"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 `{q _{1}, ..., q_{n}}` are

[Gene Ward Smith, Yahoo tuning-math group Message 6955]

. . . . . . . . .

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.)

name | factors | monzo | ratio | ~cents |

syntonic-comma | 2^{-4} 3^{4} 5^{-1} |
[-4 4, -1> | ^{81} / _{80} |
21.5062896 |

enharmonic-diesis | 2^{7} 3^{0} 5^{-3} |
[ 7 0, -3> | ^{128} / _{125} |
41.05885841 |

name | factors | monzo | ratio | ~cents |

enharmonic-diesis | 2^{7} 3^{0} 5^{-3} |
[ 7 0, -3> | ^{128} / _{125} |
41.05885841 |

maximal-diesis (porcupine-comma) | 2^{1} 3^{-5} 5^{3} |
[ 1 -5, 3> | ^{250} / _{243} |
49.16613727 |

name | factors | monzo | ratio | ~cents |

syntonic-comma | 2^{-4} 3^{4} 5^{-1} |
[-4 4, -1> | ^{81} / _{80} |
21.5062896 |

magic-comma | 2^{-10} 3^{-1} 5^{5} |
[-10 -1, 5> | ^{3,125} / _{3,072} |
29.61356846 |

name | factors | monzo | ratio | ~cents |

diaschisma | 2^{11} 3^{-4} 5^{-2} |
[11 -4, -2> | ^{2,048} / _{2,025} |
19.55256881 |

maximal-diesis (porcupine-comma) | 2^{1} 3^{-5} 5^{3} |
[ 1 -5, 3> | ^{250} / _{243} |
49.16613727 |

name | factors | monzo | ratio | ~cents |

würschmidt-comma | 2^{17} 3^{1} 5^{-8} |
[17 1, -8> | ^{393,216} / _{390,625} |
11.44528995 |

syntonic-comma | 2^{-4} 3^{4} 5^{-1} |
[-4 4, -1> | ^{81} / _{80} |
21.5062896 |

name | factors | monzo | ratio | ~cents |

kleisma | 2^{-6} 3^{-5} 5^{6} |
[-6 -5, 6> | ^{15,625} / _{15,552} |
8.107278862 |

diaschisma | 2^{11} 3^{-4} 5^{-2} |
[11 -4, -2> | ^{2,048} / _{2,025} |
19.55256881 |

name | factors | monzo | ratio | ~cents |

minimal-diesis | 2^{5} 3^{-9} 5^{4} |
[5 -9 4> | ^{20,000} / _{19,683} |
27.65984767085246 |

magic-comma | 2^{-10} 3^{-1} 5^{5} |
[-10 -1, 5> | ^{3,125} / _{3,072} |
29.61356846 |

name | factors | monzo | ratio | ~cents |

skhisma | 2^{-15} 3^{8} 5^{1} |
[-15, 8, 1> | ^{32,805} / _{32,768} |
1.953720788 |

kleisma | 2^{-6} 3^{-5} 5^{6} |
[-6 -5, 6> | ^{15,625} / _{15,552} |
8.107278862 |

name | factors | monzo | ratio | ~cents |

syntonic-comma | 2^{-4} 3^{4} 5^{-1} |
[-4 4, -1> | ^{81} / _{80} |
21.5062896 |

(unnamed?) | 2^{27} 3^{5} 5^{-15} |
[ 27, 5, -15> | ^{32,614,907,904} / _{30,517,578,125} |
115.069296354415 |

name | factors | monzo | ratio | ~cents |

skhisma | 2^{-15} 3^{8} 5^{1} |
[-15, 8, 1> | ^{32,805} / _{32,768} |
1.953720788 |

sensipent-comma | 2^{2} 3^{9} 5^{-7} |
[ 2, 9, -7> | ^{78,732} / _{78,125} |
13.39901073 |

name | factors | monzo | ratio | ~cents |

majvam | 2^{40} 3^{7} 5^{-22} |
[40, 7, -22> | ^{2,404,631,929,946,112} / _{2,384,185,791,015,625} |
14.78330103134586 |

syntonic-comma | 2^{-4} 3^{4} 5^{-1} |
[-4 4, -1> | ^{81} / _{80} |
21.5062896 |

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.

name | factors | monzo | ratio | ~cents |

skhisma | 2^{-15} 3^{8} 5^{1} |
[-15, 8, 1> | ^{32,805} / _{32,768} |
1.953720788 |

parakleisma | 2^{8} 3^{14} 5^{-13} |
[ 8, 14, -13> | ^{1,224,440,064} / _{1,220,703,125} |
5.291731873 |

name | factors | monzo | ratio | ~cents |

skhisma | 2^{-15} 3^{8} 5^{1} |
[-15, 8, 1> | ^{32,805} / _{32,768} |
1.953720788 |

