The special family of scales with no wolf intervals
If you’ve spent a lot of time exploring xenharmonic music theory, you’ve probably encountered the concept of harmonic entropy — the idea that some intervals sound smooth and resolved while others sound tense and dissonant, based on how closely they approximate simple frequency ratios. The harmonic entropy graph has valleys (low entropy, consonant-sounding intervals) and hills (high entropy, dissonant-sounding intervals), and this landscape turns out to be incredibly useful for thinking about scale design.
One concept that takes direct inspiration from this landscape is the runoff scale — a relatively new and niche but genuinely useful idea coined by xenharmonic wiki user BudjarnLambeth. This article walks through everything we currently know about runoff scales: what they are, why they matter, and which specific scales and tuning systems have been identified as runoff so far.
What Is a Runoff Scale?
A runoff scale is defined as a scale where as many of its individual intervals as possible occur inside the valleys of the harmonic entropy graph, and avoid the hills. Crucially, this condition must hold not just for the scale in its default rotation, but for all of its rotations — every mode you can derive from it must also satisfy the valley-hugging criterion as much as possible.
The term is idiosyncratic to BudjarnLambeth’s wiki page, meaning it hasn’t yet been widely adopted across the broader xenharmonic community. But the concept it describes is genuinely meaningful and potentially very useful for composers and scale designers.
A few important clarifications come directly from the definition:
Runoff does not mean low total harmonic entropy. Total harmonic entropy is calculated by finding the mean entropy of all intervals across all rotations of a scale. But runoff is determined by the median, not the mean. This distinction matters a lot. A scale could theoretically have quite high overall harmonic entropy and still qualify as runoff, as long as the majority of its intervals land in valleys rather than on hills. The scale isn’t necessarily the most concordant possible — it’s the one with the fewest “wolf” intervals, the fewest truly sour outliers.
Runoff scales are effectively scales with as few wolf intervals as possible. This is perhaps the most practical way to understand the concept. A wolf interval is one that lands on a hill of the harmonic entropy curve — not just moderately dissonant, but actively clashing in a way that stands out unpleasantly. By minimizing these, a runoff scale becomes maximally “safe” to noodle around on. You can take arbitrary subsets of it, explore it with reckless abandon, improvise freely, and you’re less likely to stumble into something that sounds shockingly sour compared to the rest of the scale.
Runoff scales are a type of omniconsonant scale. They belong to the broader family of scales designed to maximize consonance across all modes and rotations.
The concept only makes sense for scales with step sizes of at least 30 cents. When all steps are smaller than 30 cents, every feature of the harmonic entropy graph gets carpeted by notes, and the median entropy ends up being roughly the same for every scale. So runoff is a meaningful distinction only in the macrotonal-to-moderately-chromatic range.
The concept can be extended beyond ordinary octave-repeating scales: a runoff EDO is an EDO (equal division of the octave) that meets the runoff criteria when treated as a scale, a runoff EDT is an EDT (equal division of the tritave) that meets them, and so on.
Why Does This Matter for Composers?
The practical value of runoff scales is hard to overstate for a certain style of composition. If you want to write freely exploratory, improvisational, or generative music in a microtonal tuning — music where you don’t know in advance which notes you’ll be playing simultaneously — a runoff scale gives you a kind of harmonic safety net. The vast majority of intervals you land on will be in valleys of the harmonic entropy curve, meaning they’ll sound like something — like recognizable intervals with some sonic identity — rather than landing in the murky, characterless midground between two recognizable intervals.
This makes runoff scales particularly appealing for:
- Algorithmic and generative music, where intervals are produced somewhat randomly
- Free improvisation in microtonal contexts
- Ambient and drone music where arbitrary combinations of pitches might ring together
- Beginners exploring a new tuning system who want a forgiving starting point
- Composers who want to write for instruments tuned to a single fixed scale and need that scale to be harmonically versatile across all its modes
The Complete List of Known Runoff Scales
Based on the Xenharmonic Wiki page by BudjarnLambeth, here is every runoff scale or runoff tuning system identified so far, with detailed descriptions of each.
