What if you deleted the octave? — here’s a tour of four fascinating tuning systems built on the perfect twelfth instead — and why they might change how you think about harmony
The Octave Isn’t the Only Game in Town
Western music theory rests on a bedrock assumption so deep we rarely question it: the octave is special. A note an octave up from another vibrates at exactly twice the frequency, and the two sound so similar we give them the same letter name. Our entire system of keys, chords, and scales is built on this 2:1 ratio repeating forever upward and downward.
But what if consonance doesn’t have to start there?
In the early 1970s, a German microwave electronics engineer named Heinz Bohlen asked exactly that question. Reading Paul Hindemith’s analysis of tonality, he became curious about why the octave governs our scales. He turned to combination tones — the faint phantom pitches your ear generates when two notes are played together — as a guide, and derived an entirely new scale. It had 13 steps and repeated not at the octave, but at the interval of a perfect twelfth: the note you get by going up an octave and then a perfect fifth. That interval has a frequency ratio of exactly 3:1, and Bohlen’s scale was built entirely from the odd harmonics (3, 5, 7, 9…) rather than the full harmonic series.
Independently, software engineer Kees van Prooijen and later electronic engineer John R. Pierce at Bell Labs arrived at the same scale. Pierce coined the term tritave for the 3:1 interval — by analogy with “octave” (from the Latin for eight) — and the scale was eventually named the Bohlen-Pierce scale in his and Bohlen’s honour.
This article is a tour of the Bohlen-Pierce scale and three of its most important relatives, all belonging to a family called equal divisions of the tritave (or EDT). If you know your music theory but have never ventured into microtonality, you’re in the right place.
Western music theory rests on a bedrock assumption so deep we rarely question it: the octave is special. A note an octave up from another vibrates at exactly twice the frequency, and the two sound so similar we give them the same letter name. Our entire system of keys, chords, and scales is built on this 2:1 ratio repeating forever upward and downward.
But what if consonance doesn’t have to start there?
In the early 1970s, a German microwave electronics engineer named Heinz Bohlen asked exactly that question. Reading Paul Hindemith’s analysis of tonality, he became curious about why the octave governs our scales. He turned to combination tones — the faint phantom pitches your ear generates when two notes are played together — as a guide, and derived an entirely new scale. It had 13 steps and repeated not at the octave, but at the interval of a perfect twelfth: the note you get by going up an octave and then a perfect fifth. That interval has a frequency ratio of exactly 3:1, and Bohlen’s scale was built entirely from the odd harmonics (3, 5, 7, 9…) rather than the full harmonic series.
Independently, software engineer Kees van Prooijen and later electronic engineer John R. Pierce at Bell Labs arrived at the same scale. Pierce coined the term tritave for the 3:1 interval — by analogy with “octave” (from the Latin for eight) — and the scale was eventually named the Bohlen-Pierce scale in his and Bohlen’s honour.
This article is a tour of the Bohlen-Pierce scale and three of its most important relatives, all belonging to a family called equal divisions of the tritave (or EDT). If you know your music theory but have never ventured into microtonality, you’re in the right place.
First, Some Vocabulary
A few terms will come up repeatedly, so let’s define them:
The tritave is the interval 3:1 — an octave plus a perfect fifth, or about 1902 cents. In these tuning systems, it plays the same structural role that the octave plays in conventional music. Two notes a tritave apart sound like “the same note,” just as two notes an octave apart do in regular music. The tritave is called the equave (the interval of equivalence) in these systems.
EDT (Equal Divisions of the Tritave) means we divide that 1902-cent tritave into n equal steps. Just as 12-tone equal temperament (12edo) divides the octave into 12 equal semitones, 13edt divides the tritave into 13 equal steps of about 146 cents each.
JI (Just Intonation) means tuning intervals to pure frequency ratios like 5/3, 7/5, or 9/7. These are the “pure” intervals our ears tend to find most consonant. The interesting thing about these EDT systems is how well they approximate those simple ratios — in some cases, remarkably well.
The 3.5.7 subgroup is a way of saying “we only care about harmonics 3, 5, and 7 (and their combinations).” Conventional Western harmony uses the 2.3.5 subgroup (octaves, fifths, and major thirds). The Bohlen-Pierce world replaces the 2 with higher odd harmonics. The primary consonant chord in BP is 3:5:7 (analogous to the 4:5:6 major triad in conventional music).
The Lambda scale is the main 9-note “diatonic” scale within BP, analogous to our major scale. It has a pattern of large (L) and small (s) steps: sLsLsLsLs — the same kind of alternating pattern you’d find in the diatonic scale.
A few terms will come up repeatedly, so let’s define them:
The tritave is the interval 3:1 — an octave plus a perfect fifth, or about 1902 cents. In these tuning systems, it plays the same structural role that the octave plays in conventional music. Two notes a tritave apart sound like “the same note,” just as two notes an octave apart do in regular music. The tritave is called the equave (the interval of equivalence) in these systems.
