Lattice music - request for advice.

When I first started exploring lattices I was using W.A. Mathieu's
"Harmonic Experience" to assist me. I was drawn to the idea of composing
a few simple pieces using his 'Magic Mode' which is basically a section
of the 5 limit lattice. The score is given at :-

http://homepages.which.net/~alison.monteith3/image1.gif

The lattice has a spine of eight fifths, from 1/1 to 81/64 in one
direction and to 32/27 in the other. The thirds above are from 10/9 to
135/128, the thirds above that from 25/18 to 75/64. A third below the
1/1 spine there is a spine from 64/45 to 27/20.

A quick glance at the score lets you see that the piece, for guitar and
based on E so that I can have E and B drones throughout, moves around
the lattice fairly slowly, with the odd 'stolen ' tone from more distant
triads and the occasional leap. But there are no leaps big enough to
have to negotiate any commas.

I could render this in Just Intonation by simply using all 30 or so
pitches. Where I need help is to see how to find a temperament or
temperaments other than 12 suitable for playing the piece. I'm aware
that unison vectors will play a part but I'd be interested in a
'talk-through' of the processes. Any takers?

Thanks in anticipation.

Regards.

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:

> I could render this in Just Intonation by simply using all 30 or so
> pitches. Where I need help is to see how to find a temperament or
> temperaments other than 12 suitable for playing the piece. I'm aware
> that unison vectors will play a part but I'd be interested in a
> 'talk-through' of the processes. Any takers?

If you can render this in Just Intonation, that means you are not
making use of the vanishing of any unison vectors. Therefore, _any_
temperament which approximates Just Intonation will be fine. Here
we're talking about a 5-limit lattice, and your harmonic language is
based on triads, so we'll focus on the maximum error in the 5-limit
consonances. Here the ETs that give a lower maximum error than any
simpler ET (those that do so but with one exception are in
parentheses)

ET max.err.(¢)
12 16
(15) (18)
19 7
(22) (12)
31 6
34 4
(46) (5)
53 1.4
(65) (1.8)
(87) (1.6)
(99) (1.57)
118 0.39
(137) 1.23
(152) (0.68)
171 0.35
etc.

Hi Alison,

Is this the lattice?

25/18-25/24-25/16-75/64
/ \ / \ / \ / \
/ \ / \ / \ / \
10/9---5/3---5/4--15/8--45/32-135/128
/ \ / \ / \ / \ / \ / \
/ \ / \ / \ / \ / \ / \
32/27-16/9---4/3---1/1---3/2---9/8--27/16-81/64
\ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ /
64/45-16/15--8/5---6/5---9/5--27/20

> A quick glance at the score lets you see that the piece, for guitar
and
> based on E so that I can have E and B drones throughout

So which ratio represents E?

> I could render this in Just Intonation by simply using all 30 or so
> pitches. Where I need help is to see how to find a temperament or
> temperaments other than 12 suitable for playing the piece. I'm aware
> that unison vectors will play a part but I'd be interested in a
> 'talk-through' of the processes. Any takers?

Ooh yes. I just love challenges like this.

You say "all 30 or so pitches". I only count 24 pitches in the octave
equivalent lattice above. I assume you mean there are 30
(octave-specific) pitches used in the piece? Does it use all
the lattice positions?

Although it is described in 5-limit terms, there are some
close approximations to higher odd-number ratios lurking in that
lattice and some of these may be quite important since it is played
over drones. So the position of the drones on that lattice is
all-important in determining which temperaments may be suitable.

But it's already reminding me of Paul Erlich's Shrutar.

-- Dave Keenan

[Alison wrote:]
>When I first started exploring lattices I was using W.A. Mathieu's
>"Harmonic Experience" to assist me. I was drawn to the idea of composing
>a few simple pieces using his 'Magic Mode' which is basically a section
>of the 5 limit lattice. The score is given at :-
>
>http://homepages.which.net/~alison.monteith3/image1.gif
>
>The lattice has a spine of eight fifths, from 1/1 to 81/64 in one
>direction and to 32/27 in the other. The thirds above are from 10/9 to
>135/128, the thirds above that from 25/18 to 75/64. A third below the
>1/1 spine there is a spine from 64/45 to 27/20.
>
>A quick glance at the score lets you see that the piece, for guitar and
>based on E so that I can have E and B drones throughout, moves around
>the lattice fairly slowly, with the odd 'stolen ' tone from more distant
>triads and the occasional leap. But there are no leaps big enough to
>have to negotiate any commas.
>
>I could render this in Just Intonation by simply using all 30 or so
>pitches. Where I need help is to see how to find a temperament or
>temperaments other than 12 suitable for playing the piece. I'm aware
>that unison vectors will play a part but I'd be interested in a
>'talk-through' of the processes. Any takers?

