In a letter to John Chalmers, written on April 26, 1975 Erv Wilson writes the following:
“The basic structure of “moments-of-symmetry” is embarrassingly simple. Unfortunately, what is simple is not always obvious, or vice versa. And we use certain devices repeatedly, without identifying them. Or we neglect to use them when we might well have chosen to do so, had we only recognized them (Wilson, 1975).”
Wilson’s sentiment resonates with the observations that are about to be outlined here. Although it appears both simple and obvious, and presumably many other music theorists have noticed the same things, no published discourse appears to exist concerning the physical, and musical occurrences, which are about to be observed and commented upon here.
There are geometric symmetries which exist in a just intonation Tonnetz that could be used to aid the analysis, composition, and discussion of music in just intonation. James Tenney discusses the idea of geometric symmetry in just intonation in the article On Crystal Growth in Harmonic Space (Tenney 1993-1998), and he clearly explores these ideas in Chromatic Canon for two pianos (Tenney 1983). Similarly, and much more extensively, Tymoczko posits a theory of harmony based on geometry in A Geometry of Music (Tymoczko 2011). Tymoczko is aware of just intonation but does not discuss it in great detail, nor does he discuss the geometry specific to harmonic ratios in just intonation Tonnetze. His discourse centres around the geometry of chords and pitches in 12-TET. The discourse that follows herein makes explicit the specific type of symmetry which subsists in just intonation. Wilson is interested in both just intonation, and the geometric patterns which are a result of it. His variable tuning models which are available in the Wilson Archives (Wilson), and on the Wilsonic app (Satellite, 2014) demonstrate certain elements of the geometric symmetry which are about to be discussed, yet he provides no discourse on the geometric symmetry of a just intonation Tonnetz.
The Tonnetz
The Tonnetz initially appeared in Tentamen Novae Theoriae Musicae (Euler 1739, p.147) as a way to reconcile the first two distinct […] intervals generated from a resonating body (Cohn 1997, p.07). Since its first appearance in 1739, numerous theorists have presented adaptions of, and variations on the Tonnetz (Oettingen 1866, Riemann 1914-1915, Partch 1974, Vogel 1975, Tymoczko 2011, among others). The basic function of a Tonnetz is to graphically represent tonal space, and relationships between individual pitches and harmonies. Different tuning systems are situated on different surfaces; equally-tempered systems are situated on the closed unbounded surface of a torus, whereas just intonation is situated on an infinite Cartesian plane (Gollin 1998, p.196). The Tonnetze under examination in this paper are situated on an infinite Cartesian plane. Gollin observes that one of the essential aspects of a traditional Tonnetz is the arrangement of pitches along two independent axes, arranged intervallically by perfect fifths on one axis, and by major thirds on the other (Gollin 1998, p.196). Cohn provides an abstract representation of the Tonnetz, where the primary axes are generated by the generic intervals x and y, in the sense that each row increments from left to right by the value of x, and each column increments from bottom to top by the value of y (Cohn 1997, p.10). It is the geometry of this particular species of Tonnetz, described by Cohn and Gollin that is situated on an infinite Cartesian plane, that will be discussed in this paper.
Symmetries of a line in just intonation
The two rigid motions of a line, that is translation and reflection[1] are both possible in just intonation if a single axis of a Tonnetz is considered in isolation. Example 1 shows a geometric series, or a series of perfect fifths as they would appear in a just intonation matrix. The operations discussed in this section are not only limited to a series of fifths, they can be applied to any series of a given interval. A series of fifths has been used simply because 3:2 is the first generative ratio encountered in just intonation (Doty 1994, p.33).
Example 1, a series of perfect fifths.
Example 1, a series of perfect fifths.
The interval, or quotient between adjacent harmonic ratios shown in example 1 is 3:2. Consider the scenario where 1:1 shifts two places to the right. Rigidity implies that if 1:1 moves two places to the right then all other points on the line must also move two places to the right. When this operation is conducted, all relationships between adjacent ratios in the line will continue to be 3:2, assuming the scale is open-ended, therefore translation is possible. Next consider a scenario where some chosen point is fixed.[2] If a mirror were placed on the fixed point, and only one half of the line were visible, the reflection would map a row of notes which had inverse relationships to the portion of the line that were visible. For example, if 1:1 were taken as the centre point, 3:2 would be reflected as 2:3, 9:8 would be reflected as 8:9 and so on. Once the rules of just intonation are taken into account[3] it can be seen that the rigid motion of reflection is possible in just intonation.
