Yahoo Tuning Groups Ultimate Backup tuning choral tuning

I'll introduce myself - I'm a composer and poet, live in NYC, long-standing
interest in Partch, particularly his works with a substantial verbal element,
have followed more recent work in alternate tunings - including some that
people on this list have done - but only sporadically.

Since joining a chorus that sings Renaissance music I've been thinking more
about historical tuning issues. Here are a few questions I'd be curious to hear
opinions on:

1) Given the general meantone orientation of Renaissance music, what is the
status of Pythagorean tuning in the Renaissance and later? The idea seems
to linger for quite some time (even down to the present, e.g. among violinists)
that Pythagorean is the best tuning for melody (contra Partch, who thought it
"unsingable.")

2) If Pythagorean tuning has any advantage for melody, how is it to be
combined with meantone, with its larger diatonic and smaller chromatic
semitones? Is it just a matter of raising leading tones where possible, singing
pure instead of flatted fifths, but also pure thirds?

3) Are there good rules of thumb for avoiding pitch drift in a cappella singing
while attempting to maintain good tuning of the chords?

-Alan

🔗Carl Lumma <ekin@lumma.org>

>I'll introduce myself - I'm a composer and poet, live in NYC,
>long-standing interest in Partch, particularly his works with
>a substantial verbal element, have followed more recent work
>in alternate tunings - including some that people on this list
>have done - but only sporadically.

Hi Alan!

>1) Given the general meantone orientation of Renaissance music,
>what is the status of Pythagorean tuning in the Renaissance and
>later? The idea seems to linger for quite some time (even down
>to the present, e.g. among violinists) that Pythagorean is the
>best tuning for melody (contra Partch, who thought it "unsingable.")

It's important to remember, when answering this question, that
things like "pythagorean" and "meantone" are theoretical models
only outside of fixed-pitch instruments. No empirical study of
choral tuning has ever be done, to my knowledge, that would show
how applicable these models are to how choirs sing (or if any
generalization at all can be made about 'how choirs sing').

>2) If Pythagorean tuning has any advantage for melody, how is it
>to be combined with meantone, with its larger diatonic and smaller
>chromatic semitones? Is it just a matter of raising leading tones
>where possible, singing pure instead of flatted fifths, but also
>pure thirds?

We need to distinguish between harmonic and melodic intervals
here. Generally, it is possible in choirs to draw these from
different source interval sets. If that makes any sense to you.

>3) Are there good rules of thumb for avoiding pitch drift in a
>cappella singing while attempting to maintain good tuning of the
>chords?

Yeah: sing together a lot. Practice practice, listen and
practice.

There are a lot of reasons pitch drift can occur besides comma
problems.

Existing theoretical solutions to these problems ("adaptive JI")
are generally too complex to be practiced in a traditional choral
setting. One solution -- cuing singers with synthesized tones
from headphones -- has been used to good effect by Toby Twining.
Another might be to obtain a microtonal keyboard instrument for
training the choir.

In Barbershop singing, the melody is generally compared against
equal temperament and/or some intuitive sense of 'melodic
rightness' while the harmonies are sung as pure as possible. In
counterpoint this is less applicable, but hearing out dominant
melodic fragments is still a good technique. The amount of
acceptable comma drift should be determined on a per-piece
basis (if you're good enough to rule out other sources of
drift) -- pieces with lots of key changes may hide comma drift
on their own. Good beginning practices are singing harmonics
against a drone and then singing full extended otonal chords
a capella, and then known chord progressions with known amounts
of drift, and then... Bob Wendell may have some helpful advice
here, if you can reach him.

-Carl

🔗Werner Mohrlok <wmohrlok@hermode.com>

To your questions:

1. As to my opinion one should distinguish between the tuning models
of keyboards and the living tuning behavior of human musicians.
The idea of pure thirds has been invented by singers, probably by singers
in the England of the 14th century. Singers as well as human musicians with
string or wind instruments can tune in pure thirds and(!) fifths. Key
instruments
against this have to be tuned to meantone tuning models as soon as they
follow
the idea of pure thirds. The narrow meantone fifths are no ideal, they only
are a inevitable compromise.

2. Equal temperament and Pythagorean tuning are pleasant tuning models for
melodic lines, but not for multivoiced music with third and fifth intervals.
Raising leading tones is contrary to the idea of pure thirds. I don't know
who invented this nonsense, but I know that this idea is still in
the brain of many string instrument players.

3. A good compromise of combining pure thirds and poor fifths by useful
melodic steps is: Setting equal temperament as a reference line and
tune slightly upwards the frequencies of the root and the fifth in major
chords,
by tuning downwards the thirds - and inverse in minor chords: Tuning the
root and the fifth deeper and the minor third higher.
An example (integers to understand as Cent deviations from ET):
C/ E/ G = +4/ -10/ +6; A/ C/ E = -6/ +10/ -4.
This tuning behavior reduces the risk of pitch drift.
See our website:
www.hermode.com
You will find at "historical" on the end, at "software driven tuning",
a musical example with diagrams of 4 different ideas of tuning models
in just intonation, each of these 4 ideas with different pitch results.