I'm terrible at reading scores, but if you had a midi file of this,
I could recommend a fixed tuning that would tend to optimize consonance.

JdL

On Thu, 28 Jun 2001 22:16:39 +0100, Alison Monteith
<alison.monteith3@which.net> wrote:

>I could render this in Just Intonation by simply using all 30 or so
>pitches. Where I need help is to see how to find a temperament or
>temperaments other than 12 suitable for playing the piece. I'm aware
>that unison vectors will play a part but I'd be interested in a
>'talk-through' of the processes. Any takers?

Are you interested in exploiting unison vectors to reduce the number of
pitches you'll need to tune, or in avoiding scales with unison vectors to
keep all the pitches distinct?

It looks like the smallest intervals between adjacent steps in the scale
you're using are 81/80 (21.5 cents) and 2048/2025 (19.6 cents). If you
temper out the 81/80, you get the familiar meantone scales; 31-TET is a
nice one that happens to be very close to historical 1/4-comma meantone,
and other good ones include 19, 43, and 50-TET. All of the meantones have
relatively flat fifths.

2048/2025 is an interval called the "diaschisma", and tunings that
distribute the diaschisma belong to an interesting category that Graham
Breed talks about on this page: http://www.cix.co.uk/~gbreed/diaschis.htm

This kind of scale is built from two series of fifths a half an octave
apart. The fifths are slightly sharp in most cases. 34-TET has good thirds,
and fifths that are better than 31-TET; 46-TET has better fifths, but the
major thirds are only about as good as 43-TET. Diaschismic scales tend to
have better minor thirds than meantone scales.

(If you evenly distribute _both_ the 81/80 and the 2048/2025, you end up
with 12-TET.)

Probably the only other unison vector you might consider would be the
128/125. This gives you scales with major thirds as sharp as 12-TET, and
fifths that range from very sharp to very flat. 15-TET is worth exploring
(and with 15-TET, you also get the 250/243 unison vector to play with).
I've also been playing with 21-TET recently, which has some nice
properties. But these scales have a distinctly unusual flavor, and you're
probably better off starting out with the diaschismic or meantone
temperaments.

Other kinds of temperaments, such as schismic temperament, don't end up
unifying any of the notes in your scale, so then it becomes a matter of how
sharp or flat you like your intervals.

--
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

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

Assuming E is 1/1 (although I can find no Bb = 64/45 in the score)
then there are no significant approximations of higher limits to be
heard against the drones, and if your only requirement is to have
fewer notes while preserving 5-limit-JI as much as possible then you
should use a meantone (e.g. 1/4-comma or 31-EDO). Any meantone will
bring it down to 16 notes per octave (as you have notated them) or 15
since you don't seem to need the Bb.

-- Dave Keenan

I like Herman's explanation.

Yes, a diaschismic such as 1/6-diaschisma or 34-EDO may be worth
considering (for smaller errors relative to 5-limit JI), but it will
only eliminate 3 notes from the lattice of 24.

In summary:
In meantone you'll need 15 notes and can get down to 5.4c errors.
In diaschismic you'll need 21 notes and can get down to 3.3c errors.

Meantone gives you this:

(A#)--(E#)---B#----Fx
/ \ / \ / \ / \
/ \ / \ / \ / \
(F#)--(C#)--(G#)---D#----A#----E#
/ \ / \ / \ / \ / \ / \
/ \ / \ / \ / \ / \ / \
(G)---(D)---(A)----E-----B-----F#----C#----G#
\ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ /
Bb----F-----C-----G-----D-----A

The redundant notes are shown in parenthesis.
Because meantone tempers out the syntonic comma (21.5c) which is

81 3 * 3 * 3 * 3
-- = ------------- * 2^-4
80 5

or in octave-equivalent vector form [4, -1]

it conflates notes that are separated on the lattice by 4 fifths minus
a major third, like this.

(G)----*-----*-----*-----*
/
/
G

Diaschismic gives you this:

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

The redundant notes are shown in parenthesis.
Or equivalently

Bb----F-----C-----G
/ \ / \ / \ / \
/ \ / \ / \ / \
F#\---C#\---G#\---D#\---A#\---E#\
/ \ / \ / \ / \ / \ / \
/ \ / \ / \ / \ / \ / \
(G)----D-----A-----E-----B-----F#----C#----G#
\ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ /
(A#\)-(E#\)--B#\---Fx\---Cx\---Gx\

Because diaschismic tempers out the diaschisma (19.6c) which is

2048 1
---- = ----------- * 2^11
2025 3*3*3*3*5*5

or in octave-equivalent vector form [-4, -2]

it conflates notes that are separated on the lattice by 4 fifths plus
2 major thirds, like this.

