Yahoo Tuning Groups Ultimate Backup tuning [tuning] Drei Equali in Just Intonation

EUREKA!!!!!! :D:D:D:D:D:DI"VE GOT IT!!!!!!!!!!!!!!!!!!!

YESYESYES

OHH WOW
I've just solved Just Intonation!!!
Atleast the most important aspects of it.

Just intonation has the following structure.
It's an endless chain of major chords connected by pure fifths.
so for instance 16/9 10/9 4/3 5/3 1/1 5/4 3/2 15/8 9/8 45/32 27/16 etc etc
(there is never a 5/4 on a 5/4 third so no 25/x intervals ever)

There is Major tonality in here and Minor tonality.
Major tonality is when the fundamental bass is on 1/1 or any fifth connected
to it.
In major tonality the major chord is 1/1 5/4 3/2, the minor chord is 1/1
1215/1024 3/2

Minor tonality is when the fundamental bass is on 5/4 or any fifth connected
to it.
In minor tonality the major chord is 1/1 512/405 3/2, the minor chord is 1/1
6/5 3/2

I will shortly post the drei equali piece rendered in this way of Just
Intonation. It sounds PERFECT and the logic is perfect, melodies stay
perfect etc, it all works out perfect :)
I had been in the right direction for some time but had been analysing it as
Major according to the theory I just found, but it is in Minor.
(offcourse logical now that I have the theory but didn't have it before so
didn't know I was working in what I now call major)

-Marcel

🔗Marcel de Velde <m.develde@...>

First of all a correction:The Minor chord in Major tonality should be 1/1
32/27 3/2, NOT 1/1 1215/1024 3/2.
The Major chord in Minor tonality should be 1/1 81/64 3/2, NOT 1/1 512/405
3/2

So:
Major tonality is when the fundamental bass is on 1/1 or any fifth connected
to it.
In major tonality the major chord is 1/1 5/4 3/2, the minor chord is 1/1
32/27 3/2

Minor tonality is when the fundamental bass is on 5/4 or any fifth connected
to it.
In minor tonality the major chord is 1/1 81/64 3/2, the minor chord is 1/1
6/5 3/2

I've rendered the drei equali andante beginning in this new JI.
(in my previous post I had only played it from the keyboard, not rendered it
precisely and listened closely)
I'm at thesame time very happy to finally solve this piece and discover the
true structure of JI and music.
But at thesame time I'm sad because I can hear allready that it's not a
solution everybody will agree to right away.
My own ears took some adjusting, not so much with the pythagorean major
chords in minor mode, but mainly with a part of the main melody.
The melody goes F G A and this is played as F(4/3) G(3/2) A(5/3) and this
10/9 step from G to A is so different from 12tet that my ears had to adjust
to it. They did so very quickly but I'm afraid many of you will reject this.
However I'm personally sure this version is 100% correct.

Here the MIDI rendering of my Just Intonation:
/tuning/files/Marcel/DeVelde-JI_WoO30.mid

Here the MIDI rendering of 12tet as a comparison:
/tuning/files/Marcel/DeVelde_12tet_WoO30.mid

Here the Scala sequence file to see the tuning:
/tuning/files/Marcel/DeVelde-JI_WoO30.seq

-Marcel

🔗Carl Lumma <carl@...>

🔗monz <joemonz@...>

Hi Marcel,

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> EUREKA!!!!!! :D:D:D:D:D:DI"VE GOT IT!!!!!!!!!!!!!!!!!!!
>
> YESYESYES
>
> OHH WOW
> I've just solved Just Intonation!!!
> Atleast the most important aspects of it.

Sorry to "rain on your parade", but these all seems to me
to be complicated for nothing. Here's why ...

(note: to view this post correctly on the stupid Yahoo web
interface, click on the "Options" and then "Use Fixed Width Font"
links at the right)

> [Marcel:]
> Just intonation has the following structure.
> It's an endless chain of major chords connected by pure fifths.
> so for instance 16/9 10/9 4/3 5/3 1/1 5/4 3/2 15/8 9/8 45/32 27/16
> etc etc
> (there is never a 5/4 on a 5/4 third so no 25/x intervals ever)

Actually, just-intonation implies the theoretically _infinite_
planar lattice of pitches with factors of 3 and 5 (and also 2,
which is generally ignored as the ratio of octave-equivalence).

