Yahoo Tuning Groups Ultimate Backup tuning In search of unison vectors

New on the warped canon page -- collect them all! 49-TET, 59-TET, and
alternate versions of 11-TET and 17-TET.

Most of you probably know that meantone scales (in a generalized sense)
treat the interval 81/80 (the syntonic comma) as a unison; harmonic
progressions that exploit this equivalence are common in meantone tuning
(including 12-TET). This can be written as a unison vector (4 -1) (if I
have the correct notation), implying 3^4 * 5^-1, or 3*3*3*3/5 (ignoring
factors of 2 since the most widely used scales are octave-equivalent).
Meantone scales have a particular character, in part due to this unison
vector, and I'm wondering if unison vectors might be a good way to classify
other kinds of scales.

Of course, the 32805/32768 schisma (8 1) is another unison vector that's
associated with a family of similar-sounding scales. My recent experiments
with retuning my piano solo "Galticeran" illustrated that 128/125 (0 -3) is
another unison vector with harmonic implications. But there are others that
I've found in trying to classify scales by their structure. A small family
of scales with 3 large and 4 small steps (L-s-s-L-s-L-s) shares the unison
vector 25/24 (-1 2). On the other end of the spectrum, there's another tiny
group of scales with 2 large and 5 small steps (s-L-s-s-L-s-s), with the
unison vector 135/128 (3 1).

Finally, there's an interesting unison vector with 7-limit implications:
64/63 (-2 0 -1). If you listen to the canons with 12, 15, 22, 27, 37, 42,
49, and 59 equal steps to the octave (I added the 49-TET and 59-TET
versions specifically because of this discovery), I think you'll find that
the minor sevenths are nicely flat (because the fifths are sharp) and the
overall flavor of the harmony suggests a 7-limit interpretation. This is
something that might have escaped my attention if I hadn't been doing all
the different retunings of the canon and noticed similarities in the
character of the scales. I wouldn't have even considered writing anything
for 42-, 49-, or 59-TET before this. I've had some interest in 37-TET due
to its excellent 11/8 and 13/8 approximations, but I hadn't considered that
its relatively poor 3/2 might actually have an advantage.

My question is whether there might be some other unison vectors that could
unify a number of other scales, in particular some of the near-just
tunings, into a larger grouping. Is there a list of known "interesting"
unison vectors anywhere?

Incidentally, after all this listening to retuned canons, I'm now pretty
sick of the General MIDI timbres on my sound card. I've been seriously
considering Csound for a while, since I've used it before for simple tuning
experiments and such, but I never had the patience to deal with learning to
use it for real music. I've been accustomed to the real-time feedback and
editing convenience of MIDI. But sample playback is so limited compared to
synthesis. Maybe I should start writing a few simple microtonal etudes in
Csound and see how they come out.

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., Herman Miller <hmiller@I...> wrote:
> I'm wondering if unison vectors might be a good way to classify
> other kinds of scales.

You bet! Practically any interesting scale can be classified by its
unison vectors. The common-practice diatonic scale is the periodicity
block (have you digested the "Gentle Introduction"?) or lattice region
delimited by the two unison vectors 81:80 and 25:24 (you need two
because the 5-limit lattice is two-dimensional). The 81:80 is a
"commatic" unison vector and it treated as an equivalence, usually
tempered out. The 25:24 is a "chromatic" unison vector and is treated
as an alteration, and is not tempered out. My decatonic scale is the
PB defined by the three unison vectors 64:63, 50:49, and 49:48 (you
need three because the 7-limit lattice is three-dimensional). The
64:63 and 50:49 are the "commatic" unison vectors and the 49:48 the
"chromatic" unison vector in this system. All this is clearly and
colorfully depicted in my paper, _The Forms of Tonality_ (available
from me for $5) which was written for the Microfest, but sadly didn't
make it in in time.
>
> But there are others that
> I've found in trying to classify scales by their structure. A small family
> of scales with 3 large and 4 small steps (L-s-s-L-s-L-s) shares the unison
> vector 25/24 (-1 2). On the other end of the spectrum, there's another tiny
> group of scales with 2 large and 5 small steps (s-L-s-s-L-s-s), with the
> unison vector 135/128 (3 1).

Can you describe these in any more detail?
>
> My question is whether there might be some other unison vectors that could
> unify a number of other scales, in particular some of the near-just
> tunings, into a larger grouping. Is there a list of known "interesting"
> unison vectors anywhere?