Runoff EDOs
5edo
Five equal divisions of the octave, with a step size of 240 cents — the quintessential equipentatonic scale. This is the equal-tempered version of the slendro scales found in Indonesian gamelan music, and related equipentatonic traditions found across West Africa (Dagarti, Lobi, Senufo peoples), Uganda, and elsewhere. Its two interval classes — 240¢ and 480¢ — approximate the subminor third (7/6, 267¢) and the perfect fourth (4/3, 498¢) respectively, both of which sit in clear valleys. Sevish, a prominent xenharmonic composer, specifically recommends 5edo as an entry point for people newly exploring beyond 12edo, partly for exactly this reason: it’s forgiving and harmonically clear. 5edo is both a runoff EDO and the basis for many equipentatonic runoff scales.
6edo
Six equal divisions, step size 200 cents — the whole-tone scale. Its intervals are 200¢ (major second, approximating 9/8), 400¢ (major third, approximating 5/4), 600¢ (tritone, approximating 7/5 or 10/7), 800¢, 1000¢, and 1200¢. Several of these land in identifiable harmonic entropy valleys — the major second, major third, and tritone all have fairly clear just intonation identities nearby. The whole-tone scale has a dreamy, ambiguous quality in 12edo, and its runoff status reflects the fact that most of its intervals do have recognized identity even if the scale as a whole sounds harmonically unresolved.
12edo
The familiar twelve-tone equal temperament, the dominant tuning system in Western music for the past few centuries. It divides the octave into 12 parts of exactly 100 cents each. 12edo is a runoff EDO, which is consistent with its widespread appeal — the system was refined over centuries partly because it keeps most of its intervals close to recognizable just ratios. Its fifth at 700¢ is only 2 cents flat of 3/2; its major third is 13.7 cents sharp of 5/4; its minor third is 15.6 cents flat of 6/5. Most of 12edo’s interval classes land in valleys of the harmonic entropy graph, and the ones that don’t (like the tritone, which is somewhat ambiguous) are used expressively in tonal music precisely because of their tension. The runoff property of 12edo is part of why it feels so versatile and harmonically navigable across all its modes and keys.
15edo
Fifteen equal divisions, step size 80 cents. 15edo is a fascinating and somewhat unusual system — it doesn’t contain a standard meantone diatonic scale, and its fifth at 720¢ is noticeably wide. However, it supports the Blackwood decatonic temperament, has excellent approximations to 7/4 and 11/8, and its intervals across all rotations generally land in harmonic entropy valleys reasonably well. The fact that it’s a runoff EDO suggests that despite its unfamiliar character, 15edo is a harmonically generous system that rewards free exploration. It’s the smallest EDO with recognizable distinct representations of all 5-odd-limit intervals.
19edo
Nineteen equal divisions, step size approximately 63.2 cents. 19edo is one of the most beloved and widely explored xenharmonic tuning systems, and its runoff status is unsurprising. Its fifth is 694.7¢, only slightly flat of just, making it an excellent meantone tuning. Its minor third at 315.8¢ is nearly just (6/5 = 315.6¢). Its major third at 378.9¢ is close to 5/4 (386.3¢). Composers have been exploring 19edo since the 16th century — Guillaume Costeley used it in 1558. The runoff status of 19edo captures something musicians have intuited for centuries: this is a scale where you can move around freely and most intervals you land on will feel like they belong somewhere recognizable.
