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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, ..., q_n} 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 q_i is n. Then {q₁, ..., q_n} is a basis for a lattice L, consisting of every positive rational number of the form q₁^e₁ ... q₁^e_n where the e_i 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 t_i > 1 inductively as the shortest number in L independent of {t₁, ... t_i-1} and such that {t₁, ..., ti} can be extended to be a basis for L. In this way we obtain {t₁, ..., t_n}, the TM reduced basis of L. See this definition of Minkowski reduction and definitions by Gene Ward Smith.

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

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

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

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

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

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

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

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

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

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