It should come as no surprise to the reader that a geometric series is symmetrical. What is interesting is that if the pitches from a series of fifths are re-arranged into ascending order to form a musical scale, or mode, some symmetry persists. This is shown in example 2.
Example 2, the mode generated by the series of fifths from example 1.
Example 2, the mode generated by the series of fifths from example 1
The harmonic ratio of 2:1 has been included in example 2 in order to close the mode and as a result complete the symmetrical pattern which is shown by the points of symmetry. Once the harmonic ratios are rearranged into a mode, the symmetry of the line changes. The ordering of pitches shown in example 2 is much less symmetric than the ordering of pitches in example 1. In example 1, any point on the line is a possible point of reflection, whereas in example 2 the possible points of symmetry are limited to four points out of eighteen. This type of symmetrical musical scale, derived from a series of fifths can be traced back to Pythagoras, and Ling Lun. Ling Lun’s scale is shown in example 3.
Example 3, Ling Lun’s Scale, excerpt taken from Genesis of a Music (Partch 1974, p.401).
The musical scale shown in example 3 is a series of perfect fifths stacked upon one another, and then arranged in ascending order. The smaller ratios that lie between the arrows show the distance between each pitch in the scale. The points of symmetry, which have been marked in red, intercept 256:243 between 6561:4096 and 27:16, and 2187:2048 between 9:8 and 19683:16384. Although probably aware of its existence, Ling Lun omits the Pythagorean comma, instead choosing to simply close the scale with 2:1, creating an inversionally symmetrical mode. Cohn defines inversional symmetry as a property of any pitch-set which […] pairs its pitches symmetrically about a pitch-axis (Cohn 1988, p21).
Symmetries of the plane in just intonation
The symmetries of the plane, that is translation, reflection, rotation and glide reflection are also possible in just intonation. Consider the Tonnetz shown in example 4.
Example 4, a seven-limit just intonation Tonnetz.
Example 4 shows a matrix where harmonic ratios are related by intervals of fifths on the x-axis and intervals of sevenths on the y-axis. The group of nine pitches shown in example 4 can be moved to some other arbitrary co-ordinates through a translation, assuming the system is open-ended. The relationships connecting each note to each other note in the lattice will remain invariant if each pitch shifts according to a set distance. This is given, as the Tonnetz is made up of coplanar lines whose symmetries have already been shown in example 1. The three parallel lines of fifths will reflect across the vertical line 7:4 – 1:1 – 8:7 with invariant results, and the three parallel lines of sevenths will reflect across the horizontal line 4:3 – 1:1 – 3:2 also with invariant results. Reflection across the line 7:6 – 1:1 – 12:7 yields variant results; likewise, reflection across the line 32:21 – 1:1 – 21:16 also yields variant results. This indicates that the shape in question does not meet the criteria of a square as rotation by 90 or 270 degrees produces results which are variant from the identity isometry.[4] This is illustrated in examples 5 and 6.
Example 5, identity isometry of example 4, showing the intervals between the harmonic ratios shown in example 4.
Example 6, rotation of the identity isometry by 90 degrees.[5]
Example 6 shows that the results of this operation are variant, as 7:4 and 3:2 are not the inversions of each other, nor are 21:16, and 7:6. Rotation by 180 degrees produces results which are the same as that of the identity. This is shown in example 7.
Example 7, rotation of the identity isometry by 180 degrees.
The variant results produced by a 90 or 270 degree rotation, combined with the invariant results produced by a 180 degree rotation indicate that the Tonnetz is not square shaped, but rectangular. Although it seems obvious, it needs to be stated that a square is not possible in this species of just intonation Tonnetz because new dimensions are generated through introducing different prime numbers whose values are not equal to one another.
As reflection and translation are both possible in this species of just intonation Tonnetz, therefore a glide reflection is also possible, as a glide reflection is simply a reflection followed by a translation. Glide reflection is only possible, however, if the reflection does not take place through the diagonal lines 7:6 – 1:1 – 12:7 and 32:21 – 1:1 – 21:16 as it has already been established that these lines produce results which are variant from that of the identity.
Symmetry persists when the harmonic ratios in the Tonnetz shown in example 4 are arranged in ascending order to form a mode. This is shown in example 8.
Example 8, the harmonic ratios from example 4 rearranged in the form of a mode.
As with example 2, example 8 shows that once a symmetrically shaped Tonnetz is arranged into an ascending order some symmetry persists, although it is different to the symmetry of the lattice. Again, the result of this particular reordering produces a mode that has fewer symmetries than the symmetries apparent in the original layout of the Tonnetz.