Yes -- see Kees van Prooijen's web page _Searching Small Intervals_,
and click on all the links.

🔗Herman Miller <hmiller@IO.COM>

On Thu, 21 Jun 2001 03:06:19 -0000, "Paul Erlich" <paul@stretch-music.com>
wrote:

>--- In tuning@y..., Herman Miller <hmiller@I...> wrote:
>> I'm wondering if unison vectors might be a good way to classify
>> other kinds of scales.
>
>You bet! Practically any interesting scale can be classified by its
>unison vectors. The common-practice diatonic scale is the periodicity
>block (have you digested the "Gentle Introduction"?)

Yep.

> or lattice region
>delimited by the two unison vectors 81:80 and 25:24 (you need two
>because the 5-limit lattice is two-dimensional). The 81:80 is a
>"commatic" unison vector and it treated as an equivalence, usually
>tempered out. The 25:24 is a "chromatic" unison vector and is treated
>as an alteration, and is not tempered out. My decatonic scale is the
>PB defined by the three unison vectors 64:63, 50:49, and 49:48 (you
>need three because the 7-limit lattice is three-dimensional). The
>64:63 and 50:49 are the "commatic" unison vectors and the 49:48 the
>"chromatic" unison vector in this system. All this is clearly and
>colorfully depicted in my paper, _The Forms of Tonality_ (available
>from me for $5) which was written for the Microfest, but sadly didn't
>make it in in time.

Ah, the 50/49 and 49/48 are ones I hadn't found yet. It'll be interesting
to see if any other scales share either of these vectors. They both look
like they could be useful in chord progressions, which is just the sort of
thing I'm looking for. I've already noted the usefulness of the 64/63.

>> But there are others that
>> I've found in trying to classify scales by their structure. A small family
>> of scales with 3 large and 4 small steps (L-s-s-L-s-L-s) shares the unison
>> vector 25/24 (-1 2). On the other end of the spectrum, there's another tiny
>> group of scales with 2 large and 5 small steps (s-L-s-s-L-s-s), with the
>> unison vector 135/128 (3 1).
>
>Can you describe these in any more detail?

The 3L+4s scales consist of 10, 13, and 17-TET, all of which have very flat
thirds. All of these plus 6-TET, 7-TET, and 20-TET, have 25/24 as a unison
vector. These are "neutral third" scales; with the 25/24 tempered out,
there are no distinctions between sharps and flats. (The 24-TET canon uses
the same scale structure, although 25/24 isn't a unison vector in 24-TET,
but the best 24-TET tuning would just be 12-TET.)

The 2L+5s scales, with a 135/128 unison vector, are 9, 11, 16, and 23-TET.
It's a bit surprising that 11-TET fits in here, but it clearly makes sense
to group 9, 16, and 23-TET together. These three are the pelog-like scales,
with small fifths, and (as the 135/128 unison vector implies) major thirds
produced by a series of three ascending fourths rather than four fifths.

>> My question is whether there might be some other unison vectors that could
>> unify a number of other scales, in particular some of the near-just
>> tunings, into a larger grouping. Is there a list of known "interesting"
>> unison vectors anywhere?
>
>Yes -- see Kees van Prooijen's web page _Searching Small Intervals_,
>and click on all the links.

Thanks. I've got it bookmarked now, and I'll read through it as soon as I
have the time.

🔗Robert C Valentine <BVAL@IIL.INTEL.COM>

> From: "Paul Erlich" <paul@stretch-music.com>
> Subject: Re: In search of unison vectors
>
> --- In tuning@y..., Herman Miller <hmiller@I...> wrote:
> > I'm wondering if unison vectors might be a good way to classify
> > other kinds of scales.
>
> You bet! Practically any interesting scale can be classified by its
> unison vectors.

I started a program (or a spreadsheet) that approached the same
thing in a way I was more apt to understand.

Take an ED2
Create a linear naming accorfing to the best 3/2 : 4/3 (in other
words, give it a Pythagorean naming convention).

Sort (or make a data base) according to "interesting
properties" for instance

all scales where E is the best 5/4 (I would call meantones)
all scales where Fb is the best 5/4 (schismic?)
all scales where D# is the best 7/6

etc...

Just a different way of expressing unison vectors which would
match my musical intuition more.

> >
> > But there are others that
> > I've found in trying to classify scales by their structure. A small
> > family of scales with 3 large and 4 small steps (L-s-s-L-s-L-s) shares
> > the unison vector 25/24 (-1 2).

