Gene Ward Smith

Решил сделать бекап данных по нему.

Вот его музыка - https://archive.org/search.php?query=creator%3A%22Gene+Ward+Smith%22

Вот ссылки о нём

https://en.xen.wiki/w/Gene_Ward_Smith

https://microtonal.miraheze.org/wiki/Gene_Ward_Smith

https://en.xen.wiki/w/Microtonal_Music_by_Gene_Ward_Smith

Его посты в рассылках:

Subject: Wedge products and the torsion mess

http://www.robertinventor.com/tuning-math/s___3/msg_2075-2099.html#2098

Subject: Unlocking the mysteries of the wedge invariant

https://robertinventor.com/tuning-math/s___3/msg_2100-2124.html#2122

Вот то, что смог вытащить с его сайта (todo - значит, что не нашёл, но когда-то этот материал был на его сайте):

https://web.archive.org/web/20190317163716/http://www.tonalsoft.com/gws/home.aspx

Circulating Temperaments

https://web.archive.org/web/20121025200638/http://tonalsoft.com/gws/circ.html

It is a sad fact that common practice Western music is based on triads, and the thirds of these triads are woefully out of tune. The major third in 12-equal is 13.69 cents sharp, and the minor third is 15.64 cents flat. Considering that anything beyond ten cents over or under a just intonation is entering the region of major cheeziness, we can see why 12-equal was not in general use during much of the common practice period, and that other systems held sway instead. One plan for dealing with the problem of the thirds is a circulating, or well-temperament. This makes the thirds which are most likely to be used a little bit better, at the expense of other thirds, which are made a little bit worse. The circulating temperments used on this website employ a new and bolder plan: the thirds are made a lot better, and instead of trying to make up the difference with out-of-tune versions of 5/4 or 6/5, we use instead in-tune versions of xenharmonic thirds: the supermajor third 9/7, the subminor third 7/6, along with 14/11 and 13/11. The resulting chords sound strange but not repulsive. Temperaments adopting this plan are [grail], [bifrost], and [cauldron]. Another approach leading to similar results is to well-tune a scale in just intonation, where a well-tuning is a regular tuning, meaning using fifths and thirds of the same size, whicb is chosen with an eye to tempering the scale to a circulating temperament. An example of this approach is [duowell].

It is also possible to simply use a single meantone fifth for twelve notes, and bite the bullet of the resulting very sharp "wolf" fifth. One way to do this is to use a mild version of meantone, such as 1/6-comma. A way I think works better is to take a wolf fifth near or at 20/13. Systems adopting this approach are [ratwolf] and wilwolf.

todo: grail

http://tonalsoft.com/gws/grail.html

Bifrost

This has six meantone fifths, leading to three pure thirds; on each side of the chain of meantones we put two pure 3/2's, so we have four pure fifths in total; we then round out with two sharp fifths of size sqrt(2048/2025 sqrt(5)), of size 706.355 cents. This leads to three sharp major thirds which are within a cent of being pure 14/11's, being of size 416.619 cents; and two more sharp thirds of size 406.843 at E and Ab, which are the only real problems with this temperament.

! bifrost.scl

! [45/64*5^(1/4), 1/2*5^(1/2), 16/45*5^(3/4), 5/4, 2/5*5^(3/4), 15/16*5^(1/4), 5^(1/4),

! 1/2*10^(1/2), 1/2*5^(3/4), 8/15*5^(3/4), 5/4*5^(1/4), 2]

!

Six meantone fifths, four pure fifths, two on each side, two fifths of sqrt(2048/2025 sqrt(5))

12

!