Symmetry in just intonation in three-dimensions
As the most symmetric shape available in a two-dimensional just intonation matrix (of this species) is a rectangle, so the most symmetric shape in three-dimensions will be the rectangular prism, or cuboid. It should be noted that triangular based prisms, pyramids, and octahedra do exist in just intonation Tonnetze but will not be discussed here because they are all mere truncations of larger rectangular structures. The rectangular prism has three planes x, y, z of reflection, along with three axes x, y, z of rotation. Translation, reflection, rotation, glide reflection, and helical screw motion are all possible. An example translation in three-dimensions is shown in example 9.
Example 9, translation of a 7-limit lattice.
Example 9 shows that translation in three dimensions yields invariant results of identity isometry: all adjacent points on the lattice continue to be related by ratios of 3:2, 5:4, and 7:4. The translation conducted on the identity lattice was move each point east two spaces on the x-axis (starting at 1:1 to arrive at 9:8), then move north one space on the y-axis (continuing from 9:8 to arrive at 45:32), and upward one space on the z-axis (continuing from 45:32 to arrive at 315:256). Each plane of symmetry will divide the rectangular prism into congruent halves. Reflection across lines on the plane have already been shown in the previous section, so reflections across the x, y, and z planes are given.[6] Glide reflections are also given, as translation and reflection have been proven. 180-degree rotations in two-dimensions have also been proven in the previous section. 180-degree rotations are possible on any of the three axes in three dimensions; again 90, and 270 degree rotations yield results which are variant. Helical motion, that is a combination of a rotation with a parallel translation, is also possible in a three-dimensional just intonation Tonnetz (of this species).
When a three-dimensional lattice is arranged into a mode, the symmetrical pairing of pitches occurs in the following way: pitches on plane one reflect through plane two into plane three whilst undergoing a 180-degree rotation, while the pitches on plane two are paired according to a 180-degree rotation. This is illustrated in example 10, using alphabetical pairs.
Example 10, plane one reflecting through plane two, and undergoing a 180-degree rotation to form plane three.
Example 11 shows a different representation of Example 10, where harmonic ratios have been used to represent the various points on the lattice.
Example 11, the three-dimensional lattice shown in Example 10, with harmonic ratios instead of letters.
When the pitches in example 11 are arranged in ascending order the result is another symmetrical mode. The mode is shown in example 12.
Example 12, the harmonic ratios from example 11 re-arranged in ascending order as a mode.
The mode shown in example 12 is again, as with the one and two-dimensional modes, less symmetric than the arrangement of pitches as they lie in the lattice. This discourse has shown, through geometric proofs, that the particular species of Tonnetz in question are rectangular in nature, not square. Therefore, it is worth considering if perhaps the way this kind of Tonnetz is visually represented should be rectangular. At present, there is no accepted standard for how a theorist should represent each new dimension with a given length, but they may choose from at least three options. Option one is to represent all dimensions with lines that are equal in length, this is the easiest of the options, it requires little thought, and produces an arguably inaccurate visual representation of tonal space. Option two requires the theorist to either know or calculate the size of the intervals that form the building blocks of the lattice. In this option, the theorist uses interval size to determine length of lines within the lattice, for example the line from 1:1 to 3:2 would be longer than the line from 1:1 – 11:8. While it is a good choice, option two requires the most work of the three options. Option three is for the theorist to draw lattices where the length of the lines within the lattice are determined by the size of the prime numerator that is generating an interval, in this case 11:8 would appear longer than 7:4, despite 11:8 being a much narrower interval than 7:4. This option is preferable to option one as it is a more accurate representation of tonal space, and in some circumstances (e.g. when only low prime numbers are being used) requires less calculation than option two. All three options require the theorist to determine the direction of the lines.
Un-transposable Harmonic Modes (UHMs)
In the previous section of this paper, each time a Tonnetz was unfolded, and rearranged in ascending order, the resulting mode exhibited inversional symmetry. This type of mode, while certainly not a new discovery, has not been named or discussed at length, nor have theorists shown how these kinds of modes may be strategically truncated in order for composers to extract -pitch modes. The modes in question from this point will be referred to as Un-transposable Harmonic Modes, or simply UHMs. A UHM is an inversionally symmetrical mode that is derived from the pure intervals found in the harmonic series, and cannot be transposed in the tuning model within which it was conceived;[7] this is the definition that has been developed for this discussion. UHMs are conceptually derived from Messiaen’s modes of limited transposition, which were first discussed in The Technique of My Musical Language in 1944, but are actually quite different. Messiaen writes the following of his own modes:
The series is closed, it is mathematically impossible to find others, at least in our tempered system of 12 semitones (Messiaen 1944, p.58).
