This is spooky! Help!

I need a little help figuring out what is going on here.

I'm considering working with the following "7-limit JI Scale" from
the Scala archive:

7-limit 19-tone scale
0: 1/1 0.000 unison, perfect prime
1: 28/27 62.961 1/3-tone
2: 27/25 133.238 large limma, BP small semitone
3: 10/9 182.404 minor whole tone
4: 81/70 252.680 Al-Hwarizmi's lute middle finger
5: 6/5 315.641 minor third
6: 5/4 386.314 major third
7: 9/7 435.084 septimal major third, BP third
8: 4/3 498.045 perfect fourth
9: 25/18 568.717 classic augmented fourth
10: 36/25 631.283 classic diminished fifth
11: 3/2 701.955 perfect fifth
12: 14/9 764.916 septimal minor sixth
13: 8/5 813.686 minor sixth
14: 5/3 884.359 major sixth, BP sixth
15: 140/81 947.320
16: 25/14 1003.802 middle minor seventh
17: 50/27 1066.762 octave - large limma
18: 27/14 1137.039 septimal major seventh
19: 2/1 1200.000 octave

I was originally considering working with 19-tET, but change my mind
toward a just scale.

Now, when I compare this scale with 19-tET, I get the following:

I'm going to use the "meantone" enharmonic notation that people on
this list were so helpful to show me.

By the way... could a just scale like this be considered an
"adJUSTed" meantone?? I'm curious (seriously).

Now... here is the "spooky" part. Let's put the two scales together
in the following chart, and here's what we get (in cents):

PITCH 19-tET 19JI difference

C 0 0 0
C# 63 63 0
Db 126 133 +7
D 189 182 -7
D# 253 253 0
Eb 316 316 0
E 379 386 +7
E# 442 435 -7
F 505 498 -7
F# 568 568 0
Gb 631 631 0
G 695 702 +7
G# 758 765 +7
Ab 821 814 -7
A 884 884 0
A# 947 947 0
Bb 1011 1004 -7
B 1074 1069 -7
B# 1137 1137 0
C 1200 1200 0

Now, isn't this a little peculiar? Why is the difference in cents
for this 7-limit scale always either a 7 or a -7??

This seems mighty strange. What's going on here?? Help!

_____________ ________ _______ _
Joseph Pehrson

Joseph wrote,

>I'm going to use the "meantone" enharmonic notation that people on
>this list were so helpful to show me.

Well this time, you'll _have_ to include cents deviations, since, for
example, the interval between G (3/2) and D (10/9) will be quite dissonant,
and the players should expect that from the score. Remember, standard
notation works for meantone, not JI.

>Now, isn't this a little peculiar? Why is the difference in cents
>for this 7-limit scale always either a 7 or a -7??

Or 0.

>This seems mighty strange. What's going on here?? Help!

That's neat, and of course the 7 in 7 cents (rounded) being the same as the
7 in 7 limit is just a coincidence. The first 7s come because 19-tET is
essentially 1/3-comma meantone, and 1/3 comma is 7 cents. So clearly you
could construct a 5-limit scale with a maximum 7 cents deviation from 19-tET
like this:

E#-------B#
\ / \
\ / \
\ / \
\ / \
G#--------D#--------A#
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
B---------F#--------C#
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
D---------A---------E
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
F---------C---------G
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
Ab--------Eb--------Bb
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
Cb--------Gb--------Db

The central spine will have 0 deviation; the right chain will have +7 cents
deviation, and the left chain will have -7 cents deviation (since these
chains are chains of minor thirds and minor thirds are essentially just in
19-tET).

Now the 7-limit ratios in the Scala file are:

C# = 28/27
D# = 81/70
E# = 9/7
G# = 14/9
A# = 140/81
Bb = 25/14
B# = 27/14

Hmm . . . that Bb doesn't look like it belongs there. Let's lattice the
whole thing out and see:

B---------F#
\ / \
\ / \ Bb
A# \ / \ |
\ \ / \ |
\ D---------A---------E
\ |\ / \ / \
\ | \ / \ / \
C#--------G# \ / \ / \
`.\ / \ / \
F---------C---------G
\ / \ / `.
\ / \ / E#-------B#
\ / \ / \
\ / \ / \
Ab--------Eb \
\ / \ \
\ / \ D#
\ / \
\ / \
Cb--------Gb--------Db