86.802144

193.156856

299.511569

386.313714

503.421572

584.847143

696.578428

793.156857

889.735285

1001.466571

1082.892142

1200.000000

todo: cauldron

http://tonalsoft.com/gws/cauldron.html

todo: duowell

http://tonalsoft.com/gws/duowell.htm

todo: ratwolf

http://tonalsoft.com/gws/ratwolf.html

Linear Temperaments

https://web.archive.org/web/20121025200628/http://tonalsoft.com/gws/linear.htm

A rank two temperament is a [regular temperament] with two generators. If it is possible for one of the generators to be an octave, such a temperament is called a linear temperament (in the strict sense.) However, rank two temperaments in general are often called linear. The most common choice for generators is for one generator to be an octave, or some nth part of an octave for some integer n; in this case this generator is called the period and the other the generator.

A rank two temperament may be uniquely defined in various ways; one is by means of the [wedgie], another by way of the [comma sequence], and still another by means of the mapping, or [icon], for a reduced set of generators. Here we are calling a pair of generators reduced if one generator is the period, and the other is the unique generator greater than one and less than the square root of the period (or less than half the period in logarithmic terms) which together with the period gives a generator pair for the temperament. This last definition depends on the exact tuning, and hence in theory may not be uniquely determined; in practice this seldom matters but for this and other reasons when working with temperaments using computer programs the wedgie is preferable as a means of defining the temperament. The icon, or tuning map, for the reduced generator pair, listing the period first and the generator second, we may call the standard icon.

[Here is a list of seven-limit linear temperaments.]

Listed below are some of the important families of linear temperaments.

[Meantone]

[Miracle]

[Orwell]

todo:

regular temperament

http://tonalsoft.com/gws/regular.html

wedgie

http://tonalsoft.com/gws/wedgie.html

comma sequence

http://tonalsoft.com/gws/commaseq.htm

list of seven-limit linear temperaments

http://tonalsoft.com/gws/sevnames.htm

meantone

http://tonalsoft.com/gws/meantone.htm

miracle

http://tonalsoft.com/gws/miracle.htm

orwell

http://tonalsoft.com/gws/orwell.html

Planar Temperaments

http://tonalsoft.com/gws/planar.htm

A rank three temperament is a [regular temperament] with three generators. If one of the generators can be an octave, it is called a planar temperament, though the word is sometimes applied to any rank three temperament. The octave classes of notes of a planar temperament can be embedded in a plane as a lattice, hence the name.

The most elegant way to put a Euclidean metric, and hence a lattice structure, on the pitch classes of a planar temperament is to start from a lattice of pitch classes for just intonation, and orthogonally project onto the subspace perpendicular to the space determined by the commas of the temperament. For instance, 7-limit just intonation has a [symmetrical lattice structure] and a 7-limit planar temperament is defined by a single comma. If u = |* a b c> is the octave class of this comma, then v = u/||u|| is the corresponding unit vector. Then if g1 and g2 are the two generators, j1 = g1 - (g1.v)v and j2 = g2 - (g2.v)v will be projected versions of the generators, and (j1.j1)a^2 + 2(j1.j2)ab + (j2.j2)b^2 the projected quadratic form defining the metric stucture of the planar temperament lattice. Here the dot product is defined by the [bilinear form] giving the metric structure. One good, and canonical, choice for generators are the generators found by using [Hermite reduction] with the proviso that if the generators so obtained are less than 1, we take their reciprocal.

The Hermite generators for marvel (225/224 planar) temperament are {2,3,5}, and the projected lattice strucuture is defined by the norm sqrt(11a^2-14ab+11b^2), where "a" is the exponent of 3 and "b" of five. The Hermite generators for breed (2401/2400 planar) temperament are {2,49/40,10/7}, and the projected lattice structure has 49/40 perpendicular to 10/7, so it has orthogonal axes, with a norm given by sqrt(11a^2+8b^2), where now "a" is the exponent of 49/40, and "b" the exponent of 10/7.