Messiaen acknowledges that another system of tuning could potentially contain more modes of limited transposition but is content to leave them for someone else to decipher. UHMs pick up where Messiaen left off, in an alternative system of tuning, where the mathematical restrictions imposed by 12-TET are replaced by a new set of restrictions and the discovery of new modes is limited by a new set of rules. As UHMs are conceptually connected to Olivier Messiaen’s modes of limited transposition it is logical that they should share some features. Firstly, as with Messiaen’s modes, in each set of UHMs there is a maximal UHM (MUHM) from which the other UHMs in the set are derived through truncation. The maximal UHM can be defined as the UHM that exists within a given tuning model and contains the maximum possible number of pitches in that particular system, or a section of a system.[8] For example, if the tuning model under consideration is the seven-limit system that appears in example 4, repeated in example 13 for the convenience of the reader, then the MUHM of this system would be the mode shown in example 14 as it contains all of the pitches[9] from the system in example 13.
Example 13, a seven-limit just intonation Tonnetz.
Example 14, the harmonic ratios from example 13 rearranged in the form of a mode.
If the Tonnetz in example 13 were to be truncated in specific ways then further symmetrical modes could be extracted from it, for example if the ratios 7:6 and 12:7 were to be removed, the shape of the Tonnetz and the resulting mode would still be symmetrical and still create a UHM, but the MUHM of this particular system must contain all nine pitches. If the tuning system is irregularly shaped, or is multi-dimensional, then it will likely contain more than one maximal UHM, for example Partch’s 43-tone lattice contains a number of different UHM regions, and therefore this system contains more than one MUHM. Partch’s tuning system is an excellent example of this concept but it is very difficult to clearly represent it in two dimensional space so an alternative example will be used. The example taken will be La Monte Young’s The Well-Tuned Piano (Young 1964-1973-1981-Present) shown in example 15. The difficulty with Partch’s tuning system is identifying the various MUHM regions as there are many, conversely the difficulty with Young’s tuning system is determining where the MUHM lies as there is more than one possible answer.
Example 15, La Monte Young’s tuning scheme for The Well-Tuned Piano.
If the theorist attempts to force Young’s tuning model into some symmetrical form, there are four possible answers regarding what the MUHM of his system is. These are shown in examples 16 through 19.
Example 16, MUHM extracted from The Well-Tuned Piano scheme by removing 567:512 and 49:32.
Example 17, MUHM extracted from The Well-Tuned Piano scheme by removing 1:1, 3:2, 9:8, and 7:4.
Example 18, MUHM extracted from The Well-Tuned Piano scheme by removing 189:128, 567:512, and 1323:1024.
Example 19, MUHM extracted from The Well-Tuned Piano scheme by removing 567:512, 1:1, 3:2, and 9:8.
According to the provided definition of a MUHM, example 16 is technically the most accurate possibility, but as Young’s tuning system is irregularly shaped examples 17-19 show other ways that a MUHM might be found within The Well-Tuned Piano.
To reiterate, the MUHM is the UHM that exists within a given tuning model and contains the maximum possible number of pitches in that particular system, or a section of a system. All other UHMs that contain less pitches than the MUHM would be considered truncations of the MUHM of that particular system. The second feature that UHMs share with Messiaen’s modes of limited transposition is that they both exhibit inversional symmetry. Inversional symmetry, according to Cohn, can be considered “a property of any pitch-set which […] pairs its pitches symmetrically about a pitch axis” (Cohn 1988, p.21). When truncated according to specific techniques, discussed in the next section of this paper, the maximal UHMs (which is always inversionally symmetrical) yields numerous inversionally symmetrical derivative UHMs, or truncations. Despite the conceptual connection to Messiaen’s modes, UHMs do present slight differences. The first difference between UHMs and Messiaen’s synthetic modes is that UHMs do not have a limited number of transpositions. UHMs theoretically have no transpositions. It should be acknowledged here that as just intonation is an open system, any given system in just intonation could be extended indefinitely in all directions, in which case it could be argued that UHMs could be transposed indefinitely. Conversely, it could also be argued that if the system in question were to be extended indefinitely then any previous UHMs would become invalid (as they would no longer meet the criteria for what constitutes a UHM, according to the definition of a UHM), and a new set of UHMs would become the UHMs of that new system. The second difference, which is closely linked to the first difference is that UHMs inhabit rational tuning lattices whereas Messiaen’s modes, were conceived in 12-TET.
