Yahoo Tuning Groups Ultimate Backup tuning Adaptive JI

To: internet:tuning@onelist.com
Date: 04-06-99

David Mezquita wrote,
>> I have some doubts about the the major 2nd of a major scale.

Various people responded,
> Use both D1 and D2, depending on context.

I have completed version 1.000 of JI Relay, a Windows 95/98 program
which converts 12-tET MIDI to Just Intonation on the fly. It will pick
D1 or D2 (along with the tuning of all other notes) as needed for the
particular chord(s) being played.

This first version expects to find a General MIDI (GM) capable device
on output, either a sound card or an external GM MIDI synth. Tuning is
effected with pitch-bend messages, so only one voice (patch, program)
can be sounding at one time. One can, of course, change from voice to
voice while running the program.

Retuning is done in a presettable space of time; I typically use about
250 msec, so that when a continuously sounding note must change
frequency, it's not done all at once.

I'll be glad to make this program available to anyone who wants to
experience JI on the fly. Distribution is a bit of a problem, however,
since I don't have a fully functioning web site. For now, anyone may
e-mail me at 102074.2214@compuserve.com and I'll be glad to forward a
copy. If you have a PGP public key, please send it to me; otherwise,
I'll include the program, zipped, as a binary attachment.

Of course, I haven't finished many things that I hope to complete over
time, including after-the-fact retuning of a sequence with knowledge
both forward and backward in time. Standard MIDI file I/O is also not
yet supported. Nor are ET inputs other than 12-tET. Still, I think
that the program provides something of interest for tuning enthusiasts!

JdL

🔗John A. deLaubenfels <jadl@xxxxxx.xxxx>

To: tuning@onelist.com
From: John deLaubenfels
Subject: Adaptive
JI
Date: 04-07-99

Yesterday I announced that JI Relay is available.
Thanks to hints from
Manuel Op de Coul, I have succeeded in uploading it to
the file area
of the tuning list. Here's how to download it:

. Start
at http://www.onelist.com

. Click "Member center" on the left.

.
There, you must log in with your e-mail address and password. The

password will be e-mailed to you if you wish, but you must have it
to
go on to the next step.

. You are now in the Member Center Account
Manager. About half way
down is a list of lists you belong to; click
the Tuning list .

. Now you're in List center: tuning. Click on Shared
Files.

. Now you're in Shared files: tuning. Click on JIRelay.

.
Click on JIX1000.zip and save it to disk.

. Offline, unzip JIX1000.zip
in an empty directory. Run JIRelay.exe.

I'll still be glad to e-mail
anyone who has trouble downloading the
file.

By the way: Compuserve
glitched its way out of my heart, and I have a
new e-mail address:

jadl@idcomm.com

(That's "id" with a single syllable, as in ego, id,
superego).

JdL

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

🔗David Mezquita <DMEZQUI@xxxxxxxx.xxx>

🔗monz@xxxx.xxx

[Barbaro:]
> John A. deLaubenfels described how he would "slide"
> the D from 10/9 to 9/8 if the harmony changed from ii to V.
> Wouldn't a better solution for adaptive JI be to have
> the roots in, say, 1/4-comma meantone, and the chords in
> (or sliding to) JI relative to the roots? Then the maximum
> amount of slide needed would be reduced by a factor of 4,
> wouldn't it?

The thing that's so beautiful about JI is that you can
play around with the subtle differences between notes
that are this close together.

Try listening the the example of modulation that
Fox-Strangways sent Partch, and that Partch quoted
in _Genesis_.

Partch says that he played this example for a roomful
of people, with 3 different tunings for the 'second'
of the new key (the 3:2 of the old key) -

1) as a 10:9 all the way thru the modulation,
2) as a 9:8 all the way thru, and
3) as a 10:9 in the ii chord and a 9:8 in
the V chord,

and no one in the audience failed to notice that
a modulation had taken place. Partch even remarks
gleefully that one man recognized the 3 different
qualities in the examples.

To give it my subjective opinion:

1) the first method, sustaining the 10:9, sustains
a 'flatness' into the V chord that gives a sort of
lingering memory of the old key. The V chord is
noticeably out of tune but it resolves into the
new key before one can really know what happened.

2) the second method, sustaining the 9:8, makes the
'ii' consonant in the V chord, but brings it in as
a sharp (meaning higher in pitch) dissonance first
on the ii chord. This gives some sense of surprise
and anticipation at hearing the ii chord, then a
quick resolution when the V comes next as a consonance.
The new key does not come as much as a surprise as
in method 1, because the it was anticipated.

In a sense both methods give an anticipation of the
new key, but in the first method it's with *all but one*
of the notes in the chord, so that it's the one note
left over from the old key that sticks out like a
sore thumb; in the second method it's with *only one*
note. So there is a qualitative difference between
the two approaches that can certainly serve musically
expressive purposes.