EDT (Equal Divisions of the Tritave) means we divide that 1902-cent tritave into n equal steps. Just as 12-tone equal temperament (12edo) divides the octave into 12 equal semitones, 13edt divides the tritave into 13 equal steps of about 146 cents each.
JI (Just Intonation) means tuning intervals to pure frequency ratios like 5/3, 7/5, or 9/7. These are the “pure” intervals our ears tend to find most consonant. The interesting thing about these EDT systems is how well they approximate those simple ratios — in some cases, remarkably well.
The 3.5.7 subgroup is a way of saying “we only care about harmonics 3, 5, and 7 (and their combinations).” Conventional Western harmony uses the 2.3.5 subgroup (octaves, fifths, and major thirds). The Bohlen-Pierce world replaces the 2 with higher odd harmonics. The primary consonant chord in BP is 3:5:7 (analogous to the 4:5:6 major triad in conventional music).
The Lambda scale is the main 9-note “diatonic” scale within BP, analogous to our major scale. It has a pattern of large (L) and small (s) steps: sLsLsLsLs — the same kind of alternating pattern you’d find in the diatonic scale.
13edt: The Bohlen-Pierce Scale
Step size: ~146.3 cents | Xen Wiki: https://en.xen.wiki/w/13edt
This is the star of the show. Thirteen equal steps of about 146 cents each — slightly larger than a whole tone in conventional music, but smaller than a minor third — spanning a tritave.
The most important thing to understand about 13edt is why it works. Its 13 equal steps do a remarkable job of approximating the simplest odd-number frequency ratios:
- 9/7 (~435 cents, “supermajor third”): approximated at 3 steps (438.9 cents), only 3.8 cents sharp
- 7/5 (~583 cents, “septimal tritone”): approximated at 4 steps (585.2 cents), only 2.7 cents sharp
- 5/3 (~884 cents, just major sixth): approximated at 6 steps (877.8 cents), only 6.5 cents flat
- 9/5 (~1018 cents): approximated at 7 steps (1024.1 cents), 6.5 cents sharp
- 7/3 (~1467 cents): approximated at 10 steps (1463.0 cents), only 3.8 cents flat
To put that in context: the accuracy of 13edt in approximating this set of harmonics is comparable to how well 34-note equal temperament (34edo) handles the conventional 5-limit. In other words, this 13-note system is extraordinarily efficient.
The two commas that 13edt “tempers out” (treats as zero) are 245/243 (the sensamagic comma) and 3125/3087. The first of these is especially important: it means that two stacked 9/7 intervals equal 5/3. This creates a coherent internal logic for the scale, just as in conventional meantone temperament, where four stacked perfect fifths equal a major third.
Step size: ~146.3 cents | Xen Wiki: https://en.xen.wiki/w/13edt
This is the star of the show. Thirteen equal steps of about 146 cents each — slightly larger than a whole tone in conventional music, but smaller than a minor third — spanning a tritave.
The most important thing to understand about 13edt is why it works. Its 13 equal steps do a remarkable job of approximating the simplest odd-number frequency ratios:
- 9/7 (~435 cents, “supermajor third”): approximated at 3 steps (438.9 cents), only 3.8 cents sharp
- 7/5 (~583 cents, “septimal tritone”): approximated at 4 steps (585.2 cents), only 2.7 cents sharp
- 5/3 (~884 cents, just major sixth): approximated at 6 steps (877.8 cents), only 6.5 cents flat
- 9/5 (~1018 cents): approximated at 7 steps (1024.1 cents), 6.5 cents sharp
- 7/3 (~1467 cents): approximated at 10 steps (1463.0 cents), only 3.8 cents flat
To put that in context: the accuracy of 13edt in approximating this set of harmonics is comparable to how well 34-note equal temperament (34edo) handles the conventional 5-limit. In other words, this 13-note system is extraordinarily efficient.
The two commas that 13edt “tempers out” (treats as zero) are 245/243 (the sensamagic comma) and 3125/3087. The first of these is especially important: it means that two stacked 9/7 intervals equal 5/3. This creates a coherent internal logic for the scale, just as in conventional meantone temperament, where four stacked perfect fifths equal a major third.
The Lambda Scale and Its Modes
The 9-note Lambda scale (4L 5s in music theory notation) is generated by stacking 9/7 intervals within the tritave. It has two step sizes in the ratio 2:1, making it analogous to the diatonic scale in conventional music. The symmetric mode (sLsLsLsLs) is called the Cassiopeian or Lambda mode.
The note names traditionally used are J K L M N O P Q R (nine notes), with accidentals for the remaining four chromatic notes. An older alternative notation uses C D E F G H J A B (notice there’s no I, and J replaces what would be I). The 13-note “chromatic” set in BP notation functions much as the 12-note chromatic scale does in conventional music.
