Erv Wilson

The Axial Chord — A Basis for Harmony


In this paper I discuss the chord 32:33:42:44:56 as the basis for a new harmonic system, comparing it to 4:5:6 as the basis for traditional Western tonal systems. After discussing the harmonic possibilities of the chord, I introduce some simple scales built around the chord, and then more complex systems based on those scales. Most of these scales are Constant Structures, and some are based around mapping onto a traditional Halberstadt keyboard, or combinations of Combination Product Sets. Finally, I discuss tools to build new scales in the subset, and strategies for tempering in that subset.

Notes on Constant Structures:

A Constant Structure (CS, plural CS) is a scale where each interval is subtended by the same number of steps. They are useful tools in just intonation (JI), since they make keyboard and instrumental mappings much easier. Moment-of-Symmetry scales can be modeled as one-dimensional CS. In order to find the CS I use throughout this paper, I use the property of linear independence for step-sizes. Linear algebra can be used to manipulate JI intervals by putting the intervals into ket-vector notation (regular temperament theorists call this “Monzo notation” after Joseph Monzo). For a given subgroup of JI, A.B.C…, the ket-vector|k,m,n,…> = Ak•Bm•Cn•…, where A, B, C, k, m, n ∈ . By using vectors, CS can be found using linear independence. A vector is considered linearly independent from a set of vectors when it cannot be written as a sum of the other vectors. In other words, for vectors X, Y, Z… and integers k, m, n,…, if k•X+m•Y+n•Z+… = 0, then k, m, n,… = 0. To find CS, I looked for scales where step sizes are linearly independent. I am not sure if this is a necessary condition for a CS, but it is a sufficient condition. This can be proved very quickly using induction:

Let a scale have a single step, P, equal to its period. This is trivially a CS. Now, assume a scale has n different linearly independent scale steps, S1, S2, S3,…,Sn, and is a CS. Let us add a new scale step, Sn+1, such that it is linearly independent with the other steps. In order to do this, some Sk, where 0 < k ≤ n, must vanish from the set of steps. This Sk is the step split by Sn+1 and another step Sj, where 0 <j ≤ n, such that Sn+1+Sj = Sk. If this step does not vanish from the set, the set cannot be linearly independent. It is also possible that adding Sn+1 will add additional new scale steps, but these cases will play out the same way. Assume this new scale is not a CS. Then there exists some interval L in the scale where for a, b,…I; aS1+bS2+…+cSn = dS1+eS2+…+fSn+gSn+1 = I. Rearranging terms, ((a-d)S1+(b-e)S2+…+(c-f)Sn)/g = Sn+1. Thus, Sn+1 is not linearly independent, and there is a contradiction. Therefore the new scale must be a CS. Through induction, any scale with linearly independent steps must be a CS. QED.

Side Note 1: When step sizes are linearly independent, there exists a linear transformation between scale steps and harmonic generators for a CS. Therefore, when I am talking about a CS, I use “generator” to refer to both harmonic generators and the various step sizes.

Side Note 2: A quick way to check for a linearly independent scale is that any interval cycle of a scale degree coprime with the number of tones in the scale must form a complete cycle. By looking for incomplete cycles in a scale, it is easy to find that a scale is not a CS.

Part 1 – The Axial Chord:

The major triad, 4:5:6, is the foundational element for Western tonal theory. It is the simplest triad found in the harmonic series, and has constant difference tones of 1 (in other words, is an isoharmonic series segment with difference 1). JI scales built around it fit into the 2.3.5 JI subgroup. My interest is in building a system similar to the Western system harmonically, but more complex. The first decision I made is to eliminate the overly familiar 5th harmonic and replace it with the 7th and 11th harmonics. The subgroup has one more harmonic generator than 2.3.5, and is based on the next simplest prime harmonics after 5. Notably, the chord 8:11:14 is an isoharmonic series with difference 3, which is the next simplest series after the one to which 4:5:6 belongs.

Modern harmony has extended further along the 5-limit, so that the major seventh, 8:10:12:15, often supplants 4:5:6 as a main point of resolution. This chord has fascinating difference tones; 12:15 has a difference of 3, 10:15 has a difference of 5, and 8:15 has a difference of 7. However, this chord is also ambitonal, meaning its otonality and utonality are equivalent. Therefore my interest is in building a chord with similar properties, but that is distinctly otonal. By shifting the 7th and 11th harmonics by a 3:2, produces such a chord, 8:11:14:21:33. I call this chord the axial pentad, and it serves as the basis for this paper. This chord also has interesting difference tones, with 3,3,7,12 between tones, and 3,6,13,15 from the root. A recurring property of transpositions of this chord is for multiples of 3 to be more common, with simple intervals below 17 showing up as well. The transposition within an octave is 32:33:42:44:56 (with difference tones 1,10,12,24 from the root), and a transposition I have been using is 16:21:22:28:33 (with difference tones 5,6,12,17 from the root).

Side-note: Ignoring the 3rd harmonic completely, the chord 32:44:56:77 has the same construction as the major seventh chord within the 2.7.11 subgroup, and likely can be used to build its own fascinating scales. Although I do not pursue this, the chord shows up throughout my scales for those interested.