Basically what I've been looking at is exploring scale generation
in exactly the same way, or perhaps completely differently. (See
also Dans recent mail regarding LssssLsss).

The structure you show I've looked at a few ways. It obviously maps
to a neutral diatonic (LssL = 3/2, 5445454 for 31ED2 fans). Note
that the characteristic interval in this sort of mapping is a
~6/5 (5/3). By mapping various terms to to various thirds
I've gotten some other scales that show potential (for me)
(7227272 in 29ED2 is an example where Ls ~= 5/4, one
interesting rotation is 2727227 since it gives the maj7+5
sonority...)

> > On the other end of the spectrum, there's another tiny
> > group of scales with 2 large and 5 small steps (s-L-s-s-L-s-s), with the
> > unison vector 135/128 (3 1).
>
> Can you describe these in any more detail?

Ditto. And let me pass on some more voluminous thanks for the
warped canon page.

Bob Valentine

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., Herman Miller <hmiller@I...> wrote:
> On Thu, 21 Jun 2001 03:06:19 -0000, "Paul Erlich" <paul@s...>
> wrote:
>
> >--- In tuning@y..., Herman Miller <hmiller@I...> wrote:
> >> I'm wondering if unison vectors might be a good way to classify
> >> other kinds of scales.
> >
> >You bet! Practically any interesting scale can be classified by
its
> >unison vectors. The common-practice diatonic scale is the
periodicity
> >block (have you digested the "Gentle Introduction"?)
>
> Yep.
>
> > or lattice region
> >delimited by the two unison vectors 81:80 and 25:24 (you need two
> >because the 5-limit lattice is two-dimensional). The 81:80 is a
> >"commatic" unison vector and it treated as an equivalence, usually
> >tempered out. The 25:24 is a "chromatic" unison vector and is
treated
> >as an alteration, and is not tempered out. My decatonic scale is
the
> >PB defined by the three unison vectors 64:63, 50:49, and 49:48
(you
> >need three because the 7-limit lattice is three-dimensional). The
> >64:63 and 50:49 are the "commatic" unison vectors and the 49:48
the
> >"chromatic" unison vector in this system. All this is clearly and
> >colorfully depicted in my paper, _The Forms of Tonality_
(available
> >from me for $5) which was written for the Microfest, but sadly
didn't
> >make it in in time.
>
> Ah, the 50/49 and 49/48 are ones I hadn't found yet. It'll be
interesting
> to see if any other scales share either of these vectors. They both
look
> like they could be useful in chord progressions, which is just the
sort of
> thing I'm looking for. I've already noted the usefulness of the
64/63.

In case you're interested (I hope you are): The various MIRACLE
scales (in 7-limit) all have 225:224 and 2401:2400 "commatic" unison
vectors; they differ (that is, those of different cardinalities
differ) in the choice of "chromatic" unison vector: 36:35 for
blackjack, 81:80 for canasta . . .

225:224 is a very important unison vector, coming up in the work of
Fokker, Lumma, etc. etc. . . .

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., Herman Miller <hmiller@I...> wrote:

> like they could be useful in chord progressions, which is just the
sort of
> thing I'm looking for.

Hi Herman!

Since you like chord progressions, check out Graham Breed's 2401:2400
pump progression (in the blackjack scale), which Joe Monzo arranged
(near the bottom of
http://www.ixpres.com/interval/monzo/blackjack/blackjack.htm). One
cool think about this is that 2401:2400 comes out to a full semitone
in 12-tET. This fact, combined with the fact that the progression
repeats after every 7 chords and is thus metrically odd, combined to
give Joseph Pehrson the impression that the progression never repeats
itself! I helped Joseph learn to play this progression on his
blackjack-tuned keyboard. In blackjack, one can write progressions
that make use of the 2401:2400 commatic unison vector, the 225:224
commatic unison vector, or any combination of the two (most notably
1029:1024).

There is a vast field of unison vectors out there. For example, Dan
Stearns' most recent post shows how one could make use of the
761332:759375 unison vector in 20-tET (Dan wrote 8374652/8353125 but
that's not in lowest terms).

🔗Paul Erlich <paul@stretch-music.com>

🔗D.Stearns <STEARNS@CAPECOD.NET>

Paul Erlich wrote,

<<There is a vast field of unison vectors out there. For example, Dan
Stearns' most recent post shows how one could make use of the
761332:759375 unison vector in 20-tET>>

Right, and as the whole thing is riding in on the coattails of the
5-limit model, 169/165 is analogous to 25/24 in the same way that
761332/759375 is analogous to 81/80.