The 9-note Lambda scale (4L 5s in music theory notation) is generated by stacking 9/7 intervals within the tritave. It has two step sizes in the ratio 2:1, making it analogous to the diatonic scale in conventional music. The symmetric mode (sLsLsLsLs) is called the Cassiopeian or Lambda mode.
The note names traditionally used are J K L M N O P Q R (nine notes), with accidentals for the remaining four chromatic notes. An older alternative notation uses C D E F G H J A B (notice there’s no I, and J replaces what would be I). The 13-note “chromatic” set in BP notation functions much as the 12-note chromatic scale does in conventional music.
The 3:5:7 Chord
The fundamental consonant sonority of BP is the 3:5:7 chord, which works like this: if your root is at 1/1, the next note is 5/3 above (approximately 884 cents, a just major sixth), and the top note is 7/3 above the root (approximately 1467 cents, close to a minor seventh plus tritave). In 13edt, this chord falls on steps 0, 6, and 10.
A 1984 experiment found several BP chords were perceived as consonant by both musicians and untrained listeners. The 3:5:7 chord’s overtones coincide in the same way as those of a conventional major triad: the 5th partial of the root coincides with the 3rd partial of the middle note.
The full 3:5:7:9 tetrad — adding 9/1 reduced by the tritave — is also characteristic, and functions somewhat like a dominant seventh chord within this system.
The fundamental consonant sonority of BP is the 3:5:7 chord, which works like this: if your root is at 1/1, the next note is 5/3 above (approximately 884 cents, a just major sixth), and the top note is 7/3 above the root (approximately 1467 cents, close to a minor seventh plus tritave). In 13edt, this chord falls on steps 0, 6, and 10.
A 1984 experiment found several BP chords were perceived as consonant by both musicians and untrained listeners. The 3:5:7 chord’s overtones coincide in the same way as those of a conventional major triad: the 5th partial of the root coincides with the 3rd partial of the middle note.
The full 3:5:7:9 tetrad — adding 9/1 reduced by the tritave — is also characteristic, and functions somewhat like a dominant seventh chord within this system.
What Does 13edt Sound Like?
The overwhelming effect of the Bohlen-Pierce scale can hardly be described in words. It is the immediate aural impression that opens the doors to this alternate tonal world. To a conventionally trained ear, BP chords can sound alien at first — the absence of octaves and the prominence of 7-based intervals gives the music a quality that has been described as “Martian” or “crystalline.” But the system has genuine internal harmonic logic, and composers who have worked deeply with it report that it begins to feel natural.
The Bohlen-Pierce scale may sound odd due to social conditioning.
Instruments with primarily odd harmonics — clarinets (which overblow at the 12th rather than the octave), square wave synthesizers, and some wind instruments — are naturally suited to BP because their harmonic content already avoids the even harmonics that BP discards. Clarinet maker Stephen Fox created a family of BP clarinets at the request of composer Georg Hajdu, and these instruments have been used in a growing body of concert music.
The overwhelming effect of the Bohlen-Pierce scale can hardly be described in words. It is the immediate aural impression that opens the doors to this alternate tonal world. To a conventionally trained ear, BP chords can sound alien at first — the absence of octaves and the prominence of 7-based intervals gives the music a quality that has been described as “Martian” or “crystalline.” But the system has genuine internal harmonic logic, and composers who have worked deeply with it report that it begins to feel natural.
The Bohlen-Pierce scale may sound odd due to social conditioning.
Instruments with primarily odd harmonics — clarinets (which overblow at the 12th rather than the octave), square wave synthesizers, and some wind instruments — are naturally suited to BP because their harmonic content already avoids the even harmonics that BP discards. Clarinet maker Stephen Fox created a family of BP clarinets at the request of composer Georg Hajdu, and these instruments have been used in a growing body of concert music.
Rank-2 Temperaments in 13edt
Like the diatonic scale’s relationship to meantone temperament, the Lambda scale is the central MOS (moment of symmetry) scale of a rank-2 temperament called BPS (Bohlen-Pierce-Stearns). BPS is generated by a slightly sharp 9/7 interval and tempers out 245/243, making it the “meantone of the BP world.” The 3:5:7 triad plays the same role in BPS harmony that the 4:5:6 major triad plays in meantone.
Other named rank-2 temperaments available within 13edt include:
- Procyon (generator: 49/45)
- Sirius (generator: 25/21)
- Canopus (generator: 7/5)
- Arcturus (generator: 5/3)
All named after stars, which is a running theme in tritave-based temperament naming.
Like the diatonic scale’s relationship to meantone temperament, the Lambda scale is the central MOS (moment of symmetry) scale of a rank-2 temperament called BPS (Bohlen-Pierce-Stearns). BPS is generated by a slightly sharp 9/7 interval and tempers out 245/243, making it the “meantone of the BP world.” The 3:5:7 triad plays the same role in BPS harmony that the 4:5:6 major triad plays in meantone.