3) the third method tunes the 'ii' as a 10:9 in the
ii chord and as a 9:8 in the V chord, so there is
no *harmonic* dissonance. This time the dissonance
is melodic, as a perceptive listener can hear the
comma difference between the 'ii' note in the two chords.

So the point is, in 5-limit JI there are (at least)
these 3 different ways to effect a modulation, and
each one has what I would say is a different *affect*.

And of course, by going a little further along the
lattice-vectors and including higher exponents even
in 5-limit JI, you can use other 'synonomous' notes
in your modulations. And increasing the odd-/prime-limit
gives multitudes of more different possibilties.

Of course, using adaptive-JI harmonies over meantone
roots is also a viable idea - sounds like an interesting
tuning. But with so many possibilities available,
why stick limits on yourself without giving them a
musical purpose?

That's why it always bugs me when I read 'optimal tuning'.

As Parch said, and as I used to quote in my signature,

> The ability of the human ear is vastly underestimated.

I would emphasize the *vastly*.

Many scientists say that graphing experimental data
to pitched sound rather than lines or points on paper
gives one an ability to perceive differences that is
far more sensitive than what the eyes can see.

At any rate, it certainly allows us to perceive
*different* patterns in the data than what the eyes know.

-Monzo
http://www.ixpres.com/interval/monzo/homepage.html

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🔗John A. deLaubenfels <jadl@xxxxxx.xxxx>

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

I wrote,

> John A. deLaubenfels described how he would "slide" the D from 10/9 to
> 9/8 if the harmony changed from ii to V. Wouldn't a better solution
> for adaptive JI be to have the roots in, say, 1/4-comma meantone, and
> the chords in (or sliding to) JI relative to the roots? Then the
> maximum amount of slide needed would be reduced by a factor of 4,
> wouldn't it?

John A. deLaubenfels wrote:

> What's important is to achieve the proper ratios (i.e. tuning) for the
> chord being played at each moment. What my program actually does
> (see td 135 for download location, if you have a PC w/ sound card) is
> retune all the notes slightly, to achieve both:
>
> . a properly tuned chord
> . zero average shift in tuning
>
> Therefore the D will in fact move by much less than a full comma, as
> you desire.

Great! So far it sounds like you are basically answering in the affirmative.

> The exact motion of each note depends, in my program, upon
> volume: soft notes are made to move more.
>
> I resist the idea that adaptive JI would have "roots in .. meantone".
> I don't want to knock meantone or any other non-JI tuning, but JI is JI.

And that sounds like a negative. Saying "JI is JI" sounds like you're
advocating a Partch-style system where all notes are reasonably simple
ratios, but then where would an interval much less than a full comma come
from? I understand that with your system the exact tuning of the roots will
not be in JI, or in meantone, but in a tuning depending on the overall
progression and the volume of the notes involved. However, for a diatonic
chord progression which would involve a comma shift in JI, wouldn't your
desideratum of a zero average shift in tuning automatically skew the tuning
of the roots "toward" meantone tuning?

David Mezquita wrote,

>To my mind, this is not a good solution, because the relationships of
>tonality are altered. O.k. that the slide is shorter, but the tonal
>relationship is lost. I don't like to move the IV (also contained in ii)
and
>the V degrees, because they are the base of tonality.

>David.

David, what if the ii chord was leading to a modulation to V? Then the 2nd
degree becomes part of the "base of tonality" in the new key. Something's
got to give!

Also, David, you would always tune the dominant seventh chord 1/1 5/4 3/2
16/9? And John, you would have it slide toward 1/1 5/4 3/2 7/4?

🔗John A. deLaubenfels <jadl@xxxxxx.xxxx>

[I wrote:]
>> The exact motion of each note depends, in my program, upon
>> volume: soft notes are made to move more.
>>
>> I resist the idea that adaptive JI would have "roots in .. meantone".
>> I don't want to knock meantone or any other non-JI tuning, but JI is
>> JI.

[Brett Barbaro wrote:]
> And that sounds like a negative. Saying "JI is JI" sounds like you're
> advocating a Partch-style system where all notes are reasonably simple
> ratios, but then where would an interval much less than a full comma
> come from? I understand that with your system the exact tuning of the
> roots will not be in JI, or in meantone, but in a tuning depending on
> the overall progression and the volume of the notes involved. However,
> for a diatonic chord progression which would involve a comma shift in
> JI, wouldn't your desideratum of a zero average shift in tuning
> automatically skew the tuning of the roots "toward" meantone tuning?

