(Paper)Unisons

I�ve realized noone has really tried to create a full overview of the different kinds of"unison vectors"one can choose to have for different limit intervals...so here is a basic attempt to go through some,with equal tunings given as examples.Errors and inaccuracies will probably abound from here on.

6/5 = 7/6
(comma eliminated:36/35)

This does nothing more than solidify the meantone interpretation of a 7/4 as Bb.As all negative meantones have a better 7/4 in A#,it only applies to positive meantones.It usually comes together with a unison 64/63(q.v.).

Example tunings:12,17 equal

7/6 = 8/7
(comma eliminated:49/48)

This means in practice that the 7/4 approximation divides the fifth exactly in half from above.It is usually a rather poor approximation and does not gain many favorable structural properties either,except for building the"pentaenharmonic"9-tone MOS using this interval as a generator.

Example tunings:19,24,29 equal

8/7 = 9/8
(comma eliminated:64/63)

This usually comes together with a vanished 36/35,as the meantone dominant seventh.A better approximation is found in 22-tet,with a very wide 9/8(and 3/2)a better 7/4 is made.

Example tunings:12,17,22 equal

9/8 = 10/9
(comma eliminated:80/81)

Perhaps the most known of all commas,the"meantone second"constitutes the foundation of the diatonic scale.It has very favorable properties in tying together 3 and 5-limit intervals.

Example tunings:12,19,31 equal

11/10 = 10/9
(comma eliminated:100/99)

This approximation is actually better than it sounds,if you do not use the meantone 81/80.It�s difficult to make much use of though,even in 22 equal where it fits right in there are not any solid scale structures that acommodate it.

In some ways,the 12TET tritone can be thought of as approximating 11/8 in this way-although the fourth is just as close,the terribly sharp 5/4 and 7/4 biases the ear to the sharp version sounding better.

Example tunings:12(see above),22 equal

12/11 = 11/10
(comma eliminated:121/120)

In practice this means the 11/8 is exactly between the 5/4 and the 3/2,of which the near perfect 24TET rendition is a perfect example.The 31TET 11/4 is flat,but the other flat intervals in this tuning makes it acceptable.However,there aren�t many structural features to be gained by this approximation.

Example tunings:24,31 equal

13/12 = 12/11
(comma eliminated:144/143)

This is a very useful comma bridge because it allows incorporation of the 11 and 13-limit through only one new interval,usually the neutral third,which takes the function of both 11/9 and 16/13.

Example tunings:17,24,29,31 equal

14/13 = 13/12
(comma eliminated:169/168)

This puts the 13/8 right between the 7/4 and the 3/2.Difficult to get much out of by itself,but part of what makes the generic"neutral second"function in 17 and 24TET.

Example tunings:17,24

15/14 = 14/13
(comma eliminated:196/195)

Only seems to come with tunings that also discard the 225/224.In that case,it is useful because it allows a 5-7-13(possibly 11-limit with extra commatics)bridge to be expressed in one continous"line"as in 31-tet:C,B(15/8),A#(7/4),A-(13/8).

Example tunings:19,31

16/15 = 15/14
(comma eliminated:225/224)

A fundamental to 7-limit scales and harmony,to many.The 15/8 and 7/4 together with the tonic forms a continous"triad"upon wich a scale can be based.In fact almost all tunings that contain a good 7/4 approximate it this way.

Example tunings:12,19,22,31

The next part(if i will do one)will deal with more complex commas,such as 13/11=12/10,the"neutral third"concept and such.I am not really a technical writer though so if anybody who could make more sense out of it wants to continue on this,it would be appreciated.Thanks for now...

-=-=-=-=-=-=-
MATS �LJARE
http://www.angelfire.com/mo/oljare
_________________________________________________________________________
Get Your Private, Free E-mail from MSN Hotmail at http://www.hotmail.com.

Mats �ljare wrote:

> 7/6 = 8/7
> (comma eliminated:49/48)
>
> This means in practice that the 7/4 approximation divides the fifth
> exactly in half from above.It is usually a rather poor approximation
> and does not gain many favorable structural properties either,except
> for building the"pentaenharmonic"9-tone MOS using this interval as a
> generator.
>
> Example tunings:19,24,29 equal

I don't think 24 should be in this list as it's not 7-limit consistent.
But you can put 5-equal in its place. Does anybody have a name for this
interval?

