Gene's notation formula: alternate duodene?

> Message 2185
> From: "genewardsmith" <genewardsmith@j...>
> Date: Wed Dec 26, 2001 6:25 pm
> Subject: Re: Gene's notation & Schoenberg lattices
> </tuning-math/message/2185>
>
> ...
>
> For any non-zero I can define a scale by calculating for 0<=n<d
>
> step[n] = (56/55)^round(7n/d) (33/32)^round(12n/d)
> (64/63)^round(7n/d) (81/80)^round(-2n/d) (45/44)^round(5n/d)

Hmmm ... i played around with Gene's notation
formula for the simpler 5-limit case of the Duodene,
http://www.ixpres.com/interval/dict/duodene.htm

and got this:

kernel
2 3 5 ratio ~cents unison-vector

[-3 -1 2] = 25:24 70.6724269 chromatic
[ 7 0 -3] = 128:125 41.0588584 commatic
[-4 4 -1] = 81:80 21.5062896 commatic

adjoint

[12 7 3]
[19 11 5]
[28 16 7]

JI periodicity-block

degree ratio

( 12 2/1 )
11 15/8
10 9/5
9 5/3
8 8/5
7 3/2
6 36/25
5 4/3
4 5/4
3 6/5
2 9/8
1 16/15
0 1/1

triangular lattice:

A E B
5:3-----5:4-----15:8
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
F C G D
4:3-----1:1------3:2-----9:8
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
Db Ab Eb Bb
16:15----8:5-----6:5-----9:5
\ /
\ /
\ /
Gb
36:25

I was surprised to see this result instead of the
actual Duodene, which would have F# 45/32 instead
of Gb 36/25 (the difference is one of the unison-vectors,
the diesis 128:125 = [7 0 -3] ), and would thus have
a perfect parallelogram lattice.

Gene, can you explain why your formula gave this
result instead of what i expected? Is there a
"correction factor" involved?

-monz

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> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, February 03, 2002 12:43 PM
> Subject: [tuning-math] Gene's notation formula: alternate duodene?
>
>
>
> > Message 2185
> > From: "genewardsmith" <genewardsmith@j...>
> > Date: Wed Dec 26, 2001 6:25 pm
> > Subject: Re: Gene's notation & Schoenberg lattices
> > </tuning-math/message/2185>
> >
> > ...
> >
> > For any non-zero I can define a scale by calculating for 0<=n<d
> >
> > step[n] = (56/55)^round(7n/d) (33/32)^round(12n/d)
> > (64/63)^round(7n/d) (81/80)^round(-2n/d) (45/44)^round(5n/d)

Also, i just realized that this statement of the formula
already has some specific values instead of variables,
so let's generalize it:

for a set of rational unison-vectors {u1/v1, ... un/vn},
for any non-zero I can define a scale by calculating for 0<=n<d

step[n] = (u1/v1)^round(7n/d) (u2/v2)^round(12n/d)
(u3/v3)^round(7n/d) (u4/v4)^round(-2n/d) (u5/v5)^round(5n/d)

-monz

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> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, February 03, 2002 1:05 PM
> Subject: Re: [tuning-math] Gene's notation formula: alternate duodene?
>
>
> Also, i just realized that this statement of the formula
> already has some specific values instead of variables,
> so let's generalize it:
>
>
> for a set of rational unison-vectors {u1/v1, ... un/vn},
> for any non-zero I can define a scale by calculating for 0<=n<d
>
> step[n] = (u1/v1)^round(7n/d) (u2/v2)^round(12n/d)
> (u3/v3)^round(7n/d) (u4/v4)^round(-2n/d) (u5/v5)^round(5n/d)

Oops! i also just realized that "n" is used as a variable
twice now, so make that

for a set of i rational unison-vectors {u1/v1, ... ui/vi},

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> Gene, can you explain why your formula gave this
> result instead of what i expected? Is there a
> "correction factor" involved?

My formula said to round to the nearest integer, but it didn't say what to do when two integers are equally near, which can happen when you have an even number of tones in an octave. The different rounding choices then lead to different blocks, which I think are equally correct.