Let me illustrate with a concrete example, the classic "comma pump"
sequence, I-iv-ii-V-I. Assume all notes have equal volume. The
following charts how to tune JI relative to 12-tET:

Absolute Centered Retune
shift shift motion
(cents) * (cents) (cents)
-------- -------- --------
C major (I)
C 0.00 +3.91 n/a
E -13.69 -9.78 n/a
G +1.96 +5.87 n/a
A minor (iv)
A 0.00 -5.87 n/a
C +15.65 +9.78 +5.87
E +1.96 -3.91 +5.87
D minor (ii)
D 0.00 -5.87 n/a
F +15.65 +9.78 n/a
A +1.96 -3.91 +1.96
G major (V)
G 0.00 +3.91 n/a
B -13.69 -9.78 n/a
D +1.96 +5.87 +11.74
C major (I)
C 0.00 +3.91 n/a
E -13.69 -9.78 n/a
G +1.96 +5.87 +1.96

*There could easily be disagreement about which notes should be retuned
for "absolute" tuning; this is one example. No such ambiguity exists
for centered shift.

So, D moves by only about a tenth of a semitone, about half a comma,
between D minor and G major. Other transitions are considerably less.
But JI is never sacrificed.

JdL

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

> Message: 6
> Date: Sun, 11 Apr 1999 08:51:51 -0600
> From: "John A. deLaubenfels" <jadl@idcomm.com>
> Subject: Re: Adaptive JI
>
> [I wrote:]
> >> The exact motion of each note depends, in my program, upon
> >> volume: soft notes are made to move more.
> >>
> >> I resist the idea that adaptive JI would have "roots in .. meantone".
> >> I don't want to knock meantone or any other non-JI tuning, but JI is
> >> JI.
>
> [Brett Barbaro wrote:]
> > And that sounds like a negative. Saying "JI is JI" sounds like you're
> > advocating a Partch-style system where all notes are reasonably simple
> > ratios, but then where would an interval much less than a full comma
> > come from? I understand that with your system the exact tuning of the
> > roots will not be in JI, or in meantone, but in a tuning depending on
> > the overall progression and the volume of the notes involved. However,
> > for a diatonic chord progression which would involve a comma shift in
> > JI, wouldn't your desideratum of a zero average shift in tuning
> > automatically skew the tuning of the roots "toward" meantone tuning?
>
> Let me illustrate with a concrete example, the classic "comma pump"
> sequence, I-iv-ii-V-I. Assume all notes have equal volume. The
> following charts how to tune JI relative to 12-tET:
>
> Absolute Centered Retune
> shift shift motion
> (cents) * (cents) (cents)
> -------- -------- --------
> C major (I)
> C 0.00 +3.91 n/a
> E -13.69 -9.78 n/a
> G +1.96 +5.87 n/a
> A minor (iv)
> A 0.00 -5.87 n/a
> C +15.65 +9.78 +5.87
> E +1.96 -3.91 +5.87
> D minor (ii)
> D 0.00 -5.87 n/a
> F +15.65 +9.78 n/a
> A +1.96 -3.91 +1.96
> G major (V)
> G 0.00 +3.91 n/a
> B -13.69 -9.78 n/a
> D +1.96 +5.87 +11.74
> C major (I)
> C 0.00 +3.91 n/a
> E -13.69 -9.78 n/a
> G +1.96 +5.87 +1.96
>
> *There could easily be disagreement about which notes should be retuned
> for "absolute" tuning; this is one example. No such ambiguity exists
> for centered shift.
>
> So, D moves by only about a tenth of a semitone, about half a comma,
> between D minor and G major. Other transitions are considerably less.
> But JI is never sacrificed.

If you use that definition of JI (many don't!), then you would agree that in my scheme,
where the roots are tuned in meantone and the chords are tuned in JI relative to the roots,
JI is also never sacrificed. You seem to be misunderstanding me. By the way, your table
above doesn't make it easy to see what all the pitches really are -- can you produce a
comparable table with that information? Then I can show a comparison with what I've been
talking about. (Also, Am is vi, not iv. What would happen if you used IV=F major? I'm
really interested!).

🔗John A. deLaubenfels <jadl@xxxxxx.xxxx>

🔗Bob Lee <quasar@xxx.xxxx>

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

John A. deLaubenfels wrote:

> [Brett Barbaro wrote:]
> > If you use that definition of JI (many don't!)
>
> What definition of JI would tune those chords differently?

I meant that to many, the successive intervals have to be simple-integer ratios, in
addition to the simultaneous intervals. In other words, melodic JI as well as
harmonic JI.

> > then you would agree that in my scheme, where the roots are tuned in
> > meantone and the chords are tuned in JI relative to the roots, JI is
> > also never sacrificed. You seem to be misunderstanding me.
>
> Perhaps I am. But, if you are going to sound all your actual chords in
> JI, why introduce meantone into the picture?

Like your scheme, tuning the roots in meantone would be a way to avoid comma drifts
and shifts, while all simultaneities would still be in JI. I'm not yet saying it
would be a better way -- we can compare them later.

> > By the way, your table above doesn't make it easy to see what all the
> > pitches really are -- can you produce a comparable table with that
> > information? Then I can show a comparison with what I've been talking
> > about.
>
> Uhhh, to me the table is clear, but many output formats are possible.
> Exactly what format would make it clearer?