Of course in practice one must find a way to limit the number
of pitches, and there have been various schemes for this,
of which yours described here is one.

What you are describing is a very simple subset lattice
which forms the core of the theoretically infinite
5-limit just-intonation lattice.

... in the rectangular geometry which is better at showing
the prime axes of 3 and 5:

10:9 --- 5:3 --- 5:4 -- 15:8 -- 45:32
| | | | |
| | | | |
16:9 --- 4:3 --- 1:1 --- 3:2 --- 9:8 -- 27:16

... or in the triangular geometry which is better at showing
the major and minor triads:

10:9 --- 5:3 --- 5:4 -- 15:8 -- 45:32
/ \ / \ / \ / \ / \
/ \ / \ / \ / \ / \
/ \ / \ / \ / \ / \
16:9 --- 4:3 --- 1:1 --- 3:2 --- 9:8 -- 27:16

> [Marcel:]
> There is Major tonality in here and Minor tonality.
> Major tonality is when the fundamental bass is on 1/1 or
> any fifth connected to it.
> In major tonality the major chord is 1/1 5/4 3/2,
> the minor chord is 1/1 1215/1024 3/2

> Minor tonality is when the fundamental bass is on 5/4 or
> any fifth connected to it.
> In minor tonality the major chord is 1/1 512/405 3/2,
> the minor chord is 1/1 6/5 3/2

What you're describing here is a form of diaschismatic tuning,
in which you extend the two rows of the lattice so that they
include "diaschismatic equivalents", which actually should have
a different notation from the notes they are approximating:
thus to analyze the variable sizes of 3rds in your triads,
using the "root" of the chord as zero cents in all cases,
and my HEWM notation for the accidentals to the letter-names:

> In major tonality the major chord is 1/1 5/4 3/2,

C:E-:G = 0:386:702 cents

> the minor chord is 1/1 1215/1024 3/2

C:D#-:G = 0:296:702 cents

> In minor tonality the major chord is 1/1 512/405 3/2,

E-:Ab:B- = 0:406:702 cents

> the minor chord is 1/1 6/5 3/2

E-:G:B- = 0:316:702 cents

The best way to illustrate my point is to use two lattices,
one each for your "major tonality" and "minor tonality",
which have respectively C and E- as 1:1 ... i realize that
there should only be two rows to each lattice, but i
include a third row on each one, to illustrate the "implied"
just minor-3rds and major-3rds (Eb+ and G#--):

(normally i would put the 3-axis horizontally and the
5-axis vertically; here, i've switched them to fit them
into the narrow Yahoo web interface space)

"major tonality", showing three different minor-3rds:

1215:1024
D#-
296
|
405:256
G#-
|
|
135:128
C#-
|
|
45:32
F#-
|
|
3:2 --- 15:8
G B-
702 |
| |
6:5 --- 1:1 ---- 5:4
Eb+ C E-
316 0 386
|
4:3
F
|
|
16:9
Bb
|
|
32:27
Eb
294

"minor tonality", showing three different major-3rds:

81:64
G#-
408
|
27:16
C#-
|
|
9:8
F#-
|
|
6:5 ---- 3:2
G B-
316 702
| |
8:5 ---- 1:1 ---- 5:4
C E- G#--
| 0 386
| |
16:15 --- 4:3
F A-
| |
| |
64:45 -- 16:9
Bb D-
| |
| |
256:135 - 32:27
Eb G-
| |
| |
512:405 - 128:81
Ab C-
406

These lattices show that:

in "major tonality", D#- = 1215:1024 is a diaschisma
below Eb+ = 6:5, the just minor-3rd

in "minor tonality", Ab = 512:405 is a diaschisma
above G#-- = 5:4, the just major-3rd

But audibly, what you are actually doing is using
_schismatic_ substitution for the regular pythagorean
major and minor 3rds, because the skhisma is only
~2 cents and essentially an inaudible discrepancy; thus:

in "major tonality", D#- = 1215:1024 is
a skhisma above Eb = 32:27

in "minor tonality", Ab = 512:405 is
a skhisma below G#- = 81:64

I really don't follow why you're doing this. Normally,
schismatic substitution is invoked because a pythagorean
chain has "schismatic equivalents" which are only ~2 cents
away from the _just_ 3rds. Here, you're using schismatic
substitutions for _pythagorean_ 3rds, which seems pointless
to me, because you could just use an ordinary pythagorean
tuning and achieve the same audible results.