The proposal for defining blocks a while back involved defining a distance function designed to work with a particular block problem in mind. In this case, it would give

||q|| = max(|h12(q)|, |12 h7(q) - 7 h12(q)|, |12 h3(q) - 3 h12(q)|

If you take everything at a distance of less than six from the unison using this measure, and transpose to the standard octave (instead of the octave from 2^(-1/2) to 2^(1/2)) you obtain the nine note scale

1--16/15--6/5--5/4--4/3--3/2--8/5--5/3--15/8

This is the core of the block, in every version of it. If you now take everything at a distance of exactly six from one, you get
{10/9, 9/8, 25/18, 45/32, 64/45, 36/25, 16/9, 9/5}. To get a block, you add three of these to the core of the block, in such a way that the diameter of the resulting block is less than twelve: the diameter
being the maximum of all the distances between members of the block.
You can therefore add 9/5, 9/8 and 45/32, getting what you expected,
or 64/45, 16/9, 10/9, which amounts to the same thing. However, you could also add 36/25, 9/5 and 9/8, which you didn't expect, or
16/9, 10/9, and 25/18.

my generalization of Gene's periodicity-block finding formula:

> for a set of i rational unison-vectors {u1/v1, ... ui/vi},
> for any non-zero I can define a scale by calculating for 0<=n<d
>
> step[n] = (u1/v1)^round(7n/d) (u2/v2)^round(12n/d)
> (u3/v3)^round(7n/d) (u4/v4)^round(-2n/d) (u5/v5)^round(5n/d)

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, February 03, 2002 5:23 PM
> Subject: [tuning-math] Re: Gene's notation formula: alternate duodene?
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > Gene, can you explain why your formula gave this
> > result instead of what i expected? Is there a
> > "correction factor" involved?
>
> My formula said to round to the nearest integer, but it
> didn't say what to do when two integers are equally near,
> which can happen when you have an even number of tones
> in an octave. The different rounding choices then lead
> to different blocks, which I think are equally correct.

so then i guess i'm just getting the particular results i
am because of the way Excel handles the rounding, yes?

> The proposal for defining blocks a while back involved
> defining a distance function designed to work with a
> particular block problem in mind. In this case, it would give
>
> ||q|| = max(|h12(q)|, |12 h7(q) - 7 h12(q)|, |12 h3(q) - 3 h12(q)|
>
> If you take everything at a distance of less than six from
> the unison using this measure, and transpose to the standard
> octave (instead of the octave from 2^(-1/2) to 2^(1/2)) you
> obtain the nine note scale
>
> 1--16/15--6/5--5/4--4/3--3/2--8/5--5/3--15/8
>
> This is the core of the block, in every version of it.
> If you now take everything at a distance of exactly six
> from one, you get
> {10/9, 9/8, 25/18, 45/32, 64/45, 36/25, 16/9, 9/5}.
> To get a block, you add three of these to the core of the
> block, in such a way that the diameter of the resulting
> block is less than twelve: the diameter being the maximum
> of all the distances between members of the block.
>
> You can therefore add 9/5, 9/8 and 45/32, getting what you
> expected, or 64/45, 16/9, 10/9, which amounts to the same
> thing. However, you could also add 36/25, 9/5 and 9/8,
> which you didn't expect, or 16/9, 10/9, and 25/18.

in fact, i just corrected an error in my spreadsheet and
the last one is exactly what i got:

i had my spreadsheet doing an incorrect calculation because
i didn't divide the unimodular adjoint by the determinant first,
as i should have.

so, according to the way my Excel spreadsheet is handling the
rounding in Gene's formula, here's one version of the 12-tone
JI PB scale for Ellis's Duodene:

kernel:
2 3 5 ratio ~cents unison-vector

[-3 -1 2] = 25:24 70.6724269 chromatic
[ 7 0 -3] = 128:125 41.0588584 commatic
[-4 4 -1] = 81:80 21.5062896 commatic

adjoint:

[12 7 3]
[19 11 5]
[28 16 7]

determinant = | 1 |

JI periodicity-block:

2 / 1 1200
15 / 8 1088.268715
16 / 9 996.0899983
5 / 3 884.358713
8 / 5 813.6862861
3 / 2 701.9550009
25 / 18 568.717426
4 / 3 498.0449991
5 / 4 386.3137139
6 / 5 315.641287
10 / 9 182.4037121
16 / 15 111.7312853
1 / 1 0

triangular lattice:

F#
25:18
/ \
/ \
/ \
D A E B
10:9----5:3-----5:4-----15:8
/ \ / \ / \ /
/ \ / \ / \ /
/ \ / \ / \ /
Bb F C G
16:9----4:3-----1:1-----3:2
/ \ / \ /
/ \ / \ /
/ \ / \ /
Db Ab Eb
16:15----8:5-----6:5

-monz

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> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, February 03, 2002 5:23 PM
> Subject: [tuning-math] Re: Gene's notation formula: alternate duodene?
>
>
>
> > The proposal for defining blocks a while back involved
> > defining a distance function designed to work with a
> > particular block problem in mind. In this case, it would give
> >
> > ||q|| = max(|h12(q)|, |12 h7(q) - 7 h12(q)|, |12 h3(q) - 3 h12(q)|
> >
> > If you take everything at a distance of less than six from
> > the unison using this measure, and transpose to the standard
> > octave (instead of the octave from 2^(-1/2) to 2^(1/2)) you
> > obtain the nine note scale
> >
> > 1--16/15--6/5--5/4--4/3--3/2--8/5--5/3--15/8
> >
> > This is the core of the block, in every version of it.
> > If you now take everything at a distance of exactly six
> > from one, you get
> > {10/9, 9/8, 25/18, 45/32, 64/45, 36/25, 16/9, 9/5}.
> > To get a block, you add three of these to the core of the
> > block, in such a way that the diameter of the resulting
> > block is less than twelve: the diameter being the maximum
> > of all the distances between members of the block.
> >
> > You can therefore add 9/5, 9/8 and 45/32, getting what you
> > expected, or 64/45, 16/9, 10/9, which amounts to the same
> > thing. However, you could also add 36/25, 9/5 and 9/8,
> > which you didn't expect, or 16/9, 10/9, and 25/18.

i just realized that i mention exactly the same thing on
my webpage "Ellis's Duodene and a "best-fit" meantone"
http://www.ixpres.com/interval/monzo/meantone/lattices/PB-MT.htm

>> Note that this periodicity-block has three pitch-classes
>> which fall right on the eastern boundary: (2,-1) = 9/5,
>> (2,0) = 9/8 and (2,1) = 45/32. All three of these thus
>> have alternates a comma lower -- in other words, by the
>> -[4,-1] = 80:81 unison-vector--, and the alternate
>> pitch-classes fall on the western boundary: (-2,0) = 16/9,
>> (-2,1) = 10/9, and (-2,2) = 25/18, respectively.
>>
>> Also, since (-2,2) = 25/18 and (2,1) = 45/32 happen to
>> fall right on the northwest and northeast *corners* of the
>> boundary (respectively), they also have lower alternates at
>> the distance of the *other* unison-vector -[0 3] = 64:125,
>> which would place the alternates at (-2,-1) = 64/45 and
>> (2,-2) = 36/25, respectively.

you and i are talking about exactly the same structures here,
even down to exactly the same pitches.

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> > > For any non-zero I can define a scale by calculating for 0<=n<d
> > >
> > > step[n] = (56/55)^round(7n/d) (33/32)^round(12n/d)
> > > (64/63)^round(7n/d) (81/80)^round(-2n/d) (45/44)^round(5n/d)
>
>
>
> Also, i just realized that this statement of the formula
> already has some specific values instead of variables,
> so let's generalize it:
>
>
> for a set of rational unison-vectors {u1/v1, ... un/vn},
> for any non-zero I can define a scale by calculating for 0<=n<d
>
> step[n] = (u1/v1)^round(7n/d) (u2/v2)^round(12n/d)
> (u3/v3)^round(7n/d) (u4/v4)^round(-2n/d) (u5/v5)^round(5n/d)

you really think this is a correct generalization?

From: paulerlich <paul@stretch-music.com>
To: <tuning-math@yahoogroups.com>
Sent: Sunday, February 03, 2002 8:37 PM
Subject: [tuning-math] Re: Gene's notation formula: alternate duodene?

> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> >
> > for a set of rational unison-vectors {u1/v1, ... un/vn},
> > for any non-zero I can define a scale by calculating for 0<=n<d
> >
> > step[n] = (u1/v1)^round(7n/d) (u2/v2)^round(12n/d)
> > (u3/v3)^round(7n/d) (u4/v4)^round(-2n/d) (u5/v5)^round(5n/d)
>
> you really think this is a correct generalization?

no, i was responding to your previous post and realized
that it's not. it still has numbers in it which come from
a particular homomorphism.

i'm working on a real generalization, and it will be in
my other post.