I think it would be clearer to simply express each pitch at each point in time
relative to a fixed 12-tET. The information in your other columns could be easily
calculated from that.

> > What would happen if you used IV=F major? I'm really interested!.
>
> Well, maybe I'll wait till we figure out what format is clear to you;
> then I'll be glad to answer that!

Thanks!

🔗John A. deLaubenfels <jadl@idcomm.com>

[Brett Barbaro wrote:]
> I think it would be clearer to simply express each pitch at each point
> in time relative to a fixed 12-tET. The information in your other
> columns could be easily calculated from that.

But that's exactly what the table shows! To repeat part of it
(correcting my dyslexic error):

>> Let me illustrate with a concrete example, the classic "comma pump"
>> sequence, I-vi-ii-V-I. Assume all notes have equal volume. The
>> following charts how to tune JI relative to 12-tET:
>>
>> Absolute Centered Retune
>> shift shift motion
>> (cents) (cents) (cents)
>> -------- -------- --------
>> C major (I)
>> C 0.00 +3.91 n/a
>> E -13.69 -9.78 n/a
>> G +1.96 +5.87 n/a
>> A minor (vi)
>> A 0.00 -5.87 n/a
>> C +15.65 +9.78 +5.87
>> E +1.96 -3.91 +5.87
>>

After your request, I was thinking that absolute cents relative to
12-tET C might be more clear:

Absolute Centered Retune
tuning tuning motion
(cents) (cents) (cents)
-------- -------- --------
C major (I)
C 000.00 003.91 n/a
E 386.31 390.22 n/a
G 701.96 705.87 n/a
A minor (vi)
C 015.65 009.78 +5.87
E 401.96 396.09 +5.87
A 900.00 894.13 n/a
D minor (ii)
D 200.00 194.13 n/a
F 515.65 509.78 n/a
A 901.96 896.09 +1.96
G major (V)
D 201.96 205.87 +11.74
G 700.00 703.91 n/a
B 1086.31 1090.22 n/a
C major (I)
C 000.00 003.91 n/a
E 386.31 390.22 n/a
G 701.96 705.87 +1.96

Clearer or less clear?

> What would happen if you used IV=F major?

I follow the same procedure for every chord: to center the average shift
relative to 12-tET. So, F major looks very much like C major or G major
or any other major chord (again, assuming all notes of equal volume):

Absolute Centered
shift shift
(cents) (cents)
-------- --------
F major (IV)
F 0.00 +3.91
A -13.69 -9.78
C +1.96 +5.87

Or,

Absolute Centered
tuning tuning
(cents) (cents)
-------- --------
F major (IV)
C 001.96 005.87
F 500.00 503.91
A 886.31 890.22

JdL

🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

John A. deLaubenfels wrote:

> After your request, I was thinking that absolute cents relative to
> 12-tET C might be more clear:
>
> Absolute Centered Retune
> tuning tuning motion
> (cents) (cents) (cents)
> -------- -------- --------
> C major (I)
> C 000.00 003.91 n/a
> E 386.31 390.22 n/a
> G 701.96 705.87 n/a
> A minor (vi)
> C 015.65 009.78 +5.87
> E 401.96 396.09 +5.87
> A 900.00 894.13 n/a
> D minor (ii)
> D 200.00 194.13 n/a
> F 515.65 509.78 n/a
> A 901.96 896.09 +1.96
> G major (V)
> D 201.96 205.87 +11.74
> G 700.00 703.91 n/a
> B 1086.31 1090.22 n/a
> C major (I)
> C 000.00 003.91 n/a
> E 386.31 390.22 n/a
> G 701.96 705.87 +1.96

Thanks, John.

> I follow the same procedure for every chord: to center the average shift
> relative to 12-tET.

That's what I thought but I got confused because, for example, the "retune motion" being
"n/a" for the G in the V chord made me think it was the same as the G in the initial I
chord.

So John, you are relying on 12-tET in an essential way to obtain your results. You start
out with the roots in 12-tET, and then you zero the average shift with respect to 12-tET.
But the results don't seem optimal with respect to the goals you've set forth. For
example, a simple I-iii-V-I progression doesn't require any "retune motion" to be rendered
in JI, (the old-fashioned way -- just use the ratios 1/1, 9/8, 5/4, 3/2, 5/3, 15/8) while
you would render it thus:

Absolute Centered Retune
tuning tuning motion
(cents) (cents) (cents)
-------- -------- --------
C major (I)
C 000.00 003.91 n/a
E 386.31 390.22 n/a
G 701.96 705.87 n/a
E minor (iii)
B 1101.96 1096.09 n/a
E 400.00 394.13 +3.91
G 715.65 709.78 +3.91
G major (V)
D 201.96 205.87 n/a
G 700.00 703.91 -5.87
B 1086.31 1090.22 -5.87
C major (I)
C 000.00 003.91 n/a
E 386.31 390.22 n/a
G 701.96 705.87 +1.96

Now I don't consider any of those retune motions audibly significant, but the 11.74-cent
one in the first progression above might be significant under super-sensitive listening
conditions. Now let's see what happens when instead of basing everything on 12-tET, we use
1/4-comma meantone. The C-major scale, instead of 0 200 400 500 700 900 1100, is 0 193.16
386.31 503.42 696.58 889.74 1082.89.