I've been looking at a lot of these types of two-dimensional scales
lately. Remember Dave Keenan's 11-tone chain of minor thirds scale?
Well it could be seen in a similar light by way of the 3:4:5 (the
unison vectors being 15625/15552 and 16/15) in say 34-tet.

Here's the 1/6 comma meantone version with the tonic scale in the
X-VII-IV-I-IX position,

0 181 249 317 498 566 634 815 883 951 1132 1200
0 68 136 317 385 453 634 702 770 951 1019 1200
0 68 249 317 385 566 634 702 883 951 1132 1200
0 181 249 317 498 566 634 815 883 1064 1132 1200
0 68 136 317 385 453 634 702 883 951 1019 1200
0 68 249 317 385 566 634 815 883 951 1132 1200
0 181 249 317 498 566 747 815 883 1064 1132 1200
0 68 136 317 385 566 634 702 883 951 1019 1200
0 68 249 317 498 566 634 815 883 951 1132 1200
0 181 249 430 498 566 747 815 883 1064 1132 1200
0 68 249 317 385 566 634 702 883 951 1019 1200

--Dan Stearns

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:

> I've been looking at a lot of these types of two-dimensional scales
> lately.

Great -- maybe you can help us co-author a paper (go to tuning-math).
You may have missed the hypothesis I posted a while back but this
seems to relate.

> Remember Dave Keenan's 11-tone chain of minor thirds scale?
> Well it could be seen in a similar light by way of the 3:4:5 (the
> unison vectors being 15625/15552

The kleisma.

>and 16/15)

Good show! The Fokker periodicity block with unison vectors
15625/15552 and 16/15 is, in cents,

0
70.672
244.97
315.64
386.31
568.72
631.28
813.69
884.36
955.03
1129.3

with step sizes

70.6724
174.2964
70.6724
70.6724
182.4037
62.5651
182.4037
70.6724
70.6724
174.2964
70.6724

The difference between each pair of similar step sizes is the kleimsa
itself,

8.1073 cents.

So the kleisma is the commatic unison vector of this scale.

> in say 34-tet.

No need to specify an ET embedding, at this point.
>
> Here's the 1/6 comma meantone version

Let's restrict the "meantone" terminology to the meantone cases,
i.e., cases where the commatic unison vector is 81:80, shall we?

🔗D.Stearns <STEARNS@CAPECOD.NET>

Hi Paul and everyone,

<<You may have missed the hypothesis I posted a while back but this
seems to relate.>>

I guess I must've missed it. What I can do, and I've posted on this
quite a bit in the past, is find unison vectors for any given
two-stepsize scale.

<<Let's restrict the "meantone" terminology to the meantone cases,
i.e., cases where the commatic unison vector is 81:80, shall we?>>

Well that's okay with me, but it is after all a generalization of the
very same principles... so, any suggestions on what to call these
types of temperaments?

--Dan Stearns

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Paul and everyone,
>
> <<You may have missed the hypothesis I posted a while back but this
> seems to relate.>>
>
> I guess I must've missed it. What I can do, and I've posted on this
> quite a bit in the past, is find unison vectors for any given
> two-stepsize scale.

Awesome! Now we need to figure out how to do the reverse -- given a
set of unison vectors, one of which is "chromatic" while the other(s)
are "commatic", find the generator and interval of repetition of the
resulting MOS.
>
>
> <<Let's restrict the "meantone" terminology to the meantone cases,
> i.e., cases where the commatic unison vector is 81:80, shall we?>>
>
> Well that's okay with me, but it is after all a generalization of
the
> very same principles... so, any suggestions on what to call these
> types of temperaments?

"Forms of Tonality"???

🔗Paul Erlich <paul@stretch-music.com>

🔗D.Stearns <STEARNS@CAPECOD.NET>

I wrote,

"What I can do is find unison vectors for any given two-stepsize
scale."

That actually should've read,

"What I can do is find unison vectors for any given single generator
two-stepsize scale."

A simple example of this would be, say, the [4,3] neutral third 7-tone
scale which has been written about by Graham Breed and many others in
the past. The unison vectors that seem most in line with what people
interpret this scale as would be 59049/58564 and 729/704. But many
others such as 28672/28561 and 52/49 work as well.

