TM-Reduced Lattice Basis

"T" stands for Tenney [James Tenney, composer and music-theorist], "M" for Minkowski [Hermann Minkowski, mathematician]. 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]
. . . . . . . . .

5-Limit Base Examples

Here are some of the 5-limit TM-reduced lattice bases and the resulting periodicity blocks, rendered in Tonalsoft™ Tonescape™ software. The purple lines are the two unison-vectors which form the lattice basis, the spheres represent the exponents of the prime-factors 3 and 5, which designate the ratios in the just-intonation version of the tuning, the pink lines connect the ratios together along the 3-axes, and the green lines connect the ratios along the 5-axes, 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: in these images these are also colored green. Note also that the toroidal lattices cannot show the unison-vectors, because in that geometry the unison-vectors are reduced to a point. (Two higher-dimensional examples are also shown at the bottom of this page.)

12 ET/EDO

 name factors monzo ratio ~cents syntonic-comma 2-4 34 5-1 [-4 4, -1> 81 / 80 21.5062896 enharmonic-diesis 27 30 5-3 [ 7 0, -3> 128 / 125 41.05885841

15 ET/EDO

 name factors monzo ratio ~cents enharmonic-diesis 27 30 5-3 [ 7 0, -3> 128 / 125 41.05885841 maximal-diesis (porcupine-comma) 21 3-5 53 [ 1 -5, 3> 250 / 243 49.16613727

19 ET/EDO

 name factors monzo ratio ~cents syntonic-comma 2-4 34 5-1 [-4 4, -1> 81 / 80 21.5062896 magic-comma 2-10 3-1 55 [-10 -1, 5> 3,125 / 3,072 29.61356846

22 ET/EDO

 name factors monzo ratio ~cents diaschisma 211 3-4 5-2 [11 -4, -2> 2,048 / 2,025 19.55256881 maximal-diesis (porcupine-comma) 21 3-5 53 [ 1 -5, 3> 250 / 243 49.16613727

31 ET/EDO

 name factors monzo ratio ~cents würschmidt-comma 217 31 5-8 [17 1, -8> 393,216 / 390,625 11.44528995 syntonic-comma 2-4 34 5-1 [-4 4, -1> 81 / 80 21.5062896

34 ET/EDO

 name factors monzo ratio ~cents kleisma 2-6 3-5 56 [-6 -5, 6> 15,625 / 15,552 8.107278862 diaschisma 211 3-4 5-2 [11 -4, -2> 2,048 / 2,025 19.55256881

34 ET/EDO

 name factors monzo ratio ~cents minimal-diesis 25 3-9 54 [5 -9 4> 20,000 / 19,683 27.65984767085246 magic-comma 2-10 3-1 55 [-10 -1, 5> 3,125 / 3,072 29.61356846

53 ET/EDO

 name factors monzo ratio ~cents skhisma 2-15 38 51 [-15, 8, 1> 32,805 / 32,768 1.953720788 kleisma 2-6 3-5 56 [-6 -5, 6> 15,625 / 15,552 8.107278862

55 ET/EDO

 name factors monzo ratio ~cents syntonic-comma 2-4 34 5-1 [-4 4, -1> 81 / 80 21.5062896 (unnamed?) 227 35 5-15 [ 27, 5, -15> 32,614,907,904 / 30,517,578,125 115.069296354415

65 ET/EDO

 name factors monzo ratio ~cents skhisma 2-15 38 51 [-15, 8, 1> 32,805 / 32,768 1.953720788 sensipent-comma 22 39 5-7 [ 2, 9, -7> 78,732 / 78,125 13.39901073

81 ET/EDO

 name factors monzo ratio ~cents majvam 240 37 5-22 [40, 7, -22> 2,404,631,929,946,112 / 2,384,185,791,015,625 14.78330103134586 syntonic-comma 2-4 34 5-1 [-4 4, -1> 81 / 80 21.5062896

toroidal

For purposes of illustrating the lattice bases, the letter notation is largely irrelevant for cardinalities larger than 81-ed2, so all subsequent lattices will show notation only in the logarithmic ED2 degrees.

118 ET/EDO

 name factors monzo ratio ~cents skhisma 2-15 38 51 [-15, 8, 1> 32,805 / 32,768 1.953720788 parakleisma 28 314 5-13 [ 8, 14, -13> 1,224,440,064 / 1,220,703,125 5.291731873

171 ET/EDO

 name factors monzo ratio ~cents skhisma 2-15 38 51 [-15, 8, 1> 32,805 / 32,768 1.953720788 19-tone-comma 2-14 3-19 519 [ -14, -19, 19> 19,073,486,328,125 / 19,042,491,875,328 2.8155469895004357

270 ET/EDO

 name factors monzo ratio ~cents vishnuzma (semisuper-comma) 223 36 5-14 [ 23, 6, -14> 6,115,295,232 / 6,103,515,625 3.3380110846370235 vulture-comma 224 3-21 54 [24, -21, 4> 10,485,760,000 / 10,460,353,203 4.199837286203549

rectangular

Because of its immense usefulness as a simpler integer replacement for cents, the central portion of the rectangular lattice is also shown in a zoom-in view so that the logarithmic degree notation can be easily read for the ratios closest to 1/1 (n^0).