> 12/11 = 11/10
> (comma eliminated:121/120)
>
> In practice this means the 11/8 is exactly between the 5/4 and the
> 3/2,of which the near perfect 24TET rendition is a perfect example.The
> 31TET 11/4 is flat,but the other flat intervals in this tuning makes it
> acceptable.However,there aren�t many structural features to be gained
> by this approximation.
>
> Example tunings:24,31 equal

Also 38. In general, all meantone neutral-third scales have this
approximation. It follows from 9/8=10/9 and (11/9)^2=3/2. You need it
for the "wolf third" in the 7 note MOS to be a 5:6 minor third. Or for
the other neutral third scale to have a 5-limit pentatonic as a subset.
That scale also has an approximation to 8:9:10:11:12.

> 13/12 = 12/11
> (comma eliminated:144/143)
>
> This is a very useful comma bridge because it allows incorporation of
> the 11 and 13-limit through only one new interval,usually the neutral
> third,which takes the function of both 11/9 and 16/13.
>
> Example tunings:17,24,29,31 equal

That can also be written 13/8 = 18/11.

> 15/14 = 14/13
> (comma eliminated:196/195)
>
> Only seems to come with tunings that also discard the 225/224.In that
> case,it is useful because it allows a 5-7-13(possibly 11-limit with
> extra commatics)bridge to be expressed in one continous"line"as in
> 31-tet:C,B(15/8),A#(7/4),A-(13/8).
>
> Example tunings:19,31

I'm not sure either of these are consistent in the context. 19-equal
isn't consistent for 13/8, 7/4 and 14/13. I can't find an inconsistency
in 31, although it isn't consistent for the full 15-limit.

> 16/15 = 15/14
> (comma eliminated:225/224)
>
> A fundamental to 7-limit scales and harmony,to many.The 15/8 and 7/4
> together with the tonic forms a continous"triad"upon wich a scale can
> be based.In fact almost all tunings that contain a good 7/4 approximate
> it this way.
>
> Example tunings:12,19,22,31

Also 41 and 53 to be fair and treat all families equally. It's easier to
list 7-limit scales without this approximation. I came up with 26, 46 and
118.

> The next part(if i will do one)will deal with more complex commas,such
> as 13/11=12/10,the"neutral third"concept and such.I am not really a
> technical writer though so if anybody who could make more sense out of
> it wants to continue on this,it would be appreciated.Thanks for now...

I hope you will! Nobody becomes a technical writer until they write
technically. You're doing well already, to express yourself so clearly in
English.

Graham

I've had some more thoughts on this.

> > 13/12 = 12/11
> > (comma eliminated:144/143)
> >
> > This is a very useful comma bridge because it allows
incorporation of
> > the 11 and 13-limit through only one new interval,usually the
neutral
> > third,which takes the function of both 11/9 and 16/13.
> >
> > Example tunings:17,24,29,31 equal
>
> That can also be written 13/8 = 18/11.

Oh, I see the numerator of one ratio is always the denominator of the
other.

Although 31-equal isn't 13-limit consistent, I let it go. Why is
that? Well, I certainly feel that the comma is eliminated in
practice. A neutral sixth could well be heard as a 13:8 rather than
an 18:11, particularly when it's expanded to become 13:4 instead of
36:11. And the wolf fourth is more like 13:10 than any simpler ratio.
So music in 31-equal that uses either of these intervals will be
heard as 13-limit, like it or not.

It looks like 31-equal would be 15-limit consistent but for 13:11 and
13:9. 13:12 and 13:15 are also so bad that they should generally be
avoided. However, this comma does imply both inconsistend intervals,
so why is it still relevant?

You could take 1.3.5.7.9.11.15 and 1.3.5.7.13.15 as 15-limit subsets
in which 31-equal is consistent. Any chord required to be consonant
should fit one of those subsets, and ideally avoid 3 and 15 from the
second one. So the 13:8 = 18:11 comma is used to connect the two
systems.