Looks kind of like Sparky the Wonder Robot. Not a very good choice (in terms
of connectedness or number of consonances) for a 7-limit JI scale which has
all approx. ±7 or 0 deviations from 19-tET. One can do much better. But why
does it work? Well, the changes in this tuning relative to the first one
are:

A#: 140/81 ÷ 125/72 = 224/225 = -7 cents; dev. goes from +7 to 0
C#: 28/27 ÷ 25/24 = 224/225 = -7 cents; dev. goes from +7 to 0
G#: 14/9 ÷ 125/81 = 126/125 = 14 cents; dev. goes from -7 to +7
E#: 9/7 ÷ 625/486 = 4374/4375 = 0 cents; dev. goes from -7 to -7
B#: 27/14 ÷ 625/324 = 4374/4375 = 0 cents; dev. goes from 0 to 0
A#: 140/81 ÷ 125/108 = 4374/4375 = 0 cents; dev. goes from 0 to 0
Bb: 25/14 ÷ 9/5 = 125/126 = -14 cents; dev. goes from +7 to -7

There's no rhyme or reason to this scale, Joseph, other than the probable
desire to get the result you found, Joseph: the rounded deviations from
19-tET are all 0 or ±7 cents. It's clearly not a Wilson or Fokker scale --
I'd call it a hack job.

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_18807.html#18845

Thank you *SO* much, Paul, for taking the time to study over the
19-tone just scale I was considering!

>
> Hmm . . . that Bb doesn't look like it belongs there. Let's lattice
the whole thing out and see:
>
>
> B---------F#
> \ / \
> \ / \ Bb
> A# \ / \ |
> \ \ / \ |
> \ D---------A---------E
> \ |\ / \ / \
> \ | \ / \ / \
> C#--------G# \ / \ / \
> `.\ / \ / \
> F---------C---------G
> \ / \ / `.
> \ / \ / E#-------B#
> \ / \ / \
> \ / \ / \
> Ab--------Eb \
> \ / \ \
> \ / \ D#
> \ / \
> \ / \
> Cb--------Gb--------Db
>
> Looks kind of like Sparky the Wonder Robot.

This is very funny, Paul... but *NOT* a good sign...

>Not a very good choice (in terms of connectedness or number of
consonances) for a 7-limit JI scale which has all approx. ±7 or 0
deviations from 19-tET. One can do much better.

Where could I find such a scale? There are no more 7-limit just
19-tone scales in the Scala archive...

>But why does it work? Well, the changes in this tuning relative to
the first one are:
>
> A#: 140/81 ÷ 125/72 = 224/225 = -7 cents; dev. goes from +7 to 0
> C#: 28/27 ÷ 25/24 = 224/225 = -7 cents; dev. goes from +7 to 0
> G#: 14/9 ÷ 125/81 = 126/125 = 14 cents; dev. goes from -7 to +7
> E#: 9/7 ÷ 625/486 = 4374/4375 = 0 cents; dev. goes from -7 to -7
> B#: 27/14 ÷ 625/324 = 4374/4375 = 0 cents; dev. goes from 0 to 0
> A#: 140/81 ÷ 125/108 = 4374/4375 = 0 cents; dev. goes from 0 to 0
> Bb: 25/14 ÷ 9/5 = 125/126 = -14 cents; dev. goes from +7 to -7
>

I'm not sure I'm quite following what you are doing here Paul.
Rather, I'll rephrase, I absolutely don't know what you are doing
here... What are you dividing by, again??

> There's no rhyme or reason to this scale, Joseph, other than the
probable desire to get the result you found, Joseph: the rounded
deviations from 19-tET are all 0 or ±7 cents.

You know, I was a little "suspicious" of all those 7's. The first
thing that struck me was that maybe it was all "set up" so that the
7's would "mysteriously" coincide with the idea of the 7-limit...

Are you saying that this scale is ersatz, a phoney??

>It's clearly not a Wilson or Fokker scale --I'd call it a hack job.

Well, after this pronouncement, Paul... it's going to be very
difficult for me to use this scale any further. In fact, I will
certainly *NOT* use it!