289 ET/EDO

 name factors monzo ratio ~cents (unnamed?) 222 333 5-32 [ 22 33, -32> 23,316,389,970,546,096,340,992 / 23,283,064,365,386,962,890,625 2.4761848830704634 skhisma 2-15 38 51 [-15, 8, 1> 32,805 / 32,768 1.953720788

toroidal

Below is an angled view of the 289-et torus, simply to show the beauty of the spiral formations.

441 ET/EDO

 name factors monzo ratio ~cents ennealimma 21 3-27 518 [ 1, -27, 18> 7,629,394,531,250 / 7,625,597,484,987 0.861826202 lunama (hemithirds-comma) 238 3-2 5-15 [38, -2, -15> 274,877,906,944 / 274,658,203,125 1.384290297

559 ET/EDO

 name factors monzo ratio ~cents lunama (hemithirds-comma) 238 3-2 5-15 [38, -2, -15> 274,877,906,944 / 274,658,203,125 1.384290297 minortonic-comma 2-16 335 5-17 [-16, 35, -17> 50,031,545,098,999,707 / 50,000,000,000,000,000 1.091894586

612 ET/EDO

 name factors monzo ratio ~cents kwazy 2-53 310 516 [-53, 10, 16> 9,010,162,353,515,625 / 9,007,199,254,740,992 0.569430491 ennealimma 21 3-27 518 [ 1, -27, 18> 7,629,394,531,250 / 7,625,597,484,987 0.861826202

730 ET/EDO

 name factors monzo ratio ~cents minortonic-comma 2-16 335 5-17 [-16, 35, -17> 50,031,545,098,999,707 / 50,000,000,000,000,000 1.091894586 kwazy 2-53 310 516 [-53, 10, 16> 9,010,162,353,515,625 / 9,007,199,254,740,992 0.569430491

rectangular

Because of its importance as a simpler integer replacement for cents in Woolhouse 1835, the central portion of the rectangular lattice is also shown in a zoom-in view so that the logarithmic degree notation can be easily read for the ratios closest to 1/1 (n^0).

toroidal

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. In addition, ratios are omitted from the list of unison-vectors, because of the size of the numbers in their numerators and denominators, with only the factors shown.

1,171 ET/EDO and Above

 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

31 ET/EDO in 3,5,7-space

 name factors monzo ratio ~cents septimal-kleisma 2-5 32 52 7-1 [ -5, 2, 2, -1> 225 / 224 7.711522991319706 orwellisma 26 33 5-1 7-3 [ 6, 3, -1, -3> 1,728 / 1,715 13.07356932395248 syntonic-comma 2-4 34 5-1 70 [-4, 4, -1, 0> 81 / 80 21.50628959671478

triangular

In the same way that 1-dimensional (linear) lattices require a second dimension in which to warp the lattice, turning the line into a spiral or circle, 2-dimensional lattices require a third dimension in which to warp the lattice, to form a helix or (as in all examples above) a torus. Lattices which already have 3 or more dimensions in their basic geometry (rectangular or triangular) also require an additional dimension in which to warp, but these higher dimensions are not easily perceivable visually. The result is what Brian McLaren and Joe Monzo have termed "crumpled-napkin" geometry; two views of the same example are shown below.

Example of a 4-dimensional TM-basis

270-edo (tredeks) is used as an illustration here because of its excellence as a simpler integer replacement for cents in the 11-limit.

270 ET/EDO

 name factors monzo ratio ~cents kalisma 2-3 34 5-2 7-2 112 [-3 4 -2 -2 2> 9,801 / 9,800 0.1766475231436913 lehmerisma 2-4 3-3 52 7-1 112 [-4 -3 2 -1 2> 3,025 / 3,024 0.5724033938959937 breedsma 2-5 3-1 5-2 74 110 [ -5 -1 -2 4 0> 2,401 / 2,400 0.7211972814427758 vishdel 29 3-2 5-4 70 111 [9 -2 -4 0 1> 5,632 / 5,625 2.1530851746424933

rectangular

Below are two views of the 5-dimensional crumpled-napkin geometry for this tonespace.

crumpled-napkin

. . . . . . . . .

The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by selecting the highest level of financial support that you can afford. Thank you.

 support level patron: \$50 USD savior: \$100 USD angel of tuning: \$500 USD microtonal god: \$1000 USD friend: \$25 USD donor: \$5 USD