Other named rank-2 temperaments available within 13edt include:
- Procyon (generator: 49/45)
- Sirius (generator: 25/21)
- Canopus (generator: 7/5)
- Arcturus (generator: 5/3)
All named after stars, which is a running theme in tritave-based temperament naming.
17edt: The Hard Bohlen-Pierce
Step size: ~111.9 cents | Xen Wiki: https://en.xen.wiki/w/17edt
If 13edt is the “basic” or “soft” tuning of the Lambda scale, 17edt is its “hard” counterpart — analogous to the relationship between 12edo and 17edo in conventional music.
17edt divides the tritave into 17 steps of about 111.9 cents each — almost exactly the size of a conventional diatonic semitone (100 cents), but just a hair larger.
Like 13edt, 17edt tempers out 245/243, which is the key property that gives both systems their shared harmonic DNA. This means 17edt also supports the Lambda (4L 5s) scale — but now the large step is 4 EDt steps and the small step is 1, giving an L:s ratio of 4:1. This is the “hard” end of the BPS tuning spectrum, just as 17edo gives a very hard version of the diatonic scale.
The major tradeoff is accuracy: 17edt’s approximations of 5/3 and 7/3 are worse than 13edt’s. But 17edt makes up for this with some remarkable properties in higher prime limits:
- 11/7 is approximated at 7 steps (783.2 cents) with only 0.6 cents error — extraordinarily close
- 55/27 falls at 11 steps with only 1.1 cents error
- 17/15 is approximated almost exactly — the step size of 111.88 cents is only 0.15 cents sharp of the 16/15 semitone (111.73 cents)
This last property is especially interesting. Because 17/15 is so close to one step of 17edt, the scale naturally supports Dubhe temperament, which splits the 9/7 BPS generator in half so that two 17/15’s equal 9/7. This produces an 8-note (8L 1s) enneatonic scale and a 17-note chromatic scale.
17edt also supports Mintra temperament in the 3.5.7.11 subgroup, which splits the 9/7 generator into three 11/7’s. This gives access to 5L 2s (macrodiatonic) and 5L 7s (macrochromatic) MOS scales. The 5L 7s scale at 17edt behaves somewhat like a 12-note chromatic scale in conventional music but stretched to fit the tritave.
A useful way to think about 17edt’s place in the BP family: if 13edt is the analogue of 12edo — the simplest, most efficient equal temperament of this harmonic world — then 17edt is the analogue of 17edo: harder, more adventurous, with its own unique expressive character.
There is even a charming coincidence noted in the literature: 17edo and 17edt both approximate 9/7, but in opposite directions — 17edo is about 11.6 cents flat, while 17edt is about 12.4 cents sharp. Their diatonic structures are strikingly similar despite this difference.
Step size: ~111.9 cents | Xen Wiki: https://en.xen.wiki/w/17edt
If 13edt is the “basic” or “soft” tuning of the Lambda scale, 17edt is its “hard” counterpart — analogous to the relationship between 12edo and 17edo in conventional music.
17edt divides the tritave into 17 steps of about 111.9 cents each — almost exactly the size of a conventional diatonic semitone (100 cents), but just a hair larger.
Like 13edt, 17edt tempers out 245/243, which is the key property that gives both systems their shared harmonic DNA. This means 17edt also supports the Lambda (4L 5s) scale — but now the large step is 4 EDt steps and the small step is 1, giving an L:s ratio of 4:1. This is the “hard” end of the BPS tuning spectrum, just as 17edo gives a very hard version of the diatonic scale.
The major tradeoff is accuracy: 17edt’s approximations of 5/3 and 7/3 are worse than 13edt’s. But 17edt makes up for this with some remarkable properties in higher prime limits:
- 11/7 is approximated at 7 steps (783.2 cents) with only 0.6 cents error — extraordinarily close
- 55/27 falls at 11 steps with only 1.1 cents error
- 17/15 is approximated almost exactly — the step size of 111.88 cents is only 0.15 cents sharp of the 16/15 semitone (111.73 cents)
This last property is especially interesting. Because 17/15 is so close to one step of 17edt, the scale naturally supports Dubhe temperament, which splits the 9/7 BPS generator in half so that two 17/15’s equal 9/7. This produces an 8-note (8L 1s) enneatonic scale and a 17-note chromatic scale.
17edt also supports Mintra temperament in the 3.5.7.11 subgroup, which splits the 9/7 generator into three 11/7’s. This gives access to 5L 2s (macrodiatonic) and 5L 7s (macrochromatic) MOS scales. The 5L 7s scale at 17edt behaves somewhat like a 12-note chromatic scale in conventional music but stretched to fit the tritave.
A useful way to think about 17edt’s place in the BP family: if 13edt is the analogue of 12edo — the simplest, most efficient equal temperament of this harmonic world — then 17edt is the analogue of 17edo: harder, more adventurous, with its own unique expressive character.