G
/
/
*
/
/
(G)----*-----*-----*-----*

Regards,
-- Dave Keenan

From: Paul Erlich <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Thursday, June 28, 2001 2:33 PM
Subject: [tuning] Re: Lattice music - request for advice.

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:

> I could render this in Just Intonation by simply using all 30 or so
> pitches. Where I need help is to see how to find a temperament or
> temperaments other than 12 suitable for playing the piece. I'm aware
> that unison vectors will play a part but I'd be interested in a
> 'talk-through' of the processes. Any takers?

If you can render this in Just Intonation, that means you are not
making use of the vanishing of any unison vectors. Therefore, _any_
temperament which approximates Just Intonation will be fine. Here
we're talking about a 5-limit lattice, and your harmonic language is
based on triads, so we'll focus on the maximum error in the 5-limit
consonances. Here the ETs that give a lower maximum error than any
simpler ET (those that do so but with one exception are in
parentheses)

ET max.err.(�)
12 16
(15) (18)
19 7
(22) (12)
31 6
34 4
(46) (5)
53 1.4
(65) (1.8)
(87) (1.6)
(99) (1.57)
118 0.39
(137) 1.23
(152) (0.68)
171 0.35
etc.

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From: Paul Erlich <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Thursday, June 28, 2001 2:33 PM
Subject: [tuning] Re: Lattice music - request for advice.

> Here the ETs that give a lower maximum error than any
> simpler ET (those that do so but with one exception are in
> parentheses)
>
> ET max.err.(�)
> 12 16
> (15) (18)
> 19 7
> (22) (12)
> 31 6
> 34 4
> (46) (5)
> 53 1.4
> (65) (1.8)
> (87) (1.6)
> (99) (1.57)
> 118 0.39
> (137) 1.23
> (152) (0.68)
> 171 0.35
> etc.

Hmmm... 72's not on this list? Please explain where
and why it loses out. I know it gives the same good
approximation to 3 as 12-EDO, and a *much* better
approximation of 5.

???

PS - I think I sent a copy of Paul's post back by
mistake. Sorry, my bad.

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

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

In-Reply-To: <9hgtl7+ac5j@eGroups.com>
Dave Keenan wrote:

> Because diaschismic tempers out the diaschisma (19.6c) which is
>
> 2048 1
> ---- = ----------- * 2^11
> 2025 3*3*3*3*5*5
>
> or in octave-equivalent vector form [-4, -2]
>
> it conflates notes that are separated on the lattice by 4 fifths plus
> 2 major thirds, like this.
>
> G
> /
> /
> *
> /
> /
> (G)----*-----*-----*-----*

I should point out that a big chunk of Mathieu's book is about the unison
vectors of 12-equal. I don't think he calls them "unison vectors" but
they are anyway, and the diaschisma is certainly among them.

Graham

Dave Keenan wrote:

> Hi Alison,
>
> Is this the lattice?
>
> 25/18-25/24-25/16-75/64
> / \ / \ / \ / \
> / \ / \ / \ / \
> 10/9---5/3---5/4--15/8--45/32-135/128
> / \ / \ / \ / \ / \ / \
> / \ / \ / \ / \ / \ / \
> 32/27-16/9---4/3---1/1---3/2---9/8--27/16-81/64
> \ / \ / \ / \ / \ / \ /
> \ / \ / \ / \ / \ / \ /
> 64/45-16/15--8/5---6/5---9/5--27/20
>

Yes, that's pretty much the one.

> So which ratio represents E?

The 1/1

>
>
> > I could render this in Just Intonation by simply using all 30 or so
> > pitches. Where I need help is to see how to find a temperament or
> > temperaments other than 12 suitable for playing the piece. I'm aware
> > that unison vectors will play a part but I'd be interested in a
> > 'talk-through' of the processes. Any takers?
>
> Ooh yes. I just love challenges like this.
>
> You say "all 30 or so pitches". I only count 24 pitches in the octave
> equivalent lattice above. I assume you mean there are 30
> (octave-specific) pitches used in the piece? Does it use all
> the lattice positions?

Yes, the lattice in the book that set me off had a few more pitches on the line of fifths from 1/1
right and left but I didn't use these in the piece. The 24 pitches you give are the ones I use.

