GLOSSARY in progress

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Combination-Product Set (CPS)
A musical structure “generated by taking the products of n harmonic factors [or elements] m at a time and reducing the derived tones to a common octave” (Chalmers 1993, p. 205). Wilson labels these structures as m)n (“m-out-of-n”) sets. For instance, starting with a set of four harmonic factors (i.e. n = 4) such as harmonics 1, 3, 5 and 7, and multiplying two of these factors at a time (i.e. m = 2), produces the following set: 1 3, 1 5, 1 7, 3 5, 3 7, 5 7. This set of six tones is called a 2)4 set which is also known as a Hexany. Other examples of CPS include the 2)5 or 3)5 Dekany (which has ten tones), 3)6 Eikosany (twenty tones) and 4)8 Hebdomekontany (seventy tones). CPS do not imply a single tonal centre; rather, “any tone [in a CPS] is equally capable of becoming a tonic” (Grady 1986, p. 1). Wilson considers the non-centred CPS as a complement to the centred Diamond. He represents the multidimensional geometric structures of CPS as multidirectional lattice diagrams (Wilson 1989). Whilst CPS cover similar harmonic territory to Euler-Fokker genera, what is significant about Wilson’s approach is that he treats the factor 1 as a meaningful member of the generating set. This difference allows CPS to be articulated as coherent centreless structures. [TN]

Constant Structure
A scale in which “each interval occurs always subtended by the same number of steps” (Wilson cited in Grady 1999). By definition, the category of Constant Structures includes MOS scales because every interval in an MOS spans a constant number of scale degrees, but the term “Constant Structure” is usually reserved for scales in just intonation that are built not from a single generator but one that fluctuates in size. Wilson viewed Constant Structures as scales that emulate MOS because even if the intervals in a Constant Structure do not conform to a rigid cyclic pattern in the same way as an MOS, they will still be drawn towards this tendency. On the other hand, Constant Structures broaden the harmonic possibilities of MOS by including pitches from more than one limit. Wilson maps these scales onto a generalized keyboard by using his layout system for MOS scales as a guide. [TN]

Co-Prime Grid
A matrix showing pairs of numbers that are “co-prime”, or share no common factors other than 1. The Co-Prime Grid forms the basis of Wilson’s Gral Keyboard Guide (1994; 2000). [TN]

Cross-Set
A multiplication table in which the products of any two members of a set are arranged on a two-dimensional grid. See also Diamond and Lambdoma.[TN]

Diamond
A term coined by Harry Partch to describe a type of Lambdoma in which a set of harmonic ratios (usually treated as a chord) is multiplied by their reciprocal. These elements are arranged in the shape of a rhombus, hence the name “Diamond”. Diagonals sloping in one direction of the Diamond represent harmonic chords (which Partch called “Otonalities”) and diagonals sloping in the other direction represent subharmonic chords (“Utonalities”). Each parallel diagonal is a transposition of the same chord starting on a different note of its reciprocal chord, thus forming a “chordal complex consisting of interlocking harmonic and subharmonic chords” (Chalmers 1993, p. 211). Wilson rearranges the chords (i.e. each diagonal) of a Partch Diamond around a single tonic to show that all transpositions and their inversions share a common tone, and that this tone is found in different positions within each chord. Wilson uses various geometric shapes to represent each interval of the generating chord as a kind of vector with unique size and direction: a triangle for a three-note chord (e.g. harmonics 1-3-5), a triangle with a point inside for a four-note chord (e.g. 1-3-5-7), a pentagon for a five-note chord (e.g. 1-3-5-7-9), etc. Each edge or line of the basic geometric shape represents an interval, and each vertex or point represents a tone of the chord. Furthermore, every parallel line in a Wilson Diamond represents the same interval size. This form of representation shows the reciprocal harmonic and subharmonic chords as inverted shapes. Organizing the transpositions of different-sized generating chords around a central point results in various types of structures, such as the Triadic, Tetradic, Pentadic, etc., Diamonds (see Wilson 1969-1970). Augusto Novaro independently produced a schema for the 15-limit Diamond (Novaro 1927, p. 15). [TN]

Disjunction
The closing interval that is needed to complete the cycle when building a linear scale from a chain of generators. The disjunction is the ‘leftover’ interval that closes the gap between the beginning and end of the chain. It is not the same size as the generator but spans the same number of scale steps. Wilson treats the disjunction as melodically equivalent to the generator. Although it is the odd or atypical interval, it is an important signpost for determining one’s place in a scale. The term “disjunction” is also used separately in relation to ancient Greek tetrachordal scales to describe the interval that separates two tetrachords (Chalmers 1993, p. 206). [TN]

Farey series
A sequence of all fractions in lowest terms whose denominators do not exceed n, where n is the order of the Farey series, arranged in order of magnitude (Bogomolny 2012). [TN]

Generalized keyboard
A keyboard in which all identical intervals have the same geometric shape with relation to the topology of the keyboard. This enables chords or sequences of notes to be transposed to any key whilst maintaining the same hand shape or fingering pattern. For example, a major triad will always form the same hand shape on a generalized keyboard, in contrast to a traditional Halberstadt keyboard which requires the shape of the hand to change for different transpositions. Generalized keyboards were first invented by R. H. M. Bosanquet in the 1870s. [TN]