More difficult, or convoluted examples would be those with fractional
or non-octave periodicity.

A familiar non-octave example would be the [5,4] Bohlen-Pierce scale.
A beautiful set of unison vectors for this scale are 118098/117649 and
343/324 (the 1/6th comma "meantone" version of this being a personal
favorite that I've posted about several times in the past).

An example of fractional periodicity would be Messiaen's [8,2] 10-tone
scale (AKA Paul Erlich's symmetrical decatonic). One possible set of
unison vectors for this scale, notable for their relative simplicity,
are 27/25 and 352/351 where P = 1:(sqrt(2)).

--Dan Stearns

🔗D.Stearns <STEARNS@CAPECOD.NET>

Hi Paul and everyone,

<<Now we need to figure out how to do the reverse -- given a set of
unison vectors, one of which is "chromatic" while the other(s) are
"commatic", find the generator and interval of repetition of the
resulting MOS.>>

I really never think in these terms, periodicity blocks and vectors,
but I'll see if I can't reverse what I do know and try to duplicate
the same sort of thing backwards.

"Forms of Tonality" would be way too vague for what I'm specifically
doing here. In a certain sense, not calling it meantone is like not
calling any non 12-et an equal temperament because that is usually
what is thought of and meant when one says equal temperament --
present company excluded!

But I agree with you; calling them "meantone" is too confusing.
However, they are just a generalization of the meantone method of
scale construction -- hasn't anyone given this specific class of
scales a name before?

--Dan Stearns

🔗Paul Erlich <paul@stretch-music.com>

🔗D.Stearns <STEARNS@CAPECOD.NET>

Hi Paul,

<<Yes -- you must have missed my post a couple of days ago where I
suggested "linear temperament" to you in this context (it was the post
immediately following the one where I suggested "Forms of Tonality".>>

Yeah, I've apparently been having some sort of problem with the email
the last couple of days -- certain posts arriving days after they were
sent while others are just fine.

Once again I think linear temperament is too general. Why bother
discerning between Pythagorean, 12-tet and 1/4 comma meantone then?
There's got to be something a bit more specific than that that won't
be confused with the traditional use of the term meantone -- or
perhaps the time has just come for meantone to be expanded in the same
fashion that equal temperament has by contemporary microtonalist?

--Dan Stearns

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗D.Stearns <STEARNS@CAPECOD.NET>

Hi Dave,

In a way I think that depends on what you want the PB as a generalized
model to accomplish, and I could give another more reasonable
(traditional) set of unison vectors, but the whole thing is kind of a
rhetorical question anyway.

I mean of course some things are more useful than others, and
especially so if it's required that they satisfy a certain set of
conditions, this is a no-brainer. My whole line of griping is based on
my experience that satisfying or not satisfying those conditions,
however seemingly well thought out by well thinking, qualified folks,
is not always the path to *the* answer. That may seem like a
no-brainer too, and well it should be, but I think that people do
sometimes tend to take the chart's or filter's word for it --
especially when the authors of the charts and filters talk in terms of
trash and miracles!

Call this politically correct yapping if you will (it might very well
be, I'm not sure), but I'm just speaking my mind based on my
experiences. You and everyone else is more than welcome to disagree,
ignore, or what-have-you with it all.

--Dan Stearns

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> I wrote,
>
> <<perhaps the time has just come for meantone to be expanded in the
> same fashion that equal temperament has by contemporary
> microtonalist?>>
>
> Dave Keenan wrote,
>
> <<No way!>>
>
> Hmm, why the emphatic opposition? The process I'm using is a
> contemporary minded 2D generalization of the same principles, sans
any
> historical clout of course... but what's the matter with that?
>
> Perhaps you think this sort of a generalization diminishes or
> dissipates the specialness of meantone (proper)?

I agree with Dave that your analogy doesn't hold water. This is
because the term "equal temperament" in no way implies 12,
while "meantone" talks about "tone" which _specifically_ refers to
the large step in the diatonic scale.

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

You got it. Also consider the etymology "mean tone". This of course
refers to it having, if not _the_ mean, at least some kind of mean
between the major and minor tones 8:9 and 9:10. It would just be
confusing to call something a meantone whose tone (two stacked fifths)
is _outside_ that range. At least the new things called "equal
temperaments" _are_ equal and they _are_ temperaments.

Doesn't "equitone" do the job for you?

-- Dave Keenan