diagram showing meantones as JI cycles

In my previous post, I gave an example of how
fraction-of-a-comma meantones can be thought of
as cycles of JI intervals.

/tuning/topicId_30093.html#30125

I've added a diagram to the bottom of the Dictionary
entry for "meantone", showing how fraction-of-a-comma
meantones can be plotted on a lattice as cycles of
5-limit JI intervals.

http://www.ixpres.com/interval/dict/meantone.htm

The particular example in my last post, 1/6-comma as
a cycle of 64:45's, is shown on the lattice in light blue.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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> From: monz <joemonz@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, November 13, 2001 3:38 AM
> Subject: [tuning] diagram showing meantones as JI cycles
>
>
> I've added a diagram to the bottom of the Dictionary
> entry for "meantone", showing how fraction-of-a-comma
> meantones can be plotted on a lattice as cycles of
> 5-limit JI intervals.
>
> http://www.ixpres.com/interval/dict/meantone.htm

And now I've added another animated lattice applet
below that, which shows individually how these meantones
can be plotted on a 2-dimensional lattice as 5-limit JI
interval cycles, and allows easy comparison.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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--- In tuning@y..., "monz" <joemonz@y...> wrote:
> I've added a diagram to the bottom of the Dictionary
> entry for "meantone", showing how fraction-of-a-comma
> meantones can be plotted on a lattice as cycles of
> 5-limit JI intervals.
>
> http://www.ixpres.com/interval/dict/meantone.htm

Monz,

Fraction-of-a-comma meantones are not "cycles" of anything. I assume
you mean "chains".

You seem to imply that a given meantone can be considered as a single
chain of some particular large number ratio. Of course, for any
interval other than a fifth (or its octave equivalents) it must
consist of _multiple_ chains (if any). But why this should be of
interest when these ratios are not perceptibly distinct, escapes me.

Regards,
-- Dave Keenan

Hi Dave, it's been a while. Nice to have you
criticizing me again. :)

> From: Dave Keenan <D.KEENAN@UQ.NET.AU>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, November 13, 2001 5:32 PM
> Subject: [tuning] Re: diagram showing meantones as JI cycles
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > I've added a diagram to the bottom of the Dictionary
> > entry for "meantone", showing how fraction-of-a-comma
> > meantones can be plotted on a lattice as cycles of
> > 5-limit JI intervals.
> >
> > http://www.ixpres.com/interval/dict/meantone.htm
>
> Monz,
>
> Fraction-of-a-comma meantones are not "cycles" of anything. I assume
> you mean "chains".

? - Please clarify the difference. I understand fraction-of-a-comma
meantones to be constructed of cycles of "5ths" narrowed (and/or
"4ths" widened) by that fraction-of-a-comma.

I'm familiar with "chains" in tuning terminology in reference
to the multiple-gear "bike chains", as in 72-EDO. What do they
mean in meantone? (no pun intended)

> You seem to imply that a given meantone can be considered as a single
> chain of some particular large number ratio. Of course, for any
> interval other than a fifth (or its octave equivalents) it must
> consist of _multiple_ chains (if any).

I don't immediately see that. Can you explain why?

For example, the 1/6-comma meantone cycle, *if carried out far
enough*, will contain as a subset within it the cycle of 64:45
ratios, which appears on a 5-limit lattice as a linear axis.

> But why this should be of interest when these ratios are
> not perceptibly distinct, escapes me.

[Disclaimer: I haven't done controlled experiments
to validite or contradict any of the following,
other than my own listening. No claims for relation
to any actual musical practice or accepted
psychoacoustical models.]

Mostly, I'm interested in this because of the visual
effect on the lattice. But I can try to explain my
interest in the visual patterns of the tuning mathematics
in terms of my own tentative theories of interval affect.

(I've been influenced in this area by my own
reactions to music in various tunings as well
as by the considerable tuning literature I've read.
Authors who have written about interval affect
include Darreg ("moods"), Partch ("one-footed bride"),
Scott Makeig, Danielou, Terpstra, Helmholtz,
Schoenberg, Bill Wesley, and countless others.)

The 5-limit JI "cycles" (as I called them),
which represent the meantones, form linear axes
on the lattice in distinction to the regular
3 and 5 axes, somewhat similar to the way
higher prime-factors form other axes in relation
to 3 and 5 on a typical extended-JI lattice.

So my feeling is that the patterns which meantones
on the lattice may portray something about the
mood that those particular tunings convey in musical
performance. They may not... which is OK too,
because I enjoy idle theoretical speculation anyway...

Another thing I discovered about this type of mapping
is that as you view fraction-of-a-comma meantones which
approximately particular EDOs more and more closely,
you get a sense of where the EDO linear axis would fall
on the lattice. Can any of the mathematicians come up
with a formula for this? (Example: 1/6-, 2/11-, and
3/17-comma are successively closer approximations to
55-EDO. I imagine that the 55-EDO axis would fall right
between the latter two.)

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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Hello, there, Dave Keenan and Monz.