Horogram
A graphic representation of MOS scales on a series of concentric rings which are divided into small (s) and large (L) segments by radial lines that represent successive superimpositions of the generator. The size of the period of an MOS scale is represented as a full 360 degree turn and the size of the generator is represented by an angle that is proportional to the period. Starting from the innermost ring, Horograms show the process of interval division from one ring to the next, with each new ring representing a new MOS scale produced from superimpositions of the generator. [TN]

Lambdoma
A cross-set showing the results of multiplying ratios from the harmonic series (i.e. 1/1, 2/1, 3/1, …) on one axis of the table with ratios from the subharmonic series (i.e. 1/1, 1/2, 1/3, …) on the other axis. It is believed that the Lambdoma, which was used as a multiplication and division table by the ancient Greeks, was also applied to musical intervals from early times (Hero 1999, p. 61). Harry Partch’s Tonality Diamond can be seen as a variation on the Lambdoma. Wilson showed that ratios found on the Lambdoma correspond to those of the Farey series (Wilson 1996), and the intervals between each of these ratios are superparticular. Furthermore, the Lambdoma (Farey series) is embedded in the Scale Tree, and removing all repeated ratios from the Lambdoma results in the Co-Prime Grid which also forms the basis of Wilson’s Gral Keyboard Guide. [TN]

Lattice
A diagram or model for representing tuning relationships in multidimensional space. Pitches are represented as points in an array that are connected to each other along different axes. By convention, the horizontal axis represents a sequence of fifths (3/2s) ascending from left to right, the vertical axis represents thirds (5/4s) ascending from bottom to top, and the oblique axis represents sevenths (7/4s) ascending from lower left to upper right. Wilson uses other axes to represent higher tuning dimensions. Lattices are useful for seeing intervallic relationships between pitches and an important characteristic of most lattices is that parallel lines represent the same interval.[TN]

Linear scale
A generic term to describe scales built by superimposing a generating interval (such as a perfect fifth) and reducing the notes to within an interval of equivalence or period (typically an octave). [TN]

Master set
A group of harmonic factors used to generate a Combination-Product Set (CPS). For example, a Hexany generated from a master set of harmonics 1, 3, 5 and 7 would be named a 1-3-5-7 Hexany. The master set is treated like a chord and is labelled with the suffix –ad, such as a tetrad, pentad, or hexad to represent four-, five-, or six-tone chords respectively. The harmonics of the master set are arranged on the points of a geometric figure, such as a tetrahedron, and its reciprocal chord is arranged on an inverted version of the same shape, such as an upside-down tetrahedron. These geometric figures form the template for the lattice of the CPS. [TN]

Moments of Symmetry (MOS)
A linear scale “in which every interval except for the period comes in two sizes” (Xenharmonic Wiki 2012). Wilson coined the term to describe those scales resulting from a chain of intervals that produce two (and not three) different-sized intervals. These intervals are designated as the small (s) and large (L) intervals. The relative number of s and L intervals in an MOS is co-prime, i.e. they share no common factors other than 1. Fractions are used to represent MOS scales: the numerator shows the size of the generator, and the denominator shows the number of notes in the scale. The numerator and denominator of fractions representing MOS are also co-prime. Wilson organizes these fractions hierarchically on the Scale Tree. MOS are not only scales in their own right but also provide a framework or template for constructing a family of Secondary MOS scales. [TN]

A Moment of Symmetry is a scale that consists of: 
1. A generator (of any size, for example a 3/2 or a fifth which is repeatedly superimposed but reduced within
2. An Interval of Equivalence commonly called a period (of any size, for example most commonly an octave).
3. A Moment of Symmetry is formed where each scale degree or scale unit size will be represented by no more than two sizes and two sizes only (Large = L and small = s).
4.The relative number of L and s intervals is coprime, i.e. they share no common factors other than 1.
5. The numerator (generator) and denominator (period) representing MOS are also co-prime.
Wilson cited in Grady [KG]

Noble MOS
A special class of MOS scales generated from noble numbers which are irrational numbers “whose continued-fraction expansions end in infinitely many 1’s” (Schroeder 2009, p. 387). Wilson referred to these scales as “gold scales” because the ratios of L to s intervals eventually converge on the value Phi or the golden ratio. Noble MOS appear in the Scale Tree, Golden Horograms, Rabbit Sequence and Straight Line Patterns of the Scale Tree. [TN]

Rabbit Sequence
A number series that was originally suggested by Fibonacci to predict “how fast rabbits could breed in ideal circumstances” (Knott 2010) by showing how baby rabbits grow into adults and eventually have their own babies from one generation to the next. Wilson applies the same principles of the Rabbit Sequence to MOS scales in order to show how small (s) intervals become large (L) intervals, and then divide into s and L between generations. The Rabbit Sequence predicts how many L and s intervals are found in each MOS and in what order, and it also restates the fact that the relative numbers of L and s intervals in an MOS are co-prime. [TN]