22edo
Twenty-two equal divisions, step size approximately 54.5 cents. 22edo is the smallest EDO capable of representing the 11-odd-limit consistently, and it has excellent approximations to 5/4, 7/4, and 11/8. Unlike 12 and 19, it does not temper out the syntonic comma (81/80), which means it distinguishes between the major tones 9/8 and 10/9 — a more harmonically nuanced system. Its runoff status is consistent with its reputation as an exceptionally rich and harmonically rewarding tuning. 22edo is the third EDO after 12 and 19 capable of approximating 5-limit intervals to within 4 cents of Tenney-Euclidean error, and it goes well beyond the 5-limit. Theorist Paul Erlich famously developed his decatonic scale theory around 22edo.
30edo
Thirty equal divisions, step size 40 cents. 30edo is in some ways a doubled version of 15edo, and it inherits much of 15edo’s harmonic character while adding finer resolution. Its patent val through the 11-limit is twice that of 15edo’s. It also has an excellent approximation to the 13th harmonic, making it particularly useful for 13-limit harmony. The runoff status of 30edo reflects the fact that its 40-cent steps tile the harmonic entropy landscape in a way that keeps most intervals in valleys. 30edo is also notable for being what BudjarnLambeth calls a “dual-fifth” EDO — it has two viable approximations to the perfect fifth, one sharp and one flat, which opens up interesting compositional possibilities.
35edo
Thirty-five equal divisions, step size approximately 34.3 cents. 35edo is the product of 5 and 7, meaning it contains both 5edo and 7edo as subsets — you can think of it as a system that combines two of the smallest xenharmonic macrotonal EDOs in a single framework. It has very accurate approximations of harmonics 5, 7, 11, and 17, and its dual-fifth nature (with fifths at both 685.7¢ and 720¢) gives it unusual harmonic versatility. The runoff status of 35edo is perhaps its most practically meaningful property: despite the complexity of navigating its dual-fifth system, the scale as a whole tends to keep its intervals in harmonic valleys, making it safer for free exploration than one might expect from such an unusual system.
37edo
Thirty-seven equal divisions, step size approximately 32.4 cents — the largest runoff EDO identified so far. 37edo is a prime EDO with remarkably accurate approximations of harmonics 5, 7, 11, and 13, making it an excellent no-threes system. Harmonic 11 is particularly well-represented, being only 0.03 cents sharp — essentially just. 37edo supports porcupine temperament (using its sharp fifth), negri temperament (using its flat fifth), and the exotic undecimation temperament. Its runoff status is notable given how many interval classes it contains — with 37 steps, there are many opportunities to land on harmonic entropy hills, but 37edo manages to avoid most of them. This is a testament to how well its step size divides up the harmonic entropy landscape.
Runoff AFDOs
AFDOs (Arithmetic Frequency Divisions of the Octave, also called Otonal Divisions of the Octave or ODOs) divide the octave into parts of equal frequency difference rather than equal pitch difference. This produces the familiar overtone series structure — the intervals get smaller as you ascend, mirroring the way harmonics get closer together in frequency. All AFDOs are subsets of just intonation.
4afdo
Four arithmetic divisions give the scale 1/1, 5/4, 4/3, 3/2, 2/1 — adding the just major third to the mix. The 5/4 major third sits in a clear harmonic entropy valley, and the interactions between these notes produce intervals like 6/5 (minor third between 5/4 and 3/2) and 16/15 (between 1/1 and 5/4 going up to 4/3), both of which are reasonably close to valleys. 4afdo is the basis for the overtone scale Mode 4 (the 4:5:6:7:8 pentad), one of the most widely discussed chords in xenharmonic theory.
5afdo
Five arithmetic divisions: 1/1, 6/5, 5/4, 4/3, 3/2, 2/1. This adds the just minor third (6/5) to the picture. All four interior intervals — 6/5, 5/4, 4/3, 3/2 — are among the most consonant intervals in just intonation and sit in deep harmonic entropy valleys. The interactions between them (like 7/6 between 6/5 and 7/6 — wait, let me be precise: like 25/24 between 6/5 and 5/4, which is a small but identifiable interval) keep most of the generated intervals in harmonic valleys as well. 5afdo is a remarkably consonant five-note scale.