Inversion versus equidistance
Inversion in music, and mathematics implies the reversal of some relationship. Equidistance implies that a certain distance is traversed, in equal portions. In the case of UHM pitch-pairs, before moving any further ahead, the difference between these two things must be illuminated. Pitch pairs in the context of this paper and in the matrices discussed will always be equidistant from the point, line, or axis of symmetry, thus making the two notes which form a pitch pair the intervallic inversion of one another. A pitch pair may be intervallically inverse, but the ratios of the pitch pair must the reciprocals of each other in order for the pitch pair to be mathematically inverse. The factor affecting mathematical inversion of pitch pairs is the location, within the tuning model, of the fundamental ratio 1:1, as it is the placement of this ratio that will determine the value of all other ratios. This concept may be best elucidated through two examples, the tuning models of Harp of New Albion (Riley 1986), and The Well-Tuned Piano (Young 1964-1973-1981-Present). These two tuning models are reasonably similar in that they are both two-dimensional systems, and they both contain a nine-pitch segment where eight pitches pair symmetrically around a central pitch as illustrated in example 20.
Example 20, eight pitches, represented by algebraic (not musical) letter names paired around a central pitch.
The main difference between the two tuning models is that Young’s model is exclusively otonal, where Riley’s model combines otonality and utonality. Additionally, Young uses co-prime[10] ratios 3:2 and 7:4 as the generative intervals, where Riley uses co-prime ratios 3:2 and 5:4 as generators. The otonality/utonality difference influences the placement of the fundamental, 1:1. Riley places the fundamental in the centre of the Tonnetz. This is shown in example 21.
Example 21, Terry Riley’s tuning scheme for The Harp of New Albion.
Riley’s placement of the fundamental causes pitch-pairs to be intervallically inverse, mathematically inverse, and equidistant. Young, however, places the fundamental on one of the corners of the Tonnetz; this is shown in example 22.
Example 22, La Monte Young’s tuning scheme for The Well-Tuned Piano.
Young’s placement of the fundamental causes pitch-pairs to be intervallically inverse, and equidistant, but not mathematically inverse. Of course, the central portion of his model could easily be translated into the following, different ratios shown in example 23.
Example 23, A alternative expression of the central portion of the tuning scheme for La Monte Young’s Well-Tuned Piano.
The translation, one space up on the y-axis, and one space to the right on the
x-axis (with pitches 189:128, 567:512, and 1323:1024 removed) retains all relationships between pitches, and now causes all pitch-pair ratios to be mathematically inverse. In some cases, it is not possible to make pitch-pair ratios mathematically inverse. Take for example, the following two-dimensional lattice with the dimensions 2 x 3 shown in example 24.
Example 24, a two-dimensional lattice with dimensions 2 x 3.
This lattice contains three pitch-pairs which are equidistant from two points of symmetry, one point is the quotient of 7:4, and 12:7, the other point is the quotient of 21:16, and 8:7. The symmetry of the mode is shown in example 25.
Example 25, the mode resulting from the 2 x 3 lattice shown in example 17.
In such cases it is not possible to make the pitch-pair ratios mathematically inverse. This point is worth observing in circumstances such as this one where correct understanding of these terms is pertinent to the theory that is being presented.
Example of a MUHM and its truncations (derivative modes)
The Tonnetz from example 4 will be used as the starting point to demonstrate how a maximal UHM can be truncated. As shown in example 8, when this 7-limit Tonnetz is arranged into ascending order, it forms an inversionally symmetrical mode in this case the inversion is musical andmathematical. The mode shown in example 8 is the MUHM of the Tonnetz shown in example 4. For convenience, these examples are shown again in example 26.
Example 26, The 7-limit Tonnetz from example 4, and the inversionally symmetrical mode that results from example 8.
7-limit Tonnetz
MUHM of the Tonnetz
The symmetry of the Tonnetz shown in example 26 is the same as the symmetry shown in example 13, where 1:1 pairs with itself, and the ratios that are the harmonic and mathematical inversions of one another, pair with each other. The pairs are shown in table 1.
Table 1, list of pitch pairs for Tonnetz and MUHM shown in example 26.
| Ratio 1 | Ratio 2 |
| 1:1 | 1:1 (or 2:1) |
| 4:3 | 3:2 |
| 7:6 | 12:7 |
| 7:4 | 8:7 |
| 21:16 | 32:21 |
To begin to truncate the MUHM, all that is necessary is that the composer or theorist begin to remove pitch pairs from the MUHM. For example, if 21:16 and 32:21 are removed from the MUHM, a new 7-pitch truncation exists, which is also inversionally symmetrical and cannot be transposed, therefore it is a UHM. This mode is shown in example 27.