Scale
An ordered set of pitches which in Wilson’s theoretical framework is conceived melodically in contrast to a harmonic structure. [TN]

Scale Tree
A numerical configuration devised by Wilson to represent the full spectrum of MOS scales on an infinite map of hierarchically ordered fractions. Like the Stern-Brocot tree to which it is related, fractions on the Scale Tree always appear in simplest form and in ascending order from left to right. The tree can be extended indefinitely to encompass all positive fractions but each fraction will appear only once. The Scale Tree underpins much of Wilson’s work and is essential to understanding his tuning theories, including his keyboard mapping system. Ratios on the Scale Tree represent MOS scales as well as different types of keyboards in Wilson’s system. Each of these scales and their associated keyboards are arranged into nested families that are organized hierarchically on the Scale Tree. It is often assumed that the Scale Tree is identical to the Stern-Brocot tree but Wilson includes extra information such as the decimal values of fractions as well as various noble numbers upon which the zigzag patterns on the branches of the Scale Tree eventually converge. Furthermore, Wilson explores a variety of ways to reseed the Scale Tree from different fractions. [TN]

Secondary Moments of Symmetry
A subset of scales derived from a ‘parent’ MOS to form a family of variations. For example, Wilson’s paper, “The Tanabe Cycle” (1998), shows how various 5-tone scales found in Japanese music can be derived from a 7-tone MOS scale. Secondary MOS are sometimes also referred to as “sub-moments”, “nested MOS” or “bi-level MOS”. [TN]

Straight Line Patterns
A diagram representing the infinite range of MOS scales that can be generated from an interval that varies in size over a continuum. The generating interval (which can be a rational or irrational value such as a noble number) and its superimpositions are shown as diagonal lines. Any line that can be drawn horizontally to intersect the diagonals is potentially an MOS. The Straight Line Patterns specifically show Noble MOS and ET scales as horizontal lines that correspond to the Scale Tree. The diagram is drawn to scale so that the positions of ratios from the Scale Tree are shown relative to their size. An important feature of the Straight Line Patterns is that they show the L and s interval patterns of MOS scales changing gradually until they invert across an ET line to become their opposite, i.e. an L interval becomes an s, and vice versa. [TN]

Superparticular
A term describing any ratio in which the difference between the numerator and the denominator is equal to 1 (Nelson n.d.), in other words, (n + 1) / n (Chalmers 1993, p. 206). [TN]

Contributors

[TN] Terumi Narushima

[KG] Kraig Grady

References

Bogomolny, A 2012, Farey Series and Euclid's Algorithm, from Interactive Mathematics Miscellany and Puzzles, viewed 6 Feb 2006, http://www.cut-the-knot.org/blue/Farey.shtml.

Chalmers, J 1993, Divisions of the Tetrachord, Frog Peak Music, Lebanon NH.

Grady, K 1986, ‘Combination-Product Set Patterns’, Xenharmonikôn, vol. 9, 4 pages.

--- 1999, ‘Re: CS’, Alternate Tunings Mailing List, 4 Oct, viewed 23 Nov 2012, http://launch.groups.yahoo.com/group/tuning/message/5244.

Hero, B 2012, International Lambdoma Research Institute, viewed 3 Feb 2006, http://www.lambdoma.com.

Knott, R 2010, The Fibonacci Numbers and Golden Section in Nature – 1, viewed 7 Mar 2012, http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html.

Nelson, K n.d., Music-Science Glossary, viewed 6 Feb 2006, http://www.music-science.net/Glossary.html.

Novaro, A 1927, Teoría de la Musica: Sistema Natural Base del Natural-Aproximado, Mexico, D.F.

Schroeder, MR 2009, Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, Springer, Berlin, Heidelberg.

Wilson, E 1969-1970, Some Diamond Lattices (and Blanks), The Wilson Archives, viewed 31 Mar 2013, http://anaphoria.com/diamond.pdf.

--- 1989, ‘D’alessandro, Like a Hurricane’, Xenharmonikôn, vol. 12, pp. 1-38.

--- 1994, The Gral Keyboard Guide, The Wilson Archives, viewed 21 Mar 2010, http://anaphoria.com/gralkeyboard.pdf.

--- 1996, So-Called Farey Series, Extended 0/1 to 1/0 (Full Set of Gear Ratios), and Lambdoma, The Wilson Archives, viewed 6 Feb 2006, http://anaphoria.com/lamb.pdf.

--- 1998, The Tanabe Cycle and Parallelogram from the Tanabe Cycle, The Wilson Archives, pp. 11-12, viewed 22 Aug 2008, http://anaphoria.com/mos.pdf.

--- 2000, Diophantine Triplets of Temperament Derived Intervals, The Wilson Archives, viewed 6 Jan 2016, http://anaphoria.com/DiophantineTripletsTEMPER.pdf.

Xenharmonic Wiki 2012, MOS Scales, viewed 7 Oct 2012, http://xenharmonic.wikispaces.com/MOSScales.

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