Optimal Tuning of the Mixolydian and Dorian Modes

Hi, tuning people!

Paul Erlich has argued that these modes have a kind of tonality in 12-
equal because in 12-equal the diminished fifth/augmented fourth forms
a rough 7-limit tetrad with the tonic triad so there is no need for
any kind of resolution.

Now, what do you think would be the optimal tuning (or tunings) for
music in these modes? I'd think it would tend towards pythagorean.

Kalle

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> Hi, tuning people!
>
> Paul Erlich has argued that these modes have a kind of tonality in
12-
> equal because in 12-equal the diminished fifth/augmented fourth
forms
> a rough 7-limit tetrad with the tonic triad so there is no need for
> any kind of resolution.
>
> Now, what do you think would be the optimal tuning (or tunings) for
> music in these modes? I'd think it would tend towards pythagorean.
>
> Kalle

The modes are all in the same key, that of the relative natural
major. It is a misconception to think of modes as a change of key.
The tonic is always that of the natural major (the relative minor is
no exception, but its off-key-ness is acceptable - however the true
natural key of Am is C). The modes derive from the notion that a
melody should not extend beyond the compass of an octave. This is (a)
so as not to strain the singing voice and (b) to avoid melodic
crossover when more than one line is being sung. The modes merely
shift the compass of the melody, not its key. They were specific to
chants and are sometimes called the 'church modes', or 'modes' for
short, which really were sung (and 'organized') in natural temper. As
such they don't really have a place in modern music.

The modern concept of mode consists of forcibly rotating a scale,
which may be any scale structure other than full chromatic (which is
always the same no matter how much you rotate it), so as to
apparently shift the tonic. However, this has developed as a
consequence of modern ET and reveals itself as 'off-key' jazz, so
natural tunings such as the various pythagoreans are not probably not
appropriate.

Peter wrote:

> The modes are all in the same key, that of the relative natural
> major. It is a misconception to think of modes as a change of key.

When we are thinking of keys in terms of major/minor-system, yes.

> The tonic is always that of the natural major (the relative minor
is
> no exception, but its off-key-ness is acceptable - however the true
> natural key of Am is C).

That is more controversial. I agree that the tonic of the natural
major has a very strong gravitation but I also think that at least
aeolian (natural minor), mixolydian and dorian modes can be
easily "tonicized" even when the music uses triadic harmony.

The modes derive from the notion that a
> melody should not extend beyond the compass of an octave. This is
(a)
> so as not to strain the singing voice and (b) to avoid melodic
> crossover when more than one line is being sung. The modes merely
> shift the compass of the melody, not its key. They were specific to
> chants and are sometimes called the 'church modes', or 'modes' for
> short, which really were sung (and 'organized') in natural temper.
As
> such they don't really have a place in modern music.

I believe the modes were organized theoretically in pythagorean
tuning. No one really knows about the singing though. It must be
extremely difficult to sing in pythagorean tuning accurately. Also
triads were not considered stable and had to resolve.

> The modern concept of mode consists of forcibly rotating a scale,
> which may be any scale structure other than full chromatic (which
is
> always the same no matter how much you rotate it), so as to
> apparently shift the tonic. However, this has developed as a
> consequence of modern ET and reveals itself as 'off-key' jazz, so
> natural tunings such as the various pythagoreans are not probably
not
> appropriate.

Yes, modern usage was what I had in mind for these modes. I meant the
optimal tuning in terms of maximizing consonance.

Kalle

>I believe the modes were organized theoretically in pythagorean
>tuning.

Yes.

>No one really knows about the singing though. It must be
>extremely difficult to sing in pythagorean tuning accurately. Also
>triads were not considered stable and had to resolve.

Because they were treated as dissonances in the music their
precise tuning doesn't matter as much, I'd say. It doesn't
matter to get exactly 81:64 thirds, although one may naturally
be pointed there by voice leading.

