Introduction:
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 2.3.7.11 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 ∈ ℤ. 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 2.3.7.11 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,
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,
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,
and I simply call it the axial scale. This scale is an excellent bridge between traditional diatonic harmony and the 2.3.7.11 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,
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,
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 2.3.7.11 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.
I named this scale the Janus scale, because it looks back on 5 limit harmony and ahead towards 2.3.7.11 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.
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 2.3.7.11 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:
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.
Both dekanies can overlap to make a double dekany, which in this case contains 13 tones.
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:
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.
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.
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.
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.
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.
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.
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.
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:
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 2.3.7.11 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.
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.
This 47-tone fabric provides a generalized means of building JI structures in the 2.3.7.11 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
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,
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,
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,
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 2.3.7.11 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 2.3.7.11 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 2.3.7.11 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.
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 2.3.7.11 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.
Conclusion:
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 2.3.7.11 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.
Acknowledgements:
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.