# xenharmonic-bridge

[Joe Monzo]

A term coined by Margo Schulter to indicate a very small interval which allows for a fluidity of rational interpretation when listening to music, by blurring the distinctions between intervals with different prime or odd factors. Similar, but not identical, to the concept of unison vector in Fokker's theory.

[Paul Erlich has pointed out that my concept of 'bridges' refers specifically to the kinds of prime-factors and not the numbers of prime-factors, which is what Fokker described with the periodicity block and unison vector concept.]

. . . . . . . . .

NOTE FROM 2002 UPDATE:

Monzo expanded the definition of xenharmonic bridge not long after writing the above definition in 1998, to also include the small intervals which occur between just ratios and their tempered equivalents. Examples are given below.

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[Paul Erlich, Yahoo tuning, message 53929 (Sat Jul 3, 2004 5:44 pm)]

In order to take you from one prime limit to another prime limit, it [a xenharmonic bridge] has to be conceived so that one of the pitch-ratios it separates belongs to the lower prime limit. Whereas if it's simply a unison vector, it doesn't have that restriction.

[Paul Erlich, Yahoo tuning, message 53974 (Sun Jul 4, 2004 4:54 pm)]

... the vector (or "monzo") of a xenharmonic bridge must e 1 (or -1) for the highest prime, while a unison vector has no such restriction. With a xenharmonic bridge, you're never bridging from the lower prime limit up to a pitch-ratio that features a higher power of the new, higher prime, rather than just the new, higher prime itself ...

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[Gene Ward Smith, Yahoo tuning, message 54185 (Thu Jul 8, 2004 12:33 am)]

--- In tuning@yahoogroups.com, "monz" wrote:

> just to clarify: that xenharmonic-bridge in Eratosthenes
> is thus a 3==19 bridge. so it skips 5 primes in between.

These are the kind of xenharmonic bridges used in sagittal, so we might call them sagittal bridges:

Definition 1: A xenharmonic bridge is a small rational interval such that the largest prime dividing it (ie. dividing either numerator or denominator in reduced form) has an exponent of +-1.

Definition 2: A *sagittal bridge* is a xenharmonic bridge with only 2, 3 and a single prime p>3 in its factorization; that is, a small rational interval of the form 2^a 3^b p or 2^a 3^b / p.

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[Joe Monzo]

###### REFERENCE

Tanaka, Shohé.

"Studien im Gebiete der reinen Stimmung",
Vierteljahrsschrift für Musikwissenschaft, vol. 6 no. 1
Friedrich Chrysander, Philipp Spitta, Guido Adler (eds.)
Breitkopf und Härtel, Leipzig, 1890, pp. 1-90

English translation of pages 8 to 18 by Daniel J. Wolf,
"Studies in the Realm of Just Intonation",
Xenharmonikôn vol. 16, autumn 1995, pp. 118-125
on the web at The Wilson Archives

(The earliest reference to this concept found by this author.)

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[Joe Monzo, Yahoo tuning-math, message 3286 (Thu Feb 7, 2002 2:00pm)]

Below is an example of a case where I believe that xenharmonic bridges are in effect. This is an examination of the notorious "comma-pump" chord progression as it appears in 12-edo.

> From: paulerlich
> To: > 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 12-edo.

```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 12-edo diatonic scale:

```        x=

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

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

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

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 12-edo:

```		    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 12-edo 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 12-edo 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 just intonation 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 12-edo:

```		      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 12-edo 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 me about music in 12-edo 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.

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