The idea I've been trying to explain to you, tuning the roots in meantone, is equivalent
to your "absolute tuning" column, except using meantone in place of 12-tET. As shown in
the second column below, the "retune motion" involved in making each chord JI
(harmonically) is never more than 5.38 cents, or 1/4 of a comma, less than half the
maximum "retune motion" in your method. I also tried applying your centering idea with
respect to meantone, but that increases the maximum "retune motion," as shown in the
fourth column below:

Absolute Retune Centered Retune
tuning motion tuning motion
(cents) (cents) (cents) (cents)
-------- -------- -------- --------
C major (I)
C 000.00 n/a -001.79 n/a
E 386.31 n/a 384.52 n/a
G 701.96 n/a 700.16 n/a
A minor (vi)
C 005.38 +5.38 005.38 +7.17
E 391.69 +5.38 391.69 +7.17
A 889.74 n/a 889.74 n/a
D minor (ii)
D 193.16 n/a 193.16 n/a
F 508.80 n/a 508.80 n/a
A 895.11 +5.37 895.11 +5.38
G major (V)
D 198.53 +5.37 196.74 +3.58
G 696.58 n/a 694.79 n/a
B 1082.89 n/a 1081.10 n/a
C major (I)
C 000.00 n/a -001.79 n/a
E 386.31 n/a 384.52 n/a
G 701.96 +5.38 700.16 +5.38

Now for the progression where straight JI would have had no "retune motion", my way still
has a lower maximum "retune motion" than your way. And again, centering doesn't help with
my method:

C major (I)
C 000.00 n/a -001.79 n/a
E 386.31 n/a 384.52 n/a
G 701.96 n/a 700.16 n/a
E minor (iii)
E 386.31 0.00 386.31 +1.79
G 701.96 0.00 701.96 +1.79
B 1088.27 n/a 1088.27 n/a
G major (V)
D 198.53 n/a 196.74 n/a
G 696.58 -5.38 694.79 -7.17
B 1082.89 -5.38 1081.10 -7.17
C major (I)
C 000.00 n/a -001.79 n/a
E 386.31 n/a 384.52 n/a
G 701.96 +5.38 700.16 +5.38

So, are we atarting to make sense to each other?

🔗Brett Barbaro <barbaro@noiselabs.com>

🔗John A. deLaubenfels <jadl@idcomm.com>

[I wrote:]
>> I follow the same procedure for every chord: to center the average
>> shift relative to 12-tET.

[Brett Barbaro wrote:]
> That's what I thought but I got confused because, for example, the
> "retune motion" being "n/a" for the G in the V chord made me think it
> was the same as the G in the initial I chord.

I see. I didn't show it there because the G was not continuously
sounding. But a sensitive ear might still pick up the difference.

> So John, you are relying on 12-tET in an essential way to obtain your
> results. You start out with the roots in 12-tET, and then you zero the
> average shift with respect to 12-tET.

It is true that I use 12-tET to center the tuning. My reasoning is
as follows: input is 12-tET. I want an approach that does not favor one
key over another, as inevitably must happen in any different approach
as one goes out to the edges, where F# meets Gb. Therefore, modulating
from C major to G major, I "eat" 1.96 cents compared to a perhaps more
strict JI approach. Nevertheless, this method is extremely convenient
and close enough to suit my ear.

In my (as yet unwritten) leisure retuning software I intend to pay more
strict attention to retune motion, and to "drift" the average tuning
center as best suits the piece on the fly, attempting to balance the
pain of drift vs. the pain of retune motion (also vs. pain of chords
that for one reason or another can't be fully JI tuned).

The software I've been describing in these posts is real-time, with
no knowledge of how long the current set of notes will sound (nor, of
course, what will follow!). It decides what to do as if the notes now
sounding will go on for a long time, and therefore acts not to minimize
one-time transition, but rather to center and tune this chord as best
possible. I do not claim that minimum motion results, only that
less-than-comma motion results most of the time.

I do see that your approach does a good job of cutting down on retune
motion. I am curious: how do you handle rampant modulations? A circle
of fifths, for example; or a progression from C major to E major to G#
major?

> So, are we starting to make sense to each other?

Yes, I think so. By the way, do you have a web site? Please forgive
me if I've missed a posting where you have referenced it.