6afdo
Six arithmetic divisions: 1/1, 7/6, 4/3, 3/2, 5/3, 11/6, 2/1. This is Mode 6 of the harmonic series (the 6:7:8:9:10:11:12 scale), also known as the Over-6 scale. BudjarnLambeth calls it the “Freeway scale.” It introduces the septimal subminor third (7/6), the just major sixth (5/3), and the undecimal neutral seventh (11/6). The 7/6 is a well-defined interval in its own harmonic entropy valley, and the scale overall keeps its intervals in identifiable places. This is the last runoff AFDO in the list, making 6afdo the upper boundary of the runoff AFDO sequence.
Runoff IFDOs
IFDOs (Inverse Frequency Divisions of the Octave, or Utonal Divisions of the Octave) are the mirror image of AFDOs — they divide the octave according to an arithmetic progression of wavelength (string length) rather than frequency. This produces the undertone series, where intervals get larger as you ascend (the opposite of the overtone-based AFDOs). All IFDOs are also subsets of just intonation.
5ifdo
Five inverse divisions — the utonal mirror of 5afdo. This produces a scale built from the first five subharmonics, analogous to how 5afdo is built from the first five harmonics.
Carlos Beta
Carlos Beta is a non-octave equal temperament invented by the pioneering electronic music composer Wendy Carlos and first used on her 1986 album Beauty in the Beast. It has a step size of approximately 63.833 cents — almost exactly every fifth step of 94edo — and does not repeat at the octave. Instead, 11 steps of Carlos Beta approximate the perfect fifth (3/2), 6 steps approximate the major third (5/4), and 5 steps approximate the minor third (6/5).
Carlos Beta is the result of minimizing the mean square deviation across those three target intervals simultaneously, producing a scale that is optimized for smooth approximation of 5-limit harmony without being constrained to octave repetition. The step size can be derived precisely by putting the perfect fifth and major third in an 11:6 ratio.
Carlos Beta’s runoff status is a natural consequence of its design: a tuning built to approximate 5-limit just intervals as closely as possible will naturally keep most of its intervals in harmonic entropy valleys, since those valleys are precisely centered on simple just ratios. One advantage Carlos Beta has over its sibling Carlos Alpha is a better approximation of the seventh harmonic (7:4) — 15 steps comes to about 957.5 cents, close to the 968.8-cent harmonic seventh.
The scale is related to the sycamore family of temperaments (betic and 5-limit sycamore in particular), and has lookalikes in 19edo, 11edf, 30edt, every 5th step of 94edo, and every 4th step of 75edo.
Moon Dust
Moon dust is a 16-tone just intonation scale invented by BudjarnLambeth in 2025, with a period of 16/1 (four octaves). It is a non-octave, non-tritave scale — quite an unusual structure. Its intervals are: 17/14, 19/14, 3/2, 9/4, 19/8, 11/4, 7/2, 4/1, 9/2, 11/2, 6/1, 9/1, 19/2, 11/1, 57/4, 16/1.
The scale was inspired by a JI chord shared by Maeve Gutierrez in the Xenharmonic Alliance Discord server in September 2025. Lambeth moved the chord down an octave, added a 4/1 equivalence and 16/1 period, and filled in the remaining notes using Scale Workshop and his ears.
Through the lens of primodality, the moon dust scale is an over-7 scale, occurring above the tonic in the 112th mode of the harmonic series (112afdo). Written as a JI chord, it is 56:68:76:84:126:133:154:196:224:252:308:336:504:532:616:784:896.
Moon dust is described as having “distinctly dynamic melodic steps with interspersed bendy little steps and big stark leaps,” and containing a mixture of familiar 3-limit harmonies with very xenharmonic no-5s no-13s 19-limit harmonies. It has many dissonances to create tension alongside many consonances to resolve it.
