[Gene Ward Smith, Yahoo tuning-math message 12906 (Sun Oct 16, 2005 9:27 pm)]

Given a projection matrix defining a tuning for a rank-two temperament, there will be a left eigenspace of dimension two corresponding to the eigenvalue 1. If the projection map gives pure octaves, one of the eigenvectors will be the octave, that is, |1 0 0 ... 0>. Ignoring 2, the rest of the eigenspace is defined as multiples of a single vector. Choosing a multiple such that the coefficients are integers, and therefore the result is a monzo, is always possible if the projection matrix is defined over the rationals; ie, has rational coefficients. In this case we can choose a monzo so that the corresponding interval `q` satisfies `1<q<2`. There will be two of these; we can make the choice unique by requiring that `1<q<sqrt(2)`, but I prefer choosing the one with the least Tenney height. We may call this, and the corresponding monzo, the eigenmonzo for the temperament tuning. Given a temperament, the eigenmonzo determines the tuning.

For least squares tunings, the projection matrix defined over the rationals and the corresponding eigenmonzo is readily obtained. The minimax tuning also gives a rational projection matrix, and if the two tunings are distinct, then all the other tunings can be defined in terms of `t*(least sqares) + (1-t)*(minimax)`, where "least squares" and "minimax" denote matrices. The minimax matrix, for someone who has a simplex algorithm over the rationals implemented, is easy to find. Those, like me, who only have a floating point version have alternative methods. One involves the fact that the set of possible eigenmonzos for a minimax tuning is finite.

For a minimax tuning with respect to a consonance set C, there will be two elements u and v of C such that the errors of u and v are equal, and maximal among all errors for C. Hence, of u/v and v/u reduced to the octave, the one with minimal Tenney height will be the eigenmonzo. This gives us a finite set of possibilities for any C, and we can find the minimax tuning simply by checking all the possible eigenmonzos. For the 7-limit, for instance, possible eigenmonzos for minimax tunings are the 21 values 49/48, 25/24, 21/20, 15/14, 35/32, 7/6, 25/21, 49/40, 5/4, 9/7, 21/16, 7/5, 35/24, 3/2, 49/30, 5/3, 7/4, 25/14, 9/5, 15/8, and 35/18.

. . . . . . . . .

[Gene Ward Smith, Yahoo tuning-math message 12907 (Sun Oct 16, 2005 10:23 pm)]

Decimal | 3/2 |

Dominant seventh | 7/5 |

Diminished | 21/16 |

Blackwood | 25/21 |

Augie | 7/5 |

Pajara | 35/24 |

Mavila | 7/4 |

Negri | 3/2 |

Kemun | 3/2 |

Augmented | 7/6 |

Godzilla | 5/4 |

Meantone | 5/4 |

Injera | 35/24 |

Doublewide | 3/2 |

Porcupine | 5/4 |

Unidec | 7/5 |

Superpyth | 7/5 |

Mothra | 5/4 |

Muggles | 7/6 |

Beatles | 5/3 |

Flattone | 7/5 |

Magic | 3/2 |

Myna | 3/2 |

Sensi | 7/4 |

Orwell | 7/5 |

Miracle | 5/3 |

Valentine | 7/6 |

Mothra | 5/4 |

Garibaldi | 7/6 |

Superkleismic | 7/5 |

Squares | 5/4 |

Semififths | 5/4 |

Diaschismic | 7/4 |

Octacot | 5/4 |

Tritonic | 7/6 |

Rodan | 7/5 |

Shrutar | 7/4 |

Catakleismic | 7/4 |

Hemiwuerschmidt | 3/2 |

Hemikleismic | 7/6 |

Hemithirds | 7/6 |

Wizard | 7/5 |

Waage | 35/24 |

Slender | 5/3 |

Amity | 7/5 |

Hemififths | 5/4 |

Pontiac | 7/5 |

Tertiaseptal | 7/5 |

Enneadecal | 7/4 |

Ennealimmal | 5/4 |

. . . . . . . . .

[Gene Ward Smith, Yahoo tuning-math message 12912 (Mon Oct 17, 2005 2:10 pm)]

--- In tuning-math@yahoogroups.com, Carl Lumma

> You lost me on this one. You might say what you're trying to

> do here.

Some ideas to keep in mind from this are that rank two temperament tunings can often be defined by giving a single interval which the tuning tunes exactly. This interval is easily found for least squares and minimax tunings, and for minimax tunings, the intervals are simple. The method works when the tuning uses fractional monzos, that is, monzos with integer coefficients. Of course, this allows one to get arbirarily close to any given tuning.

Hence, for instance, instead of 1/4-comma meaantone we could speak of 5/4-eigenmonzo meantone. Similarly, we might want to consider 5/3-eigenmonzo miracle, or 9/5-eigenmonzo miracle, etc.

We have, for meantone,

1/3-comma = 5/3-eigenmonzo |

2/7-comma = 25/24-eigenmonzo |

1/4-comma = 5/4-eigenmonzo |

1/5-comma = 16/15-eigenmonzo |

1/6-comma = 45/32-eigenmonzo |

1/7-comma = 135/128-eigenmonzo |

1/11-comma = 10935/8192-eigenmonzo |

0-comma = 3/2-eigenmonzo |

7/26-comma = 78125/73728-eigenmonzo |

. . . . . . . . .

[Gene Ward Smith, Yahoo tuning-math message 12915 (Mon Oct 17, 2005 3:19 pm)]

It might be better yet to point out how these are related in the 5-limit meantone case. For q-comma meantone, the corresponding projection matrix is

[|1 0 0>, |4q 1-4q q>, |16q-4 4-16q 4q>]

The non-octave 1 left eigenvectors are multiples of |0 1-4q q>. If we pick a value for q, let's say q=5/17, then we can susbstitute and get a rational monzo: |0 -3/17 5/17>. Clearing denominators gives |0 -3 5>, or 3125/27. Reducing to the octave gives v=3125/1728. Since v has a lower Tenney height than 2/v, v is the eigenmonzo.

. . . . . . . . .

[Paul Erlich, Yahoo tuning-math message 12921 (Mon Oct 17, 2005 4:05 pm)]

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" wrote:

> There will be two of these; we can make the choice

> unique by requiring that 1
> with the least Tenney height.

Maybe this is a nitpick, but I think this is an unfortunate mixing of octave-equivalent and non-octave-equivalent paradigms, so to speak. In the former, I believe Kees expressibility (aka "odd-limit") is relevant, and Tenney HD isn't. After all, we're talking about interval *classes* here, right?

. . . . . . . . .

The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. 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)