Part 2 – Some Basic Scales:

To start making music with the axial pentad, we need some scales. The simplest scale of all would be to simply use the tones of the axial chord, which itself is a CS. This scale is:

1/1, 33/32, 21/16, 11/8, 7/4, 2/1,

Fig 1: Lattice for the axial pentatonic

and I call it axial pentatonic. It bears an auditory similarity to a number of other pentatonic scales, such as the Japanese Iwato scale, and as a transposition of the 5 note meta-pelog scale, although the quarter tone intervals make the axial pentatonic more extreme in its scale sizes.  

The next simple scale is a hexatonic scale patterned after the major seventh chord. The major seventh is ambitonal; the major triad builds up from the root and its inversion builds down. By adding the tone 231/128 to the axial pentad, a CS scale with the same property is made:

1/1, 33/32, 21/16, 11/8, 7/4, 231/128, 2/1,

Fig 2: Lattice for the primal scale

and I call it the primal scale. Note that the primal scale can also be described as a 7-11-77-231 hexany, which is a type of Combination Product Set (see Part 4).  On its own it is nice compositionally, and looking ahead the primal scale is a great tool for building larger scales.

A different path from the primal scale is to build a heptatonic scale analogous to the major scale. A heptatonic CS very similar to the diatonic locrian mode is:

1/1, 33/32, 7/6, 21/16, 11/8, 99/64, 7/4, 2/1,

Fig 3: Lattice for the axial scale

and I simply call it the axial scale. This scale is an excellent bridge between traditional diatonic harmony and the subgroup. It sounds very similar to the diatonic, but the main mode is locrian, the least used diatonic mode. Also, interval classes vary far more in consonance and dissonance than the Ptolemaic sequence, meaning new considerations for counterpoint must be taken.

Combining the primal and axial scales gives an octatonic scale, and by adding two additional tones we get this decatonic CS scale:

1/1, 33/32, 7/6, 77/64, 21/16, 11/8, 3/2, 99/64, 7/4, 231/128, 2/1,

Fig 4: Lattice for the primaxial scale

and I named this the primaxial scale. This scale is really nice for music based strongly around a single tonal center, such as drone music. The added tones make it easy to reinforce the axial scale.

One final simple scale is a just intonation CS version of 15EDO’s Blackwood scale. It is:

1/1, 33/32, 8/7, 77/64, 21/16, 11/8, 3/2, 11/7, 7/4, 231/128, 2/1,

Fig 4: Lattice for the Blackstar scale

and I call it the Blackstar scale (this name will make more sense later). Like Blackwood, this scale is built from two 5EDO-adjacent chains, which for Blackstar are entirely within the 2.3.7 subgroup:

1/1, 8/7, 21/16, 3/2, 7/4, 2/1.

The second chain is transposed by 11:8, making 11/8 the root. This scale is a fantastic scale to compose with, as it contains both the primal scale and a 7-limit pentatonic. Like Blackwood, you can form chains of major and minor seventh chords, but in JI the character of these chords vary wildly.

Part 3 – Scales for a 12 Tone Halberstadt Keyboard:

For people new to JI or xenharmonic music in general, scales that fit onto traditional Halberstadt keyboards are very valuable. Hopefully these two scales will be helpful for people who are unable to build or uncomfortable with new keyboards. Both are Constant Structures.

The first of these two scales places the axial scale within the full 11 limit:

1/1, 33/32, 9/8, 7/6, 5/4, 21/16, 11/8, 3/2, 99/64, 5/3, 7/4, 15/8, 2/1.

Fig 6: Lattice for the Janus scale

I named this scale the Janus scale, because it looks back on 5 limit harmony and ahead towards harmony simultaneously. Mapping C to 1/1, the ionian mode on C is almost the Ptolemaic sequence (with 21/16 instead of 4/3), and the locrian on C is the axial scale. This scale is built by adding two 3/2s around each of the prime harmonics in the limit, which makes building chords easy for people familiar with the Western tonal system. It also approximates the 12EDO scale to within 50 cents, making it approachable for people new to xenharmony. 

The second 12-tone scale is a CS extension of primaxial, called primaxial[12]:

1/1, 33/32, 9/8, 7/6, 77/64, 21/16, 11/8, 3/2, 99/64, 77/48, 7/4, 231/128, 2/1.

Fig 7: Lattice for primaxial[12]

Notice that this scale differs from the last by just 3 tones. It lacks a recognizable ionian mode however, as 77/64 is closer to a minor third. This scale is within the subgroup, so is good for people who do not want any 5th harmonic intervals.

Part 4 – Combination Product Sets and the Astral Scale:

The scales up to this point have a clear tonal center, and various modes based around transposing from that center. However, many musicians will want to work with different tonal centers (or an atonal fabric), and Erv Wilson’s Combination Product Sets (CPS) are designed to do just that. For integers n,k where n,k ≥ 0; a k)n CPS is the set of all possible k-term combinations of the m generators. The number of elements of a CPS is equal to n choose k, or the corresponding value on Pascal’s triangle. CPS have a symmetrical structure, so that k)n is the mirror of k–n)n

Side Note: The primal scale is a 2)4 CPS (a hexany) generated by 7-11-77-231:

Fig 8. Primal Scale as a 7-11-77-231 hexany

The axial pentad can act as the basic structure for the k)5 family of CPS, seeding them with 1-7-11-21-33. This chord would be the 1)5 CPS. The 2)5 and 3)5 are called dekanies, and normally contain 10 tones. However, because 11•21 = 7•33, these dekanies only contain 9 tones. This fact is key to scales later in this section.