19-tone-comma | 2^{-14} 3^{-19} 5^{19} |
[ -14, -19, 19> | ^{19,073,486,328,125} / _{19,042,491,875,328} |
2.8155469895004357 |

name | factors | monzo | ratio | ~cents |

vishnuzma (semisuper-comma) | 2^{23} 3^{6} 5^{-14} |
[ 23, 6, -14> | ^{6,115,295,232} / _{6,103,515,625} |
3.3380110846370235 |

vulture-comma | 2^{24} 3^{-21} 5^{4} |
[24, -21, 4> | ^{10,485,760,000} / _{10,460,353,203} |
4.199837286203549 |

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}).

name | factors | monzo | ratio | ~cents |

(unnamed?) | 2^{22} 3^{33} 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} 3^{8} 5^{1} |
[-15, 8, 1> | ^{32,805} / _{32,768} |
1.953720788 |

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

name | factors | monzo | ratio | ~cents |

ennealimma | 2^{1} 3^{-27} 5^{18} |
[ 1, -27, 18> | ^{7,629,394,531,250} / _{7,625,597,484,987} |
0.861826202 |

lunama (hemithirds-comma) | 2^{38} 3^{-2} 5^{-15} |
[38, -2, -15> | ^{274,877,906,944} / _{274,658,203,125} |
1.384290297 |

name | factors | monzo | ratio | ~cents |

lunama (hemithirds-comma) | 2^{38} 3^{-2} 5^{-15} |
[38, -2, -15> | ^{274,877,906,944} / _{274,658,203,125} |
1.384290297 |

minortonic-comma | 2^{-16} 3^{35} 5^{-17} |
[-16, 35, -17> | ^{50,031,545,098,999,707} / _{50,000,000,000,000,000} |
1.091894586 |

name | factors | monzo | ratio | ~cents |

kwazy | 2^{-53} 3^{10} 5^{16} |
[-53, 10, 16> | ^{9,010,162,353,515,625} / _{9,007,199,254,740,992} |
0.569430491 |

ennealimma | 2^{1} 3^{-27} 5^{18} |
[ 1, -27, 18> | ^{7,629,394,531,250} / _{7,625,597,484,987} |
0.861826202 |

name | factors | monzo | ratio | ~cents |

minortonic-comma | 2^{-16} 3^{35} 5^{-17} |
[-16, 35, -17> | ^{50,031,545,098,999,707} / _{50,000,000,000,000,000} |
1.091894586 |

kwazy | 2^{-53} 3^{10} 5^{16} |
[-53, 10, 16> | ^{9,010,162,353,515,625} / _{9,007,199,254,740,992} |
0.569430491 |

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}).

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 | 2^{37} 3^{25} 5^{-33} |
2^{54} 3^{2} 5^{-37} |

1,783 ET / EDO | 2^{54} 3^{2} 5^{-37} |
2^{-90} 3^{-15} 5^{49} |

2,513 ET / EDO | 2^{-107} 3^{47} 5^{14} |
2^{-17} 3^{62} 5^{-35} |

4,296 ET / EDO | 2^{71} 3^{37} 5^{-99} |
2^{-90} 3^{-15} 5^{49} |

6,809 ET / EDO | 2^{-178} 3^{146} 5^{23} |
2^{-90} 3^{-15} 5^{49} |

16,572 ET / EDO | 2^{92} 3^{191} 5^{-170} |
2^{161} 3^{-81} 5^{-12} |

20,868 ET / EDO | 2^{161} 3^{-81} 5^{-12} |
2^{21} 3^{290} 5^{-207} |

25,164 ET / EDO | 2^{-111} 3^{-305} 5^{256} |
2^{161} 3^{-81} 5^{-12} |

52,841 ET / EDO | 2^{21} 3^{290} 5^{-207} |
2^{-412} 3^{153} 5^{73} |

73,709 ET / EDO | 2^{21} 3^{290} 5^{-207} |
2^{-573} 3^{237} 5^{85} |

78,005 ET / EDO | 2^{140} 3^{195} 5^{-374} |
2^{-573} 3^{237} 5^{85} |

name | factors | monzo | ratio | ~cents |

septimal-kleisma | 2^{-5} 3^{2} 5^{2} 7^{-1} |
[ -5, 2, 2, -1> | ^{225} / _{224} |
7.711522991319706 |

orwellisma | 2^{6} 3^{3} 5^{-1} 7^{-3} |
[ 6, 3, -1, -3> | ^{1,728} / _{1,715} |
13.07356932395248 |

syntonic-comma | 2^{-4} 3^{4} 5^{-1} 7^{0} |
[-4, 4, -1, 0> | ^{81} / _{80} |
21.50628959671478 |

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.

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

name | factors | monzo | ratio | ~cents |

kalisma | 2^{-3} 3^{4} 5^{-2} 7^{-2} 11^{2} |
[-3 4 -2 -2 2> | ^{9,801} / _{9,800} |
0.1766475231436913 |

lehmerisma | 2^{-4} 3^{-3} 5^{2} 7^{-1} 11^{2} |
[-4 -3 2 -1 2> | ^{3,025} / _{3,024} |
0.5724033938959937 |

breedsma | 2^{-5} 3^{-1} 5^{-2} 7^{4} 11^{0} |
[ -5 -1 -2 4 0> | ^{2,401} / _{2,400} |
0.7211972814427758 |

vishdel | 2^{9} 3^{-2} 5^{-4} 7^{0} 11^{1} |
[9 -2 -4 0 1> | ^{5,632} / _{5,625} |
2.1530851746424933 |

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

. . . . . . . . .

