 10 | D♯4 | |
| 17 | A♯4 | |
| 5 | 5 | A♯3 |
| 12 | F4 |
The 8th connections are:
| Note | N % 7 |
(N + 2) % 7 |
||
|---|---|---|---|---|
| Chain | Octave | Chain | Octave | |
| G | 0 | 4 | 2 | 3 |
| A | 2 | 4 | 4 | 3 |
| B | 4 | 4 | 6 | 3 |
| F♯ | 6 | 4 | 1 | 3 |
| G♯ | 1 | 4 | 3 | 3 |
| A♯ | 3 | 4 | 5 | 3 |
The 8th tunings cross from octave 4 to 3, from F to B, connecting each chain 0-6, so the 2nd harmonic of the 1st group is bound to the 1st harmonic of the 3rd group. The 2nd group is bound by 5ths at the 1st harmonic to the 1st group.
That establishes the middle temperament.
Placing such temperaments contiguously against each other is the following series of ranges, where each temperament is a leaf node, and each 8th connection forms a trunk edge:
| Leaf | Leaf Node | Trunk Edge | ||
|---|---|---|---|---|
| 1st | Lowest | D0 | Anchor | A♯0 |
| Highest | A1 | Ground | A♯1 @ 2nd | |
| 2nd | Lowest | A♯1 | Anchor | F2 |
| Highest | E3 | Ground | F3 @ 3nd | |
| 3rd | Lowest | F3 | Anchor | C4 |
| Highest | B4 | Ground | C4 @ 256Hz | |
| 4th | Lowest | C5 | Anchor | G5 |
| Highest | F♯6 | Ground | G4 @ 3rd | |
| 5th | Lowest | G6 | Anchor | D7 |
| Highest | C♯8 | Ground | D6 @ 4th | |
The 2nd, 3rd (middle), and 4th temperaments can be implemented similarly. The 5th temperament just needs to work around the missing C♯ which is trivial.
The 1st temperament is impractical. That should just be A0-A1 tuned by 8ths from A2, descending. The harmonics are more complex on these wider strings with brighter tones. Just tune from top to bottom, from A1 to A0, using 8ths from A2 to A1. A1 will then be tuned by the time A0 is tuned. It would be much harder to tune 5ths this low, 8ths being hard enough to hear. And music going this low will more likely line up on 8ths than 5ths.
Note that each temperament leaf is not actually a copy of other leaves. Although each leaf uses the same method, due to differences in inharmonicity, the results are different, and tailored to each temperament range. The ratios between corresponding chains of 5ths vary in each leaf. That is, the ratio between chain 1 and 2 in the 4th temperament may differ from the ratio between chain 1 and 2 in the 3rd temperament, because the inharmonicity coefficient may vary. That is by design.
That is a benefit of the recursion in the method, rather than the recursion of the results of a single temperament. The method can be recursed because it is actually no more complex than extrapolating a reference temperament because consonance is being measured either way.
The starting note of each leaf is the 8th lowest note. The 3rd temperament starts at C4, the 8th note above F3, its lowest note. Each contiguous temperament needs to be connected to the neighboring temperaments in some way. Connecting by a 5th would have a side effect of connecting two 5th chains together, but a design requirement is that the 5th chains be a short as possible. Therefore, temperaments should be connected by 8ths.
Connecting to the first chain in a sequence by an 8th is free, interfering with no other aspects of the tuning. So, connect the 8th inward toward to the 3rd temperament to the starting note of each other temperament.
However, each temperament flexes the widths in between its 5th chains to accommodate differences in inharmonicity coefficient across the keyboard. There are 8ths connecting within a temperament range, but the other 8ths are disconnected, so the end of a sequence can drift significantly from the beginning of a sequence. That can happen even from just small errors in tuning.
The typical solution is to extend 8ths from a single reference tuning, and rely on check-tuning to keep the non-8ths acceptable. Recall that, in PATH, 8th connections can be arbitrarily placed as long as the constraint is satisfied that each 5th chain can only have a single intersecting 8th. So instead of only connecting 8ths within a small area, the 8th connections within a temperament can be staggered, odd 8ths connecting from outside the temperament, and even 8ths connecting within the temperament.
The tuning system can be seen as a graph of tunings connecting each interval. PATH limits this graph to consonant 5ths and 8ths. In this view, the staggered connections have the effect of limiting the distance of tuned interval edges connecting the intervals. Check-tuning can be seen as adding more edges beyond what is rationally possible to align with consonance, whereas PATH simply optimizes the consonant tuning paths between intervals.