Fig 9: Axial 2)5 and 3)5 dekanies

Both dekanies can overlap to make a double dekany, which in this case contains 13 tones.

Fig 10: Axial double dekany

The 0)5, 1)5, 4)5, and 5)5 CPS can then fit into the double dekany to add only 2 more tones, resulting in a 15 tone scale made from the complete CPS family:

Fig 11: Complete axial k)5 family

1/1, 33/32, 147/128, 4851/4096, 77/64, 2541/2048, 21/16, 693/512, 11/8, 363/256, 1617/1024, 53361/32768, 7/4, 231/128, 7623/4096, 2/1.

Fig 12: Axial k)5 as a rectangular lattice

Unfortunately, this 15 tone scale is not a CPS. In order to make it constant, I used linear independence. Notice the 15 tone scale has 5 different step sizes. 8192/7623 is not linearly dependent, as it can be deconstructed into 128/121•64/63. Introducing the tones 441/256 and 128/64 splits the two instances of this interval to make a CS scale with four different step sizes (which is the smallest possible number since this scale has four harmonic generators: 2, 3, 7, and 11):

1/1, 33/32, 147/128, 4851/4096, 77/64, 2541/2048, 21/16, 693/512, 11/8, 363/256, 1617/1024, 53361/32768, 441/256, 7/4, 231/128, 7623/4096, 121/64, 2/1.

Fig 13: Astral[17] in lattice form

I have named this scale Astral[17] after the star pattern of the double dekany. Looking at the lattice, the common tones in the dekanies (3•7•11=231) overlap in the center of the lattice, giving a center point to a normally atonal structure. This scale is not very useful; the size of the 49/44 interval makes it unwieldy. However, extensions of this scale produce amazing results.

Notice that 441=212, and 128=112. Adding the intervals 49/32 (72) and 1089/1024 (332) completes the symmetry of the lattice, and splits the scale step 49/44 into 33/32 • 392/363. The resulting scale is a 19-tone CS with 4 step sizes:

1/1, 33/32, 1089/1024, 147/128, 4851/4096, 77/64, 2541/2048, 21/16, 693/512, 11/8, 363/256, 49/32, 1617/1024, 53361/32768, 441/256, 7/4, 231/128, 7623/4096, 121/64, 2/1.

Fig 14: Astral[19] lattice

This scale, Astral[19], is structurally the most fascinating in this paper. It contains 6 transpositions of the primal scale, each sharing a common tone at 231/128.  

For one, this scale is a logical extension of the 13-note double dekany. According to Wilson, the 2)5 and 3)5 dekanies can be stellated, adding tones rotationally symmetric to 1/1. Interestingly, the 2)5 stellate and 3)5 stellate add an identical 4 tones of 6 total, equal to the 4 extra tones in Astral[19] from the 15-tone k)5 CPS family. This means Astral[19] is simply the double dekany stellate for 1-7-11-21-33.

Fig 15: Axial double dekany stellate with hexany overlap. Each dekany stellate adds 6 tones

Astral[19] can also make sense going the opposite direction. The inverse conceptually of a CPS is what Harry Partch called a diamond, or Wilson called a Lambdoma. A diamond for a set of harmonic generators shows every possible division of one generator by another. The 1-7-11-21-33 diamond has 17 tones, and is not a Constant Structure:

1/1, 33/32, 22/21, 8/7, 33/28, 14/11, 21/16, 4/3, 11/8, 16/11, 3/2, 32/21, 11/7, 56/33, 7/4, 21/11, 64/33, 2/1.

Fig 16: Axial tonality diamond

Adding the tones 256/231 and 231/128 makes this into a CS, and is in fact identical to Astral[19]. What these two facts mean is that the primal scale can transpose to any tone within the scale, making Astral[19] perfect for modulating music.

Fig 17: Axial diamond with added tones in red gives Astral[19]

There is one additional Astral extension I want to point out. I initially found it through what I call simple decomposition. This technique involves decomposing a scale step into two scale steps, where one is a preexisting step size and the other is less complex than the initial interval. In this case, 392/363 can be decomposed to 128/121 • 49/48. Adding the tones 3/2 and 693/512 does exactly this, resulting in Astral[21], another CS scale with 4 different scale steps:

1/1, 33/32, 1089/1024, 693/512, 147/128, 4851/4096, 77/64, 2541/2048, 21/16, 693/512, 11/8, 363/256, 3/2 49/32, 1617/1024, 53361/32768, 441/256, 7/4, 231/128, 7623/4096, 121/64, 2/1.

Fig 18: Astral[21] lattice

This scale is much more consistent in scale step sizes, and contains two new 7-11-77-231 tetrads (the chord that generates the primal hexany). Additionally, Blackstar can be built from 231/128. This explains the name of Blackstar: a combination of Blackwood and Astral. Using Blackstar gives a nice Halberstadt mapping for Astral[21]. The white keys give Blackstar, and all other tones are relegated to the black keys. Overall I think Astral[21] makes more sense written symmetrically, with 231/128 = 1/1:

1/1, 33/32, 22/21, 256/231, 8/7, 33/28, 77/64, 14/11, 21/16, 4/3, 11/8, 16/11, 3/2, 32/21, 11/7, 128/77, 56/33, 7/4, 231/128, 21/11, 64/33, 2/1.