-Carl

> From: "Kalle Aho" <kalleaho@mappi.helsinki.fi>
> Subject: Re: Optimal Tuning of the Mixolydian and Dorian Modes
>
> Peter wrote:
>
> > The tonic is always that of the natural major (the relative minor
> is
> > no exception, but its off-key-ness is acceptable - however the true
> > natural key of Am is C).
>
> That is more controversial. I agree that the tonic of the natural
> major has a very strong gravitation but I also think that at least
> aeolian (natural minor), mixolydian and dorian modes can be
> easily "tonicized" even when the music uses triadic harmony.
>

This is the view I take inthis response (and in life).

So, assuming that you have decided you
are going to make music in one of these modes, now you want to
optimize your tuning of that mode for the music that you are going to make.

This may be very different than maximizing consonance, or in having
a good "major scale" on the ionian found as the seventh mode of your
dorian etc....

dorian : LsL L LsL

dorian seems kind of trivial since you can start a chain of fifths
from the third scale degree and get lots of good properites.

1/1 9/8 32/27 4/3 3/2 27/16 16/9

If I think of Dorian harmony, (and you don't have to think of a harmonic
music, but it is a kind of music one can think of) then the two cadences
that define dorian mode to the exclusion of others are II- I- and IV I-.

The II- I- cadence here is very nice. The IV has a pythagorean major
third, which might be just the buzz you were looking for.

But maybe not. Making the IV into a 4:5:6 and trying to preserve the
omniterachodal nature leads to

dorian : MsL L MsL

1/1 10/9 32/27 4/3 3/2 5/3 16/9

and now the II- I- cadence is a little flavorful.

Note that this isn't the standard JI "major" from the second degree.

mixolydian : LLs L LsL

problem 1, if we want a JI major triad and a diatonic scale, we;re
heading towards meantone (and 1/4 comma may not be a bad choice for
mixolydian. The harmonic cadences are bVII I or V- I and they'd sound
great). Of course, if a more active major third is desired, with better
fifths, then the comma can be dialed till you reach 12-tet, or pythagorean,
or you can go out the other side, but thats getting a little strange.

so now for a more JI approach.

mixolydian : LMs L MsL

This is the same pattern as above, again, not hte trad JI pattern in
its mixo rotation.

1/1 9/8 5/4 4/3 3/2 5/3 16/9

This might work out, nice V- and a pythagorean bVII cadenceing to I
may really work.

So, these two approaches certainly aren't mathematically optimal
for generating consonances (some sort of meantone probably is
though), but if I were approaching tuning a piece in these modes,
these are the games I might play.

I dunno, poked my head out after a year or so and found things very
busy, in a somewhat disturbed way. (Not the usual arguments). Glad
to know a lot of you folks are still pluggin away here and
(I hope) in your studios, on your instruments etc...

Happy Holidays

Bob

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> > I believe the modes were organized theoretically in pythagorean
> > tuning.
>
> Yes.

the standard reference for 1000 years (from c.505 to c.1500)
was Book 4 of Boethius's _de institutione musica_:

http://www.music.indiana.edu/tml/6th-8th/BOEMUS4_TEXT.html

> > No one really knows about the singing though. It must be
> > extremely difficult to sing in pythagorean tuning accurately.
> > Also triads were not considered stable and had to resolve.
>
> Because they were treated as dissonances in the music their
> precise tuning doesn't matter as much, I'd say. It doesn't
> matter to get exactly 81:64 thirds, although one may naturally
> be pointed there by voice leading.

the musical context always presented the "perfect" consonances
("4ths" and "5ths") as resolutions, which would have been
very easy to sing and to tune by ear, but presented the
"imperfect" consonances ("2nds", "3rds", "6ths", and "7ths")
as dissonances.

as Carl points out, the tuning of the dissonances would
allow for some margin of error. for example, it would be
probably be very easy for someone singing _a capella_ to
sing a 15:19 ratio (~409.2443014 cents) instead of a
64:81 (~407.8200035 cents) without anyone noticing a
difference.

on the other hand, keep in mind that in _a capella_
vocal performances of church music during the medieval
period, the singers were highly trained monks who
specialized in singing musical pieces, and they were
trained with the monochord. so the performers would
have had a very well-developed sense of Pythagorean
tuning for the diatonic scale.