So either I have to go "back" to 19-tET, find a "better" 7-limit JI
19-tone scale somehow, OR consider the following 5-limit JI
19-tone scale from Scala:

5-limit 19-tone scale
0: 1/1 0.000 unison, perfect prime
1: 25/24 70.672 classic chromatic semitone
2: 135/128 92.179 major limma, large chroma
3: 16/15 111.731 minor diatonic semitone
4: 9/8 203.910 major whole tone
5: 75/64 274.582 classic augmented second
6: 6/5 315.641 minor third
7: 5/4 386.314 major third
8: 4/3 498.045 perfect fourth
9: 27/20 519.551 acute fourth
10: 45/32 590.224 tritone
11: 3/2 701.955 perfect fifth
12: 25/16 772.627 classic augmented fifth
13: 8/5 813.686 minor sixth
14: 5/3 884.359 major sixth, BP sixth
15: 27/16 905.865 Pythagorean major sixth
16: 225/128 976.537 augmented sixth
17: 9/5 1017.596 just minor seventh, BP seventh
18: 15/8 1088.269 classic major seventh
19: 2/1 1200.000 octave

Does this one work out any better?? There are no immediate "cutsie"
numerical correspondences with 19-tET and perhaps, given the former
scale, that's a good sign...

Any advice on a GOOD just 19-tone scale to use?? or would I be better
off with 19-tET? Remember, though, that means my music won't be
played in California.... [That's just a JOKE, list!]

__________ _______ ______ ___
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

> >Not a very good choice (in terms of connectedness or number of
> consonances) for a 7-limit JI scale which has all approx. ±7 or 0
> deviations from 19-tET. One can do much better.
>
> Where could I find such a scale? There are no more 7-limit just
> 19-tone scales in the Scala archive...

Hi Joseph -- perhaps I'll construct some when I get back to the
office.
>
> I'm not sure I'm quite following what you are doing here Paul.
> Rather, I'll rephrase, I absolutely don't know what you are doing
> here... What are you dividing by, again??

I'm dividing by the ratios implied by the first lattice I posted,
which is all 5-limit, three chains of minor thirds . . . go back and
read the post very carefully, I tend to be concise . . .
>
> Well, after this pronouncement, Paul... it's going to be very
> difficult for me to use this scale any further. In fact, I will
> certainly *NOT* use it!

Well, I wouldn't want theoretical considerations to ever override the
purely auditory appraisal of how it sounds . . . but frankly the
whole idea of writing a piece in 19-tET and then picking a random 19-
tone JI scale from Scala to play it in always struck me as, well,
analogous to performing Bach on a piano set up for one of Cage's
prepared piano pieces . . . any finite JI scale will have lots
of "wolves" relative to the corresponding ET, so one has to carefully
choose the tuning based on the composition, or else carefully compose
according to the tuning.
>
> So either I have to go "back" to 19-tET, find a "better" 7-limit JI
> 19-tone scale somehow, OR consider the following 5-limit JI
> 19-tone scale from Scala:
>
>
> 5-limit 19-tone scale
> 0: 1/1 0.000 unison, perfect prime
> 1: 25/24 70.672 classic chromatic semitone
> 2: 135/128 92.179 major limma, large chroma
> 3: 16/15 111.731 minor diatonic semitone
> 4: 9/8 203.910 major whole tone
> 5: 75/64 274.582 classic augmented second
> 6: 6/5 315.641 minor third
> 7: 5/4 386.314 major third
> 8: 4/3 498.045 perfect fourth
> 9: 27/20 519.551 acute fourth
> 10: 45/32 590.224 tritone
> 11: 3/2 701.955 perfect fifth
> 12: 25/16 772.627 classic augmented fifth
> 13: 8/5 813.686 minor sixth
> 14: 5/3 884.359 major sixth, BP sixth
> 15: 27/16 905.865 Pythagorean major sixth
> 16: 225/128 976.537 augmented sixth
> 17: 9/5 1017.596 just minor seventh, BP seventh
> 18: 15/8 1088.269 classic major seventh
> 19: 2/1 1200.000 octave
>
> Does this one work out any better?? There are no
immediate "cutsie"
> numerical correspondences with 19-tET and perhaps, given the former
> scale, that's a good sign...

But the deviations from 19-tET are very large, so your piece will
sound very different, both melodically and harmonicallymay wreak.
>
> Any advice on a GOOD just 19-tone scale to use?? or would I be
better
> off with 19-tET? Remember, though, that means my music won't be
> played in California.... [That's just a JOKE, list!]