Using this notation, the Blackstar step pattern can be rotated through all 21 tones of Astral[21], and because Astral is a CS scale, tones will always show up in the same place. The 6 primal scales can be found on the subharmonic primal tones:

Fig 19: Matrix of Blackstar transpositions in Astral[21], arranged symmetrically. Scales that contain the primal scale built on the root are highlighted. Non-CS scales have a single 33/28 interval that is inconsistent.

These scales show a symmetrical pattern around 3/2, where each mirror scale has an inverted lattice (although they might not share 1/1 as the root). This means that an inverted Blackstar scale can be found on 256/231. Note that all of these decatonic scales are CS except for the scale built on 8/7 and its mirror on 64/33, which each have a single instance of an inconsistent interval of 33/28 (and thus through inversion 56/33).

Part 5 – Extensions of the Axial Scale:

The Astral scales provide transpositions for the primal scale. The following scales are models for transposing the axial scale throughout pitch space.

The first scale I made has a very simple construction. I transposed the axial scale lattice onto each tone of the axial scale, allowing any chord to transpose throughout the scale. This results in a 22-tone scale:

1/1, 2079/2048, 49/48, 33/32, 1089/1024, 147/128, 7/6, 9801/8192, 77/64, 21/16, 693/512, 49/36, 11/8, 363/256, 49/32, 99/64, 3267/2048, 77/48, 441/256, 7/4, 231/128, 121/64, 2/1.

Removing the “tails” on the lattice reduces the scale to 17 tones, which only transposes the axial scale to 7/4 or 11/8:

1/1, 49/48, 33/32, 1089/1024, 147/128, 7/6, 77/64, 21/16, 693/512, 11/8, 363/256, 49/32, 99/64, 77/48, 7/4, 231/128, 121/64, 2/1.

Fig 20: Chromaxial[17] in lattice form. Chromaxial[22] with added red tones

Unfortunately, neither of these scales are Constant Structures. I had named these scales chromaxial[22] and chromaxial [17] (as they relate to the axial scale the way the chromatic relates to the major scale), but if I find a similar scale that is a CS I feel that would be a better use for that name. Because the axial scale is a CS itself, I think this scale could be useful on a Halberstadt keyboard with the 1/1 axial on white keys and extra tones on black keys.

The next step I took was to decompose chromaxial[22] into a larger scale so that it has 4 linearly independent step sizes. The scale steps I found are 896/891, 99/98, 2079/2048, and 128/121. The resulting CS scale has 16 896/891 steps, 14 99/98 steps, 12 2079/2048 steps, and 5 128/121 steps, for a total of 47 tones. I call any scale of this type, that extends chromaxial[22], an enharmaxial scale, after the enharmonic. From the above proof, any arrangement of these steps will be a CS, but only certain arrangements line up with chromaxial[22]. I found 2 different scales of this type. What I found is that the scale can be structured so that there are only 3 possible step sequences after each 128/121; two of which repeat. The first I built using repetitive sequences of steps. It is:

1/1, 2079/2048, 49/48, 33/32, 68609/65536, 539/512, 1089/1024, 9/8, 18711/16384, 147/128, 297/256, 7/6, 33/28, 9801/8192, 77/64, 14/11, 1323/1024, 343/264, 21/16, 43659/32768, 343/256, 693/512, 49/36, 11/8, 22869/16384, 539/384, 363/256, 3/2, 6237/4096, 49/32, 99/64, 14/9, 11/7, 3267/2048, 77/48, 56/33, 441/256, 343/198, 7/4, 14533/8192, 343/192, 231/128, 49/27, 11/6, 7623/4096, 539/288, 121/64, 2/1.

The second scale has a less pleasant pattern of scale steps, but the lattice is more compact and replaces chains of 7/4s and 11/8s in favor of 3/2s. Out of the two this scale contains the most instances of primaxial[10]:

1/1, 2079/2048, 49/48, 33/32, 68609/65536, 539/512, 1089/1024, 9/8, 18711/16384, 147/128, 297/256, 7/6, 4851/4096, 9801/8192, 77/64, 160083/131072, 1323/1024, 2673/2048, 21/16, 43659/32768, 88209/65536, 693/512, 49/36, 11/8, 22869/16384, 539/384, 363/256, 3/2, 6237/4096, 49/32, 99/64, 14/9, 1617/1024, 3267/2048, 77/48, 53361/32768, 441/256, 891/512, 7/4, 14533/8192, 29403/16384, 231/128, 49/27, 11/6, 7623/4096, 539/288, 121/64, 2/1.

Fig 21: Lattice for the first and second Enharmaxial scale.

This 47-tone fabric provides a generalized means of building JI structures in the limit. For example, breaking down Astral[21] into the four elemental steps results in the same set of 46 intervals. I used these intervals to build a symmetrical 47-tone scale that contains Astral[21]. This scale, Astral[47], contains many of the smaller scales presented in this paper, and is ideal for the most ambitious JI composers:

1/1, 99/98, 64/63, 33/32, 28/27, 22/21, 256/231, 9/8, 112/99, 8/7, 297/256, 7/6, 33/28, 32/27, 77/64, 14/11, 9/7, 128/99, 21/16, 297/224, 4/3, 693/512, 49/36, 11/8, 16/11, 72/49, 1024/693, 3/2, 448/297, 32/21, 99/64, 14/9, 11/7, 128/77, 27/16, 56/33, 12/7, 512/297, 7/4, 99/56, 16/9, 231/128, 21/11, 27/14, 64/63, 63/32, 196/99, 2/1