>
> Although it is described in 5-limit terms, there are some
> close approximations to higher odd-number ratios lurking in that
> lattice and some of these may be quite important since it is played
> over drones. So the position of the drones on that lattice is
> all-important in determining which temperaments may be suitable.

Do you work this out by looking at the cent equivalents of the 5 limit ratios and finding close
higher limit ratios? Or is there another method?

>
> But it's already reminding me of Paul Erlich's Shrutar.
>
> -- Dave Keenan

Thanks Dave.

"John A. deLaubenfels" wrote:

>
> I'm terrible at reading scores, but if you had a midi file of this,
> I could recommend a fixed tuning that would tend to optimize consonance.
>
> JdL

Thanks, John. Here's a midi file attached. This should be interesting. It'll take me a while to
listen to the results because I'll have to convert the .wav files to something a Mac can use.

Best wishes.

Herman Miller wrote:

> On Thu, 28 Jun 2001 22:16:39 +0100, Alison Monteith
> <alison.monteith3@which.net> wrote:
>
> >I could render this in Just Intonation by simply using all 30 or so
> >pitches. Where I need help is to see how to find a temperament or
> >temperaments other than 12 suitable for playing the piece. I'm aware
> >that unison vectors will play a part but I'd be interested in a
> >'talk-through' of the processes. Any takers?
>
> Are you interested in exploiting unison vectors to reduce the number of
> pitches you'll need to tune, or in avoiding scales with unison vectors to
> keep all the pitches distinct?

Both really. I'm interested in the process that a more adept theoretician than myself would use to
achieve these two objectives. I'm drawn to the idea of using lattices for some compositions and
then looking at the various ET and JI possibilities, a microtonal take on 'theme and variations'

>
> It looks like the smallest intervals between adjacent steps in the scale
> you're using are 81/80 (21.5 cents) and 2048/2025 (19.6 cents). If you
> temper out the 81/80, you get the familiar meantone scales; 31-TET is a
> nice one that happens to be very close to historical 1/4-comma meantone,
> and other good ones include 19, 43, and 50-TET. All of the meantones have
> relatively flat fifths.
>
> 2048/2025 is an interval called the "diaschisma", and tunings that
> distribute the diaschisma belong to an interesting category that Graham
> Breed talks about on this page: http://www.cix.co.uk/~gbreed/diaschis.htm
>
> This kind of scale is built from two series of fifths a half an octave
> apart. The fifths are slightly sharp in most cases. 34-TET has good thirds,
> and fifths that are better than 31-TET; 46-TET has better fifths, but the
> major thirds are only about as good as 43-TET. Diaschismic scales tend to
> have better minor thirds than meantone scales.
>
> (If you evenly distribute _both_ the 81/80 and the 2048/2025, you end up
> with 12-TET.)
>
> Probably the only other unison vector you might consider would be the
> 128/125. This gives you scales with major thirds as sharp as 12-TET, and
> fifths that range from very sharp to very flat. 15-TET is worth exploring
> (and with 15-TET, you also get the 250/243 unison vector to play with).
> I've also been playing with 21-TET recently, which has some nice
> properties. But these scales have a distinctly unusual flavor, and you're
> probably better off starting out with the diaschismic or meantone
> temperaments.
>
> Other kinds of temperaments, such as schismic temperament, don't end up
> unifying any of the notes in your scale, so then it becomes a matter of how
> sharp or flat you like your intervals.

Thanks. I appreciate you running through this for me. This will keep me busy for a wee while but I
can see the process clearly.

Best Wishes.

>

Paul Erlich wrote:

> --- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:
>
> > I could render this in Just Intonation by simply using all 30 or so
> > pitches. Where I need help is to see how to find a temperament or
> > temperaments other than 12 suitable for playing the piece. I'm aware
> > that unison vectors will play a part but I'd be interested in a
> > 'talk-through' of the processes. Any takers?
>
> If you can render this in Just Intonation, that means you are not
> making use of the vanishing of any unison vectors. Therefore, _any_
> temperament which approximates Just Intonation will be fine. Here
> we're talking about a 5-limit lattice, and your harmonic language is
> based on triads, so we'll focus on the maximum error in the 5-limit
> consonances. Here the ETs that give a lower maximum error than any
> simpler ET (those that do so but with one exception are in
> parentheses)
>
> ET max.err.(�)
> 12 16
> (15) (18)
> 19 7
> (22) (12)
> 31 6
> 34 4
> (46) (5)
> 53 1.4
> (65) (1.8)
> (87) (1.6)
> (99) (1.57)
> 118 0.39
> (137) 1.23
> (152) (0.68)
> 171 0.35
> etc.