I find the concept of xenharmonic bridges to be ideal here. Think of
1.3.5.7.9.11.15 and 1.3.5.7.13.15 as islands which are connected by
the 13:8=18:11 bridge. You can only be on one at a time, but the
bridge will carry you between them. More accurately, they are two
peninsulas jutting out from the 1.3.5.7.15 land mass.

So we require the intervals each side of the = to belong to a
consistent system, but not for them both to belong to the same system.

You can also add 41-equal to the list as a fully consistent tuning
with this comma.

> > 15/14 = 14/13
> > (comma eliminated:196/195)

<snip>

> > Example tunings:19,31
>
> I'm not sure either of these are consistent in the context.
19-equal
> isn't consistent for 13/8, 7/4 and 14/13. I can't find an
inconsistency
> in 31, although it isn't consistent for the full 15-limit.

This comma does indeed lie within the 1.3.5.7.13.15 consistency island
of 31-equal. But as 19-equal is inconsistent for 1.7.13, that still
shouldn't be on the list. Put 41-equal in its place.

41-equal also works with 11:10 = 10:9 which was in Mats' original post
but not my reply.

Graham

Hello, there, Matt Oljare and everyone, and please let me apologize
for the very incomplete ASCII character set. While the first letter of
your name, Matt, appears in my text-based applications as a kind of
Greek "pi" symbol or the like, I take it from your e-mail address that
it may be some variety of "O" with an appropriate diacritical mark.
Please let me know about the proper diacritical mark, if you like, so
that I can at least approximate it if possible with some ASCII
character.

Your discussion of commas and "unison vectors" prompts me to add some
remarks regarding these ratios and compromises as they affect
neo-Gothic tunings and music. One benefit of such a dialogue is that
it may suggest how commas or dieses and the like may take on very
different qualities and meanings depending on the stylistic setting.

For example, in many styles of European music from around 1450 to the
present, a "major third" tends to suggest a ratio of around 5:4
(~386.31 cents); in neo-Gothic music, a "major third" tends to suggest
a ratio somewhere between around 81:64 (~407.82 cents) and 9:7
(~435.08 cents).

As a result, people looking at the same tuning from these two points
of view are very likely often to choose different intervals as "the
major third of this scale." In 46-tone equal temperament (46-tET), for
example, from a Renaissance-Romantic viewpoint we would take 15/46
octave or 21 fifths up (~391.30 cents) as the usual major third; from
a neo-Gothic viewpoint, we would take the regular major third of 16/46
octave or four fifths up (~417.39 cents), a virtually pure 14:11.

If viewpoints concerning basic definitions of interval categories can
diverge in this manner, so may viewpoints on commas, dieses, or unison
vectors. Here it may be helpful to propose a few definitions or
clarifications of terms in a neo-Gothic setting.

In comparing regular tunings including equal divisions of the octave,
we encounter an issue sometimes debated on the Tuning List: can the
term "meantone" properly or felicitously be applied to positive
tunings (with fifths larger than 700 cents), and especially to tunings
with fifths wider than pure (e.g. 17-tET, 46-tET, 22-tET)?

People such as Dave Keenan and Paul Erlich have persuaded me that some
other term for these very characteristic neo-Gothic tuning systems may
be preferable, because historically the term "meantone" tends to imply
specifically a _negative_ tuning designed to achieve or approximate
regular major and thirds at 5:4 and 6:5.

Here I would like to propose the related term _equitone_ for a
positive, and especially Pythagorean or supra-Pythagorean tuning,
where four fifths up produce a regular neo-Gothic major third,
typically in the neighborhood ranging from around 81:64 to 9:7.
As in meantones, so in equitones, a regular major third is equal to
four fifths (minus two octaves), or two regular whole-tones (the
"mean-tones" or "equi-tones" giving these tunings their name).

Also, I might propose the general term _lineotunings_ for negative
meantones and positive equitones alike: these tunings share a
"lineolate" structure with all intervals within a given octave derived
from a chain of identical fifths, and these chains in different
octaves form a kind of two-dimensional grid or network.

In the special case of an equal division of the octave, e.g. a
meantone at 19-tET or 31-tET, or an equitone at 29-tET or 46-tET, we
might speak of a _unilineotuning_, since here the octave is also
derived from a single chain or "line" of fifths.