Fig 22: Lattice for Astral[47]

Notes on Pentatonic Scales:

Scales in the enharmaxial fabric all contain 5 instances of 128/121, and because intervals between these tones can be grouped into three unique step patterns, they imply pentatonic scales with 3 different step sizes. Taking the tones immediately following each 128/121, the first 47-tone scale makes the pentatonic:

1/1, 9/8, 14/11, 3/2, 56/33, 2/1,

Fig 23: First pentatonic scale in lattice form. Uses nonstandard generators 3 and 7/11.

with step sizes 9/8, 112/99, and 33/28. The second 47-tone scale makes this pentatonic, entirely within the 2.3.7 limit:

1/1, 9/8, 1323/1024, 3/2, 441/256, 2/1,

Fig 24: Second pentatonic scale in lattice form. Uses nonstandard generators 3 and 7^2

with step sizes 9/8, 147/128, and 512/441. This process can be applied to the Astral scales. Astral[21] (and thus Astral[41]). produces a pentatonic scale in the 2.3.7 limit:

1/1, 8/7, 21/16, 32/21, 7/4, 2/1,

Fig 25: Third pentatonic scale in lattice form.

with step sizes 8/7, 147/128, and 512/441. This scale is symmetrical, and is one tone away from the 2.3.7 pentatonic found in Blackstar. These pentatonic scales make a nice starting point towards building new JI structures in the limit, by decomposing the steps into larger scales. 

Part 6: Tempering:

I am personally more interested in JI than temperaments, but I feel this paper would be incomplete without discussing tempering. Unlike 2.3.5, there is not a convenient small EDO like 12 or 19 that strongly supports axial harmony. 22EDO has a usable axial pentad, and 17EDO has a lower TE error in than 22, but they do not stand out the way 12 or 19 do. However, using regular temperaments there are many effective strategies for scale design.  

Something interesting about the enharmaxial fabric is that all possible step sizes other than 128/121 are tempered out in 5EDO. This could explain the pentatonic scales discussed in the last section. Therefore, staggered chains of 5EDO provide a starting point for tempering. The closest approximation to the limit with two chains (a tuning of Blacksmith, a generalized version of the Blackwood scale discussed above), has a very high adjusted error of 18 cents, and is not very useful. However, by using scales that approximate 5EDO, there are many great candidates for regular temperaments. 5 degrees of 8/7 for a Moment-of-Symmetry (MOS) scale that acts as a JI “5EDO”. Therefore I took a look at scales with a tempered 8/7 as the generator:

The Miracle temperament, first discovered by the late George Secor, can be found from combining 31&72, and splits a tempered 8/7 in half for a generator. The 31-tone MOS works great for axial harmony, although the 21-note MOS known as blackjack does not contain an axial pentad. By trading accuracy for utility, the scale found from combining 41&46 has multiple axial pentads in the 21-tone MOS, and has a 26-tone MOS Miracle does not have. I have worked out a notational system for the 26 MOS, along with a Halberstadt keyboard layout. Below are the spellings of various 13-limit chords and scales in this system.

Fig 26: Notation, Halberstadt layout, and chord locations for 26-tone MOS of 41&46. Blue highlight represents the first subset scale in parenthesis; yellow the second. Generator order is shown on the right.

Moving beyond Rank-1 temperaments, the enharmaxial fabric can be used as a model to find Fokker periodicity blocks. Using the 896/891, 99/98, or 2079/2048 as unison vectors gives three Rank-3 tunings of varying accuracy and complexity. Tempering all three gives 5EDO.

As for EDOs, a good EDO to work with and axial harmony is 41EDO. 41 tempers out 896/891, and maps both 99/98 and 64/63 onto 1 step. Although a pure 8/7 generator does not have a 41-tone MOS, many significant temperments using a tempered 8/7 do, such as Miracle and 41&46.  Plus, because the 3/2 is so accurate, 41 gives a method of experimenting with tritave scales in the 3.7.11 subgroup.


The axial chord has been a breakthrough for me compositionally, allowing me to use rich sounds like those in extended tonal harmony, while ignoring the 5th harmonic. The Constant Structure scales in this paper offer different ways of approaching this type of harmony, and hopefully readers now have the tools at their disposal to make space easy and approachable. By naming the Constant Structures, I hope to encourage their use by musicians interested in JI. I would like to go deeper on this topic, both in terms of the acoustic properties of the axial chord and performance strategies. I will likely write a followup showing generalized keyboard mappings for all CS scales presented. I would also like to write a followup on instrument building. The Blackstar scale is ideal for guitars, and Astral[21] would also work well, although the 64/63 interval becomes difficult in high registers.


I had a lot of help working out the ideas in this paper, and it is necessary to point out where that help came from. First off is Kraig Grady, who has helped me so much to understand the theories and structures of Erv Wilson. For this paper after it was reviewed also helped me turn my hand drawings into computer graphics. I also thank Paul Erlich for helping me understand regular temperament theory. My use of it in Part 6 is thanks to conversations with him.

I should also note that most of the ideas presented here build directly from Erv Wilson’s writings. His scope of work is humbling. I thought I had found something new with the double dekany, but a day later I saw sketches of his with two, three, and even four overlapping dekanies. Anyone interested in Just Intonation should explore his writings at Anaphoria.