Each 8th connection that staggers from outside the temperament can be placed anywhere on the chain to which it connects (as a sink, forward in the progression). That provides some ambiguity, or variability, in the connections. These links should be made where the tuning paths are shortest. That is likely the middle link of the chain.
With respect to the implementation of the tuning, the chains and connections may be seen as a directed graph of note nodes, where notes are connected by 5ths or 8ths, where each connection has a source side and a sink side. That is, while tuning, start at a reference source, and tune 5ths or 8ths in sequence, each next note becoming a sink of its source note.
Although the graph is logically bidirectional, and undirected, the implementation needs to flow in a particular direction, since it must be expressed as a tuning sequence, since the actual intervals, affected by discovered inharmonicity, are evaluated during the tuning process itself. That evaluation needs to flow in a sequence.
Since the tuning sequence is completely serial, there are constraints to the directed graph:
| Source | Sink | ||
|---|---|---|---|
| Modulo | Note | Modulo | Note |
| 6 | A2 | A1 | |
| 5 | G♯2 | G♯1 | |
| 4 | G2 | G1 | |
| 3 | F♯2 | F♯1 | |
| 2 | F2 | F1 | |
| 1 | E2 | E1 | |
| 0 | D♯2 | D♯1 | |
| 6 | D2 | D1 | |
| 5 | C♯2 | C♯1 | |
| 4 | C2 | C1 | |
| 3 | B1 | B0 | |
| 2 | A♯1 | A♯0 | |
| A1 | A0 | ||
| Modulo | Source | Note | Sink |
|---|---|---|---|
| 2 | A♯1 | A♯0@1 | |
| F3@3 | F2 | F1@1 | |
| C3 | C2 | ||
| 4 | C3 | C2 | C1@1 |
| G2 | G1@1 | ||
| D3 | D2 | ||
| 6 | D3 | D2 | D1@1 |
| A2 | A1@1 | ||
| E3 | E2 | ||
| 1 | E3 | E2 | E1@1 |
| B2 | B1 | ||
| 3 | B2 | B1 | B0@1 |
| F♯2 | F♯1@1 | ||
| C♯3 | C♯2 | ||
| 5 | C♯3 | C♯2 | C♯1@1 |
| G♯2 | G♯1@1 | ||
| D♯3 | D♯2 | ||
| 0 | D♯3 | D♯2 | D♯1@1 |
| A♯2 |
| Modulo | Source | Note | Sink |
|---|---|---|---|
| 0 | F3 | F2@2 | |
| Ref | C4 | ||
| G4 | G3, G5@4 | ||
| 2 | G4 | G3 | |
| D4 | |||
| A4 | A3 | ||
| 4 | A4 | A3 | |
| E4 | |||
| B4 | B3 | ||
| 6 | B4 | B3 | |
| F♯4 | F♯3 | ||
| 1 | F♯4 | F♯3 | |
| C♯4 | |||
| G♯4 | G♯3 | ||
| 3 | G♯4 | G♯3 | |
| D♯4 | |||
| A♯4 | A♯3 | ||
| 5 | A♯4 | A♯3 | |
| F4 |
| Modulo | Source | Note | Sink |
|---|---|---|---|
| 5 | C5 | ||
| G4@3 | G5 | ||
| D6 | D5, D7@5 | ||
| 0 | D6 | D5 | |
| A5 | |||
| E6 | E5 | ||
| 2 | E6 | E5 | |
| B5 | |||
| F♯6 | F♯5 | ||
| 4 | F♯6 | F♯5 | |
| C♯6 | C♯5 | ||
| 6 | C♯6 | C♯5 | |
| G♯5 | |||
| D♯6 | D♯5 | ||
| 1 | D♯6 | D♯5 | |
| A♯5 | |||
| F6 | F5 | ||
| 3 | F6 | F5 | |
| C6 |
| Modulo | Source | Note | Sink |
|---|---|---|---|
| 3 | G6 | ||
| D6@4 | D7 | ||
| A7 | A6 | ||
| 5 | A7 | A6 | |
| E7 | |||
| B7 | B6 | ||
| 0 | B7 | B6 | |
| F♯7 | C♯7 | ||
| 2 | F♯7 | C♯7 | |
| G♯7 | G♯6 | ||
| 4 | G♯7 | G♯6 | |
| D♯7 | |||
| A♯7 | A♯6 | ||
| 6 | A♯7 | A♯6 | |
| F7 | |||
| C8 | C7 | ||
| 1 | C8 | C7 | |
| G7 |
There are several parallel paths in the graph. Tuning is executed in serial, so a single path through the graph may be expressed.