Again, I can create all sorts of beautiful 19-tone JI scales, but
none of that will mean anything if you just stick your already-
written piece into those tunings. What I need to know is, in your
piece, which notes are you using with which other notes in harmonic
simultaneity. It may be that no JI scale will work, but remember, JI
and ET are two extremes, and there are a lot of possibilities in-
between.

At your service,
Paul

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_18807.html#18887

> --- In tuning@y..., jpehrson@r... wrote:
>
> > >Not a very good choice (in terms of connectedness or number of
> > consonances) for a 7-limit JI scale which has all approx. ±7
or 0 deviations from 19-tET. One can do much better.
> >
> > Where could I find such a scale? There are no more 7-limit just
> > 19-tone scales in the Scala archive...
>
> Hi Joseph -- perhaps I'll construct some when I get back to the
> office.

It would be great to hear some good ones. However, now, suddenly,
I'm really excited about 19-tET because of the consistency of the
thirds and fifths. I think, particularly with a live acoustic
instrument -- in this case the trombone -- that would be a better
scale to use... The player could really HEAR the consistent
intervals, eventually...

I'm pretty set on it... (unless I decide to change my mind! :))

> >
> > I'm not sure I'm quite following what you are doing here Paul.
> > Rather, I'll rephrase, I absolutely don't know what you are doing
> > here... What are you dividing by, again??
>
> I'm dividing by the ratios implied by the first lattice I posted,
> which is all 5-limit, three chains of minor thirds . . . go back
and read the post very carefully, I tend to be concise . . .
> >

Actually, I got that much... but I could't figure out how you derived
the ratios for the 5-limit. How does one do that??

> > Well, after this pronouncement, Paul... it's going to be very
> > difficult for me to use this scale any further. In fact, I will
> > certainly *NOT* use it!
>
> Well, I wouldn't want theoretical considerations to ever override
the purely auditory appraisal of how it sounds . . .

It *did* seem to work pretty well for my electronic piece VERKLARTE
NEUNZEHN... but that piece was pretty "dense" (I mean, in TEXTURE)
and also didn't use a live player... I may need something a bit more
"precise" for this next project...

>but frankly the whole idea of writing a piece in 19-tET and then
picking a random 19-tone JI scale from Scala to play it in always
struck me as, well, analogous to performing Bach on a piano set up
for one of Cage's prepared piano pieces . . .

When I was a teenager, one of my virtuoso young lady friends played
Chopin on my Cage-prepared grand piano. The result?? You can
imagine...

>any finite JI scale will have lots of "wolves" relative to the
corresponding ET, so one has to carefully choose the tuning based on
the composition, or else carefully compose according to the tuning.
>

This is a very interesting topic, since I would be inclined to
almost ALWAYS do the latter... That means USING the wolves and all
the rest simply by AUDITORY logic in a piece. I'm sure you might
find that kind of compositional method "intellectually suspect" and
possibly it is, but from my experience some of the results, I feel,
come out sounding SUPERIOR to the compositions where certain
intervals were CALCULATED and then the composer MADE himself use them
in certain places, even if the compositional logic didn't call for
it. In a way, that kind of compositional approach is a little like
the 12-tone method... INSISTING on using certain pitches even if they
aren't necessarily appropriate.

I suppose a lot of what works and what doesn't work in a composition
really just comes from the experience of COMPOSING and having
performances. Probably quite a bit of it doesn't make QUANTIFIABLE
sense! But, that might not necessarily mean that there is NO LOGIC
to it... Supposedly, there are several kinds of intelligences or
talents, and perhaps they all have separate logics... (!?) Experience
and listening to a lot of new music over the years, I believe, also
counts for something...

Regarding the SCALA scales... maybe it would be a nice idea to
sometime start some kind of "readme" file for the SCALA archive that
has more information about some of the scales. Perhaps Manuel op de
Coul would also like that idea.

The point is, there was ANOTHER 19-tone scale derived from a 31 tone
just scale that had been included in the scale archive and Manuel
ALSO said that it was TERRIBLE. He "improved" it, finally, so it made
some sense.

I have the impression that there are probably many "questionable"
scales in that archive that make no sense and are poorly constructed,
along with some fine ones.

It might be nice to have some kind of "readme" file that points out
some of the features of the better scales and which points people
towards one rather than another. I don't mean somebody has to go
through them all... but I mean just as a place to put certain
information, like the latticing that you did, so that people can
understand what's in there as people involve themselves with various
scale.