Forward (excerpt) to ‘Microtonality and The Tuning Systems of Erv Wilson’

Ervin M. Wilson was arguably the most creative music theorist in the world of alternative scales and tunings. He was also one of the least accessible because he chose to work alone and to share his prodigious discoveries and inventions personally, rather than teaching courses, writing formal papers, or publishing textbooks on his remarkably fruitful musical theories.

I first met Erv, as he preferred to be called, in the early 1960s when I was a graduate student in Biology at the University of California, San Diego. At that time, UCSD was located on the beach at the Scripps Institution of Oceanography, and the atmosphere was so intense that graduate students were given keys to the university library stacks. One evening I got tired of reading about molecular biology and genetics, so I decided to peruse the holdings of the Journal of the Acoustical Society of America. Soon I discovered that a certain Tillman Schafer had constructed a 19-tone electromechanical musical instrument at Mills College in the late 1940s, and that he had worked for the U.S. Navy in Point Loma, a suburb of San Diego. Hoping that he still lived in the San Diego area, I called the facility and asked to speak to him.

As it turned out, Schafer was still in San Diego and invited me to his home where I saw the instrument, but more importantly, was given the names of other people interested in microtonal music. One of these was Ivor Darreg, who then lived in Los Angeles; so I drove to LA and met Ivor, who in turn referred me to Ervin Wilson. Just prior to this meeting, I had attended a seminar at the Salk Institute by David Rothenberg, a composer and theorist from New York City, who is best known for his theories on the perception of musical tones in scalar contexts. Remarkably, David also knew Wilson and was visiting him the same time I was. So, through this network of associations, I met Erv Wilson and became one of the fortunate few who have had the opportunity to learn firsthand of his work.

As a graduate student, I had access to a large mainframe computer, so in 1968 I asked Erv if there were any computations I could do for him. We decided that a table of all the equal temperaments from 5 to 120 tones per octave would be useful, especially if I also computed the errors in a set of small-number ratios for each system. I did this, and at Erv’s suggestion, sent a copy to Professor Fokker in The Netherlands. Fokker was the leading proponent of 31-tone equal temperament in Europe, and the author of a number of articles as well as a composer in that system. Having finished this study, Erv and I also compiled a very large table of just intervals and distributed multiple copies to other workers. Other projects included equal divisions of 3/1 and other integers, as these can approximate divisions of the octave with improved representations of certain harmonically important intervals. In recent years, other composers such as Heinz Bohlen, John Pierce, Kees van Prooijen, Enrique Moreno, and others have also become interested in divisions of 3/1.

At this point, I had to stop and finish my dissertation, which was on the genetics of the tryptophan pathway in Neurospora crassa, if I were ever going to graduate from UCSD. (One of my graduate advisors had started referring to me as “graduate student emeritus”.) After getting my doctorate, I moved around the country, and became involved in research and teaching in the fields of microbial genetics, industrial microbiology, biochemistry, and biotechnology, eventually returning to UCSD. Through all of these peregrinations, I continued to correspond with Erv and pored over his sometimes puzzling letters, intriguing keyboard diagrams, and cryptic worksheets. An invitation from Larry Polansky to spend a summer writing at Mills College followed by a part-time research position again at U.C. Berkeley allowed me to complete my book, Divisions of the Tetrachord, a task that was originally suggested by Lou Harrison. Much of this book was directly concerned with Erv’s theories and musical discoveries. Central to Erv’s work and among his first discoveries are his keyboard designs. To play microtonal music with the same skill and expression as is done in 12-tone equal temperament, one needs instruments that are designed for alternative tunings.

Wilson was also one of the most intuitive mathematicians one is likely ever to meet outside of academia, though he had little or no formal training in the subject. In addition to being able to visualize and geometrically plot musical scales as objects in higher dimensional space, he rediscovered or reinvented a method equivalent to the approximation of irrational numbers by continued fractions (The Scale Tree), and characteristically applied it as an organized system for discovering new musical scales of the type he terms Moments of Symmetry (MOS). Dr Narushima’s explanation of this mathematical process is crystal clear without losing sight of its musical significance, particularly as it applies to keyboard design.

Other key concepts of Wilson’s musical theories are Constant Structures, which are scales in which every occurrence of a given interval is always divided by the same number of smaller intervals. These may be considered as a generalization of MOS and have comparable structural stability. They exist in both equal temperaments and ratiometric tunings (extended Just Intonation) as do another class of scales, the Combination-Product Sets.

CPS are found by multiplying a set of n harmonic generators, m at a time. The prototype is the Hexany, a six-note set generated from four integers representing harmonic functions such as 1, 3, 5, and 7, two at a time. The resulting set of pitches is partitionable into four pairs of (generalized) triads and their inversions. Others are the Dekany (two out of five or three out of five), and the Eikosany, four out of six. These in turn may be divided into smaller CPS: Dekanies into Hexanies, and Eikosanies into both. These structures are especially fascinating because they are harmonic without being centric as any note or none can function as the tonic. They are also defined in equal temperaments, but are generated by addition rather than multiplication.

Numerous composers have composed innovative and aesthetically significant music based on materials invented and discovered by Ervin M. Wilson, thus proving that his work is not empty speculation and audibly imperceptible theoretical invention.

John H. Chalmers, PhD.