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> so, according to the way my Excel spreadsheet is handling the
> rounding in Gene's formula, here's one version of the 12-tone
> JI PB scale for Ellis's Duodene:
>
>
> kernel:
> 2 3 5 ratio ~cents unison-vector
>
> [-3 -1 2] = 25:24 70.6724269 chromatic
> [ 7 0 -3] = 128:125 41.0588584 commatic
> [-4 4 -1] = 81:80 21.5062896 commatic

25:24 is not a unison vector of the 12-tone scale at all. it is a
step vector. so if it's a step vector you want to use in gene's
formula that you generalized, 441:440 would sure seem like a poor
choice for the schoenberg case!

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, February 03, 2002 9:11 PM
> Subject: [tuning-math] Re: Gene's notation formula: alternate duodene?
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > so, according to the way my Excel spreadsheet is handling the
> > rounding in Gene's formula, here's one version of the 12-tone
> > JI PB scale for Ellis's Duodene:
> >
> >
> > kernel:
> > 2 3 5 ratio ~cents unison-vector
> >
> > [-3 -1 2] = 25:24 70.6724269 chromatic
> > [ 7 0 -3] = 128:125 41.0588584 commatic
> > [-4 4 -1] = 81:80 21.5062896 commatic
>
> 25:24 is not a unison vector of the 12-tone scale at all. it is a
> step vector. so if it's a step vector you want to use in gene's
> formula that you generalized, 441:440 would sure seem like a poor
> choice for the schoenberg case!

ok ... i'm s t i l l confused about equivalence interval,
period, and unison-vector, but i thought i had the difference
between commatic and chromatic unison-vectors.

now, what's the difference between a chromatic unison-vector
and a step-vector?

in the 12-tone scale, of which the duodene is an example,
there is a difference between C and C#, and in this case
that difference is most likely to be 25:24.

so why is that a step (which i would equate with a
"diatonic semitone") and not a chromatic unison-vector
(which i would equate with a "chromatic semitone")?

more confused ...

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> ok ... i'm s t i l l confused about equivalence interval,
> period, and unison-vector, but i thought i had the difference
> between commatic and chromatic unison-vectors.
>
> now, what's the difference between a chromatic unison-vector
> and a step-vector?

a chromatic unison vector corresponds to zero steps in the PB -- it
just takes you from one variant of a note to another.

> in the 12-tone scale, of which the duodene is an example,
> there is a difference between C and C#, and in this case
> that difference is most likely to be 25:24.

but 25:24 is a chromatic unison vector with respect to the 7-tone PB
that is the diatonic scale, but not in the case of a 12-tone PB.

> so why is that a step (which i would equate with a
> "diatonic semitone") and not a chromatic unison-vector
> (which i would equate with a "chromatic semitone")?

clearly in this case, a 12-tone closed system, there is no difference
between a diatonic semitone and a chromatic semitone. in a 7-tone PB,
step vectors would include 16:15, 10:9, and 9:8.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, February 03, 2002 9:22 PM
> Subject: [tuning-math] Re: Gene's notation formula: alternate duodene?
>
>

> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > in the 12-tone scale, of which the duodene is an example,
> > there is a difference between C and C#, and in this case
> > that difference is most likely to be 25:24.
>
> but 25:24 is a chromatic unison vector with respect to the 7-tone PB
> that is the diatonic scale, but not in the case of a 12-tone PB.
>
> > so why is that a step (which i would equate with a
> > "diatonic semitone") and not a chromatic unison-vector
> > (which i would equate with a "chromatic semitone")?
>
> clearly in this case, a 12-tone closed system, there is no difference
> between a diatonic semitone and a chromatic semitone. in a 7-tone PB,
> step vectors would include 16:15, 10:9, and 9:8.

ok ... i think i got it, but it sure is confusing stuff ...

Paul, i think you might be confusing some of the posts
i sent in today. some are about the Schoenberg 12-tone PB,
which is definitely meant to be 12-edo with all unison-vectors
tempered out, but others are about the duodene, which Ellis
meant to be JI with nothing tempered out.