todo:

symmetrical lattice structure/7-limit lattice

http://tonalsoft.com/gws/sevlat.htm

Symmetric Bilinear Form

http://mathworld.wolfram.com/SymmetricBilinearForm.html

Hermite Normal Form

http://mathworld.wolfram.com/HermiteNormalForm.html

Crystal Balls

http://tonalsoft.com/gws/crystal.htm

We may define the nth q-limit Hahn shell as the octave classes at exactly Hahn distance n from the unison in terms of the q-odd-limit [Hahn norm]. The number of notes in the 5-limit Hahn shell is (for n>0) 6n, and in the 7-limit Hahn shell n has 10n^2+2 notes. If we take the union of the Hahn shells up to shell n we obtain the q-limit crystal ball; the reason behind that name is that the number of notes in the 7-limit crystal balls are called crystal ball numbers or magic numbers in some chemical and crystallographic contexts. The number of notes in the nth 5-limit crystal ball is 3n^2 + 3n + 1 and in the nth 7-limit crystal ball is (2n + 1)(5n^2 + 5n + 3)/3. Because of the way they are formed crystal balls are not especially regular as scales, but they are abundently supplied with chords.

Here are the first few 5-limit Hahn shells:

Shell 0

[1]

Shell 1 -- the 5-limit consonances

[6/5, 5/4, 4/3, 3/2, 8/5, 5/3]

Shell 2

[25/24, 16/15, 10/9, 9/8, 32/25, 25/18, 36/25, 25/16, 16/9, 9/5, 15/8, 48/25]

Shell 3

[128/125, 27/25, 144/125, 125/108, 75/64, 32/27, 125/96, 27/20, 45/32, 64/45,

40/27, 192/125, 27/16, 128/75, 216/125, 125/72, 50/27, 125/64]

Shell 4

[81/80, 648/625, 135/128, 625/576, 256/225, 625/512, 768/625, 100/81, 81/64,

162/125, 512/375, 864/625, 625/432, 375/256, 125/81, 128/81, 81/50, 625/384,

1024/625, 225/128, 1152/625, 256/135, 625/324, 160/81]

Here are the first three 7-limit Hahn shells:

Shell 0

[1]

Shell 1 -- the 7-limit consonances

[8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4]

Shell 2

[50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, 28/25,

9/8, 25/21, 60/49, 49/40, 32/25, 9/7, 64/49, 21/16, 49/36, 48/35,

25/18, 36/25,35/24, 72/49, 32/21, 49/32, 14/9, 25/16, 80/49, 49/30,

42/25, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18,

96/49, 49/25]

Here are the first two 7-limit crystal ball scales:

Crystal ball 1 13 notes -- the 7-limit Tonality Diamond

[1, 8/7, 7/6, 6/5, 5/4, 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4]

Crystal ball 2 55 notes

[1, 50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9,

28/25,9/8, 8/7, 7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 32/25, 9/7, 64/49,

21/16, 4/3, 49/36, 48/35, 25/18, 7/5, 10/7, 36/25, 35/24, 72/49, 3/2,

32/21, 49/32, 14/9, 25/16, 8/5, 80/49, 49/30, 5/3, 42/25, 12/7, 7/4,

16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 49/25]

Crystal ball one can also be described as Cube[2], the 2x2x2 [cube scale], which consists of the notes of the eight chords [i, j, k] with i, j, and k either -1 or 0. Crystal ball two consists of Cube[4], the 4x4x4 cube with i, j, and k from -2 to 1, minus the eight chords [-2 -2 1], [-2 1 -2], [-2 1 1], [1 -2 -2], [1 -2 1], [-2 -2 -2], [1 1 -2], [1 1 1].

The first two crystal balls can also equally well be described as Euclidean ball scales; they began to diverge with the third crystal ball. If we take everything within a radius of one of the unison, we get crystal ball one; if we take everything within a radius of two, we get crystal ball two. This means we also have two intermediate scales, Euclidean balls of radius sqrt(2) and sqrt(3).

