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relationship between the points it connects, no matter where it occurs in the lattice. Repeatedly adding the same vector (repeatedly stacking the same interval) moves you further in the same direction. Lattices in just intonation (limited to intervals comprising primes, their powers, and their products) are theoretically infinite (because no power of any prime equals any power of another prime). However, lattices are sometimes also used to notate limited subsets that are particularly interesting (such as an
Eikosany illustrated further below or the various ways to extract particular scale shapes from a larger lattice).
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241:(each vector representing a ratio of 1/n or n/1 where n is a prime) has a unique spacing, avoiding clashes even when generating lattices of multidimensional, harmonically based structure. Wilson would commonly use 10-squares-to-the-inch graph paper. That way, he had room to notate both ratios and often the scale degree, which explains why he didn't use a template where all the numbers where divided by 2. The scale degree always followed a period or dot to separate it from the ratios.
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has made significant headway with developing lattices than can represent higher limit harmonics, meaning more than 2 dimensions, while displaying them in 2 dimensions. Here is a template he used to generate what he called an "Euler" lattice after where he drew his inspiration. Each prime harmonic
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The points in a lattice represent pitch classes (or pitches if octaves are represented), and the connectors in a lattice represent the intervals between them. The connecting lines in a lattice display intervals as vectors, so that a line of the same length and angle always has the same intervalic
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on one dimension and a series of major thirds on those fifths in the second (horizontal and vertical), with the option of imagining depth to model octaves:
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A lattice showing Erv Wilson's
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is the highest prime number used in the ratios that define the intervals used by a tuning.
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with respect to some other point on the lattice). The lattice can be two-, three-, or
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including equal temperament (12-tone equal temperament = 2 (or 2), 24-tet = 2,
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of 2 and 3), is represented through a two-dimensional lattice (or, given
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of points in a periodic multidimensional pattern. Each point on the
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141:. Many multi-dimensional higher-limit tunings have been mapped by
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Tonnetz, pitches are connected by lines if they are separated by
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399:, edited by Bob Gilmore. Urbana: University of Illinois Press.
203:| | | | | | | | (Db—)-Ab-—-Eb—--Bb
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187:| | | | | | | | F----C----G----D =
129:. Musical intervals in just intonation are related to those in
84:-dimensional, with each dimension corresponding to a different
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The Music of James Tenney, Volume 1: Contexts and
Paradigms
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440:(University of Illinois Press, 2021), 155-65.</ref>
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Johnston, Ben (2006). "Rational
Structure in Music",
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60:"is a way of modeling the tuning relationships of a
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225:Wilson template for mapping higher limit systems
395:Gilmore, Bob (2006). "Introduction", p.xviii,
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421:"Maximum Clarity" and Other Writings on Music
397:"Maximum Clarity" and Other Writings on Music
121:and the tuning systems of composer-theorists
334:7-limit just intonation (3/2, 5/4, and 7/4)
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107:Examples of musical lattices include the
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298:5-limit just intonation (3/2 and 5/4)
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96:a lattice may be referred to as a
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72:corresponds to a ratio (i.e., a
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369:-limit tuning are equal to the
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283:{\displaystyle {\sqrt{4}}}
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514:Fokker periodicity blocks
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139:Fokker periodicity blocks
92:]." When listed in a
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318:{\displaystyle \varphi }
251:Pythagorean tuning (3/2)
371:prime-counting function
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434:Wannamaker, Robert,
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255:Musical temperaments
529:Pitch constellation
451:The Wilson Archives
534:Spiral array model
331:Three dimensional
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162:octave equivalence
154:Pythagorean tuning
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524:Pitch class space
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64:system. It is an
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117:(1739) and
94:spreadsheet
90:pitch class
30:major third
26:minor third
568:Categories
373:minus one.
244:Examples:
238:Erv Wilson
143:Erv Wilson
313:φ
341:See also
326:and 3/2)
78:interval
76:, or an
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381:Sources
110:Tonnetz
70:lattice
58:lattice
20:On the
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152:Thus
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115:Euler
74:pitch
66:array
28:(/),
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