Example 27, a seven-pitch truncation of the MUHM shown in example 26.
This process can be continued, where pitch pairs, or groups of pitch pairs are removed to create a number of inversionally symmetrical truncations of the MUHM. In total the Tonnetz shown in example 26 contains thirty-one inversionally symmetrical modes, including the MUHM, but not including a mode with zero pitches. Of these thirty-one modes, there are five modes that can be transposed within the Tonnetz: the remaining twenty-six modes cannot be transposed within the tuning system. The complete list of symmetrical modes contained within this lattice are shown in table 2, with the five transposable modes being highlighted in red.
Table 2, list of modes contained within the 7-limit lattice shown in example 26.
The characteristic that all of the transposable modes shown in table 2 share is that each of these modes does not contain a corner pitch-pair. Each mode containing a corner pair will be un-transposable: this rule holds in all dimensions. The final mode in the table, with a single pitch is musically and mathematically valid, a zero pitch mode is also valid in both fields but has been omitted here.
With smaller lattices it is reasonably easy for the composer to determine how many possible modes exist within the system without performing any calculations. Larger lattices require an equation in order to determine the number of possible -pitch modes. Very large lattices pose the problem of combinatorial explosions. In order for the composer or theorist (i.e. a human) to manage and categorize modes from very large lattices it is necessary to impose a series of parameters on the lattice in order to keep the number of modes to a manageable level. The specific set of limitations will obviously vary from one problem or piece of music to the next. Alternatively, computer software could be used to categorize all modes within a larger system of for example, 46 pitches.
Method and equation for truncating MUHMs
As shown in the previous section, a MUHM is truncated by removing pairs of pitches which are equidistant from a point, line, or axis of symmetry. When a single pair is removed, this does not have any impact upon the symmetry of other pairs, as the symmetry of each pitch pair is completely independent of any other pitch pairs which may be included in the tuning model. Consequently, the removal of a single pair has little effect on the overall symmetrical structure of a mode. The following equation shows that is equal to all possible combinations of equidistant pairs that can exist in any lattice.[11]
All the composer, theorist or analyst needs to do to use this equation is input two or more numbers (depending on how many dimensions there are) in place of nd in order to determine how many equidistant pitch pair combinations are available in a lattice. nd is equal to the product of the size of the lattice. For example, if the lattice forms a 3 x 3, 4 x 4, or 5 x 12 grid, then nd will be 9, 16, or 60 respectively. If the lattice exists in three or more dimensions, then nd will be the product of the size of the three or more dimensions of the lattice.
For example, if one wanted to find all possible combinations of pitch-pairs from the Tonnetz shown in example 26, then the equation would be used as follows.
The -1 at the end of the equation accounts for the fact that zero notes, although technically valid, generally do not constitute a musical mode. This equation does however allow two-pitch, and even single-pitch combinations to constitute a mode. Some compositions such as 4’33 (Cage 1960), Quattro Pezzi (Scelsi 1959), and 1960 #7 (Young, 1960) contain no pitches, a single pitch, and two pitches, respectively. These compositions are artistically valid, however, they can be considered the exception to the rule. Many composers would consider zero, single, and two note modes too restrictive to suit their needs. For this reason a second equation has been included in this paper which allows composers, theorists or analysts to determine the number of -pitch modes that a lattice contains. The equation allowing this is as follows.
Where f is equal to:
By assigning a value to , then a number of
-pitch modes can be shown to exist within a lattice. The value of
will be between 0 and
, and
can be a defined value. If the seven-limit lattice from example 26 is used, then
can have values between 0 and 5. So
can now be represented as:
To find how many zero-pitch modes are contained in the system, input a value of 0 for and as it has been established that
has a value of 5, input a value of 5 for
. This will appear as:
Once calculated, this equates to 120/120 = 1. So there is one zero-pitch mode. To find the number of modes containing one and two pitches, input a value of 1 for . This will appear as:
Once calculated, this equates to 120/24 = 5. This shows that there are five modes containing one or two pitches. The find the number of modes containing three and four pitches, input a value of 2 for . This can be continued until a total of 32 modes is reached. Table 3 illustrates how the remaining values are found.
Table 3, illustration of how the equation can be applied to extract -pitch modes from a seven-limit lattice with dimensions 3×3.