There is even a charming coincidence noted in the literature: 17edo and 17edt both approximate 9/7, but in opposite directions — 17edo is about 11.6 cents flat, while 17edt is about 12.4 cents sharp. Their diatonic structures are strikingly similar despite this difference.
26edt: Double Bohlen-Pierce
Step size: ~73.2 cents | Xen Wiki: https://en.xen.wiki/w/26edt
26edt is simply 13edt doubled — each step of 13edt is split into two equal steps of about 73 cents each. This means every interval of 13edt is still present exactly, just accessed at even-numbered steps. But the extra notes between open up new harmonic possibilities.
The step size of ~73 cents is roughly three-quarters of a conventional semitone. To give a sense of how fine this is: the difference between a major and minor second in conventional music is 100 cents, so 26edt’s step is considerably smaller.
Because it contains 13edt, 26edt tempers out the same 3.5.7 commas (245/243 and 3125/3087). But the doubled resolution primarily pays off in higher prime limits:
- Prime 17 is added accurately for the first time. 26edt tempers out 2025/2023, meaning two steps of 17/15 exactly equal 9/7. This makes 26edt the canonical tuning of Dubhe temperament.
- 21/17 is approximated with extraordinary accuracy — the tuning forms a 21-note consistent circle through this interval, allowing for rich modulation.
- 26edt achieves consistency to the no-twos 21-odd-limit, meaning it consistently and accurately represents all odd harmonics up to 21. It is the first EDT to achieve this.
The primary new scale structure in 26edt (beyond the inherited 13edt scales) is the Dubhe[9] scale — an 8L 1s MOS scale generated by 17/15. This is a fascinating structure: 8 large steps and 1 small step, like a nearly-equal scale with one slight impurity, similar in feel to a Pythagorean tuning of the diatonic scale.
26edt also introduces a curious numerical coincidence: many of its intervals happen to be very close (within ±6.5 cents) to the corresponding intervals of 26-tone equal temperament (26edo), despite the two systems having completely different theoretical foundations. The reason is that 26edt corresponds to approximately 16.4edo, and 26/16.4 ≈ 1.585, which happens to keep many intervals tightly aligned.
For composers, 26edt represents a practical expansion of the BP harmonic palette: you get the full BP harmonic world, plus a well-integrated prime-17 dimension, all in a 26-note system that’s manageable for instrument design and notation.
Step size: ~73.2 cents | Xen Wiki: https://en.xen.wiki/w/26edt
26edt is simply 13edt doubled — each step of 13edt is split into two equal steps of about 73 cents each. This means every interval of 13edt is still present exactly, just accessed at even-numbered steps. But the extra notes between open up new harmonic possibilities.
The step size of ~73 cents is roughly three-quarters of a conventional semitone. To give a sense of how fine this is: the difference between a major and minor second in conventional music is 100 cents, so 26edt’s step is considerably smaller.
Because it contains 13edt, 26edt tempers out the same 3.5.7 commas (245/243 and 3125/3087). But the doubled resolution primarily pays off in higher prime limits:
- Prime 17 is added accurately for the first time. 26edt tempers out 2025/2023, meaning two steps of 17/15 exactly equal 9/7. This makes 26edt the canonical tuning of Dubhe temperament.
- 21/17 is approximated with extraordinary accuracy — the tuning forms a 21-note consistent circle through this interval, allowing for rich modulation.
- 26edt achieves consistency to the no-twos 21-odd-limit, meaning it consistently and accurately represents all odd harmonics up to 21. It is the first EDT to achieve this.
The primary new scale structure in 26edt (beyond the inherited 13edt scales) is the Dubhe[9] scale — an 8L 1s MOS scale generated by 17/15. This is a fascinating structure: 8 large steps and 1 small step, like a nearly-equal scale with one slight impurity, similar in feel to a Pythagorean tuning of the diatonic scale.
26edt also introduces a curious numerical coincidence: many of its intervals happen to be very close (within ±6.5 cents) to the corresponding intervals of 26-tone equal temperament (26edo), despite the two systems having completely different theoretical foundations. The reason is that 26edt corresponds to approximately 16.4edo, and 26/16.4 ≈ 1.585, which happens to keep many intervals tightly aligned.
For composers, 26edt represents a practical expansion of the BP harmonic palette: you get the full BP harmonic world, plus a well-integrated prime-17 dimension, all in a 26-note system that’s manageable for instrument design and notation.
39edt: Triple Bohlen-Pierce
Step size: ~48.8 cents | Xen Wiki: https://en.xen.wiki/w/39edt
39edt is 13edt tripled — each step of 13edt is divided into three equal steps of about 48.8 cents each. This is roughly half a conventional semitone, putting it in the territory of “quarter-tone” music (though these are not conventional quarter-tones, since the system is tritave-based).