Euclid 2 19 notes

[1, 21/20, 15/14, 8/7, 7/6, 6/5, 5/4, 4/3, 48/35, 7/5,

10/7, 35/24, 3/2, 8/5, 5/3, 12/7, 7/4, 28/15, 40/21]

Euclid 3 43 notes

[1, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, 28/25, 8/7,

7/6, 25/21, 6/5, 60/49, 49/40, 5/4, 9/7, 21/16, 4/3, 48/35, 7/5, 10/7, 35/24,

3/2, 32/21, 14/9, 8/5, 80/49, 49/30, 5/3, 42/25, 12/7, 7/4, 25/14, 9/5, 64/35,

28/15, 15/8, 40/21, 48/25, 35/18, 96/49]

todo:

Hahn norm/Hahn Distance

http://tonalsoft.com/gws/hahn.htm

Chord cubes/cube scale

http://tonalsoft.com/gws/cube.htm

A cube scale is a 7-limit scale (we might also consider the 9 odd limit) whose notes are derived from a cube in the [7-limit lattice] of chords. For odd n, we will call the octave-reduced set of notes deriving from all chords of the form [i, j, k], (1-n)/2 <= i, j, k <= (n-1)/2, Cube[n]. If n is even, we will use Cube[n] to refer to the notes of [i, j, k] with 1-n/2 <= i, j, k < n/2. If n is odd, Cube[n] has (n+1)^3/2 notes to it; if n is even, its growth is more complicated but still approximately cubic. For odd n, the inversion of the scale gives another scale, centered around a minor rather than a major tetrad. Here are the first three cube scales:

Cube[2] -- the stellated hexany, 14 notes

[1, 21/20, 15/14, 35/32, 9/8, 5/4, 21/16, 35/24, 3/2, 49/32, 25/16, 105/64, 7/4, 15/8]

Cube[3] 32 notes

[1, 49/48, 25/24, 21/20, 15/14, 35/32, 9/8, 8/7, 7/6, 6/5, 60/49, 49/40, 5/4, 9/7, 21/16, 4/3, 7/5, 10/7, 35/24, 3/2, 49/32, 25/16, 8/5, 105/64, 5/3, 42/25, 12/7, 7/4, 25/14, 9/5, 15/8, 35/18]

Cube[4] 63 notes

[1, 50/49, 49/48, 36/35, 25/24, 21/20, 16/15, 15/14, 35/32, 10/9, 28/25, 9/8, 8/7, 7/6, 25/21, 6/5, 128/105, 60/49, 49/40, 5/4, 32/25, 9/7, 35/27, 64/49, 21/16, 4/3, 168/125, 49/36, 48/35, 25/18, 480/343, 7/5, 10/7, 343/240, 36/25, 35/24, 72/49, 125/84, 3/2, 32/21, 49/32, 54/35, 14/9, 25/16, 8/5, 80/49, 49/30, 105/64, 5/3, 42/25, 12/7, 7/4, 16/9, 25/14, 9/5, 64/35, 28/15, 15/8, 40/21, 48/25, 35/18, 96/49, 49/25]

Dwarves

http://tonalsoft.com/gws/dwarf.htm

Dwarves

Suppose v is a val such that the gcd of the coefficients is 1, v(2)>0 and for every odd prime p, v(p)>=0. According to my original definition of val at any rate, vals are defined for all positive rational numbers, but for all but a finite number of primes they map the prime to 0. Bearing the precise definition in mind, we can define the dwarf scale for v, dwarf(v), as the reduction to the octave of the integers n of minimal Tenney height which form a complete set of residues v(n) mod v(2). The reason for the name "dwarf" is that height is as small as possible. The results are not extremely sensitive to this exact definition, as sorting by Hahn distance from unity and using Tenney height to break ties seems to usually lead to the same result.

Because v(2n) mod v(2) = v(n) mod v(2), the integers n will always be odd. Because if v(q) = 0 then v(qn) = v(n), the integers n will always be in the p-limit, where p is the largest prime for which v(p)>0. If r is any prime for which v(r)=0, then r does not appear in the factorization of the integers n, so this definition also covers subgroup situations, such as {2,3,7}-scales, so long as 2 is in the picture and we are using octave equivalence.