-monz

On Fri, 12 Dec 2003 21:30:31 -0000 Monz wrote

>the standard reference for 1000 years (from c.505 to c.1500)
>was Book 4 of Boethius's _de institutione musica_:

>http://www.music.indiana.edu/tml/6th-8th/BOEMUS4_TEXT.html

Thanks Monz,

This is a work I have thought I would like to read at some point -
if I could. Do you have a reference to a translation, please?

Peter Frazer,
www.midicode.com

hi Peter,

--- In tuning@yahoogroups.com, Peter Frazer <paf@e...> wrote:
> On Fri, 12 Dec 2003 21:30:31 -0000 Monz wrote
>
> > the standard reference for 1000 years (from c.505 to c.1500)
> > was Book 4 of Boethius's _de institutione musica_:
>
> > http://www.music.indiana.edu/tml/6th-8th/BOEMUS4_TEXT.html
>
> Thanks Monz,
>
> This is a work I have thought I would like to read at
> some point - if I could.

if you're interested in the history of European music,
you *should* read it.

> Do you have a reference to a translation, please?

Boethius, Anicius Manlius Torquatus Severinus. c.505 AD.
_De institutione musica libri quinque_.
Godofredus Friedlein (ed.), Leipzig, 1867, Frankfurt a.M., 1966.
Italian translation by G. Marzi, Cremona.
English translation _Fundamentals of music_ by Calvin M. Bower,
Yale University Press, New Haven, 1989. ASIN: 0300039433.
L.o.C.#: MT5.5 .B613 1989.

amazon.com says that the Bower translation is out of print.
but it will be in any decent college or university library.

-monz

Hello!

I'd like to thank everyone who replied! Let me explain more clearly
what I have in mind:

The tuning would be a chain of seven fifths i.e. some form of
meantone. The dominant seventh chord on the mixolydian tonic would
have to approximate 4:5:6:7 and the minor sixth chord on the dorian
tonic would have to approximate its utonal counterpart in root
position, that is 1/1:6/5:3/2:12/7 (although 16:19:24:27 might be
interesting too). I'd think maximizing consonance for these with
something like minimized unweighted RMS-errors or minimax would make
the fifth much wider than meantone fifth, something close to
pythagorean (around 702 or 703 cents). But that makes major thirds
wide and I think these modes still sound better in 12-equal. Why
won't the optimal tuning sound better than 12-equal?

Kalle

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> > Hello!
> >
> > I'd like to thank everyone who replied! Let me explain more
clearly
> > what I have in mind:
> >
> > The tuning would be a chain of seven fifths i.e. some form of
> > meantone. The dominant seventh chord on the mixolydian tonic
would
> > have to approximate 4:5:6:7 and the minor sixth chord on the
dorian
> > tonic would have to approximate its utonal counterpart in root
> > position, that is 1/1:6/5:3/2:12/7 (although 16:19:24:27 might be
> > interesting too). I'd think maximizing consonance for these with
> > something like minimized unweighted RMS-errors or minimax would
> make
> > the fifth much wider than meantone fifth, something close to
> > pythagorean (around 702 or 703 cents). But that makes major
thirds
> > wide and I think these modes still sound better in 12-equal. Why
> > won't the optimal tuning sound better than 12-equal?
>
> I calculated an rms-optimum at 701.797 cents for 4:5:6:7 and
> 16:19:24:27. I'm not sure if this is correct but the tetrads sound
a
> bit better than in 12-equal. But the thirds sound worse in
isolation.

Then I got 702.226 cents for 7-limit consonances only. The minor
sixth chord sounds pretty much similar to the above.

Kalle

--- In tuning@yahoogroups.com, Bob Valentine <BVAL@I...> wrote:

> But maybe not. Making the IV into a 4:5:6 and trying to preserve
the
> omniterachodal nature leads to
>
> dorian : MsL L MsL
>
> 1/1 10/9 32/27 4/3 3/2 5/3 16/9

The modes of this would be

MsL L MsL -- tetrachordal
sLLMsLM -- not tetrachordal
L LMs LMs -- tetrachordal
LMs LMs L -- tetrachordal
MsL MsL L -- tetrachordal
sLMsLLM -- not tetrachordal
LMs L LMs -- tetrachordal

So it's almost omnitetrachordal -- it only fails in 2 octave species
out of 7.