This tuning was first seriously proposed as a musical system by music theorist Paul Erlich, and is sometimes called the Triple Bohlen-Pierce scale or Triple BP. Erlich noted that 39edt adds accurate approximations of harmonics 11 and 13 to the BP framework — something neither 13edt nor 26edt accomplishes well.
The key new intervals are:
- 11/9 (~347 cents): approximated at 7 steps (341.4 cents), about 6 cents flat
- 13/9 (~637 cents): approximated at 13 steps (634.0 cents), about 2.6 cents flat
- 15/11 (~537 cents): approximated at 11 steps (536.4 cents), only 0.5 cents flat — essentially just
- 11/7 (~783 cents): approximated at 16 steps (780.3 cents), about 2.2 cents flat
This is what makes 39edt special: it is described in the Xenharmonic Wiki as having a better no-twos 13-limit relative error than any other EDT up to 914edt. In plain English: 39edt is extraordinarily efficient at approximating a rich palette of odd-number harmonics simultaneously.
Step size: ~48.8 cents | Xen Wiki: https://en.xen.wiki/w/39edt
39edt is 13edt tripled — each step of 13edt is divided into three equal steps of about 48.8 cents each. This is roughly half a conventional semitone, putting it in the territory of “quarter-tone” music (though these are not conventional quarter-tones, since the system is tritave-based).
This tuning was first seriously proposed as a musical system by music theorist Paul Erlich, and is sometimes called the Triple Bohlen-Pierce scale or Triple BP. Erlich noted that 39edt adds accurate approximations of harmonics 11 and 13 to the BP framework — something neither 13edt nor 26edt accomplishes well.
The key new intervals are:
- 11/9 (~347 cents): approximated at 7 steps (341.4 cents), about 6 cents flat
- 13/9 (~637 cents): approximated at 13 steps (634.0 cents), about 2.6 cents flat
- 15/11 (~537 cents): approximated at 11 steps (536.4 cents), only 0.5 cents flat — essentially just
- 11/7 (~783 cents): approximated at 16 steps (780.3 cents), about 2.2 cents flat
This is what makes 39edt special: it is described in the Xenharmonic Wiki as having a better no-twos 13-limit relative error than any other EDT up to 914edt. In plain English: 39edt is extraordinarily efficient at approximating a rich palette of odd-number harmonics simultaneously.
Mintra Temperament and the Macrodiatonic
The most important musical framework unlocked by 39edt is Mintra temperament in the 3.5.7.11 subgroup (with a 13-limit extension also available). Mintra splits the BPS generator of 9/7 into three 11/7’s, meaning the interval 11/7 (about 783 cents) serves as the generator of the scale. This creates a family of MOS scales:
- 5L 2s (macrodiatonic): A 7-note scale generated by 11/7, analogous to the conventional diatonic scale but with all intervals stretched to fit the tritave. The “fourth” of this scale is an 11/7, and the “fifth” is a 27/11. This scale contains the 7:9:11 chord as its primary consonance.
- 5L 7s (macrochromatic): A 12-note scale, analogous to the chromatic scale in conventional music but again stretched to the tritave. This is the primary working scale in Mintra.
- 5L 12s: A 17-note scale for fully chromatic work.
These scales are sometimes called “macrodiatonic” because all their intervals are much larger than their conventional counterparts — what would be a diatonic fifth (about 702 cents) is now an 11/7 (about 783 cents), stretched by being measured against the tritave rather than the octave.
The 7:9:11 triad — the primary consonance of the 3.7.11 subgroup — plays the role in Mintra that the 4:5:6 major triad plays in meantone. In 39edt specifically, this chord falls on steps 0, 9, 16.
The most important musical framework unlocked by 39edt is Mintra temperament in the 3.5.7.11 subgroup (with a 13-limit extension also available). Mintra splits the BPS generator of 9/7 into three 11/7’s, meaning the interval 11/7 (about 783 cents) serves as the generator of the scale. This creates a family of MOS scales:
- 5L 2s (macrodiatonic): A 7-note scale generated by 11/7, analogous to the conventional diatonic scale but with all intervals stretched to fit the tritave. The “fourth” of this scale is an 11/7, and the “fifth” is a 27/11. This scale contains the 7:9:11 chord as its primary consonance.
- 5L 7s (macrochromatic): A 12-note scale, analogous to the chromatic scale in conventional music but again stretched to the tritave. This is the primary working scale in Mintra.
- 5L 12s: A 17-note scale for fully chromatic work.
These scales are sometimes called “macrodiatonic” because all their intervals are much larger than their conventional counterparts — what would be a diatonic fifth (about 702 cents) is now an 11/7 (about 783 cents), stretched by being measured against the tritave rather than the octave.
The 7:9:11 triad — the primary consonance of the 3.7.11 subgroup — plays the role in Mintra that the 4:5:6 major triad plays in meantone. In 39edt specifically, this chord falls on steps 0, 9, 16.