Its runoff status means that despite its unusual structure and its 19-limit harmonic content, most of its intervals still land in identifiable harmonic entropy valleys — a remarkable property for such an exotic scale. Lambeth recommends it in 24edo, 30edo, 31edo, or 72edo as approximations.
The 1–5–11–13 Hexany
A hexany is a six-note scale built from all possible 2-element combinations of a set of four intervals — the simplest non-trivial case of a combination product set, a concept developed by Ervin Wilson. The 1–5–11–13 hexany is a specific hexany using the factors 1, 5, 11, and 13, producing a 13-odd-limit scale with no factors of 3 or 7.
This creates the scale: 1/1, 11/10, 13/10, 11/8, 13/8, 143/80, 2/1, with steps of 11/10, 13/11, 55/52, 13/11, 11/10, 160/143. The suggested keyboard mapping is C D E F# G# A# C.
What’s notable about this hexany is that the ratio between its smallest and largest steps is approximately 3:1, but even the smallest step is large enough that playing all the notes together is not overly dissonant — a practical consideration for a six-note chord as well as a scale.
Its runoff status reflects the fact that the 1–5–11–13 combination product set happens to hit interval sizes that sit comfortably in harmonic entropy valleys. The 11/10 steps (approximately 165 cents) and 13/11 steps (approximately 289 cents) both approximate reasonably identifiable just ratios. The hexany ends up being a surprisingly accessible and harmonically coherent scale despite its unusual construction from 11- and 13-limit factors.
All Phoenix Tunings
The phoenix tuning continuum is a family of non-octave equal temperaments ranging from 16ed9/5 (lower bound, approximately 63.6 cents per step) to 11edf (upper bound, approximately 63.814 cents per step). All phoenix tunings are runoff scales.
Phoenix tunings are characterized by a step size of around 63.5–63.8 cents, which stretches the octave by approximately 8–12 cents. The defining feature of phoenix is that prime-numbered harmonics are, on average, approximated more reliably than composite ones — a highly unusual property that concentrates tuning error around composite harmonics.
The name was chosen by Mason Green because these scales approximate most small intervals reasonably well but have a noticeable weakness at the 8th harmonic (8:1), which falls almost exactly between two scale degrees. The scale, like a phoenix, “dies” at harmonic 24 and “rises from the ashes” again at 31 or 32.
Phoenix tunings include 16ed9/5, 30edt, 11edf, and other scales in the continuum. The fact that all of them are runoff scales reflects the remarkable design philosophy behind phoenix: by focusing approximation accuracy on prime harmonics (2, 3, 5, 7, 11, 13…) rather than composite ones, phoenix naturally puts most of its intervals into harmonic entropy valleys, since those valleys are defined by simple prime-ratio relationships. Notable phoenix scales include 30edt(30 equal divisions of the tritave, sometimes called Bohlen-Pierce-adjacent) and 11edf (11 equal divisions of the perfect fifth).
Most Equipentatonic Scales (Including Slendro)
Equipentatonic scales are pentatonic scales with five roughly equally-spaced tones per octave. They are usually not exactly equally spaced, but deviate from perfect equality by small amounts — typically to improve the tuning of 3/1 (the perfect twelfth or tritave). The canonical exactly-equal version is 5edo, already discussed above as a runoff EDO.
Equipentatonic scales appear in a wide range of musical traditions: Dagarti, Lobi, and Senufo music from West Africa; Indonesian gamelan (slendro scales); southern Ugandan music; and Baganda music, among others. The fact that so many independent musical traditions have converged on this general scale structure is musically significant.
Slendro specifically refers to the family of Indonesian gamelan equipentatonic scales, which vary between gamelans but generally feature five tones roughly equally spaced across the octave, with slight irregularities that vary from one gamelan set to another. Each individual gamelan has its own unique tuning, but all fall within the general equipentatonic range.