When working with a lattice that has an odd number of pitches, such as the Tonnetz appearing in example 26, the one and two-pitch modes, and three and four-pitch modes, and so on will be categorized simultaneously as the central pitch in the lattice pairs with itself, and therefore frequently needs to be counted as two notes. The equation is useful as it provides the user with a quick way to determine how many -pitch modes exist within a lattice. The equation does not take into account that certain transposable pitch-pairs, and larger sets of transposable pitches (which are inversionally symmetrical) typically exist within a lattice, as was shown in table 2. So although the equations shown here provide us with a way to truncate MUHMs in a somewhat methodical fashion, further work is necessary in order to eliminate or at least systematically identify the transposable sets of pitches.
A possible method for classifying UHMs
If a composer or theorist takes the information in this paper and applies it, then they will require a method to classify the modes. The suggestion being presented here is just that, a suggestion: others may wish to adopt another method for classification. The information that might be included in the label of a UHM includes the number of pitches that the mode contains, how many of those n-pitch modes exist in that particular Tonnetz, which number (out of the total n-pitch modes) the mode is, information regarding the prime numerators (or denominators) that make up the converging geometric series, and the name of the piece of music the modes are being used in. Example 28 illustrates how such a classification method might appear.
Example 28, a potential method for classifying UHMs.
The diagram in example 28 shows that there are two geometric series that converge to form this UHM, one generated by seven, the other generated by five. It shows that this mode has 30 pitches in it, and that this mode is one of one 30-pitch modes in this system. The benefit of using the system presented in example 28 is that it presents a reasonably large amount of information in a small space. The downside of this system is that it is a little more awkward than simply labelling something an octatonic or pentatonic mode.
Modes of Synthesized Inversional Symmetry (MOSIS)
There exists a method for generating a mode that will be guaranteed to be inversionally symmetrical. This is achieved through a technique where the composer or theorist synthesizes the symmetry, as opposed to discovering the symmetry that is naturally occurring within a Tonnetz.
Any set of intervals can be infinitely replicated in just intonation; this can be achieved by translating a set of ratios into a new dimension. Alternatively, a translation can be performed within the same dimensions in which the first set of ratios exists. Simply translating a set of ratios to a different position on an existing lattice, or into a new dimension does not guarantee that combining the two sets will generate a symmetrical mode. In order to guarantee symmetry, the replication process must simultaneously incorporate a 180-degree rotation about a given point. To move to a new dimension, a new co-prime ratio is necessary, and the value of the new co-prime ratio must be asserted. The ratios in set one can then be multiplied by the reciprocal of the new co-prime ratio in order to achieve the 180 rotation. This is one way of finding the value of the ratios in set two. If set one is composed of pitches with the values a, b, c, d which are related by the intervals n, q, r then set one can be represented as the mode shown in example 29.
Example 29, set one.
If a1 is found through the operation of multiplying a by x/y, then the remaining values of set two can be found by multiplying the remaining values of set one by y/x. This is illustrated in example 30.
Example 30, diagram showing the relationship between set one and set two.
An alternative method for finding the values of set two is to assert the value of x/y and then calculate the value of b1, c1, d1 using the values of the intervals n, q, r, whose values are known. This is a theoretical way to synthesize symmetry in a mode; its usefulness is yet to be tested as a compositional tool. As a concept Modes of Synthesized Inversional Symmetry (MOSIS) is valuable because it provides the composer or theorist with a quick way to find a corresponding symmetrical set in a defined tonal space, when such a set may not be immediately apparent. Consider the scenario where a composer has an asymmetrical mode, and the ratios which form the mode are multi-dimensional and distantly related. In such a scenario it may be difficult to determine a corresponding set of ratios that would unify the asymmetrical mode, making it symmetrical. MOSIS provides a way of doing this at a time where it seems there is no method explaining how to unify an asymmetric set, with another similarly asymmetric set (with inverse identity isometry), forming an inversionally symmetrical mode. There appears to be an inverse link between UHMs and MOSIS, and using one model to predict the other is an ill-posed problem. If one were to use MOSIS in order to produce a particular shape on a lattice, it would prove rather difficult, perhaps impossible to produce a specific polygon or polyhedron without already knowing the dimensions of that shape. Conversely, it is also particularly difficult to determine the way in which a shape in a lattice will be broken into two congruent halves which form a MOSIS, without having already constructed the MOSIS. This is demonstrated in examples 31 through 36. Example 31 shows a 7-limit lattice with twenty pitches.
Example 31, a 7-limit lattice with 20 pitches.
Because we can see the matrix in example 31, then the geometric shape of the Tonnetz is known. What is unknown at this point is exactly how the shape will break into two congruent halves that will be self-similar when mapped upon one another. By arranging the Tonnetz into ascending order to find the MUHM, the congruent halves can be discovered. The MUHM for this Tonnetz is shown in example 32.
