It draws on familiar principles from traditional tuning systems—particularly the stacking and folding of small integer ratios. In place of the typical 4:5:6 major triad of Western harmony, 13ED3 builds its internal structure from odd harmonics: 3:5:7. It also appeals to a long tradition of tuning systems based on prime number structures and the ideal of "harmonic purity".
However, from a practical and perceptual perspective, this system is functionally indistinguishable from a much simpler and more usable system: 8EDO (eight equal divisions of the octave).
Let us consider why.
Step Size: Almost Identical
Accumulation & Structural Divergence
Where these scales do diverge is not in their basic intervallic content, but in how these intervals accumulate over a wide range. In 13ED3, the eighth step produces a pitch ratio of \[3^{8/13} \approx 1.96:1 \quad (\text{≈1170 cents}) \]
This is very close to an octave (1200 cents), but not quite. As such, this "false octave" introduces a structural dissonance when mapped to instruments that expect 2:1 periodicity (like guitars or pianos). Yet melodically, it often passes as an octave—especially when accompanied by timbral masking.
This is where timbre manipulation becomes relevant. When the spectrum is shaped appropriately, the brain can re-interpret the 1170-cent interval as a legitimate octave by dissolving its internal dissonance (according to spectral models like dissonance curves). The outcome is that listeners treat this as an octave, with a nearly "extratonal" timbre. Normal.
This process points to the larger truth: the underlying internal structure is what the brain recognizes first (see Tonal Constancy). Both 13ED3 and 8EDO initially present the same internal hierarchy—but one does so with physical and perceptual structural coherence (8EDO), while the other does not (13ED3).
A compelling comparison can be made by examining how each system approximates the octave (1200 cents):
An acoustic instrument like the guitar, with a standard size, scale, and string gauge setup, typically provides a usable range of 4000–4500 cents. So that near-12ed2 closure rarely happens in practice.
This leads to chromatic inflation: more and more pitches emerge across the range, lacking any repeatable pattern or stable sense of class identity. The system cannot "fold back" in any meaningful way (except for the 'Tritave,' which is not a perceptual equivalence—try playing any music, like Clair de Lune, with tritave substitutions...).
In contrast, 8EDO allows immediate octave closure. Its pitches "repeat" every 1200 cents, enabling ensemble coherence (bass, guitar, voice, etc.) and perceptual tractability. Musicians don't have to memorize hundreds of slightly misaligned chromas. They work with eight analytically coherent classes, all compatible with existing instruments timbres.
Philosophical Aside: A Problem Inherited
Why, then, does 13ED3 persist as a popular theoretical model?
In part, because of its symbolic allure: it seems to fulfill a Pythagorean ideal—the construction of harmony via integer ratios, prime structures, and “pure” divisions of the "tritave". It inherits the same philosophical impulse that drives some mathematical idealists to square the circle—a noble, if ultimately impractical, pursuit.
But just as certain geometric problems turned out to be impossible not because of mathematics, but because of the wrong initial constraints, so too here: the desire to define harmony by fixed ratios (e.g., “stacking fifths” or “folding 3:5:7”) locks composers into unnecessarily complex and unusable systems.
13ED3, while elegant in theory and historically significant in music, addresses an artificial issue—the quest for rational-based harmony that fails to resolve meaningfully within practical instrument ranges. In contrast, 8EDO tackles the perceptual and structural problem head-on, offering the same internal relationships in a more practical and accessible way.
Final Note: The Illusion of 13 Classes
In case it wasn't clear, 13ED3 only appears to have 13 distinct pitch classes. In reality, those classes don’t loop. You must continually calculate the interval relationships between new notes and the root—there is no consistent identity function. The scale produces infinite chromas, which are perceptually ambiguous, difficult to teach, and nearly impossible to internalize in ensemble performance.
In contrast, 8EDO provides a stable octave, perceptual symmetry, practical novel intervals(quarter-tones), and a meaningful class system—all within reach of existing instruments and musicians.
However, from a practical and perceptual perspective, this system is functionally indistinguishable from a much simpler and more usable system: 8EDO (eight equal divisions of the octave).
Let us consider why.
Step Size: Almost Identical
- 8EDO has a step size of exactly 150 cents.
- 13ED3 has a step size of approximately 146.3 cents.
The difference between the two is 3.7 cents—almost exactly equal to the average just-noticeable difference (JND) in pitch discrimination for trained listeners. In other words, they are perceptually interchangeable for nearly all musical applications.