These are new terms, as far as I know, and may or may not be clear to
interested readers, so I warmly invite questions, dialogue, and
suggestions for refining concepts and definitions.

Also, in a modal neo-Gothic style, a given tuning does not necessarily
have any definite "tonic" or "1/1," but rather presents an assortment
of natural or transposed finals (notes of repose) and cadential
progressions. Thus the use of interval ratios in the style "7:6"
rather than "7/6" may generally be preferable, although one might say,
for example, "in 22-tET, the mode of D Dorian has close approximations
of a 7/6 and 12/7 in relation to the final."

An important point is that neo-Gothic music often prefers complex
ratios, not necessarily related to harmonic partials but simply to
"interesting" sonorities, for example "supraminor/submajor" thirds at
or near 14:17:21. The concept is often one of a continuum of rational
or irrational ratios.

Further, while supraminor and submajor thirds are often thus referred
to as "17-flavor" intervals as a convenient shorthand, 14:17:21 is
only one possible rational representation of a region of the continuum
with various shadings and quite "fuzzy" borders. Other possible
shadings could be represented by integer ratios such as 38:46:57
(approximated in 29-tET) or 200:243:300 (not too far from 46-tET).

Here I focus on two ratios of special interest: 49:48 and 64:63.

-----

49:48 (~35.70 cents)
7:6 vis-a-vis 8:7
12:7 vis-a-vis 7:4

The presence of intermediate intervals somewhere between 8:7 and 7:6
(e.g. ~15:13, a very large major second or very small minor third), or
between 12:7 and 7:4 (e.g. ~26:15, a very large major sixth or very
small minor seventh) is an intriguing and highly treasured property of
a characteristic neo-Gothic tuning such as 29-tET, and also certain
"not-so-characteristic" tunings such as 20-tET and 24-tET.

The ability of these intermediates to act either as a large major
second or sixth expanding respectively to a fourth or octave, or as a
small minor third or seventh contracting to a unison or fifth, makes
possible various kinds of musical "puns." Also, the sheer sonority and
"ambiguous" quality of these intervals has an appeal all its own.

In 29-tET or 24-tET, where these intervals have ratios near 250 and
950 cents (or ~15:13, ~26:15), they seem ideally to fill this
ambiguous role. This is the typical neo-Gothic "13-flavor" exemplified
by 29-tET.

However, in other tunings, intervals may lend themselves to both
musical roles (8:7-7:6 or 12:7-7:4) while having a decided leaning or
"bias" in one direction or the other.

A fine example is the intervals of 240 cents and 960 cents in 20-tET,
which typically serve as large major seconds or small minor sevenths,
but can also take the role of "quasi-thirds" contracting to unisons or
"quasi-sixths" expanding to octaves. When used in the latter role,
these intervals might be said to constitute a kind of "super-13
flavor."

With a large enough tuning set, a single tuning might offer both close
approximations of 7-flavor ratios (e.g. 8:7 and 7:6, 12:7 and 7:4) and
13-flavor alternatives. In Pythagorean tuning, while chains of 14
fifths up and 15 fifths down give close approximations of 8:7 and 7:6,
chains of 26 fifths up or 27 fifths down yield intervals of around
250.83 cents and 247.21 cents, comparable to 24-tET and 29-tET.

-----

64:63 (~27.26 cents)
9:8 vis-a-vis 8:7
32:27 vis-a-vis 7:6
81:64 vis-a-vis 9:7
128:81 vis-a-vis 14:9
27:16 vis-a-vis 12:7
16:9 vis-a-vis 7:4

Just as negative meantones feature regular whole-tones somewhere
between 9:8 and 10:9, so neo-Gothic equitones feature such whole-tones
somewhere between 9:8 and 8:7.

Typical neo-Gothic tunings range from around Pythagorean to 22-tET. In
Pythagorean, the whole-tone is a pure 9:8 and the septimal comma of
64:63 may be approximated in larger tuning sets by the Pythagorean
comma of 531441:524288 (~23.46 cents, a 3-7 schisma of ~3.80 cents
smaller). In 22-tET, it is very slightly larger than the mean of 9:8
and 8:7, producing regular major thirds (~436.36 cents) very close to
a pure ratio of 9:7.