Author of Divisions of the Tetrachord

Founding Editor of Xenharmonikôn

Rancho Santa Fe, CA, USA



Erv Wilson Remembered

ERV WILSON, COSMIC DREAMER (1928-2016) by Gary David



I knew Erv Wilson for over fifty years starting in 1964. He was my guide into the world of pitch fields, his fields of dreams. Each one who knew him has a different story. This one is mine. In the 1960s and early 70s, I led an experimental vocal-instrumental jazz group called The Sound of Feeling. I had been interested in tunings that deviated from the accepted 12 equal system the prevailed in Western cultures. Erv stewarded me into the methods of making scale formation part of the creative act of composition. I feathered into our music what I learned with Erv, and it was the first time arrays such as hexanies were featured by a musical ensemble on a major label. The many hours of theory and ear-training with Erv changed the meaning of music for me. The great psychologist-researcher Silvan Tomkins said, “The world we perceive is a dream we learn to have from a script we have not written.” Erv Wilson was a weaver of dreams that embodied previously unheard or unrecognized webs of frequencies. In the process, I got to know one of the most coherent and unusual individuals of my lifetime.

Here is a short account of my experience of how that unfolded.

Is it knowing to contemplate while dreaming? Is it understanding? The eye which dreams does not see, or at least it sees with another vision. . . Cosmic reverie makes us live in a state (where) the communication between the dreamer and his world is very close in reverie; it has no distance, not that distance which marks the perceived world. — Gaston Bachelard

As I follow the curve of space-time, I spiral into my own movement, and there I become an eddy of space-time itself and I “disappear” into my own vision. —Gary David

On Dec. 8, Erv Wilson took his leave from planet earth. While I feel a sense of grief, I don’t mourn. My past with him lives now as I write. He is inextricably lodged as a basic element in my being. George Spencer-Brown wrote, “Each being experiences what he, she, or it regards as the world as if of a dream of one’s own creation, and each of us is also an appearance in the ‘dream world’ of another. When the other dies, we too are lost from the dream. If we were prominent in that dream, we feel the loss acutely, and call it ‘grief.’”

“I myself prefer to be known as a “natural approximationist of (the) Augusto Novaro school (circa 1927). I will not be the one to split hairs in this matter; it’s the way one splits the tones that really matter, and I prefer to split them lengthwise, as opposed to those who split them sideways, or not to split them at all, for that matter. My entire philosophy of musical intonation can be wrapped up in one succinct little quote: “Nothing that exists is un-natural.” This includes the arts and artifices and artificiality of the musical imagination. Or, indeed, “why settle for the real thing when one might as well be having artificial?” (Lou Harrison, the 2nd funniest thing I ever heard him say). All scales are artificial, as are all other great works of art and products of the human imagination. You want to know what “natural” is? Let me tell you what “natural is. You don’t want to know. Well, I’ll tell you anyway.”

Erv Wilson

He never did tell us. He showed us. It didn’t matter if it was pitch fields or corn fields, he was a unifying weaver of meanings. He uncovered unity wherever he found it. A friend of mine who met Erv said, “I never thought someone could render me so fascinated by the potential of individual corns on a cob! And his flying micro-tonal scales hanging from his ceiling amazed me.”

Once you become aware of this force for unity in life, you can’t ever forget it. It becomes part of everything you do …. my conception of that force keeps changing shape.

John Coltrane

I have written elsewhere about Erv Wilson’s work, but here I give you a mere taste of my sense, feeling, and experience of the man. I’ve told the story of my first meeting with Erv Wilson many times, and I never tire of repeating that memorable moment. Harry Partch, was among the first of modern American composers to systematize microtonal music. In 1963, he was living in a chicken hatchery in Petaluma California with all of his instruments. I was living in San Francisco, and Emil Richards urged me to visit him. I did, and we spent the day talking about music and tuning (and drinking). I told Harry I was moving to Los Angeles and he said, “You should look up Erv Wilson.” In 1964 I moved to Los Angeles from San Francisco. One of my priorities was to meet Erv. One day I went to his apartment on Poinsettia Drive in Hollywood. I knocked on the door, but no one answered. The door was ajar, so I pushed it open and walked in. No one was there, but there were some unusual instruments in the living room of the small apartment. I picked up some mallets and began to play a microtonally tuned marimba. I was engrossed in the sound when behind me I heard a gentle voice begin to describe to me what I was doing. We started talking about the tunings the sounds and numbers. No introductions. That initiated a 30-year conversation.
In all the years I worked with Erv and enjoyed our friendship, I learned that there was no difference between who he was and what he was. As far as I could tell, there was no division. I experienced an aura of emotional safety whenever we engaged. He treated the novice I was in 1964 the same way he would relate to a master. That made it both easy to learn, as well as to accept my inability to grasp his meanings beyond my experience. There was little shame in not understanding.

If. . . the being I love most in the world came and asked me one day what choice he should make– what refuge is the most profound, the sweetest, the most immune to attack — I would advise him to shelter his destiny in the haven of a soul devoted to noble growth. — Maurice Maeterlinck, Sagesse et destinee, 1902

One of the innate traits of human life is curiosity. Those who lose access to that live an impoverished life. That fate never befell Erv, even later in life when his resources began to fail. His curiosity and his humor and his minimal egoism were hallmarks of both who and what he was.

He was what I call a fluid learner. Human awareness is both explicate and implicate – focal and subsidiary. Erv relied heavily on subsidiary awareness, while his focal awareness was a guide. He took literally, “the world is a dream.”

