> [Dave Keenan, TD 529.21]
> ... the common-as-dirt I IV V7 I chord progression, with notes
> sustained between chords. e.g.
>
>     F---F
> E   |       E
>     |   D
> C---C       C
>         B   |
>     A       |
> G       G---G
>
> That implies that G:F is 9:16, and assuming we also want
> G:B to be 4:5, we get a size of 45:64 (610 c) for the
> B:F dim 5th.
>
> In 5-limit JI the D:F minor third is broken anyway and so we
> go with the latter result.


Huh?  What do you mean by 'broken'?


> [Dave]
> They [Jerry's singers] must somehow hide the not inconsiderable
> difference between the 9:16 and the 4:7 (27.3 cents). With the
> above progression, I think that the best they can do is to
> gliss the C and F down by 9.1 cents and the G up by 9.1 cents.


Dave means here that the tuning of 'C' in the C chord should
be 1/1, but that in the F chord it should be lowered by 9.1 cents
along with the 'F' root, and 'G' should be 3/2 in the C chord but
raised by 9.1 cents as the root of the G chord.  All other
notes are to be tuned in JI ratios from those roots.


          F 4/3 -9.1¢  ---F 4:7 over G
 E 5/4    |                               E 5/4
          |               D 2:3 over G
 C 1/1 ---C 1/1 -9.1¢                     C 1/1
          [=2:3 over F]   B 4:5 over G    |
          A 4:5 over F                    |
 G 3/2                    G 3/2 +9.1¢  ---G 3/2


in approximate cents-values:


            F  488.9...F  479.9
 E  386.3   |                     E 386.3
            |          D  213.0
 C    0.0...C 1190.9              C   0.0
                       B 1097.4   |
            A  875.3              |
 G  702.0              G  711.1...G 702.0



which would give a maximum distribution of the difference
between any two tones of 9.1 cents, or 1/3 the total difference.



I've made a webpage with MIDI-files of several different
tunings for the illustration given by Dave, and I intend to
add more tunings to it.


[listen to the MIDI-file of Dave's adaptive-JI version]


12-EDO
------

All notes in the example come from the basic scale:

B  2^(11/12)
A  2^( 9/12)
G  2^( 7/12)
F  2^( 5/12)
E  2^( 4/12)
D  2^( 2/12)
C  2^( 0/12)

The usual familiar 12-EDO/tET scale.

[listen to the MIDI-file]


5-limit 9:16
------------

The ratios used in the example:

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

The I and IV triads are tuned to the 5-limit JI 4:5:6 proportions,
and the V7 to 36:45:54:64 = 4:5:6|27:32.

[listen to the MIDI-file]



5-limit 5:9
-----------

The ratios used in the example:

F        4/3  27/20  
E  5/4               5/4
D              9/8
C  1/1   1/1         1/1
B             15/8
A        5/3
G  3/2         3/2   3/2


The I and IV triads are tuned to the 5-limit JI 4:5:6 proportions,
and the V7 to 20:25:30:36 = 4:5:6|5:6.

[listen to the MIDI-file]


7-limit
-------

The ratios used in the example:

F        4/3  21/16 
E  5/4               5/4
D              9/8
C  1/1   1/1         1/1
B             15/8
A        5/3
G  3/2         3/2   3/2

The I and IV triads are tuned to the 5-limit JI 4:5:6 proportions,
and the V7 to 4:5:6:7.

[listen to the MIDI-file]


Pythagorean
-----------

The ratios used in the example:

F        4/3   4/3  
E  5/4               5/4
D              9/8
C  1/1   1/1         1/1
B            243/128
A        5/3
G  3/2         3/2   3/2

The I and IV triads are tuned to the 5-limit JI 4:5:6 proportions,
and the V7 to 576:729:864:1024 == 3^2 : 3^6 : 3^3 : 3^0.

[listen to the MIDI-file]


7-limit 64:81
-------------

The ratios used in the example:

F        4/3  21/16 
E  5/4                5/4
D              9/8
C  1/1   1/1          1/1
B            243/128
A        5/3
G  3/2         3/2    3/2

The I and IV triads are tuned to the 5-limit JI 4:5:6 proportions,
and the V7 to 576:729:864:1008 == 3^2 : 3^3 : 3^6 | 6:7


The idea here is that the 'leading-tone' in the V7 chord is
raised, to have it approach its resolution to 1/1 more closely,
and the '7th' in the V7 is lowered, both to have it approach its
resolution to 5/4 more closely and to make it a 'harmonic 7th'.

[listen to the MIDI-file]


1/4-c m-t
---------

Abbreviation for '1/4-comma meantone'.
Each '5th' is narrowed by (81/80)^(1/4).

All notes used in the example come from the basic scale:

B  ( (2^-7)*(3^ 5) ) / ( (81/80)^( 5/4) ) 
E  ( (2^-6)*(3^ 4) ) / ( (81/80)^( 4/4) )
A  ( (2^-4)*(3^ 3) ) / ( (81/80)^( 3/4) )
D  ( (2^-3)*(3^ 2) ) / ( (81/80)^( 3/4) )
G  ( (2^-1)*(3^ 1) ) / ( (81/80)^( 1/4) )
C  ( (2^ 0)*(3^ 0) ) / ( (81/80)^( 0/4) )
F  ( (2^ 2)*(3^-1) ) / ( (81/80)^(-1/4) )

In this tuning, the G:B '3rd' is exactly a 4:5,
the G:D '5th' is almost exactly a 107:160 (107 is a prime),
and the B:F 'diminished 5th' is nearly a 7:10.

[listen to the MIDI-file]


-monz

Joseph L. Monzo    Philadelphia     monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
 |"...I had broken thru the lattice barrier..."|
 |                            - Erv Wilson     |
--------------------------------------------------