the I-IV-V7-I cadence in different tunings

This post is copied from a new webpage I just made
exploring several different tunings of the I-IV-V7-I cadence
complete with MIDI-files:
http://www.ixpres.com/interval/td/monzo/i-iv-v7-i/i-iv-v-i.htm

> [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.

Here's the MIDI-file of Dave's adaptive-JI version:
http://www.ixpres.com/interval/td/monzo/i-iv-v7-i/keenan.mid

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.

Here's the MIDI-file of the 12-EDO version:
http://www.ixpres.com/interval/td/monzo/i-iv-v7-i/12edo.mid

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.

Here's the MIDI-file of this 'standard' 5-limit JI version:
http://www.ixpres.com/interval/td/monzo/i-iv-v7-i/5l-9-16.mid

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.

Here's the MIDI-file of this alternate 5-limit JI version:
http://www.ixpres.com/interval/td/monzo/i-iv-v7-i/5l-5-9.mid

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.

Here's the MIDI-file of the 7-limit rational version:
http://www.ixpres.com/interval/td/monzo/i-iv-v7-i/7l.mid

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.

Here's the MIDI-file of this version with a Pythagorean V7:
http://www.ixpres.com/interval/td/monzo/i-iv-v7-i/pythag.mid

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'.

Here's the MIDI-file of this version, where the V7 has a
Pythagorean 64:81 '3rd' and a 4:7 'harmonic 7th':
http://www.ixpres.com/interval/td/monzo/i-iv-v7-i/7l-64-81.mid

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 G:F '7th' is nearly a 7:10.

Here's the MIDI-file of the 1/4-comma meantone version:
http://www.ixpres.com/interval/td/monzo/i-iv-v7-i/1-4c-mt.mid

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

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Oops! In my last post, near the end, under the description
of the 1/4-comma meantone version, I wrote:

> [me, monz]
> 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 G:F '7th' is nearly a 7:10.

That last line is incorrect and should say:

> and the B:F 'diminished 5th' is nearly a 7:10.

-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 |
--------------------------------------------------

________________________________________________________________
YOU'RE PAYING TOO MUCH FOR THE INTERNET!
Juno now offers FREE Internet Access!
Try it today - there's no risk! For your FREE software, visit:
http://dl.www.juno.com/get/tagj.

Joe Monzo wrote,

>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

>Here's the MIDI-file of Dave's adaptive-JI version:
>http://www.ixpres.com/interval/td/monzo/i-iv-v7-i/keenan.mid

This could be the version that could satisfy both Dave Keenan and Gerald
Eskelin -- it has a 4:5:6:7 dominant seventh chord, but all the scale
degrees "adjust" by less than 10� at a time. However, the 4:5:6:7 did sound
a bit "alien" to me in this context. I feel (along with Mathieu) that
4:5:6:7 serves well as a tonic I7 chord in jazz/R&B, and even a subdominant
IV7 chord, but for the dominant which needs to "pull" toward the tonic, a
more unstable (and diatonic) sonority is called for.

>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

>Here's the MIDI-file of this 'standard' 5-limit JI version:
>http://www.ixpres.com/interval/td/monzo/i-iv-v7-i/5l-9-16.mid

This one sounded very right and good to me; the dominant seventh chord
seemed to resolve in a most natural way.

>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.

The comma-up shift in scale degree 4 sounds poor. I don't think anyone would
advocate this version.

Here's the MIDI-file of this alternate 5-limit JI version:
http://www.ixpres.com/interval/td/monzo/i-iv-v7-i/5l-5-9.mid

>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

>Here's the MIDI-file of the 7-limit rational version:
>http://www.ixpres.com/interval/td/monzo/i-iv-v7-i/7l.mid

That's the David Doty/Harold Fortuin version, and it made me nauseous. The
"low seventh" of the V at 3/2 sounds too flat to represent any kind of
"scale degree 4 in major", sounding instead like a microtonal experiment.

>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

>Here's the MIDI-file of this version with a Pythagorean V7:
>http://www.ixpres.com/interval/td/monzo/i-iv-v7-i/pythag.mid

The V triad sounds a bit too dissonant here, but if played more expressively
(unlike a computer), somthing like this might be effective.

>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'.

>Here's the MIDI-file of this version, where the V7 has a
>Pythagorean 64:81 '3rd' and a 4:7 'harmonic 7th':
>http://www.ixpres.com/interval/td/monzo/i-iv-v7-i/7l-64-81.mid

This combines the worst features of the two previous versions: dissonance
and nausea.

>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) )

>Here's the MIDI-file of the 1/4-comma meantone version:
>http://www.ixpres.com/interval/td/monzo/i-iv-v7-i/1-4c-mt.mid

This shows that the dominant seventh chords used by Handel, etc. were
noticeably more dissonant than the ones we use today. However, there isn't
the slightest bit of motion-sickness about this one, due to the fixed
pitches.

I hope Gerald Eskelin, Mark Nowitzky, and others will listen to these files
and share their honest impressions. Thanks again, Monz!!!

That's great Joe! To me, the two most musical versions are the 5-limit
16/9, and the Keenan 7-limit ones. I could never stomach the 9/5 in this
context, and the 1/4 comma version sounds anemic compared to the 16/9 just
one. The non-adaptive 7-limit versions sound very strange.

-Carl