JdL

🔗rtomes@xxxxx.xxx.xxxxxxxxxxxxx)

OK, I have gone and searched for "Adaptive JI" in the archives and read many of
these messages (although the search facility is a bit weird).

I like what John A. deLaubenfels is describing there which seems to be along the
same lines as what I am suggesting.

Paul Erlich wrote:
>We've discussed several reasons why compromises are necessary --
>to prevent wandering tonics,

I didn't find the reason for this. In my scheme the method of adjusting notes
is to take all the scales off of C in each direction as follows.

C has ratios 24 27 30 32 36 40 45 for the white notes and whatever seems
reasonable for the black (separate issue to this one). To determine the
frequencies for G multiply all the above by 3/2 and the only note that actually
shifts is A which shifts by 81/80 ratio which I assume is the 9/8 vs 10/9
discussion. I say that this note should shift by that ratio. Then carry on for
each extra sharp and do the reverse process for each flat. The final result has
a discrepancy between 6 sharps and 6 flats and so they are quite different.

That means that you cannot modulate around the corner from 6# to 5b and I
frankly don't care. I hate it when they have B# and Cb because my piano doesn't
have black notes there. Actually there is a solution to even this but it isn't
worth bothering with.

Where is the problem? Please give an example which has a problem.

>to tune chords with contradictory consonances like major
>add 6 add 9 and augmented,

Can you give me that as a set of notes so that my simple brain can cope - I
don't think in standard music theory terms. Then I will give my answer to what
the ratios should be from my AJI chord ratio calculator.

>and to balance the conflicting demands of harmonic and melodic
>perfection.

Again an example of some actual music (preferable the actual notes in here
rather than a reference to a piece of music). If the analysis method is getting
it right then there should be no conflict (he said in a not quite certain tone).

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
Cycles email list -- http://www.kcbbs.gen.nz/users/af/cyc.htm
Alexandria eGroup list -- http://www.kcbbs.gen.nz/users/af/alex.htm
Boundaries of Science http://www.kcbbs.gen.nz/users/af/scienceb.htm

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

Ray Tomes wrote:

>OK, I have gone and searched for "Adaptive JI" in the archives and read
many of
>these messages (although the search facility is a bit weird).
>
>I like what John A. deLaubenfels is describing there which seems to be
along the
>same lines as what I am suggesting.
>
>Paul Erlich wrote:
>>We've discussed several reasons why compromises are necessary --
>>to prevent wandering tonics,
>
>I didn't find the reason for this. In my scheme the method of adjusting
notes
>is to take all the scales off of C in each direction as follows.
>
>C has ratios 24 27 30 32 36 40 45 for the white notes and whatever seems
>reasonable for the black (separate issue to this one). To determine the
>frequencies for G multiply all the above by 3/2 and the only note that
actually
>shifts is A which shifts by 81/80 ratio which I assume is the 9/8 vs 10/9
>discussion. I say that this note should shift by that ratio. Then carry
on for
>each extra sharp and do the reverse process for each flat. The final
result has
>a discrepancy between 6 sharps and 6 flats and so they are quite different.
>
>That means that you cannot modulate around the corner from 6# to 5b and I
>frankly don't care. I hate it when they have B# and Cb because my piano
doesn't
>have black notes there. Actually there is a solution to even this but it
isn't
>worth bothering with.

Please do bother . . .

>Where is the problem? Please give an example which has a problem.

I don't think you've read the relevant "Adaptive JI" posts in the archives.
The problem arises in just one key, as in the I-IV-ii-V-I and I-vi-ii-V-I
progressions (observe all common tones and for added fun, try adding a 6th
to the IV chord and/or a 7th to the ii chord).

>
>>to tune chords with contradictory consonances like major
>>add 6 add 9 and augmented,
>
>Can you give me that as a set of notes so that my simple brain can cope - I
>don't think in standard music theory terms. Then I will give my answer to
what
>the ratios should be from my AJI chord ratio calculator.

We're already discussing the first example; see Kraig Grady and Joe Monzo's
recent posts. I advocate meantone, not ET, to best handle wandering tonics
and 6/9 chords. The second example is an augmented triad, which is more
controversial but is a stack of two major thirds (5:4) and in 12-ET the
outer notes form a tolerable minor sixth (8:5). I don't think it's too
horrible in LucyTuning, though, where one of the major thirds must be 436
cents, very nearly a 9:7 (435 cents).

>
>>and to balance the conflicting demands of harmonic and melodic
>>perfection.
>
>Again an example of some actual music (preferable the actual notes in here
>rather than a reference to a piece of music). If the analysis method is
getting
>it right then there should be no conflict (he said in a not quite certain
tone).

This is, for example, what John A. deLaubenfels refers to as balancing the
pain of retuning held notes against the pain of mistuned harmonies. Also, I
find identical tetrachords melodically "perfect" as opposed to ones in which
one note is displaced by a comma. Again, see the archives for extensive
discussions on these issues.

