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Encyclopedia of Microtonal Music Theory

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TM-Reduced Lattice Basis

"T" stands for Tenney, "M" for Minkowski. A method for reducing the basis of a lattice. First we need to define Tenney height. if p / q is a positive rational number in reduced form, then the Tenney height is TH(p / q) = p · q. Now suppose {q1, ..., qn} are n multiplicatively linearly independent positive rational numbers. Linear independence can be equated, for instance, with the condition that rank of the matrix whose rows are the monzos for qi is n. Then {q1, ..., qn} is a basis for a lattice L, consisting of every positive rational number of the form q1e1 ... q1en where the ei are integers and where the log of the Tenney height defines a norm. Let t1 > 1 be the shortest (in terms of Tenney height) rational number in L greater than 1. Define ti > 1 inductively as the shortest number in L independent of {t1, ... ti-1} and such that {t1, ..., ti} can be extended to be a basis for L. In this way we obtain {t1, ..., tn}, the TM reduced basis of L. See this definition of Minkowski reduction and definitions by Gene Ward Smith.

[Gene Ward Smith, Yahoo tuning-math group Message 6955]
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5-Limit Base Examples

Here are some of the 5-limit lattice bases and the resulting periodicity blocks, rendered in Tonalsoft™ Tonescape™. The thickest pink lines are the two unison-vectors which form the lattice basis, the cubes represent the exponents of the prime-factors 3 and 5, which designate the ratios in the just intonation version of the tuning, and the thin pink lines connect the ratios together, showing the periodicity-block structure. Note that in the rectangular lattices the pink connectors represent only the 3 and 5 axes, while in the triangular lattices there is also a third connector which represents an axis containing both 3 and 5. Note also that the toroidal lattices cannot show the unison-vectors, because in that geometry the unison-vectors are reduced to a point.

12 ET/EDO

Enharmonic Diesis 27 30 5-3 [ 7, 0, -3> 128 / 125 41.05885841
Syntonic Comma 2-4 34 5-1 [-4 4, -1> 81 / 80 21.5062896

rectangular

Rectangular Lattice 12EDO Rectangular Lattice 12EDO

triangular

Triangular Lattice 12EDO Triangular Lattice 12EDO

toroidal

Toroidal Lattice 12EDO Toroidal Lattice 12EDO

15 ET/EDO

Maximal Diesis 21 3-5 53 [ 1 -5, 3> 250 / 243 49.16613727
Enharmonic Diesis 27 30 5-3 [ 7, 0, -3> 128 / 125 41.05885841

rectangular

Rectangular Lattice 15EDO Rectangular Lattice 15EDO

triangular

Triangular Lattice 15EDO Triangular Lattice 15EDO

toroidal

Toroidal Lattice 15EDO Toroidal Lattice 15EDO

19 ET/EDO

Magic Comma 2-10 3-1 55 [-10 -1, 5> 3125 / 3072 29.61356846
Syntonic Comma 2-4 34 5-1 [-4 4, -1> 81 / 80 21.5062896

rectangular

Rectangular Lattice 19EDO Rectangular Lattice 19EDO

triangular

Triangular Lattice 19EDO Triangular Lattice 19EDO

toroidal

Toroidal Lattice 19EDO Toroidal Lattice 19EDO

31 ET/EDO

Syntonic Comma 2-4 34 5-1 [-4 4, -1> 81 / 80 21.5062896
  217 31 5-8 [17 1, -8> 393,216 / 390,625 11.44528995

rectangular

Rectangular Lattice 31EDO Rectangular Lattice 31EDO

triangular

Triangular Lattice 31EDO Triangular Lattice 31EDO

toroidal

Toroidal Lattice 31EDO Toroidal Lattice 31EDO

34 ET/EDO

Diaschisma 211 3-4 5-2 [11 -4, -2> 2048 / 2025 19.55256881
Kleisma 2-6 3-5 56 [-6 -5, 6> 15,625 / 15,552 8.107278862