In the case of meantones, two tunings may especially receive attention
as producing regular thirds with pure 5-based ratios:

1/4-comma meantone pure 5:4 major third (31-tET close)
1/3-comma meantone pure 6:5 minor third (19-tET close)

By analogy, corresponding neo-Gothic equitones would be:

1/4-septimal-comma equitone pure 9:7 major third (22-tET close)
1/3-septimal-comma equitone pure 7:6 minor third (27-tET close)

However, while 22-tET is notable as a neo-Gothic tuning combining
approximate ratios of 3 and 7 in a regular "equitonal" structure (four
fifths up are very close to 9:7, and three fourths up quite close to
7:6), three factors may make it simply one temperament of interest,
rather than a central focus of interest in the way that 1/4-comma
meantone or 31-tET is as a negative tuning.

First, since the septimal comma is somewhat larger than the syntonic
comma, an absolutely larger amount of temperament is required for
1/4-septimal-comma (or 22-tET) than for 1/4-(syntonic)-comma (or
31-tET) -- in fact, 22-tET (fifths ~7.14 cents wide) is comparable to
the 19-tET approximation (fitths ~7.22 cents narrow) of 1/3-comma
meantone.

Secondly, fifths and fourths are the primary concords of neo-Gothic
music, so that their temperament may be felt more decidedly than in a
meantone setting where thirds and sixths are the primary concords.

Thirdly, while meantone styles generally favor regular thirds quite
close to the simple ratios of 5:4 and 6:5, neo-Gothic styles often
favor such thirds with complex ratios, e.g. the Pythagorean 81:64 and
32:27, or 21st-century variants such as 14:11 and 13:11, etc.

In a tuning such as 17-tET, which you mention, we have complex ratios
which might be viewed as intermediate between Pythagorean and 7-based
intervals, for example the regular minor third at 4/17 octave or
~282.35 cents (~20:17), somewhere between 32:27 (~294.13 cents) and
7:6 (~266.87 cents).

In other characteristic systems such as 29-tET, or a regular 12-note
set of either 46-tET or the almost identical pure 14:11 equitone,
complex thirds and sixths of various shadings are also the norm.

In larger tuning sets, "7-flavor" intervals differing from the regular
ones by some variety of comma or diesis are common.

In a 24-note Pythagorean tuning, for example, the Pythagorean comma of
531441:524288 (~23.46 cents) can also serve as an approximation of the
64:63 comma (~27.26 cents), with the difference equal to a 3-7 schisma
of around 3.80 cents. Thus 16 fifths up (~431.28 cents) serves as a
near-9:7 major third, and 15 fourths up (~270.67 cents) as a near-7:6
minor third, in addition to the regular 81:64 and 32:27.

In the e-based equitone with fifths at around 704.61 cents, where the
whole-tone and diatonic semitone have a ratio of sizes equal to
Euler's e (~2.71828), regular major and minor thirds have sizes of
around 418.42 cents (a bit larger than 14:11) and 286.18 cents
(somewhere between 33:28 and 13:11). In a 24-note tuning, we also have
available "7-flavor" thirds formed from 13 fourths up (~440.11 cents)
or 14 fifths up (~264.50 cents).

Thus while dispersing the 64:63 comma, as in 22-tET, is one available
solution for obtaining 7-flavor ratios, and the ideal solution in a
12-note tuning, other types of solutions involving some kind of comma
or diesis are often favored for larger tuning sets, with 24 notes as a
typical size. Here the multiple varieties of intervals, semitones, and
cadential "flavors" are seen as a positive advantage.

However, given that tunings such as 20-tET (fifths at 720 cents,
~18.04 cents wide) can be very effective for neo-Gothic music,
especially in the right timbres, equitone temperaments at or near
1/4-septimal-comma (e.g. 22-tET) or 1/3-septimal-comma (e.g. 27-tET)
remain viable options along with more complex 7-flavor schemes.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., "Mats Öljare" <oljare@h...> wrote:

/tuning/topicId_20312.html#20312

Thanks so much, Mats, for posting this "unison vectors" material...
it really summarizes some of the "periodicity block" procedures for
me!

________ ______ ____
Joseph Pehrson