39edt as Tritave-Period Mavila
Another surprising property: if you insert conventional octaves into 39edt (treating every 25 steps as an octave, since 25/39 of a tritave ≈ 1200 cents), the resulting tuning functions as a very flat version of the conventional fifth-generated scale — specifically, a tuning of mavila temperament, which uses a ~683-cent fifth. This is the same kind of tuning used in the music of some African traditions (Ugandan amadinda, etc.), but here arrived at from a completely different theoretical direction.
Another surprising property: if you insert conventional octaves into 39edt (treating every 25 steps as an octave, since 25/39 of a tritave ≈ 1200 cents), the resulting tuning functions as a very flat version of the conventional fifth-generated scale — specifically, a tuning of mavila temperament, which uses a ~683-cent fifth. This is the same kind of tuning used in the music of some African traditions (Ugandan amadinda, etc.), but here arrived at from a completely different theoretical direction.
Multiple Uses of 39edt
One of 39edt’s most appealing features for composers is its flexibility. You can use it as:
- Standard Bohlen-Pierce (accessing only every 3rd step)
- A 13-limit harmonic system using Mintra temperament
- An interlocking triple BP — three interlocked copies of the BP scale, offset by one 39edt step each, creating smooth quarter-tone-like transitions between BP harmonies
- A mavila-like diatonic system (with inserted octaves)
As the Xen Wiki article notes, 39edt “can be used in a variety of ways, for both just intonation chords and harmonies, as standard Bohlen-Pierce scale interlocking three times with calm-sounding quarter-tones, and for various JI modulations.”
One of 39edt’s most appealing features for composers is its flexibility. You can use it as:
- Standard Bohlen-Pierce (accessing only every 3rd step)
- A 13-limit harmonic system using Mintra temperament
- An interlocking triple BP — three interlocked copies of the BP scale, offset by one 39edt step each, creating smooth quarter-tone-like transitions between BP harmonies
- A mavila-like diatonic system (with inserted octaves)
As the Xen Wiki article notes, 39edt “can be used in a variety of ways, for both just intonation chords and harmonies, as standard Bohlen-Pierce scale interlocking three times with calm-sounding quarter-tones, and for various JI modulations.”
How These Four Systems Relate
It helps to see all four as part of a single family, with 13edt as the parent:
SystemSteps in tritaveStep sizeLike… (conventional analogy)13edt13~146 ¢12edo — the basic, most efficient form17edt17~112 ¢17edo — harder, with unique higher-limit properties26edt26~73 ¢24edo (quarter-tones) — doubles 13edt, adds prime 1739edt39~49 ¢31edo or 36edo — triples 13edt, adds primes 11 and 13
All four temper out the same 3.5.7 commas (245/243 and 3125/3087), meaning they all share the same fundamental BP harmonic logic. The higher systems add new harmonic dimensions on top.
The family also shares a set of named rank-2 temperaments that make musical use of various generators:
- BPS — the core BP temperament (generator: ~9/7)
- Dubhe — splits 9/7 into two 17/15’s (supported by 26edt and 17edt)
- Mintra — splits 9/7 into three 11/7’s (supported by 39edt and 17edt)
- Deneb — 3.5.11 subgroup, generator ~11/9 (supported by 39edt)
- Electra — 3.5.11.13 subgroup, generator ~15/11 (supported by 39edt)
It helps to see all four as part of a single family, with 13edt as the parent:
SystemSteps in tritaveStep sizeLike… (conventional analogy)13edt13~146 ¢12edo — the basic, most efficient form17edt17~112 ¢17edo — harder, with unique higher-limit properties26edt26~73 ¢24edo (quarter-tones) — doubles 13edt, adds prime 1739edt39~49 ¢31edo or 36edo — triples 13edt, adds primes 11 and 13
All four temper out the same 3.5.7 commas (245/243 and 3125/3087), meaning they all share the same fundamental BP harmonic logic. The higher systems add new harmonic dimensions on top.
The family also shares a set of named rank-2 temperaments that make musical use of various generators:
- BPS — the core BP temperament (generator: ~9/7)
- Dubhe — splits 9/7 into two 17/15’s (supported by 26edt and 17edt)
- Mintra — splits 9/7 into three 11/7’s (supported by 39edt and 17edt)
- Deneb — 3.5.11 subgroup, generator ~11/9 (supported by 39edt)
- Electra — 3.5.11.13 subgroup, generator ~15/11 (supported by 39edt)
Getting Started: Practical Tips for Musicians
Notation: The standard BP notation uses 9 nominals (J K L M N O P Q R) for the Lambda scale, with sharps and flats for the remaining chromatic notes. This is the notation you’ll encounter in most Xen Wiki articles and BP-specific software. Some older sources use the letter names C D E F G H J A B.