So your method is to look at the limit of the piece, as defined by the lattice, then to look at
temperaments which offer the best 5-limit (in this case) consonances? Looks like it might lay out
well on a 31 or 34 tone guitar. This is most helpful and I appreciate the time you've taken to
answer my question. Time permitting, I'll see if I can score it out for 31 or 34, though I'm back
to notation issues for that.

Best Wishes.

>

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:

> So your method is to look at the limit of the piece, as defined by
the lattice, then to look at
> temperaments which offer the best 5-limit (in this case)
consonances?

Well of course Herman and Dave are right that those temperaments
which absorb the syntonic comma (such as 19 and 31) will require much
fewer pitches, and those that absorb the diaschisma (such as 22 and
34) will require a bit fewer pitches, than those that absorb neither
(such as 53). But you said you were interested in both cases --
reducing the number of pitches, and not reducing the number of
pitches. So my method gives you all the interesting possibilities
(assuming you want the purest possible consonances for a given
tightness of fret spacing, and assuming you want an ET!).

> Looks like it might lay out
> well on a 31 or 34 tone guitar. This is most helpful and I
appreciate the time you've taken to
> answer my question. Time permitting, I'll see if I can score it out
for 31 or 34, though I'm back
> to notation issues for that.

31 is notated conventionally, since it's a meantone. 34 will cause
problems . . . why not use tablature?

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> From: Paul Erlich <paul@s...>
> To: <tuning@y...>
> Sent: Thursday, June 28, 2001 2:33 PM
> Subject: [tuning] Re: Lattice music - request for advice.
>
>
> > Here the ETs that give a lower maximum error than any
> > simpler ET (those that do so but with one exception are in
> > parentheses)
> >
> > ET max.err.(¢)
> > 12 16
> > (15) (18)
> > 19 7
> > (22) (12)
> > 31 6
> > 34 4
> > (46) (5)
> > 53 1.4
> > (65) (1.8)
> > (87) (1.6)
> > (99) (1.57)
> > 118 0.39
> > (137) 1.23
> > (152) (0.68)
> > 171 0.35
> > etc.
>
>
>
> Hmmm... 72's not on this list? Please explain where
> and why it loses out. I know it gives the same good
> approximation to 3 as 12-EDO, and a *much* better
> approximation of 5.
>
> ???

I posted a response but it looks like the response got lost. Anyway,
53 and 65 _both_ beat 72 for max. 5-limit error, so it would only
have made it had I included doubly-parenthesized entries in the
table . . .

Here's the same table but for 7-limit:
ET max.err.(¢)
19 21
22 17
(26) (17)
27 14
31 6.0
(37) 12
(41) 6.3
(53) (6.2)
68 3.9
72 3.0
99 1.6
(103) (2.9)
(118) (2.9)
(130) 1.8
140 1.5
171 0.4
etc.

for 9-limit:

ET max.err.(¢)
19 21
22 19
(26) (20)
27 18
31 11
41 6.794
(46) (9)
53 6.167
(58) (6.790)
72 3.910
(77) (5.889) -- there it is!
(94) (4.722)
99 2.151
(118) (2.851)
(130) (2.212)
171 0.405
etc.

for 11-limit:

ET max.err.(¢)
22 20
(26) (22)
29 20
31 11.1
41 10.6
46 8.6
58 7.3
72 3.9
(80) (6.2)
(94) (4.7)
(111) (4.1)
118 2.9
(130) (3.4)
152 2.2
(159) 2.8
etc.

[I wrote:]
>>I'm terrible at reading scores, but if you had a midi file of this,
>>I could recommend a fixed tuning that would tend to optimize
>>consonance.

[Alison wrote:]
>Thanks, John. Here's a midi file attached. This should be interesting.
>It'll take me a while to listen to the results because I'll have to
>convert the .wav files to something a Mac can use.

Hey, this is the first time I've successfully brought down an attachment
from the list! I got your .mid file, and this is what I find (note that
this analysis is restricted in that it assumes _only_ 12 pitches classes
per octave; sorry Dave Keenan et al!). My recommended fixed tuning, in
circle-of-fifths order (values in cents relative to 12-tET):

Bb: -7.6
F : -12.4
C : -2.3
G : +15.5
D : +12.8
A : +4.4
E : +4.3
B : +5.7
F#: -2.6
C#: -2.0
G#: -4.9
D#: -11.1

This is based upon a 5-limit tuning-file free analysis of the intervals.
I'll post boring mathematical details in tuning-math.