It could even be in the form of a text file in the archive with the
same name as the Scala file. In fact, that might be the best way, if
Manuel would approve of it...

Just an idea...

>
> Again, I can create all sorts of beautiful 19-tone JI scales, but
> none of that will mean anything if you just stick your already-
> written piece into those tunings. What I need to know is, in your
> piece, which notes are you using with which other notes in harmonic
> simultaneity. It may be that no JI scale will work, but remember,
JI and ET are two extremes, and there are a lot of possibilities in-
> between.
>

Well, as I mentioned, I would never think of "sticking a piece" into
a certain tuning. On the other hand, I would rarely know what notes
were going to be sounding together until I started composing and
"messing around" with it by ear... The tuning would come first, and
the ideas of simultaneities and structure would come AUDITORILY from
the tuning...

At least that's the way I would work with it.

Is that "intellectually suspect??" Dunno. I would be interested in
hearing other peoples' comments on this matter.

It might end up sounding pretty good, though...

> At your service,
> Paul

And I *greatly* appreciate it. For the time being, I'm sticking to
19-tET for this piece, but if you have time and can come up with some
great just 19-tone scales it would be nice if they could be added to
the archive, and I would be very interested in listening to them and
"messing around" with them...

Thanks again!!!!!

________ _____ ______ _
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

> Actually, I got that much... but I could't figure out how you derived
> the ratios for the 5-limit. How does one do that??

Those ratios were derived from the first lattice I posted. Since you understand the 5-limit lattice
very well, I'm sure you could reproduce those ratios yourself by looking at that lattice. Yes?
>
> >any finite JI scale will have lots of "wolves" relative to the
> corresponding ET, so one has to carefully choose the tuning based on
> the composition, or else carefully compose according to the tuning.
> >
>
> This is a very interesting topic, since I would be inclined to
> almost ALWAYS do the latter... That means USING the wolves and all
> the rest simply by AUDITORY logic in a piece. I'm sure you might
> find that kind of compositional method "intellectually suspect"

Not at all -- both methods are fine. One good way of doing the former might be to compose in
an ET and then use an adaptive tuning program (John deLaubenfels' is only set up for 12-tone
now, but perhaps eventually 19-tone will be feasible) to determine the final tuning.

> and
> possibly it is, but from my experience some of the results, I feel,
> come out sounding SUPERIOR to the compositions where certain
> intervals were CALCULATED and then the composer MADE himself use them
> in certain places, even if the compositional logic didn't call for
> it. In a way, that kind of compositional approach is a little like
> the 12-tone method... INSISTING on using certain pitches even if they
> aren't necessarily appropriate.

Wait a minute . . . now it sounds like you're criticizing the latter method, rather than supporting it . .
.?

>
> > Again, I can create all sorts of beautiful 19-tone JI scales, but
> > none of that will mean anything if you just stick your already-
> > written piece into those tunings. What I need to know is, in your
> > piece, which notes are you using with which other notes in harmonic
> > simultaneity. It may be that no JI scale will work, but remember,
> JI and ET are two extremes, and there are a lot of possibilities in-
> > between.
> >
>
> Well, as I mentioned, I would never think of "sticking a piece" into
> a certain tuning. On the other hand, I would rarely know what notes
> were going to be sounding together until I started composing and
> "messing around" with it by ear... The tuning would come first, and
> the ideas of simultaneities and structure would come AUDITORILY from
> the tuning...
>
> At least that's the way I would work with it.
>
> Is that "intellectually suspect??"

Of course not . . . anyhow, besides anything I might most, perhaps Dave Keenan can come up
with some "micro-tempered" 19-tone tunings?

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_18807.html#18946

> --- In tuning@y..., jpehrson@r... wrote:
>
> > Actually, I got that much... but I could't figure out how you
derived the ratios for the 5-limit. How does one do that??
>
> Those ratios were derived from the first lattice I posted. Since
you understand the 5-limit lattice very well, I'm sure you could
reproduce those ratios yourself by looking at that lattice. Yes?
> >

Thank you so much, Paul, for your optimism. However, I'm not getting
the numbers to work out, so I must be doing something wrong.