With regard to "meantone cycles," my concept of a "virtual cycle"
might be more pragmatically musical than mathematical: if a
fractional-comma meantone can serve as a circulating tuning under
given stylistic and timbral conditions, then it may be called a
"cycle," although it offers no _precise_ and _obligatory_ mathematical
closure.

Please note that I'm not referring to JI here, only to "cycle" as in
"approximate circle of fifths" or "circulating temperament."

A classic example would be a 31-note cycle of 1/4-comma meantone,
where 31 regular fifths actually fall short of 18 pure octaves by
about 6.07 cents.

It makes a very nice circulating temperament, one which someone in the
16th or early 17th century might have tuned on an archicembalo or
_Sambuca Lincea_ without necessarily noticing the slight asymmetries.

Of course, if we assume a mathematically precise realization of this
"cycle," what we actually have are 30 regular fifths at 1/4-comma
narrow, plus one "odd" near-pure fifth about 6.07 cents wider
generated when we tune the octave of the starting note.

Also, there is nothing that prevents us from continuing to use the
tempered fifths and getting new intervals beyond our 31-note "cycle."

However, for historical or stylistic purposes, I find it convenient to
speak of a "31-note meantone cycle," which would include either
1/4-comma or a precise 31-tET -- and also any of the related shadings
that might arise in tuning an instrument by ear.

Maybe the term "virtual cycle" is most descriptive: we have an
optional "musical closure," but not a precise mathematical one.

This kind of question has had special interest for me because of the
question of how Vicentino (1555) or Colonna (1618) might have tuned
their 31-note "enharmonic" meantone cycles.

With Vicentino, 31-tET would fit his division of the whole-tone into
five apparently equal dieses, while 1/4-comma would fit the apparent
intent of his alternative adaptive JI tuning with two 19-note meantone
chains at about 1/4-comma apart that the major thirds on each keyboard
be "perfect," that is, at a pure 4:5.

Similarly, while it is often and very correctly stated that no
Pythagorean tuning of any size can form a precise mathematical
"cycle," since the powers of 3:2 cannot form any exact power of 2:1,
nevertheless there is a very nice "virtual cycle" of 53 notes, with
the "odd" fifth narrowed by about 3.62 cents.

Here I would say that the term "virtual cycle" generally tends to
imply near-symmetrical circulating systems of this kind, with
discrepancies like the 6.07 cents of 1/4-comma meantone in 31 notes or
the 3.62 cents of Pythagorean in 53 notes.

Under certain timbral conditions, a Pythagorean or 1/4-comma meantone
tuning in only 12 notes might serve as a kind of circulating tuning,
but here the asymmetries are great enough that we might want to use a
term such as "well-timbrement" to distinguish from a usual "virtual
cycle" of the kind I have described.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., "monz" <joemonz@y...> wrote:
> Hi Dave, it's been a while. Nice to have you
> criticizing me again. :)

Hopefully I can speak for Dave, as we tend to think alike.
>
>
> > From: Dave Keenan <D.KEENAN@U...>
> > To: <tuning@y...>
> > Sent: Tuesday, November 13, 2001 5:32 PM
> > Subject: [tuning] Re: diagram showing meantones as JI cycles
> >
> >
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > > I've added a diagram to the bottom of the Dictionary
> > > entry for "meantone", showing how fraction-of-a-comma
> > > meantones can be plotted on a lattice as cycles of
> > > 5-limit JI intervals.
> > >
> > > http://www.ixpres.com/interval/dict/meantone.htm
> >
> > Monz,
> >
> > Fraction-of-a-comma meantones are not "cycles" of anything. I
assume
> > you mean "chains".
>
>
> ? - Please clarify the difference. I understand fraction-of-a-comma
> meantones to be constructed of cycles of "5ths" narrowed (and/or
> "4ths" widened) by that fraction-of-a-comma.

A cycle closes; a chain may simply go on and on forever or until it
is broken.
>
> > You seem to imply that a given meantone can be considered as a
single
> > chain of some particular large number ratio. Of course, for any
> > interval other than a fifth (or its octave equivalents) it must
> > consist of _multiple_ chains (if any).
>
>
> I don't immediately see that. Can you explain why?

Because meantone is _generated_ by the fifth!
>
> For example, the 1/6-comma meantone cycle, *if carried out far
> enough*, will contain as a subset within it the cycle

chain

> of 64:45
> ratios, which appears on a 5-limit lattice as a linear axis.

Right -- but having the entire meantone tuning requires that there be
6 such chains, which is what Dave is getting at.

> So my feeling is that the patterns which meantones
> on the lattice may portray something about the
> mood that those particular tunings convey in musical
> performance. They may not... which is OK too,
> because I enjoy idle theoretical speculation anyway...

What about meantones like LucyTuning and Golden Meantone. Clearly
they don't convey a mood distinct from their a/b-comma meantone
cousins -- yet their patterns (could you even construct them) would
look totally different.
>
> Another thing I discovered about this type of mapping
> is that as you view fraction-of-a-comma meantones which
> approximately particular EDOs more and more closely,
> you get a sense of where the EDO linear axis would fall
> on the lattice. Can any of the mathematicians come up
> with a formula for this? (Example: 1/6-, 2/11-, and
> 3/17-comma are successively closer approximations to
> 55-EDO. I imagine that the 55-EDO axis would fall right
> between the latter two.)