If you're able to tune with more than 12 pitch classes per octave, that
is certainly preferable.

JdL

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> I got your .mid file, and this is what I find (note that
> this analysis is restricted in that it assumes _only_ 12 pitches
classes
> per octave; sorry Dave Keenan et al!). My recommended fixed
tuning, in
> circle-of-fifths order (values in cents relative to 12-tET):
>
> Bb: -7.6
> F : -12.4
> C : -2.3
> G : +15.5
> D : +12.8
> A : +4.4
> E : +4.3
> B : +5.7
> F#: -2.6
> C#: -2.0
> G#: -4.9
> D#: -11.1
>
> This is based upon a 5-limit tuning-file free analysis of the
intervals.

Can we hear it??

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:
> > Although it is described in 5-limit terms, there are some
> > close approximations to higher odd-number ratios lurking in that
> > lattice and some of these may be quite important since it is
played
> > over drones. So the position of the drones on that lattice is
> > all-important in determining which temperaments may be suitable.
>
> Do you work this out by looking at the cent equivalents of the 5
limit ratios and finding close
> higher limit ratios? Or is there another method?

Yes that's how. But after a while you get to know where the common
unison vectors (commas) go on the lattice.

[I wrote:]
>>I got your .mid file, and this is what I find (note that this analysis
>>is restricted in that it assumes _only_ 12 pitches classes per octave;
>>sorry Dave Keenan et al!). My recommended fixed tuning, in
>>circle-of-fifths order (values in cents relative to 12-tET):

>>Bb: -7.6
>>F : -12.4
>>C : -2.3
>>G : +15.5
>>D : +12.8
>>A : +4.4
>>E : +4.3
>>B : +5.7
>>F#: -2.6
>>C#: -2.0
>>G#: -4.9
>>D#: -11.1

>>This is based upon a 5-limit tuning-file free analysis of the
>>intervals.

[Paul E:]
>Can we hear it??

Not until you listen to the Chopin and the Gershwin! ;-> And eat your
peas. No, but seriously, folks... I'm still long-distance from my web
site, so I'll e-mail you a copy, Paul, unless there's lots of interest.
Alison, I'll send you a copy too.

JdL

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:
>Time permitting, I'll see if I can score it out
for 31 or 34, though I'm back
> to notation issues for that.

Alison,

You've already notated it just fine for 31. I gave 2 notations for 34
(Diaschismic) in
/tuning/topicId_25767.html#25794

But here's another that will require the minimum changes to your
existing score.

A#\\--E#\\--B#\\--Fx\\
/ \ / \ / \ / \
/ \ / \ / \ / \
F#\---C#\---G#\---D#\---A#\---E#\
/ \ / \ / \ / \ / \ / \
/ \ / \ / \ / \ / \ / \
(G)----D-----A-----E-----B-----F#----C#----G#
\ / \ / \ / \ / \ / \ /
\ / \ / \ / \ / \ / \ /
(Bb/)-(F/)---C/----G/----D/----A/

Nominals, sharps and flats are based on a chain of fifths as usual.
Unlike meantone, but like Pythagorean, A# is sharper than Bb. Change /
and \ to whatever you like. Say up and down arrows. :-) What they mean
is up or down by one step of 34-EDO.

G and Fx\\ are the same pitch. So are
Bb/ and A#\
F/ and E#\

Regards,
-- Dave Keenan

In a message dated 6/29/2001 2:41:56 PM Eastern Daylight Time,
alison.monteith3@which.net writes:

>
>
>
> "John A. deLaubenfels" wrote:
>
> >
> > I'm terrible at reading scores, but if you had a midi file of this,
> > I could recommend a fixed tuning that would tend to optimize consonance.
> >
> > JdL
>
> Thanks, John. Here's a midi file attached. This should be interesting.
> It'll take me a while to
> listen to the results because I'll have to convert the .wav files to
> something a Mac can use.
>
>

What are you converting Alison? Macintoshes can read .wav files. I played a
.wav file on my mac, in plain old quick time, a few minutes ago. Perhaps the
file just got corrupted and thats why it wont play?(just guessing)

Cheers,
Andy

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> From: Paul Erlich <paul@s...>
> To: <tuning@y...>
> Sent: Thursday, June 28, 2001 2:33 PM
> Subject: [tuning] Re: Lattice music - request for advice.
>
>
> > Here the ETs that give a lower maximum error than any
> > simpler ET (those that do so but with one exception are in
> > parentheses)
> >
> > ET max.err.(¢)
> > 12 16
> > (15) (18)
> > 19 7
> > (22) (12)
> > 31 6
> > 34 4
> > (46) (5)
> > 53 1.4
> > (65) (1.8)
> > (87) (1.6)
> > (99) (1.57)
> > 118 0.39
> > (137) 1.23
> > (152) (0.68)
> > 171 0.35
> > etc.
>
>
>
> Hmmm... 72's not on this list? Please explain where
> and why it loses out. I know it gives the same good
> approximation to 3 as 12-EDO, and a *much* better
> approximation of 5.
>
> ???