| 3rd Temperament Leaf | ||||
|---|---|---|---|---|
| From | To | Chain | Segment | Memo |
| Ref | C4 | 0 | 3 | |
| C4 | F3 | Down a 5th | ||
| C4 | G4 | |||
| G4 | G3 | 2 | 3 | Down an 8th |
| G3 | D4 | |||
| D4 | A4 | |||
| A4 | A3 | 4 | 3 | Down an 8th |
| A3 | E4 | |||
| E4 | B4 | |||
| B4 | B3 | 6 | 3 | Down an 8th |
| B3 | F♯4 | |||
| F♯4 | F♯3 | 1 | 3 | Down an 8th |
| F♯3 | C♯4 | |||
| C♯4 | G♯4 | |||
| G♯4 | G♯3 | 3 | 3 | Down an 8th |
| G♯3 | D♯4 | |||
| D♯4 | A♯4 | |||
| A♯4 | A♯3 | 5 | 3 | Down an 8th |
| A♯3 | F4 | |||
| 2nd Temperament Leaf | ||||
| From | To | Chain | Segment | Memo |
| F3@3 | F2 | 2 | 2 | |
| F2 | A♯1 | Down a 5th | ||
| F2 | C3 | |||
| C3 | C2 | 4 | 2 | Down an 8th |
| C2 | G2 | |||
| G2 | D3 | |||
| D3 | D2 | 6 | 2 | Down an 8th |
| D2 | A2 | |||
| A2 | E3 | |||
| E3 | E2 | 1 | 2 | Down an 8th |
| E2 | B2 | |||
| B2 | B1 | 3 | 2 | Down an 8th |
| B1 | F♯2 | |||
| F♯2 | C♯3 | |||
| C♯3 | C♯2 | 5 | 2 | Down an 8th |
| C♯2 | G♯2 | |||
| G♯2 | D♯3 | |||
| D♯3 | D♯2 | 0 | 2 | Down an 8th |
| D♯2 | A♯2 | |||
| 4th Temperament Leaf | ||||
| From | To | Chain | Segment | Memo |
| G4@3 | G5 | 5 | 4 | |
| G5 | C5 | Down a 5th | ||
| G5 | D6 | |||
| D6 | D5 | 0 | 4 | Down an 8th |
| D5 | A5 | |||
| A5 | E6 | |||
| E6 | E5 | 2 | 4 | Down an 8th |
| E5 | B5 | |||
| B5 | F♯6 | |||
| F♯6 | F♯5 | 4 | 4 | Down an 8th |
| F♯5 | C♯6 | |||
| C♯6 | C♯5 | 6 | 4 | Down an 8th |
| C♯5 | G♯5 | |||
| G♯5 | D♯6 | |||
| D♯6 | D♯5 | 1 | 4 | Down an 8th |
| D♯5 | A♯5 | |||
| A♯5 | F6 | |||
| F6 | F5 | 3 | 4 | Down an 8th |
| F5 | C6 | |||
| 5th Temperament Leaf | ||||
| From | To | Chain | Segment | Memo |
| D6@4 | D7 | 3 | 5 | |
| D7 | G6 | Down a 5th | ||
| D7 | A7 | |||
| A7 | A6 | 5 | 5 | Down an 8th |
| A6 | E7 | |||
| E7 | B7 | |||
| B7 | B6 | 0 | 5 | Down an 8th |
| B6 | F♯7 | |||
| F♯7 | C♯7 | 2 | 5 | Down a 4th (5th-8th) |
| C♯7 | G♯7 | |||
| G♯7 | G♯6 | 4 | 5 | Down an 8th |
| G♯6 | D♯7 | |||
| D♯7 | A♯7 | |||
| A♯7 | A♯6 | 6 | 5 | Down an 8th |
| A♯6 | F7 | |||
| F7 | C8 | |||
| C8 | C7 | 1 | 5 | Down an 8th |
| C7 | G7 | |||
| 1st Temperament Leaf | ||||
| From | To | Chain | Segment | Memo |
| A2@2 | A1 | |||
| G♯2@2 | G♯1 | |||
| G2@2 | G1 | |||
| F♯2@2 | F♯1 | |||
| F2@2 | F1 | |||
| E2@2 | E1 | |||
| D♯2@2 | D♯1 | |||
| D2@2 | D1 | |||
| C♯2@2 | C♯1 | |||
| C2@2 | C1 | |||
| B1@2 | B0 | |||
| A♯1@2 | A♯0 | |||
| A1 | A0 | |||
All 88 keys should now be in tune.
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 an efficient method for calculating a particular logarithm, it is an efficient way to approximate an entire 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).
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:
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… |