For instance, if I start with E# as 1/1 then from there to B# is 3/2.
Then I continue downward to the D#, so that should be 5/4 X 3/2, and
then to get to the A#, I would multiply by 3/2 again, so I get:

A# = 3/2 * 5/4 * 3/2 = 45/16

And YOU say that A# should be 125/72 so, clearly, I am doing
something wrong... :(

> > >any finite JI scale will have lots of "wolves" relative to the
> > corresponding ET, so one has to carefully choose the tuning based
on the composition, or else carefully compose according to the
tuning.
> > >
> >
> > This is a very interesting topic, since I would be inclined to
> > almost ALWAYS do the latter... That means USING the wolves and
all the rest simply by AUDITORY logic in a piece. I'm sure you might
> > find that kind of compositional method "intellectually suspect"
>
> Not at all -- both methods are fine. One good way of doing the
former might be to compose in an ET and then use an adaptive tuning
program (John deLaubenfels' is only set up for 12-tone now, but
perhaps eventually 19-tone will be feasible) to determine
the final tuning.
>

Sure... That sounds like it would be an interesting way to go...

> > and possibly it is, but from my experience some of the results, I
feel, come out sounding SUPERIOR to the compositions where certain
> > intervals were CALCULATED and then the composer MADE himself use
them in certain places, even if the compositional logic didn't call
for it. In a way, that kind of compositional approach is a little like
> > the 12-tone method... INSISTING on using certain pitches even if
they aren't necessarily appropriate.
>
> Wait a minute . . . now it sounds like you're criticizing the
latter
method, rather than supporting it . .
> .?
>

I'm just criticizing an approach where either a certain "lattice" or
12-tone structure is set up beforehand and the composition has to do
such and so at a certain point to satisfy the THEORY, rather than
using what I am now calling "sound logic!"

_________ ______ __ _
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
>
> For instance, if I start with E# as 1/1

Why would you do that? I'm following the convention of the Scala tuning file and calling 1/1 "C"
-- I'm not sure why you would use E# of all things!
>
> I'm just criticizing an approach where either a certain "lattice" or
> 12-tone structure is set up beforehand and the composition has to do
> such and so at a certain point to satisfy the THEORY, rather than
> using what I am now calling "sound logic!"

Well, if the theory doesn't agree with the "sound logic", then you need a better theory!

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_18807.html#18845

>
> E#-------B#
> \ / \
> \ / \
> \ / \
> \ / \
> G#--------D#--------A#
> \ / \ / \
> \ / \ / \
> \ / \ / \
> \ / \ / \
> B---------F#--------C#
> \ / \ / \
> \ / \ / \
> \ / \ / \
> \ / \ / \
> D---------A---------E
> \ / \ / \
> \ / \ / \
> \ / \ / \
> \ / \ / \
> F---------C---------G
> \ / \ / \
> \ / \ / \
> \ / \ / \
> \ / \ / \
> Ab--------Eb--------Bb
> \ / \ / \
> \ / \ / \
> \ / \ / \
> \ / \ / \
> Cb--------Gb--------Db

>
> A#: 140/81 ÷ 125/72 = 224/225 = -7 cents; dev. goes from +7 to 0
> C#: 28/27 ÷ 25/24 = 224/225 = -7 cents; dev. goes from +7 to 0
> G#: 14/9 ÷ 125/81 = 126/125 = 14 cents; dev. goes from -7 to +7
> E#: 9/7 ÷ 625/486 = 4374/4375 = 0 cents; dev. goes from -7 to -7
> B#: 27/14 ÷ 625/324 = 4374/4375 = 0 cents; dev. goes from 0 to 0
> A#: 140/81 ÷ 125/108 = 4374/4375 = 0 cents; dev. goes from 0 to 0
> Bb: 25/14 ÷ 9/5 = 125/126 = -14 cents; dev. goes from +7 to -7
>

Hi Paul...

Finally I'm getting this.

To get the Bb for the 5-limit lattice as 9/5:

It's simply going from C (1/1) to G (3/2) times the minor third (6/5):

3/2 * 6/5 = 18/10 = 9/5

Ok... so far so good.

And to get to the C# at 25/24..

First I go from C 1/1 to the G (3/2)

THEN, I DIVIDE since I'm going UP the lattice.

I have to go up to an E (5/6) and then again to a C# (multiplying
again by 5/6):

3/2 * 5/6 * 5/6 = 75/72 = 25/24 (C#)

GOT IT! Thank for not "giving it away" immediately... it was fun to
figure out!

______ ____ _____ _____ _
Joseph Pehrson

*One can't argue with "sound logic"*