Software: Most microtonal DAW plugins support arbitrary tuning via Scala (.scl) files. The Scala files for all four tuning systems are freely available online. Surge XT, ZynAddSubFX, and VCV Rack all support Scala tuning import. The Xen Wiki has Scala files linked from each EDT article.
Instruments: Instruments with odd-partial timbres work best — clarinets, square wave synths, and certain brass timbres. For electronic music, designing timbres whose partials align with 3, 5, 7, 9 (and 11 for 39edt) will maximize the consonance of BP harmony.
Listening: A large body of BP music exists and is growing. The Bohlen-Pierce scale music page on Xen Wiki lists dozens of composers and pieces. For 39edt specifically, look for works by Francium (Strange Juice, 2025) and Phanomium (Polygonal, 2025).
Starting point for composition: The 9-note Lambda scale in 13edt is the most accessible entry point, just as the 7-note diatonic scale is the entry point to conventional tonal music. Learn the 3:5:7 chord (steps 0–6–10) and its inversions. The four “diatonic” triads built on scale degrees of the Lambda scale function similarly to major and minor triads in conventional music.
Notation: The standard BP notation uses 9 nominals (J K L M N O P Q R) for the Lambda scale, with sharps and flats for the remaining chromatic notes. This is the notation you’ll encounter in most Xen Wiki articles and BP-specific software. Some older sources use the letter names C D E F G H J A B.
Software: Most microtonal DAW plugins support arbitrary tuning via Scala (.scl) files. The Scala files for all four tuning systems are freely available online. Surge XT, ZynAddSubFX, and VCV Rack all support Scala tuning import. The Xen Wiki has Scala files linked from each EDT article.
Instruments: Instruments with odd-partial timbres work best — clarinets, square wave synths, and certain brass timbres. For electronic music, designing timbres whose partials align with 3, 5, 7, 9 (and 11 for 39edt) will maximize the consonance of BP harmony.
Listening: A large body of BP music exists and is growing. The Bohlen-Pierce scale music page on Xen Wiki lists dozens of composers and pieces. For 39edt specifically, look for works by Francium (Strange Juice, 2025) and Phanomium (Polygonal, 2025).
Starting point for composition: The 9-note Lambda scale in 13edt is the most accessible entry point, just as the 7-note diatonic scale is the entry point to conventional tonal music. Learn the 3:5:7 chord (steps 0–6–10) and its inversions. The four “diatonic” triads built on scale degrees of the Lambda scale function similarly to major and minor triads in conventional music.
Why Does Any of This Matter?
The Bohlen-Pierce family isn’t just a curiosity. It represents something genuinely new: a complete, self-consistent harmonic world built on different physical foundations than conventional Western music.
“The Bohlen-Pierce scale really is fundamentally different, and requires a fundamentally new music theory. This theory is not trivial or obvious.”
For composers, these systems offer harmonic resources that simply don’t exist in 12-tone music: pure septimal intervals, 11-limit chords, and tonal progressions built on 3:5:7 rather than 4:5:6. For theorists, they offer a fascinating parallel universe to study — one where meantone’s role is played by BPS, the diatonic scale’s role is played by the Lambda scale, and the chromatic scale’s role is played by BP’s 13-note system.
And for listeners? The music is genuinely striking. Whether it sounds alien or beautiful — or both at once — depends partly on the composer and partly on your ears. But the more you listen, the more the internal logic reveals itself, and the more rewarding the experience becomes.
The tritave is waiting. The only question is whether you’re curious enough to step into it.
Further reading and listening: Xenharmonic Wiki, The Bohlen-Pierce Site, Xen Wiki: 13edt, Xen Wiki: 17edt, Xen Wiki: 26edt, Xen Wiki: 39edt, Xen Wiki: BPS temperament
The Bohlen-Pierce family isn’t just a curiosity. It represents something genuinely new: a complete, self-consistent harmonic world built on different physical foundations than conventional Western music.
“The Bohlen-Pierce scale really is fundamentally different, and requires a fundamentally new music theory. This theory is not trivial or obvious.”
For composers, these systems offer harmonic resources that simply don’t exist in 12-tone music: pure septimal intervals, 11-limit chords, and tonal progressions built on 3:5:7 rather than 4:5:6. For theorists, they offer a fascinating parallel universe to study — one where meantone’s role is played by BPS, the diatonic scale’s role is played by the Lambda scale, and the chromatic scale’s role is played by BP’s 13-note system.
And for listeners? The music is genuinely striking. Whether it sounds alien or beautiful — or both at once — depends partly on the composer and partly on your ears. But the more you listen, the more the internal logic reveals itself, and the more rewarding the experience becomes.
The tritave is waiting. The only question is whether you’re curious enough to step into it.
Further reading and listening: Xenharmonic Wiki, The Bohlen-Pierce Site, Xen Wiki: 13edt, Xen Wiki: 17edt, Xen Wiki: 26edt, Xen Wiki: 39edt, Xen Wiki: BPS temperament
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