The list includes ETs that are better that all simpler ETs (and in
parentheses, ETs that are better than all simpler ETs but one),
where "better" means "lower max. 5-limit error". 72 is not on the
list because both 53 and 65 are "better" in this particular sense.

Here's the same list for 7-limit instead of 5-limit:

ET max.err.(¢)
19 21
22 17
(26) (17)
27 14
31 6.0
(37) (12)
(41) (6.3)
(53) (6.2)
68 3.9
72 3.0
99 1.6
(103) (2.9)
(118) (2.8)
(130) (1.8)
140 1.5
171 0.4
etc.

_Now_ 72 shows up (as it would for 9-limit, 11-limit, and so on up to
17-limit).

Andy wrote:

> What are you converting Alison? Macintoshes can read .wav files. I played a
> .wav file on my mac, in plain old quick time, a few minutes ago. Perhaps the
> ffile just got corrupted and thats why it wont play?(just guessing)

Well, .wav files are a real kettle of worms. They support an open-ended set of
codecs, and if the right one isn't installed on your machine, they won't play.
Cross-platform applications that "work with .wav files" often only understand
straight PCM.

--

Graham

"I toss therefore I am" -- Sartre

> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, June 29, 2001 2:29 PM
> Subject: [tuning] 77-tET (was: Re: Lattice music - request for advice.)
>
>
> I posted a response but it looks like the response got lost. Anyway,
> 53 and 65 _both_ beat 72 for max. 5-limit error, so it would only
> have made it had I included doubly-parenthesized entries in the
> table . . .

Duh! Thanks, Paul. Of course, I already knew that 53-EDO
provides by far the lowest error from 5-limit JI for its cardinality
range. I think I know what confused me... read on...

>
> Here's the same table but for 7-limit:

Yup... 72 is there in all three of the subsequent tables you posted,
and has much lower error for its cardinality range than anything
before it.

I've made tables like this long ago (actually, I turned the
tables into graphs and looked at *those*... you know me...),
and realized that 72-EDO was a terrific canditate for representing
lots of JI pitches with an EDO. Guess I just forgot about 53 for
a moment. Thanks.

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

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

From: Alison Monteith <alison.monteith3@which.net>
To: <tuning@yahoogroups.com>
Sent: Friday, June 29, 2001 11:36 AM
Subject: Re: [tuning] Re: Lattice music - request for advice.

>
>
> Paul Erlich wrote:
>
> > ET max.err.(�)
> > 12 16
> > (15) (18)
> > 19 7
> > (22) (12)
> > 31 6
> > 34 4
> > (46) (5)
> > 53 1.4
> > (65) (1.8)
> > (87) (1.6)
> > (99) (1.57)
> > 118 0.39
> > (137) 1.23
> > (152) (0.68)
> > 171 0.35
> > etc.
>
> So your method is to look at the limit of the piece, as
> defined by the lattice, then to look at temperaments which
> offer the best 5-limit (in this case) consonances? Looks
> like it might lay out well on a 31 or 34 tone guitar.
> This is most helpful and I appreciate the time you've taken
> to answer my question. Time permitting, I'll see if I can
> score it out for 31 or 34, though I'm back to notation
> issues for that.

Hi Allison,

Considering only this much of Paul's table:

> > ET max.err.(�)
> > 12 16
> > 19 7
> > 31 6
> > 34 4
> > 53 1.4

I'd think that 19 would give you the most "bang for the buck",
in terms of close approximations to your JI scale with minimum
number of "extra" notes.

The max.err. for 19 is only 1 cent more than for 31, and is
much better than 12, while introducing only 7 "extra" frets.

Plus, the notation presents no problem at all: one size of
semitone uses a sharp, the other uses a flat.