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Sunday, February 03, 2002 9:22 PM
> > Subject: [tuning-math] Re: Gene's notation formula: alternate
duodene?
> >
> >
>
> > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > > in the 12-tone scale, of which the duodene is an example,
> > > there is a difference between C and C#, and in this case
> > > that difference is most likely to be 25:24.
> >
> > but 25:24 is a chromatic unison vector with respect to the 7-tone
PB
> > that is the diatonic scale, but not in the case of a 12-tone PB.
> >
> > > so why is that a step (which i would equate with a
> > > "diatonic semitone") and not a chromatic unison-vector
> > > (which i would equate with a "chromatic semitone")?
> >
> > clearly in this case, a 12-tone closed system, there is no
difference
> > between a diatonic semitone and a chromatic semitone. in a 7-tone
PB,
> > step vectors would include 16:15, 10:9, and 9:8.
>
>
>
> ok ... i think i got it, but it sure is confusing stuff ...
>
> Paul, i think you might be confusing some of the posts
> i sent in today. some are about the Schoenberg 12-tone PB,
> which is definitely meant to be 12-edo with all unison-vectors
> tempered out, but others are about the duodene, which Ellis
> meant to be JI with nothing tempered out.

25:24 is still a step vector in the duodede. look! you can find it
between distinct pitches in the duodene!

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, February 03, 2002 9:32 PM
> Subject: [tuning-math] Re: Gene's notation formula: alternate duodene?
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > Paul, i think you might be confusing some of the posts
> > i sent in today. some are about the Schoenberg 12-tone PB,
> > which is definitely meant to be 12-edo with all unison-vectors
> > tempered out, but others are about the duodene, which Ellis
> > meant to be JI with nothing tempered out.
>
> 25:24 is still a step vector in the duodede. look! you can find it
> between distinct pitches in the duodene!

but 25:24 only occurs between notes with the same letter-name
and a change of accidental! in the JI duodene, the sharps
and flats have distinct spellings, and 25:24 is indeed
functioning as a chromatic change on the same letter-name,
which is not what i would call a step.

i do understand, however, how it is a step-vector in 12-edo.

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> but 25:24 only occurs between notes with the same letter-name
> and a change of accidental! in the JI duodene, the sharps
> and flats have distinct spellings, and 25:24 is indeed
> functioning as a chromatic change on the same letter-name,
> which is not what i would call a step.

conventional diatonic notation is based on a 7-tone periodicity
block. the duodene mixes this notation with a 12-tone periodicity
block.

if an interval takes you from one pitch to another _within the
block_, that interval is *not* a unison vector of that block. if it
takes you between two adjacent, in pitch, notes in the block, it is a
step vector of the block.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, February 03, 2002 10:12 PM
> Subject: [tuning-math] Re: Gene's notation formula: alternate duodene?
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > but 25:24 only occurs between notes with the same letter-name
> > and a change of accidental! in the JI duodene, the sharps
> > and flats have distinct spellings, and 25:24 is indeed
> > functioning as a chromatic change on the same letter-name,
> > which is not what i would call a step.
>
> conventional diatonic notation is based on a 7-tone periodicity
> block. the duodene mixes this notation with a 12-tone periodicity
> block.

hmmm ... that's interesting.

> if an interval takes you from one pitch to another _within
> the block_, that interval is *not* a unison vector of that
> block. if it takes you between two adjacent, in pitch, notes
> in the block, it is a step vector of the block.

a h ! ! ! thanks, Paul, now it's clear as a mountain stream.

the "within the block" bit is the key that unlocked that
puzzle for me.

but now i'm really curious -- why is it necessary to put a
step-vector into the kernel to derive a notation. intuitively,
it makes sense to me, even as i ask it, but i'd still like
a good explanation of how and why it works.

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> but now i'm really curious -- why is it necessary to put a
> step-vector into the kernel to derive a notation.

because otherwise, you'd never get past the unison, to the second,
third, etc.!

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, February 03, 2002 11:36 PM
> Subject: [tuning-math] Re: Gene's notation formula: alternate duodene?
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > but now i'm really curious -- why is it necessary to put a
> > step-vector into the kernel to derive a notation.
>
> because otherwise, you'd never get past the unison, to the second,
> third, etc.!

right, that makes sense ... but how does only one step-vector
give you the whole scale? all the other steps can be derived
from that and the commas, apparently. ?

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> right, that makes sense ... but how does only one step-vector
> give you the whole scale? all the other steps can be derived
> from that and the commas, apparently. ?

yes. and not only all the other steps, but all possible JI ratios.