There are no just intervals whatsoever in 55-EDO (except the octave).
So how do you assign it an "axis"?

--- In tuning@y..., "monz" <joemonz@y...> wrote:
> Hi Dave, it's been a while. Nice to have you
> criticizing me again. :)

Tee hee. Sorry.

> > Fraction-of-a-comma meantones are not "cycles" of anything. I
> > assume you mean "chains".
>
> ? - Please clarify the difference.

A cycle is a circle, a chain is open-ended. If you follow a cycle you
come back to where you started. If you follow a chain you
(generally) fall off the end.

> I'm familiar with "chains" in tuning terminology in reference
> to the multiple-gear "bike chains", as in 72-EDO. What do they
> mean in meantone? (no pun intended)

The bike chain is a red herring. Most chains are open-ended.

> > You seem to imply that a given meantone can be considered as a
single
> > chain of some particular large number ratio. Of course, for any
> > interval other than a fifth (or its octave equivalents) it must
> > consist of _multiple_ chains (if any).
>
> I don't immediately see that. Can you explain why?

Take any linear temperamant which is a single chain of some
generator. e.g.

Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B#
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+

If some interval I is approximated well by a chain of N generators,
e.g. a 4:5 by 4 generators. Then there must be N places you could
start a chain of Is and have no notes in common with any other such
chain. e.g.

Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B#
...+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+...
...+-----------+-----------+-----------+-----------+...
...+-----------+-----------+-----------+-----------+...
...+-----------+-----------+-----------+-----------+...
...+-----------+-----------+-----------+-----------+...

> For example, the 1/6-comma meantone cycle, *if carried out far
> enough*, will contain as a subset within it the cycle of 64:45
> ratios, which appears on a 5-limit lattice as a linear axis.

The 1/6-comma meantone chain, *if carried out far enough*, will
contain as subsets within it 6 chains of 64:45 ratios, which appear on
a 5-limit lattice as straight lines.

> The 5-limit JI "cycles" (as I called them),
> which represent the meantones, form linear axes
> on the lattice in distinction to the regular
> 3 and 5 axes, somewhat similar to the way
> higher prime-factors form other axes in relation
> to 3 and 5 on a typical extended-JI lattice.

A chain of any 5-prime-limit ratio will form a straight line on a
5-limit lattice. I don't think that should come as a surprise to
anyone. But they don't, in any sense I can see, form new _axes_ in the
way that higher primes do.

Regards,
-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> > For example, the 1/6-comma meantone cycle, *if carried out far
> > enough*, will contain as a subset within it the cycle of 64:45
> > ratios, which appears on a 5-limit lattice as a linear axis.
>
> The 1/6-comma meantone chain, *if carried out far enough*, will
> contain as subsets within it 6 chains of 64:45 ratios, which appear
on
> a 5-limit lattice as straight lines.

Actually, only one of them can appear on the 5-limit lattice at all.
Placing the others on it involves an arbitrary (and generally,
musically inappropriate) decision on how to plot irrational numbers.

Hi Dave, sorry to take so long to respond.

> From: Dave Keenan <D.KEENAN@UQ.NET.AU>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, November 14, 2001 3:39 PM
> Subject: [tuning] Re: diagram showing meantones as JI cycles
>
> monz wrote:
>
> > > Fraction-of-a-comma meantones are not "cycles" of anything. I
> > > assume you mean "chains".
> >
> > ? - Please clarify the difference.
>
> A cycle is a circle, a chain is open-ended. If you follow a cycle you
> come back to where you started. If you follow a chain you
> (generally) fall off the end.
>
> <etc. -- snip>

I want to thank you for clarifiying for me the
difference between "chain" and "cycle". Got it now.

Also -- that is, in addition to the as-yet uncorrected
description of these meantones as "cycles" instead of
"chains" -- I realize that I've mistakenly characterized
the actual lattice-plots of the linear meantone chains at
<http://www.ixpres.com/interval/dict/meantone.htm>.

In order even to qualify as "chains", the meantones I plotted
would all have to have several pitches in between the ones
I plotted, i.e., 1/3-comma would have two pitches between
each one that I plotted, 1/4-comma would have three, etc.

Also, I originally intended to draw the bounding limits of
those meantone chains more meaningfully, ending each one at
the points where pitches on either end of the cycle would
become audibly indistinguishable, i.e., 19 for 1/3-comma,
31 for 1/4-comma, 43 for 1/5-comma, 55 for 1/6-comma.

In order to portray what I describe in both of the above
two paragraphs, I would have to strech the scales of both
the 3- and 5-axes of my lattice.

So I'm writing this post both to acknowledge your response
to me before any more time passes, and to mention the changes
that I hope to make to the "meantone" Dictionary page someday
but probably won't any time soon (because it means quite a
bit of work).

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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