🔗rtomes@xxxxx.xxx.xxxxxxxxxxxxx)

"Brett Barbaro" <barbaro@noiselabs.com> wrote:
>The problem arises in just one key, as in the I-IV-ii-V-I and I-vi-ii-V-I
>progressions (observe all common tones and for added fun, try adding a 6th
>to the IV chord and/or a 7th to the ii chord).

Thanks Brett, I understand the problem now. IMO the correct frequencies
are as follows (I will use 240 as the tonic for ease of calculation):
240-300-360, 320-400-480, 266.7-320-400, 360-450-540, 240-300-360
which of course has no wandering. However the problem is obviously that
one is inclined to double the 266.7 to get 533.3 rather than 540.
IMO that is a mistake because even though that is the only common note
between the two chords, the other two are quite clearly much better
aligned with the earlier notes in the way that I did it and so they
should carry the weight of argument.

Of course the question remains whether an automatic system can achieve
this result, and I think that it can. For the purposes of this
discussion I am not sure whether the argument is about the doability of
it or about the correct result.

>This is, for example, what John A. deLaubenfels refers to as balancing the
>pain of retuning held notes against the pain of mistuned harmonies.

This is equivalent to the above case if the ii note in the third chord
is held over to the fourth chord. It makes no difference. If the music
does a curly thing like change the meaning of a note then the frequency
has to change on the fly.

The best example of this is The Well-Tempered Clavichord where each
arpeggio has at least two meanings and sometimes three. In each case
the first arpeggio of each repeated set echoes the note meanings of the
last repeat of the set before so the whole thing is wonderously funny as
a musical pun. Of course this was designed for ET or something like it,
as a sort of show piece for it in fact. All the same, it could be
played in an adaptive JI. In that case the tuning for the last repeat
of one arpeggio would match the first repeat of the next one and then
the exact same notes would be played with a different tuning in order to
fully demonstrate the nature of the pun by actually "saying" it both
ways.

If the composer does things that require this then generally they have
created a tension somewhere through punniness. The tension can either
be resolved by a small break at the weakest point (as I did above) or it
can be evenly distributed throughout the passage for maximum disguise.
If this is required, then there is another way of preventing too much
tonic wandering and that is simply to keep pulling the error back by say
10% per chord played. That would be almost imperceptible and prevent
the tonic getting very far from where it started.

🔗monz@xxxx.xxx

[as always with my posts, use a fixed-width font, such as
Courier New, to view the diagrams and tables correctly]

[Ray Tomes, TD 179.1]
>
> In my scheme the method of adjusting notes is to take all the
> scales off of C in each direction as follows.
>
> C has ratios 24 27 30 32 36 40 45 for the white notes and
> whatever seems reasonable for the black (separate issue to this
> one).

On a Monzo lattice:

27 D
/
45 B /
/ '-._ /
/ 36 G
/ /
30 E /
/ '-._ /
/ 24 C
/ /
40 A /
'-._ /
32 F

[Tomes]
> To determine the frequencies for G multiply all the above
> by 3/2 and the only note that actually shifts is A which shifts
> by 81/80 ratio which I assume is the 9/8 vs 10/9 discussion.
> I say that this note should shift by that ratio.

Following your math to the letter:

40.5 A
/
67.5 F# /
/ '-._ /
/ 54 D
/ /
45 B /
/ '-._ /
/ 36 G
/ /
60 E /
'-._ /
48 C

First of all, you surely don't mean to have decimal points
in your *ratios* (not that it's going to affect the final
result, but still...). By your own description of the 81/80,
the original numbers would have to be multiplied by 3, and
not by 3/2, to give the ratios you would use in your description
of the 'G major' scale:

81 A
/
135 F# /
/ '-._ /
/ 108 D
/ /
90 B /
/ '-._ /
/ 72 G
/ /
120 E /
'-._ /
96 C

[Tomes]
> Then carry on for each extra sharp and do the reverse process
> for each flat. The final result has a discrepancy between 6
> sharps and 6 flats and so they are quite different.

You'd run into a similar problem going in the 'flat' direction,
as the numbers would contain decimals and would get smaller
and smaller the farther you transpose.

Since you're assuming 'octave'-equivalence anyway (i.e.,
ignoring powers of 2), I find it much simpler to use the
prime-factor notation:

['fac.' means 'prime-factor', the numbers within the table
are the exponents of those factors]

'C major':

fac. 3 5
-----------
B | 2 1 |
A | 0 1 |
G | 2 0 |
F | 0 0 |
E | 1 1 |
D | 3 0 |
C | 1 0 |

'G major':

fac. 3 5
------------
F# | 3 1 |
E | 1 1 |
D | 3 0 |
C | 1 0 |
B | 2 1 |
A | 4 0 |
G | 2 0 |

A comparison of these two tables makes it very easy
to see which notes have changed and which haven't:

'C major': 'G major':

fac. 3 5 fac. 3 5
----------- ------------
B | 2 1 | B | 2 1 |
A | 0 1 | A | 4 0 |
G | 2 0 | G | 2 0 |
F# | 3 1 |
F | 0 0 |
E | 1 1 | E | 1 1 |
D | 3 0 | D | 3 0 |
C | 1 0 | C | 1 0 |

There are two notes which 'shift' here:

The obvious metamorphosis (which you didn't even mention)
is the shift of F ( 0 0 ) to F# ( 3 1 ).