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.

a&b temperament [a&b are numbers]

55-edo (comma) (Mozart's tuning)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

a&b temperament [a&b are numbers]

apotome (Greek interval)

aristoxenean (temperament family)

atomic (temperament family)

augmented / diesic (temperament family)

augmented-2nd / aug-2 / #2 (interval)

augmented-4th / aug-4 / #4 (interval)

augmented-5th / aug-5 / #5 (interval)

augmented-6th / aug-6 / #6 (interval)

augmented-9th / aug-9 / #9 (interval)

blackjack (tuning)

cent / ¢ (unit of interval measurement)

centitone / iring (unit of interval measurement)

chromatic-semitone / augmented-prime (interval)

daseian (musical notation)

dekamu / 10mu (MIDI-unit)

diapason (Greek interval)

diapente (Greek interval)

diatessaron (Greek interval)

diatonic semitone (minor-2nd) (interval)

diesic (temperament family)

diezeugmenon (Greek tetrachord)

diminished-5th / dim5 / -5 / b5 (interval)

diminished-7th / dim7 / o7 (interval)

doamu / 2mu (MIDI-unit)

dodekamu / 12mu (MIDI-unit)

dominant-7th (dom-7, x7) (chord)

dorian (mode)

eleventh / 11th (interval)

enamu / 1mu (MIDI-unit)

endekamu / 11mu (MIDI-unit)

enharmonic semitone (interval)

ennealimmal (temperament family)

enneamu / 9mu (MIDI-unit)

farab (unit of interval measurement)

fifth / 5th (interval)

flu (unit of interval measurement)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

fourth / 4th (interval)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

grad (unit of interval measurement)

hexamu / 6mu (MIDI-unit)

Hurrian Hymn (Monzo reconstruction)

hypate (Greek note)

hypaton (Greek tetrachord)

hyperbolaion / hyperboleon (Greek tetrachord)

hypophrygian (Greek mode)

imperfect (interval quality)

iring / centitone (unit of interval measurement)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

jot (unit of interval measurement)

JustMusic: A New Harmony [Monzo's book]

JustMusic prime-factor notation [Monzo essay]

kwazy (temperament family)

leimma / limma (Greek interval)

lichanos (Greek note)

limma / leimma (Greek interval)

locrian (mode)

lydian (mode)

magic (temperament family)

Mahler 7th/1 [Monzo score and analysis]

marvel (temperament family)

meantone (temperament family)

mem (unit of interval measurement)

meride (unit of interval measurement)

mese (Greek note)

meson (Greek tetrachord)

millioctave / m8ve (unit of interval measurement)

mina (unit of interval measurement)

minerva (temperament family)

miracle (temperament family)

mixolydian (mode)

monzo (prime-exponent vector)

Monzo, Joe (music-theorist)

morion / moria (unit of interval measurement)

mutt (temperament family)

mystery (temperament family)

octamu / oktamu / 8mu (MIDI-unit)

octave (interval)

oktamu / octamu / 8mu (MIDI-unit)

orwell (temperament family)

p4, perfect 4th, perfect fourth (interval)

p5, perfect 5th, perfect fifth (interval)

pantonality of Schoenberg [Monzo essay]

paramese (Greek note)

paranete (Greek note)

parhypate (Greek note)

pentamu / 5mu (MIDI-unit)

prime-factor notation (JustMusic) [Monzo essay]

proslambanomenos (Greek note)

savart (unit of interval measurement)

schismic / skhismic (temperament family)

Schoenberg's pantonality [Monzo essay]

second / 2nd (interval)

semisixths (temperament family)

semitone (unit of interval measurement)

seventh / 7th (interval)

sixth / 6th (interval)

sk (unit of interval measurement)

skhismic / schismic (temperament family)

sruti tuning [Monzo essay]

studloco (tuning)

subminor 3rd (interval)

Sumerian tuning [speculations by Monzo]

synemmenon (Greek tetrachord)

temperament-unit / tu (unit of interval measurement)

tenth / 10th (interval)

tetrachord-theory tutorial [by Monzo]

tetradekamu / 14mu (MIDI-unit)

tetramu / 4mu (MIDI-unit)

third / 3rd (interval)

thirteenth / 13th (interval)

tina (unit of interval measurement)

tone (interval, and other definitions)

tredek (unit of interval measurement)

triamu / 3mu (MIDI-unit)

tridekamu / 13mu (MIDI-unit)

trihemitone (Greek interval)

trite (Greek note)

tu / temperament-unit (unit of interval measurement)

Türk sent (unit of interval measurement)

twelfth / 12th (interval)

whole-tone (interval)

woolhouse-unit (unit of interval measurement)