Neil Haverstick's book "19 Tones: A New Beginning" presents
a very clear illustration of it. And you should find this
webpage invaluable:
http://eceserv0.ece.wisc.edu/~sethares/tet19/guitarchords19.html

For more on 19-EDO as a representation of meantone, read the
section about it in my Woolhouse page, under:
http://www.ixpres.com/interval/monzo/woolhouse/essay.htm#temp

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

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

In a message dated 6/30/2001 6:06:23 AM Eastern Daylight Time,
graham@microtonal.co.uk writes:

> > What are you converting Alison? Macintoshes can read .wav files. I played
> a
> > .wav file on my mac, in plain old quick time, a few minutes ago. Perhaps
> the
> > ffile just got corrupted and thats why it wont play?(just guessing)
>
> Well, .wav files are a real kettle of worms. They support an open-ended
> set of
> codecs, and if the right one isn't installed on your machine, they won't
> play.
> Cross-platform applications that "work with .wav files" often only
> understand
> straight PCM.
>
> --
>
> Graham
>
>

well yes of course one has to have the right software to run them ... but it
is available for mac.

Andy

JoJoBuBu@aol.com wrote:

> In a message dated 6/29/2001 2:41:56 PM Eastern Daylight Time,
> alison.monteith3@which.net writes:
>
>
>
>>
>
> What are you converting Alison? Macintoshes can read .wav files. I
> played a
> .wav file on my mac, in plain old quick time, a few minutes ago.
> Perhaps the
> file just got corrupted and thats why it wont play?(just guessing)
>
> Cheers,
> Andy
>

I reckon there must be something up with the files because I have an all
singing and dancing system. I'll try a freeware converter of some sort.

Best Wishes.

>
>
> I'd think that 19 would give you the most "bang for the buck",
> in terms of close approximations to your JI scale with minimum
> number of "extra" notes.
>
> The max.err. for 19 is only 1 cent more than for 31, and is
> much better than 12, while introducing only 7 "extra" frets.
>
> Plus, the notation presents no problem at all: one size of
> semitone uses a sharp, the other uses a flat.
>
> Neil Haverstick's book "19 Tones: A New Beginning" presents
> a very clear illustration of it. And you should find this
> webpage invaluable:
> http://eceserv0.ece.wisc.edu/~sethares/tet19/guitarchords19.html
>
> For more on 19-EDO as a representation of meantone, read the
> section about it in my Woolhouse page, under:
> http://www.ixpres.com/interval/monzo/woolhouse/essay.htm#temp
>
> -monz
> http://www.monz.org
> "All roads lead to n^0"

Thanks Monz. I have Neil's book and will read up on your as ever comprehensive website.

Best Wishes.

Paul Erlich wrote:

> --- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:
>
> > So your method is to look at the limit of the piece, as defined by
> the lattice, then to look at
> > temperaments which offer the best 5-limit (in this case)
> consonances?
>
> Well of course Herman and Dave are right that those temperaments
> which absorb the syntonic comma (such as 19 and 31) will require much
> fewer pitches, and those that absorb the diaschisma (such as 22 and
> 34) will require a bit fewer pitches, than those that absorb neither
> (such as 53). But you said you were interested in both cases --
> reducing the number of pitches, and not reducing the number of
> pitches. So my method gives you all the interesting possibilities
> (assuming you want the purest possible consonances for a given
> tightness of fret spacing, and assuming you want an ET!).
>
> > Looks like it might lay out
> > well on a 31 or 34 tone guitar. This is most helpful and I
> appreciate the time you've taken to
> > answer my question. Time permitting, I'll see if I can score it out
> for 31 or 34, though I'm back
> > to notation issues for that.
>
> 31 is notated conventionally, since it's a meantone. 34 will cause
> problems . . . why not use tablature?
>

I will use tab for guitar pieces but I need to get a firm handle on
other notations because some
of my music will be re written for other instruments. Thanks again.

Regards.

>

In a message dated 6/30/2001 5:55:49 PM Eastern Daylight Time,
alison.monteith3@which.net writes:

> > What are you converting Alison? Macintoshes can read .wav files. I
> > played a
> > .wav file on my mac, in plain old quick time, a few minutes ago.
> > Perhaps the
> > file just got corrupted and thats why it wont play?(just guessing)
> >
> > Cheers,
> > Andy
> >
>
> I reckon there must be something up with the files because I have an all
> singing and dancing system. I'll try a freeware converter of some sort.
>
>

Sounds like a plan theres tons of them out there as I'm sure you already
know. Just out of curiosity though what is a singing and dancing system? A
system that can play just about anything I assume?(I can get mine to sing but
I've never seen it square dance :)

Andy

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:
>
> I will use tab for guitar pieces but I need to get a firm handle on
> other notations because some
> of my music will be re written for other instruments. Thanks again.

What instruments do you have in mind??? You're not likely to be able to easily get an accurate
performance in any ET other than 12 on most instruments . . . well maybe 24 . . .