Because of the way they are constructed, dwarves are always permutation epimorphic and have a bias towards otonality over utonality. Here are some examples:

<7 11 16| 1, 9/8, 5/4, 45/32, 3/2, 27/16, 15/8

transposes to: 1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8

<7 11 16 20| 1, 9/8, 5/4, 21/16, 3/2, 27/16, 7/4

transposes to: 1, 9/8, 7/6, 4/3, 3/2, 5/3, 7/4

<10 16 23 28| 1, 35/32, 9/8, 5/4, 21/16 45/32, 3/2, 105/64, 7/4, 15/8

transposes to: 1, 16/15, 7/6, 6/5, 4/3, 7/5, 3/2, 8/5, 7/4, 15/8  

Marvelous dwarves

A marvelous dwarf is a scale with the following attributes:

(1) It is a marvel tempering of a 5-limit dwarf.

(2) It has the same number n of otonal tetrads, otonal pentads, utonal tetrads and utonal pentads. As a consequence of this it also has n subminor and n supermajor tetrads.

(3) It is covered by its pentads--that is, every note is harmonized by a pentad, and the scale is the union of its pentads.

(4) It has more 5-limit triads than pentads.

(5) It has no approximate tetrads deriving from anything but marvel; in the 5-limit scale which is tempered the smallest comma which produces approximate tetrads is 225/224.

If every condition but the third--the covering condition--is satisfied, I call it a semimarvelous dwarf. Why there are these scales exhibiting such regularity as a result of finding the 5-limit dwarf is an interesting question. Whatever the reason for it, the marvelous dwarves--of size 12, 15, 18, 19, 20, 21, and 25--seem like excellent scales for instrumentalists and composers interested in 9-limit harmony and scales in this size range. The 25-note scale, whose 5-limit preimage is the Euler genus of 15^4, is particularly striking from the point of view of the quantity of pentads it supplies.

There is a marvelous or semimarvelous dwarf for each scale size from 11 to 22, and then the 25 note scale. So far as I know this is the complete list.

Here is a brief description; the numbers are pentad number, numbers of major-minor triads, and size/pentad ratio.

<11 17 26| 1 6-6 11 semimarvelous

<12 19 28| 2 6-6 6 marvelous

<13 20 30| 1 7-6 13 semimarvelous

<14 22 33| 2 7-6 7 semimarvelous

<15 24 35| 3 8-8 5 marvelous

<16 25 37| 2 7-6 8 semimarvelous

<17 27 40| 4 10-9 4.25 semimarvelous

<18 29 42| 4 10-10 4.5 marvelous

<19 30 44| 5 12-11 3.8 marvelous

<20 32 47| 6 12-12 3.333 marvelous

<21 33 49| 5 12-12 4.2 marvelous

<22 35 51| 6 14-13 3.667 semimarvelous

<25 40 58| 9 16-16 2.778 marvelous

todo:

Composers

http://tonalsoft.com/gws/composers.html

Colleagues

http://tonalsoft.com/gws/coll.htm

Meantone Music

http://tonalsoft.com/gws/meanmus.htm

Modern Masters of Meantone

http://tonalsoft.com/gws/mmm.htm

Popular

http://tonalsoft.com/gws/pops.html

Mad Science Tuning

http://tonalsoft.com/gws/mad.html

Intervals and Vals

http://tonalsoft.com/gws/intval.html

The Wedge Product

http://tonalsoft.com/gws/wedge.html

The Brat

http://tonalsoft.com/gws/brat.html

Bosanquet Lattices

http://tonalsoft.com/gws/bosanquet.html

TOP and Tenney Space

http://tonalsoft.com/gws/top.htm

Kees space and Kees tuning

http://tonalsoft.com/gws/kees.htm

The Hermite Basis

http://tonalsoft.com/gws/hermbas.htm

Composing in Meantone

http://tonalsoft.com/gws/composing.htm

Names of seven-limit commas

http://tonalsoft.com/gws/commalist.htm