PS - Thanks to your post, i've just made a comprehensive and
long-needed update to my Encylopedia page on just-intonation:

http://tonalsoft.com/enc/j/just.aspx

-monz
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

🔗joemonz <joemonz@...>

🔗Marcel de Velde <m.develde@...>

>
> By the way, I've found your periodic toiling with this
> Beethoven piece quite amusing, since it has no comma problem
> and should go into JI just fine. I picked it just because
> I had a 7-limit rendering of it that I thought would drive
> you up a wall (which I subsequently discovered I'd lost),
> not as a challenge to strict JI, like the Lassus piece.
>

It does have a comma problem even in normal 5-limit JI.
Where the melody goes F G A and then the bass goes F G A the bass gives 1/1
40/27 64/27 40/9 on G
To drop the bass a syntonic comma to avoid this 40/27 fifth destroys the
melody and you'd probably also then drop the higher melody to thesame note
destroying that melody too.
I beleive the 1/1 40/27 64/27 40/9 on G is correct.
It's thesame solution as I gave for the Lassus piece.

I do however no longer think my previous version is correct.
It should indeed be normal 5-limit JI, realised this yesterday.
The normal 5-limit JI version I allready made on my second try of this
piece, within a week of receiving it about 4/3 year ago.
I then rejected it immediately mainly because of how the F G A sounds with G
- A 10/9.
I had interpreted the piece differently when hearing it in 12tet and when
played by a real trombone quartet.
But this interpretation which is pretty much my rendering 2 versions ago I
think destroys the melodies and makes no sense.
I'm not sure I like the way normal 5-limit JI does this piece in several
ways, but I've learned to hear it as correct now.
Also don't like that there can be 1/1 5/4 3/2 on 1/1 and thesame major on
5/4, making syntonic commma shifting music easy it seems to me, but it seems
like I should look to other solutions to this.
Too bad I didn't hear it this way when I first started working on this
piece, though I did learn a lot working on all my crazy interpretations of
this piece.

Here the "semi-normal" 5-limit rendering (with my 40/27 fifth) of the drei
equali beginning:
/tuning/files/Marcel/DeVelde-JI3_WoO30.mid

The scala sequence file and 12tet comparison you'll find in my folder.

I'll make the full version of this piece, and the other 2 drei equali pieces
soon.
However I won't be posting them on this list anymore as I'm leaving this
list.
Carl has decided to moderate me and while I have all the respect for Carl
and his knowledge of tuning - it's nothing I have against Carl - I can't
live with beeing moderated.
So this is goodbye tuning list.
I'll be hanging out in the JustIntonation list and may also post full pieces
sometimes in MMM.

Also thanks Joe for your answer, and I do finally agree with normal 5-limit
(except maybe for some comma shift handling etc)

-Marcel

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> > By the way, I've found your periodic toiling with this
> > Beethoven piece quite amusing, since it has no comma problem
> > and should go into JI just fine. I picked it just because
> > I had a 7-limit rendering of it that I thought would drive
> > you up a wall (which I subsequently discovered I'd lost),
> > not as a challenge to strict JI, like the Lassus piece.
>
> It does have a comma problem even in normal 5-limit JI.
> Where the melody goes F G A and then the bass goes F G A the
> bass gives 1/1 40/27 64/27 40/9 on G
> To drop the bass a syntonic comma to avoid this 40/27 fifth
> destroys the melody and you'd probably also then drop the
> higher melody to the same note destroying that melody too.

I don't know how you're getting those numbers. I think
you're talking about bars 8-10, but I don't know why you're
talking about G -- the version I have is in Dmin. One does
need two different Gs (10/9 and 9/8) and two different
Ds (5/3 and 27/16) over the course of the excerpt, but they
do not occur consecutively and there's no comma pump.
The scale is just

!
andante
11
!
10/9 ! G
9/8 ! G
75/64 ! Ab
5/4 ! A
45/32 ! B
3/2 ! C
25/16 ! C#
5/3 ! D
27/16 ! D
15/8 ! E
2/1 ! F
!

-Carl