Example 32, MUHM for the Tonnetz shown in example 31.
In example 32, we can see the two points of symmetry, and these points at 160:147, and 12,288:12,005 show where the two congruent halves of the mode are (that is from 3:2 to 1029:1024 and from 35:32 to 12005:8192). Examples 33 and 34 show the two congruent halves of the Tonnetz.
Example 33, Congruent half one.
Example 34, Congruent half two.
The way in which to map example 33 onto example 34 is to perform a 180-degree rotation on the xz plane, followed by a reflection of the shape across the yz plane.
If the first of the two congruent halves, shown in example 33, is represented as half of a mode that the composer wants to replicate in a different space to form an inversionally symmetrical mode, example 35 illustrates how this would appear.
Example 35, one congruent half of the Tonnetz from example 31, in ascending order.
From here, the composer or theorist could choose any co-prime ratio to multiply the ratios in example 35 by. Provided that the instructions from the diagram shown in example 30 were followed, an inversionally symmetrical mode would result. To re-create the Tonnetz shown in example 31 however, 1:1 must be multiplied by 36,015:32,768 and the remainder of the mode shown in example 35 must be multiplied by the reciprocal of this ratio (32:768:36,015). Example 36 shows the resulting MOSIS.
Example 36, the Tonnetz from example 31, expressed as two congruent halves of a MOSIS divided by two points of symmetry 160:146 and 12,288:12,005.
In order to select the precise ratio, in this case 36,015:32,768, to multiply the first half of the mode by, which produces a second congruent half that happens to form a perfect rectangular prism in 7-limit, the shape of the Tonnetz must first be known. If the composer or theorist works in the opposite direction, where one half of a mode is known, and then its identity isometry replicated at a 180-degree rotation in some arbitrary tonal space, then the geometric shape of the resulting Tonnetz may be difficult to predict without first mapping the first half of the mode onto a tonal space.
Conclusion
Through understanding the symmetry of an abstract Tonnetz that is situated on an infinite Cartesian plane, composers and theorists can extract inversionally symmetrical -pitch sets from symmetrical sections of tuning systems. Furthermore, by identifying the points of symmetry in a mode, the congruent halves of a Tonnetz can be found and conversely if working with an asymmetric pitch set, there exists a simple method for replicating the identity isometry of the set in any dimension to generate a symmetrical mode.
At present, to categorize all the UHMs of a very large lattice, computer software would be required. Additionally, there is still the problem of identifying and separating transposable and un-transposable modes (both symmetrical). Although just intonation offers a way to hear different versions of modes that are popular in 12-tone equal temperament, it also offers infinite gradations of new harmony and these areas of new harmony are often in unmapped tonal space. Accessing large areas of tonal space that are unknown or infrequently traversed requires some kind of road map. The information presented here allows composers a way to manage large symmetrical pitch sets, generate their own symmetrical pitch sets, and organize their search for symmetrical sets with desirable acoustic qualities.
Acknowledgements
I am grateful to Dr. Matthew Sorell for his help developing the equations regarding the truncation of MUHMs which are presented in this article, and for highlighting the symmetrical qualities of a Tonnetz.
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[1] The musical terms transposition and inversion correspond to the geometric operations of translation and reflection.
[2] A point can either be a pitch, or the interval between two pitches.
[3] What is meant by ‘the rules of just intonation’ is that harmonic ratios are frequently represented as fractions with values between one and two, as this is the way to neatly frame an interval inside a single octave. Ratios yielding values below one, or above two are altered in order to keep them within the desired range, unless explicit discussion of intervals greater than an octave is taking place.
[4] An identity is a mathematical relationship equating one quantity to another. Isometry is a mapping of a metric space onto another or onto itself so that the distance between any two points in the original space is the same as the distance between their images in the second space (Definitions from Merriam-Webster Dictionary).
[5] Rotation by 270 degrees yields the same results as a 90 degree rotation when looking at the distance between ratios which are shown in red.
[6] i.e. reflection across the x, y, and z planes are an established fact.
[7] In this case the tuning model, although it could technically be continued infinitely in all directions, is closed.
[8] This is my own definition of a maximal UHM.
[9] i.e. the maximal amount of pitches.
[10] i.e. a ratio where the denominator and numerator are derived from two different prime numbers, for example (2,3), (3,5), (7,13), (17, 3) and so on.
[11] m can be thought of as number of modes, nd can be thought of as number of dimensions, and the special brackets that encapsulate nd/2indicate that the quotient of nd/2 should always be rounded up.