Accumulation & Structural Divergence
Where these scales do diverge is not in their basic intervallic content, but in how these intervals accumulate over a wide range. In 13ED3, the eighth step produces a pitch ratio of \[3^{8/13} \approx 1.96:1 \quad (\text{≈1170 cents}) \]
This is very close to an octave (1200 cents), but not quite. As such, this "false octave" introduces a structural dissonance when mapped to instruments that expect 2:1 periodicity (like guitars or pianos). Yet melodically, it often passes as an octave—especially when accompanied by timbral masking.
This is where timbre manipulation becomes relevant. When the spectrum is shaped appropriately, the brain can re-interpret the 1170-cent interval as a legitimate octave by dissolving its internal dissonance (according to spectral models like dissonance curves). The outcome is that listeners treat this as an octave, with a nearly "extratonal" timbre. Normal.
This process points to the larger truth: the underlying internal structure is what the brain recognizes first (see Tonal Constancy). Both 13ED3 and 8EDO initially present the same internal hierarchy—but one does so with physical and perceptual structural coherence (8EDO), while the other does not (13ED3).
A compelling comparison can be made by examining how each system approximates the octave (1200 cents):
- In 13ed3, the 33rd step equals \(33 \cdot 146.3 \approx 4820 \, \text{cents},\) which is very close to 4 octaves (4800 cents).
- In 8edo, the 32nd step equals \(32 \cdot 150 = 4800 \, \text{cents},\) exactly 4 octaves.
Practical Considerations: Range and Usability
The tuning instability of 13ED3 becomes even more evident when one attempts to perform it on traditional instruments. For example:
The tuning instability of 13ED3 becomes even more evident when one attempts to perform it on traditional instruments. For example:
An acoustic instrument like the guitar, with a standard size, scale, and string gauge setup, typically provides a usable range of 4000–4500 cents. So that near-12ed2 closure rarely happens in practice.
This leads to chromatic inflation: more and more pitches emerge across the range, lacking any repeatable pattern or stable sense of class identity. The system cannot "fold back" in any meaningful way (except for the 'Tritave,' which is not a perceptual equivalence—try playing any music, like Clair de Lune, with tritave substitutions...).
In contrast, 8EDO allows immediate octave closure. Its pitches "repeat" every 1200 cents, enabling ensemble coherence (bass, guitar, voice, etc.) and perceptual tractability. Musicians don't have to memorize hundreds of slightly misaligned chromas. They work with eight analytically coherent classes, all compatible with existing instruments timbres.
Philosophical Aside: A Problem Inherited
Why, then, does 13ED3 persist as a popular theoretical model?
In part, because of its symbolic allure: it seems to fulfill a Pythagorean ideal—the construction of harmony via integer ratios, prime structures, and “pure” divisions of the "tritave". It inherits the same philosophical impulse that drives some mathematical idealists to square the circle—a noble, if ultimately impractical, pursuit.
But just as certain geometric problems turned out to be impossible not because of mathematics, but because of the wrong initial constraints, so too here: the desire to define harmony by fixed ratios (e.g., “stacking fifths” or “folding 3:5:7”) locks composers into unnecessarily complex and unusable systems.
13ED3, while elegant in theory and historically significant in music, addresses an artificial issue—the quest for rational-based harmony that fails to resolve meaningfully within practical instrument ranges. In contrast, 8EDO tackles the perceptual and structural problem head-on, offering the same internal relationships in a more practical and accessible way.
Final Note: The Illusion of 13 Classes
In case it wasn't clear, 13ED3 only appears to have 13 distinct pitch classes. In reality, those classes don’t loop. You must continually calculate the interval relationships between new notes and the root—there is no consistent identity function. The scale produces infinite chromas, which are perceptually ambiguous, difficult to teach, and nearly impossible to internalize in ensemble performance.
In contrast, 8EDO provides a stable octave, perceptual symmetry, practical novel intervals(quarter-tones), and a meaningful class system—all within reach of existing instruments and musicians.
It's important to clarify that this analysis is from and for the perspective of a practical musician. Both Bohlen and Pierce made significant contributions to advancing music theory and opening doors that were quite closed in Western theory. This doesn't mean we should disregard historical and theoretical integrity. The system, often introduced as a discovery rather than an invention , is a clear demonstration of mathematics serving the arts, and humanity exploring logic and creativity.
(why a discovery? Has it been cognitively validated as perceptually significant? Not even 12EDO has)
EXAMPLES:
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