I have always been attracted to those with both a high and vast sense of life. To the overly focused, they appear either crazy or dangerous. Jimi Hendrix depicted it this way:

I can see my rainbow calling me through the misty breeze of my waterfall.

Some people say daydreaming’s for the lazy minded fools with nothing else to do.

What Erv saw in his waterfall was no fantasy. He made it available to all of us. He created a unified field that we can mine for generations to come. I am ever grateful for his gift, and to have participated with him.

“I begin by imagining
The impossible
And end by accomplishing
The impossible.” ― Sri Chinmoy

“The earth has music for those who listen.” – William Shakespeare

Does not everything depend on our interpretation of the silence around us? — — Lawrence Durrell

Ervin Wilson

I consider myself one of those composers who is never happy with the established order of things. So when Erv Wilson invited me into his world of numbers and symbols and sonic codes, I was entranced with the vast array of stars in his sky, the zillions of unexplored planets out there with jewels just lying on the ground. His mode of distribution was straight up and simple,… he was very old school. He drew his treasure maps by hand, ( diagrams/erv-wilson-diagrams/)  xeroxed them, and would mail them to whomever he thought would be interested. But he would leave it up to you to determine what to do once you get into the vicinity of each treasure site. Much of what was so unique about his approach to master pitch selection is reflected by his rejection of the notion of “The Truth”, (i.e. that there exists some ultimately superior scale or tuning system). He believed you don’t have to stay in any one system but should mine the resources of whichever ones you might need for a particular compositional purpose. Hence instead of handing people pre-drawn circles, he offered a larger context in which you draw your own circles. He advocated a process by which what you intuit can be systematized into a master set of musical pitches, then evaluated, changed and even discarded…. ultimately creating scales based on what you “feel” more than what you “think”. This now broadens a composer’s capacity to create color, novelty and interest. Ultimately the back and forth interaction between imaginative play and it’s systematization renders a dynamic flow of innovation that never gets stuck in a “way of doing things”.


Someone once told me Erv stole fire from the gods. I agree. Yet why would one believe such a thing? The light of the gods burns hot and cannot be held in the hand. For as soon as the hand opens to reveal the stolen prize, the fire immediately erupts, spreading itself into the room and beyond, revealing its infinite nature. Erv’s musical frequency structures contain this property of stolen fire. They are self-propagating yantras… the Scale Tree, Holograms, Pascal’s Triangle, The Coprime Grid, the Combination Product Sets (,… their numbers spill off the page as they go forth and multiply.  Most everyone who met Erv and saw his work, musician or not, would sense the presence of that fire., feeling this intuitively without having to understand the work itself. All of us who have explored his structures were drawn to him and his material not because of some intellectual curiosity but due to this feeling of “the emerging unexpected” that gets triggered when in the presence of one who has stolen fire.




The website,, is dedicated to explaining the musical breakthroughs of Erv Wilson and the larger cultural narratives implied by his work. Wilson can be described as a “musical cosmologist” who, by creating/discovering brand new pitch fields. has cracked open a vast unexplored mother lode of new musical note relationships for composers to use freely.

Many of his dynamic tuning systems have the organic property of spawning an infinite number of previously unknown musical scales nested within each other like Chinese boxes. The video interviews and other content on this site attest to the broad impact of his work.



From the Editor



It is with great pleasure that we relaunch Xenharmonikon. It has been a long journey of over a year and a half of working, planning and rethinking.

I am taking this opportunity to explain how Xenharmonikon will not only continue the work of the past but also be different in various ways. Most of these changes stem from a shift in format from print to online which makes possible new ways of organizing the journal. Innovation is a vital aspect of microtonality and intonational practice, thus the subject calls for innovative forms of presentation.

The new Xenharmonikon moves from an informal print journal to a single-blind peer-reviewed publication. Why single-blind as opposed to double-blind? Many individuals working in the field of microtonality have unique practices that would be difficult to discuss without being identified in some way. Contributors may still request to submit their work anonymously for the peer review process, but this will not be a requirement. Unlike some peer-reviewed publications, we will not be charging a fee to publish articles. We hope to be able to rely on donations to cover our costs.

While retaining the idea of volumes, these will be identified by tags along with keywords, which over time will be more useful in accessing articles on related subjects. Articles will be published online when the review process is complete. This prevents work being delayed due to the status of others. We also plan to take advantage of the online format by expanding Xenharmonikon to announce and cover microtonal activity around the world, especially the many festivals focusing on tuning as well as the latest developments in software and technology. Some of these will be dealt with in more detailed reviews.

It should remain clear that Xenharmonikon does not represent any single approach to the subject of microtonality or intonational practices and plans to be proactive in representing the field.

We do begin our first issue with a tribute to one of the important contributors to the original Xenharmonikon, Erv Wilson, who passed away while we were planning the relaunch of the journal. For the second issue, we are calling for articles relating to Heinz Bohlen who also passed away not long ago. For future issues, we plan to focus on the Helmholtz-Ellis notation and the school of composers who use it. Contributions beyond these specific themes are also encouraged and we welcome suggestions for other topics.

Lastly, thanks to John Chalmers, the former editor of Xenharmonikon, for making the transition possible through his support and suggestions. Also, giant thanks to Lucija Kordic who designed the website over many weeks and months and envisioned its possibilities, and to my fellow editor Terumi Narushima who provided invaluable feedback and criticism.

We now welcome the community to make Xenharmonikon their own. Comments and suggestions can be sent to


Kraig Grady