xenharmonic bridges in the 12edo comma pump (was: exactly what...)

ok, i've redone as much as i could of the post that disappeared.
actually, some of it turned out better this time. enjoy!

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, February 06, 2002 9:36 PM
> Subject: [tuning-math] Re: exactly what is a xenharmonic bridge?
>
>
> questions for monzo:
>
> (note i'm using the new notation now)
>
> are 80;81 and 128;125 and 648;625 and 2048;2025 and 32805;32768
> xenharmonic bridges in 12-tET?
>
> is 80;81 a xenharmonic bridge in all meantones?
>
> just trying to figure out what you mean by xenharmonic bridge.

it's hard for me to reason about this stuff abstractly,
because i don't know enough about the algebra. so i'll
have to use an example to illustrate.

how about if i pick what's probably the most meaningful
example? -- the "comma pump" progression in 12edo.

comma pump

I - vi - ii - V - I
C - Am - Dm - G - C

where "ratio" = 2^(x/12), the chords in this
progression are all subsets of notes in the
12edo diatonic scale:

x=

B 11
A 9
G 7
F 5
E 4
D 2
C 0

C [0 4 7] I
\ \
Am [9 0 4] vi
` - .
Dm [2 5 9] ii
` - .
G [7 11 2] V
` - .
C [0 4 7] I

i've used slashes and other marks to show the
common-tone relationships between pairs of chords.

the fact that every pair of chords here has at
least one common-tone is what enables the circular
progression in 12edo:

I = C
_______
.-' 0 `-.
,' 11 .. 1 `.
/ . . \
/ 10 . . 2 \
; . . :
| 9 . . 3 |
: . . ;
\ 8 . . 4 /
\ . ' /
`. 7 5,'
`-.___6___.-'

vi = Am
_______
.-' 0 `-.
,' 11 . . 1 `.
/ . . \
/ 10 . . 2 \
; . . :
| 9 . . 3 |
: ' . . ;
\ 8 ' . 4 /
\ /
`. 7 5,'
`-.___6___.-'

ii = Dm
_______
.-' 0 `-.
,' 11 1 `.
/ \
/ 10 . 2 \
; . ' ' :
| 9 .' ' 3 |
: . ' ;
\ 8 . ' 4 /
\ . ' /
`. 7 ' 5,'
`-.___6___.-'

V = G
_______
.-' 0 `-.
,' 11 . 1 `.
/ . ' . \
/ 10 . ' 2 \
; . ' :
| 9 . ' 3 |
: . ' ;
\ 8 . ' 4 /
\ . ' /
`. 7 5,'
`-.___6___.-'

I = C
_______
.-' 0 `-.
,' 11 .. 1 `.
/ . . \
/ 10 . . 2 \
; . . :
| 9 . . 3 |
: . . ;
\ 8 . . 4 /
\ . ' /
`. 7 5,'
`-.___6___.-'

this comma pump progression would need two
different D's to be tuned beat-free in JI:
a 10/9 for the Dm chord (= 5/3 utonality) and
a 9/8 for the G chord (= 3/2 otonality):

10:9----5:3-----5:4----15:8
D A E B
\ / \ / \ / \
\ / \ / \ / \
4:3-----1:1-----3:2-----9:8
F C G D

so assuming that the musical context implies
this JI structure, we note that since 12edo offers
only one D, the syntonic comma 81:80 is being
tempered out:

3==5 bridge

[2 3 5] [-3 2 0] = 9:8 Pythagorean D
- [2 3 5] [ 1 -2 1] = 10:9 JI D
--------------------
[2 3 5] [-4 4 -1] = 81:80 syntonic comma

but then we must also note that there are two
xenharmonic bridges in effect for the relationship
of the 12edo D to each of the JI D's :

12edo==Pythagorean bridge for 9/8

[2 3 5] [ -3 2 0] = 9/8 Pythagorean D
- [2 3 5] [ 1/6 0 0] = 12edo D
-----------------------
[2 3 5] [-19/6 2 0] = ~3.910001731 cents

12edo==JI bridge for 10/9

[2 3 5] [ 1/6 0 0] = 12edo D
- [2 3 5] [ 1 -2 1] = 10/9 JI D
----------------------
[2 3 5] [-5/6 2 -1] = ~17.59628787 cents

so our matrix for these three bridges is:

[2 3 5] [-4 4 -1] = 81:80 syntonic comma
[-19/6 2 0] = 12edo==9/8 bridge = ~3.910001731 cents
[ -5/6 2 -1] = 12edo==10/9 bridge = ~17.59628787 cents

and note that these three bridges are linearly dependent.

now, most likely in "common-practice" repertoire a
meantone harmonic paradigm is intended at least part
of the time. the equivalent meantone to 12edo is
1/11-comma meantone, and the two tunings are very
close indeed:

12edo==1/11cmt bridge

[2 3 5] [ 1/6 0 0 ] = 12edo D
- [2 3 5] [-25/11 14/11 2/11] = 1/11cmt D
-------------------------------
[2 3 5] [161/66 -14/11 -2/11] = ~0.000232741 = ~1/4300 cent

and here are the 1/11-comma meantone bridges to the
two JI pitches:

1/11cmt==Pythagorean bridge for 9/8

[2 3 5] [ -3 2 0 ] = 9/8 Pythagorean D
- [2 3 5] [-25/11 14/11 2/11] = 1/11cmt D
-------------------------------
[2 3 5] [ -8/11 8/11 -2/11] = ~3.910234472 cents

1/11cmt==JI bridge for 10/9

[2 3 5] [-25/11 14/11 2/11] = 1/11cmt D
- [2 3 5] [ 1 -2 1 ] = 10/9 JI D
-------------------------------
[2 3 5] [-36/11 36/11 -9/11] = ~17.59605512 cents

and again, note that these last two bridges and the
syntonic comma are linearly dependent.

so here's the entire list of bridges which i would say
are in effect for the comma pump in 12edo:

2 3 5 ~cents

[ -4 4 -1 ] = 81:80 syntonic comma = 21.5062896
[ -5/6 2 -1 ] = 12edo==10/9 bridge = 17.59628787
[-36/11 36/11 -9/11] = 1/11cmt==10/9 bridge = 17.59605512
[ -8/11 8/11 -2/11] = 1/11cmt==9/8 bridge = 3.910234472
[-19/6 2 0 ] = 12edo==9/8 bridge = 3.910001731
[161/66 -14/11 -2/11] = 12edo==1/11cmt bridge = 0.000232741

now, i'm not claiming that any listener is consciously
aware of all of these xenharmonic bridges at any given time.

but any intelligent harmonic analysis of 12edo performance
of "common-practice" repertoire (a good example is the
thousands of MIDI files of this repertoire -- without
any pitch-bend -- which are in existence), must take
these bridges into account.

so if anyone wants to say something to m e about music
in 12edo which features the comma pump, it would be a
good idea to mention something about this batch of intervals.

so paul, should i be using the new notation (\ and ;) for
the 81:80 here?

i could find similar sets of dependent vectors using
128;125, 648;625, 2048;2025, and 32805;32768 instead
of the syntonic comma.

-monz

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

> so our matrix for these three bridges is:
>
> [2 3 5] [-4 4 -1] = 81:80 syntonic comma
> [-19/6 2 0] = 12edo==9/8 bridge = ~3.910001731 cents
> [ -5/6 2 -1] = 12edo==10/9 bridge = ~17.59628787 cents
>
>
> and note that these three bridges are linearly dependent.

The second bridge is just P^(1/6), where P is the Pythagorean comma.
The Pythagorean and syntonic commas together define the 12-et in the
5-limit; one way to express that is 81/80 ^ P = 81/80 X P (where the wedge product in this case becomes the vector cross-product) =
h12, the [12, 19, 28] val. Your other bridge is W^(1/6), where
W = (81/80)^6 P^(-1). Is there anything gained by taking roots of commas? I don't see it.

> so here's the entire list of bridges which i would say
> are in effect for the comma pump in 12edo:
>
> 2 3 5 ~cents
>
> [ -4 4 -1 ] = 81:80 syntonic comma = 21.5062896
> [ -5/6 2 -1 ] = 12edo==10/9 bridge = 17.59628787
> [-36/11 36/11 -9/11] = 1/11cmt==10/9 bridge = 17.59605512
> [ -8/11 8/11 -2/11] = 1/11cmt==9/8 bridge = 3.910234472
> [-19/6 2 0 ] = 12edo==9/8 bridge = 3.910001731
> [161/66 -14/11 -2/11] = 12edo==1/11cmt bridge = 0.000232741

It seems to me the only one on this list which is important for the comma pump is 81/80, and that defines it. The others define 12-et
and 1/11 comma meantone, and have nothing to do with the pump so far as I can see.

> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Thursday, February 07, 2002 6:00 AM
> Subject: [tuning-math] xenharmonic bridges in the 12edo comma pump (was:
exactly what...)
>
> ...
>
> so here's the entire list of bridges which i would say
> are in effect for the comma pump in 12edo:
>
> 2 3 5 ~cents
>
> [ -4 4 -1 ] = 81:80 syntonic comma = 21.5062896
> [ -5/6 2 -1 ] = 12edo==10/9 bridge = 17.59628787
> [-36/11 36/11 -9/11] = 1/11cmt==10/9 bridge = 17.59605512
> [ -8/11 8/11 -2/11] = 1/11cmt==9/8 bridge = 3.910234472
> [-19/6 2 0 ] = 12edo==9/8 bridge = 3.910001731
> [161/66 -14/11 -2/11] = 12edo==1/11cmt bridge = 0.000232741

oops ... that really should say "are in effect for the note D
for the comma pump in 12edo". i didn't examine the bridges
which are in effect for any of the other notes.

-monz

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