The runoff status of “most equipentatonic scales” — not all, but most — reflects a deeper pattern: equipentatonic scales in EDOs are often runoff EDOs (the Xenharmonic Wiki notes that EDOs with equipentatonic scales “are often runoff edos”), they are often dual-fifth EDOs, and they often support superpyth or a related temperament. The equipentatonic structure naturally places its five tones in positions that avoid harmonic entropy hills, partly because the roughly 240-cent step size approximates the subminor third (7/6, 267¢) or the major second (9/8, 204¢), both of which are in valleys, depending on exactly how the spacing falls.
Equipentatonic scales found in EDOs between 1edo and 40edo that are often runoff include those in 22, 27, 32, 34, 37, and 39edo. Between 40 and 60edo: 42, 44, 47, 49, 52, 54, 57, and 59edo. Between 60 and 80edo: 62, 64, 67, 69, 72, 74, 77, and 79edo.
The Broader Pattern
Looking at this list as a whole, a few patterns emerge.
First, simple just intonation systems tend to be runoff. The AFDOs and IFDOs in the list are all small, low-complexity JI scales built from the first few members of the harmonic or subharmonic series. The harmonic entropy valleys are defined by simple frequency ratios, so a scale built directly from those ratios will naturally fall into the valleys.
Second, historically beloved EDOs tend to be runoff. 5edo, 12edo, 19edo, and 22edo are all in the list, and all of them have long histories of musical use and theoretical attention. This is not a coincidence — the reason these EDOs have attracted so much interest is precisely that their interval classes feel “identifiable” and harmonically rich, which is exactly what the runoff criterion captures.
Third, the concept rewards unusual and exotic designs. Carlos Beta, Moon Dust, and the 1–5–11–13 Hexany are not famous or widely used scales — they are niche, specialized, and unusual. Their runoff status suggests that despite their exotic construction, they share a deep harmonic intelligibility with simpler systems. This is one of the most intriguing aspects of the runoff concept: it can identify which strange, non-standard scales are nevertheless “safe” to play on freely.
Fourth, the phoenix continuum shows that runoff status can characterize entire families rather than just individual scales. All tunings within a defined range share the property, suggesting that runoff is in some cases a regional feature of the tuning landscape rather than a pinpoint one.
Could There Be More?
Almost certainly yes.
The list of known runoff scales is explicitly presented as preliminary and incomplete. BudjarnLambeth’s wiki page is a working document, and the concept of runoff scales is recent enough that systematic searches have not been conducted across all possible scales and tuning systems.
There are vast spaces of potential runoff scales that have likely not been checked:
- Higher EDOs beyond 37 might be runoff, either in their entirety or as specific MOS subscales
- Non-octave equal temperaments other than Carlos Beta and the phoenix family may qualify
- JI scales with more complex structures — higher-limit hexanies, other combination product sets, scales drawn from higher modes of the harmonic series — may satisfy the runoff criterion in ways that haven’t been explored
- Irregular temperaments and well temperaments might have runoff properties in some of their modes
- AFDO and IFDO systems beyond 6afdo and 5ifdo may qualify, especially at values that happen to align well with the harmonic entropy landscape
- Scales from specific musical traditions outside the Western and gamelan contexts might turn out to be runoff when analyzed through this lens
The runoff concept is also purely theoretical at this point — there’s no algorithmic search tool that automatically identifies runoff scales from a given search space. Such a tool would require a reliable, high-resolution harmonic entropy calculation, a definition of “valley” precise enough to implement computationally, and a search algorithm capable of checking all rotations of candidate scales. Building that tool would be a significant project, but it would likely turn up many new runoff scales.
In the meantime, the concept offers a useful framework for composers and theorists thinking about scale design. Whether you’re working in a familiar system like 19edo or an exotic one like the Moon Dust scale, asking whether your scale is runoff — whether most of its intervals, in all its modes, sit in harmonic entropy valleys — is a powerful way to predict how freely playable and forgiving it will be.
The landscape of runoff scales is still being mapped. There is almost certainly more out there, waiting to be discovered.
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