The subtle one, which you did point out, is the shift of
A ( 0 1 ) to A ( 4 0 ).

Of course, since I'm using prime-factors here, I'd restore
our basic reference note of 'C' to its proper place as 1:1,
or ( 0 0 ), and shift all the other exponents accordingly.
I purposely kept the factoring in agreement with the numbers
you used in your posting.

[Tomes]
> <snip>
> Where is the problem? Please give an example which has a problem.

OK - the most famous example I can think of using only the
diatonic 'major' scales you're illustrating here, is the
'comma pump':

I - vi - ii - V

or in letter names [key of 'C'],

C maj - A min - D min - G maj

What ratios would you give for the ii (D min) chord?

According to your 'C major' scale, the ratios for the
D minor chord would be 27:32:40.

The article on 'music' in the 1981 Encyclopaedia Britannica
calls this tuning of the ii chord 'quite unacceptable',
or some such nonsense, and you've inferred the same.

That argument would insist that the ii chord would have to
be a 10:12:15 proportion to be 'correctly' in-tune, which
would have it incorporating notes which lie outside the
given scale.

How do you do that?

Do you make the 'D' a 3:2 below your 'A 40'? If you do that,
you've got to shift all the numbers of your scale, so that the
16:9 Bb would become 32, and according to your theory, what's
happened then is you've 'modulated' into 'F'!

Or do you keep the 'D 27', and shift the 'A' to 81? Then
you have to move along the 5-axis, and make either the
16:15 Db or the 8:5 Ab your 32, in order to accomodate the 'F',
so you're 'modulating' even further into the 'flat' keys.

I'm not at all convinced that there's anything wrong with
that 27:32:40 chord.

Sure, it's very dissonant, the 40:27 between 'A' and 'D'
being very disagreeable in itself, and taken out of any
real musical context. BUT THAT'S EXACTLY MY POINT.

My feeling is that if you used this tuning of this chord in
'C major' *music*, it would only help to reinforce the virtual
fundamental of 'C', because all the notes in the entire
progression ultimately come from the 'C major' scale.

An important thing that's often been overlooked by theorists
of harmony is that music requires the temporal dimension.

Sure, if you're making an 'objective' consideration of
chords as static entities, then you'll probably want to
use the smallest-integer ratios that will describe the notes.

But in real music, which unfolds over time, there are melodic
considerations that play an important part in deciding what
the 'correct' ratios should be. And I'm not just talking
about the 'melody' itself - I'm talking about the movement
of the bass 'roots' and the inner voices as well.

I've frequently created JI chord progressions that looked
good on paper, and I thought would sound great, and the
individual chords *did* sound great. But put them all together,
and... yuk!

It was because of 'melodic dissonances' in the notes of the
inner voices.

So it's not so simple.

Those are just a few of the problems Paul Erlich was addressing.

And I promised I'd look at your work before writing anymore,
and I went back on that promise, so now I'll give it a look.
My initial glance at it shows that you do consider prime
factoring, so I'm interested (that's *my* pet theory). :)

-monz

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

Adaptive JI

🔗John deLaubenfels <102074.2214@xxxxxxxxxx.xxxx>

🔗John A. deLaubenfels <jadl@xxxxxx.xxxx>

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

🔗David Mezquita <DMEZQUI@xxxxxxxx.xxx>

🔗monz@xxxx.xxx

🔗John A. deLaubenfels <jadl@xxxxxx.xxxx>

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

🔗John A. deLaubenfels <jadl@xxxxxx.xxxx>

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

🔗John A. deLaubenfels <jadl@xxxxxx.xxxx>

🔗John A. deLaubenfels <jadl@xxxxxx.xxxx>

🔗Bob Lee <quasar@xxx.xxxx>

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

🔗John A. deLaubenfels <jadl@idcomm.com>

🔗Paul H. Erlich <PErlich@Acadian-Asset.com>

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

🔗Brett Barbaro <barbaro@noiselabs.com>

🔗John A. deLaubenfels <jadl@idcomm.com>

🔗rtomes@xxxxx.xxx.xxxxxxxxxxxxx)

🔗Brett Barbaro <barbaro@xxxxxxxxx.xxxx>

🔗rtomes@xxxxx.xxx.xxxxxxxxxxxxx)

🔗monz@xxxx.xxx