rectangular

Rectangular Lattice 34EDO Rectangular Lattice 34EDO

triangular

Triangular Lattice 34EDO Triangular Lattice 34EDO

toroidal

Toroidal Lattice 34EDO Toroidal Lattice 34EDO

53 ET/EDO

Kleisma 2-6 3-5 56 [-6 -5, 6> 15,625 / 15,552 8.107278862
Schisma 2-15 38 51 [-15, 8, 1> 32,805 / 32,768 1.953720788

rectangular

Rectangular Lattice 53-edo, letter notation Rectangular Lattice 53-edo, 53-edo degree notation

triangular

Triangular Lattice 53-edo, letter notation Triangular Lattice 53-edo, 53-edo degree notation

toroidal

Toroidal Lattice 53-edo, letter notation Toroidal Lattice 53-edo, 53-edo degree notation

65 ET/EDO

  22 39 5-7 [ 2, 9, -7> 78,732 / 78,125 13.39901073
Schisma 2-15 38 51 [-15, 8, 1> 32,805 / 32,768 1.953720788

118 ET/EDO

  28 314 5-13 [ 8, 14, -13> 1,792,620 / 1,787,149 5.291731873
Schisma 2-15 38 51 [-15, 8, 1> 32,805 / 32,768 1.953720788

rectangular

Rectangular Lattice 118EDO

triangular

Triangular Lattice 118EDO

toroidal

Toroidal Lattice 118EDO

171 ET/EDO

Schisma 2-15 38 51 [-15, 8, 1> 32,805 / 32,768 1.953720788
  21 3-27 518 [ 1, -27, 18> 4,711,802 / 4,711,802 0.861826202

rectangular

Rectangular Lattice 171EDO

triangular

Triangular Lattice 171EDO

toroidal

Toroidal Lattice 171EDO

289 ET/EDO

  27 341 5-31 [ 7, 41, -31> 5,146,069 / 5,132,918 4.429905671
Schisma 2-15 38 51 [-15, 8, 1> 32,805 / 32,768 1.953720788

441 ET/EDO

  238 3-2 5-15 [38, -2, -15> 6,719,816 / 6,714,445 1.384290297
  21 3-27 518 [ 1, -27, 18> 4,711,802 / 4,711,802 0.861826202

559 ET/EDO

  238 3-2 5-15 [38, -2, -15> 6,719,816 / 6,714,445 1.384290297
  2-16 335 5-17 [-16, 35, -17> 6,437,705 / 6,433,646 1.091894586

612 ET/EDO

  21 3-27 518 [ 1, -27, 18> 4,711,802 / 4,711,802 0.861826202
  2-53 310 516 [-53, 10, 16> 4,758,837 / 4,757,272 0.569430491

toroidal top view

Toroidal Top View 612EDO

toroidal close up

Toroidal Close Up 612EDO

toroidal side-view close up

Toroidal Close Up 612EDO

When the cardinality of the EDO gets this high, it is difficult to see a difference in the geometry of their toruses, so graphics for the following are omitted.

730 ET/EDO

  2-16 335 5-17 [-16, 35, -17> 6,437,705 / 6,433,646 1.091894586
  2-53 310 516 [-53, 10, 16> 4,758,837 / 4,757,272 0.569430491

1,171 ET/EDO and Above

additional 5-limit lattice bases
1,171 ET / EDO 237 325 5-33 254 32 5-37
1,783 ET / EDO 254 32 5-37 2-90 3-15 549
2,513 ET / EDO 2-107 347 514 2-17 362 5-35
4,296 ET / EDO 271 337 5-99 2-90 3-15 549
6,809 ET / EDO 2-178 3146 523 2-90 3-15 549
16,572 ET / EDO 292 3191 5-170 2161 3-81 5-12
20,868 ET / EDO 2161 3-81 5-12 221 3290 5-207
25,164 ET / EDO 2-111 3-305 5256 2161 3-81 5-12
52,841 ET / EDO 221 3290 5-207 2-412 3153 573
73,709 ET / EDO 221 3290 5-207 2-573 3237 585
78,005 ET / EDO 2140 3195 5-374 2-573 3237 585
TM-basis 31-et 7-limit: 126/125, 81/80, 1029/1024 no lattice points.gif
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