Reply to Joseph Pehrson on adaptive JI

>Thanks to Paul Erlich and Daniel Wolf for the comments and suggestions
>regarding Just Intonation Schubert. I don't mean to be so far "behind the
>curve," but if you have time could someone explain a bit more fully what
>"adaptive" JI is?? (Ratio and pitch examples might help, too.) I don't
>seem to find that term here in my Doty JI primer which is, unfortunately,
>about the limit of my limits. Thanks!

OK, Joe. Let's take the progression I-IV-ii-V-I with common tones wherever
possible. In JI (strict _or_ adaptive), a major triad is is 4:5:6 and a
minor triad is 1/6:1/5:1/4. So that determines the vertical constitution of
the chords. But their horizontal relationship is not so straighforward. A
Doty-style JI advocate might render the progression as follows:

ratios:

1/1----------1/1 10/9----x-----9/8 5/4
3/2 5/3----------5/3 3/2----------3/2
5/4 4/3 10/9 15/8 2/1
1/1 4/3----------4/3 3/2 1/1
I IV ii V I

cents:

0-----------0 182----x----204 386
702 884---------884 702---------702
386 498 182 1088 1200
0 498---------498 702 0
I IV ii V I

while a JI rendition that observes common tones would go:

ratios:

1/1----------1/1 10/9---------10/9 100/81
3/2 5/3----------5/3 40/27--------40/27
5/4 4/3 10/9 50/27 160/81
1/1 4/3----------4/3 40/27 80/81
I IV ii V I

cents:

0------------0 182----------182 365
702 884----------884 680----------680
386 498 182 1067 1178
0 498----------498 680 -22
I IV ii V I

The first rendition is problematic because the shift of a full comma from
10/9 to 9/8 (from 182 to 204 cents) is large enough to sound like a
performer uncertainty in melodic intonation, particularly when the motivic
structure of the piece calls for an exact repetition and when the voice in
question is the soprano. The second rendition is problematic because if it
repeats enough times during the course of a piece, the "I" will drift so far
from the original "I" (one comma per repetition) that you lose the sense of
"home key".

The adaptive tuning solution that Vicentino would have chosen using his
second tuning of 1555, and that I've been trying to pitch to John
deLaubenfels, involves tuning the roots horizontally in meantone temperament
while maintaining the same vertical proportions as JI. While no longer
expressible in ratios, the progression becomes (in cents):

0-----.------5.5 193----.-----198.5 386
702 889.5---.----895 696.5---.----702
386 503.5 195 1083.5 1200
0 503.5---.----509 696.5 0
I IV ii V I

Here every common tone undergoes a shift of only 1/4-comma, an interval too
small to hear melodically. Meanwhile, there is no overall drift; the "I"
remains where it started. Essentially, one has distributed the comma
shift/drift equally over the four chord changes. Remember, each chord, in
itself, is in the same JI simple-integer proportions as before, so harmonic
consonance is not sacrificed at all.

In TD453.24 "Paul H. Erlich" <PErlich@Acadian-Asset.com> wrote:
>OK, Joe. Let's take the progression I-IV-ii-V-I with common tones wherever
>possible.
... [excellent explanation of a "comma pump" chord progression]
>The adaptive tuning solution that Vicentino would have chosen using his
>second tuning of 1555, and that I've been trying to pitch to John
>deLaubenfels, involves tuning the roots horizontally in meantone
>temperament while maintaining the same vertical proportions as JI.

That is a good solution for the comma pumps and will work for most music, but it won't save you from the diesis pumps (major and minor). i.e. a repeated sequence of either 3 chords whose major thirds stack or 4 chords whose minor thirds stack. These shift you by about 40 or 60 cents per iteration (much worse than comma pumps). The only solution I see to these is to leave the roots in 12-tET. Oh dear.

-- Dave Keenan
http://dkeenan.com

I wrote,

>>OK, Joe. Let's take the progression I-IV-ii-V-I with common tones wherever
>>possible.
... [excellent explanation of a "comma pump" chord progression]
>>The adaptive tuning solution that Vicentino would have chosen using his
>>second tuning of 1555, and that I've been trying to pitch to John
>>deLaubenfels, involves tuning the roots horizontally in meantone
>>temperament while maintaining the same vertical proportions as JI.

Dave Keenan wrote,

>That is a good solution for the comma pumps and will work for most music,
but it won't save you from the >diesis pumps (major and minor). i.e. a
repeated sequence of either 3 chords whose major thirds stack or 4 >chords
whose minor thirds stack.

Exactly. Dave, you may not have been following this discussion very closely,
but what I've been telling Joseph is that this latter trick is an important
feature of Schubert's music (though rare in composers before Schubert), and
that Schubert's music would therefore require greater sacrifices to put it
in adaptive JI:

>These shift you by about 40 or 60 cents per iteration (much worse than
comma pumps). The only solution >I see to these is to leave the roots in
12-tET. Oh dear.