4. An additional pitfall is: A major chord in just intonation is an
unequivocal
acoustic structure, easy-to-hear with its 4 : 5 : 6 frequency ratios.
But for minor chords there exist two different ideal frequency ratios:
The 10 : 12 : 15 frequency ratios, showing the same third intervals
like a major chord in just intonation - and the 16 : 19 : 24 frequency
ratios. The last one translated in Cent deviations from ET:
root / minor third/ fifth = 0 / -2 / +2. This is very tight to ET and
therefore minor chords in ET sound agreeable.
Now, if the choir would follow naive its ears, it could happen as follows:
C/ E/ G = +4/ -10/ +6; E/ G / B(H) = -10/ -12/ -8 (the second E in
correspondence to the first one, the minor chord following
the 16 : 19 : 24 frequency ratios). As a result of this the pitch
drifts immediately downwards.
With other words: Minor chords in just intonation have to be
trained to the high minor thirds of the 10 : 12 : 15 frequency ratios,
one cannot trust to a naive intonation behavior.

Do you possess a MAC and LOCIC7? Sorry for making PR -
but there you will find Hermode Tuning, which is a living,
self-correcting tuning model so that you could listen to the
acoustic result of pure thirds and fifths, following the line
of ET as best as possible.

Werner

-----Ursprï¿½ngliche Nachricht-----
Von: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com]Im Auftrag von
afrshaw
Gesendet: Samstag, 30. April 2005 23:42
An: tuning@yahoogroups.com
Betreff: [tuning] choral tuning

I'll introduce myself - I'm a composer and poet, live in NYC,
long-standing
interest in Partch, particularly his works with a substantial verbal
element,
have followed more recent work in alternate tunings - including some that
people on this list have done - but only sporadically.

Since joining a chorus that sings Renaissance music I've been thinking
more
about historical tuning issues. Here are a few questions I'd be curious to
hear
opinions on:

1) Given the general meantone orientation of Renaissance music, what is
the
status of Pythagorean tuning in the Renaissance and later? The idea seems
to linger for quite some time (even down to the present, e.g. among
violinists)
that Pythagorean is the best tuning for melody (contra Partch, who thought
it
"unsingable.")

2) If Pythagorean tuning has any advantage for melody, how is it to be
combined with meantone, with its larger diatonic and smaller chromatic
semitones? Is it just a matter of raising leading tones where possible,
singing
pure instead of flatted fifths, but also pure thirds?

3) Are there good rules of thumb for avoiding pitch drift in a cappella
singing
while attempting to maintain good tuning of the chords?

-Alan

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🔗afrshaw <alan.shaw@verizon.net>

Carl Lumma wrote:

>It's important to remember . . . that things like "pythagorean" and "meantone"
are theoretical models only outside of fixed-pitch instruments. No empirical
study of choral tuning has ever been done, to my knowledge, that would show
how applicable these models are to how choirs sing (or if any generalization
at all can be made about 'how choirs sing').<

Yes, and any such generalization would not be of much interest anyway, at
least not to me. The point is not how the average choir sings, but how to do it
better. And certainly, I don't expect "better" to follow any one model. That was
my question: given these different (and often conflicting) models, which is
more helpful in which contexts?

>Existing theoretical solutions to these problems ("adaptive JI") are generally
too complex to be >practiced in a traditional choral setting. One solution --
cuing singers with synthesized tones from headphones -- has been used to
good effect by Toby Twining. Another might be to obtain a microtonal
keyboard instrument for training the choir.<

Twining's work is impressive, but it's extended JI, isn't it? To use a similar
approach for traditional choral music you would have to first determine the
exact pitch of every note, whether by using one of these complex "adaptive"
algorithms or otherwise. I for one wouldn't want to try to follow such rigid cues
in actual performance, where intonation, like other elements, needs a certain
interpretive leeway. For training, though, I suppose it could be useful.

A question about barbershop: is it true that pitch drift is not of much concern
there, or is it, as you say, on a piece-by-piece basis? Certainly it is not
considered acceptable by most performers of traditional a cappella music,
though a lot depends on how much over how long a piece. A seven-minute
piece that ends a quartertone lower than it began is not going to bother
anyone except maybe those with absolute pitch, provided the fall is not all in
one place. On the other hand a tone and a half over, say, two minutes, can
begin to make you a little queasy.

Werner Mohrlok wrote:

>Singers as well as human musicians with string or wind instruments can
tune in pure thirds and(!) fifths. Key instruments against this have to be tuned
to meantone tuning models as soon as they follow the idea of pure thirds. The
narrow meantone fifths are no ideal, they only are a inevitable compromise.

>. . . Equal temperament and Pythagorean tuning are pleasant tuning models
for melodic lines, but not for multivoiced music with third and fifth intervals.
Raising leading tones is contrary to the idea of pure thirds. I don't know who
invented this nonsense, but >I know that this idea is still in the brain of many
string instrument players.<

Well, the "nonsense" must come from using pure rather than narrow fifths,
which is already Pythagorean. It's true that the resulting melodic intervals
differ considerably from those of meantone, hence my question about
reconciling them. Renaissance choral music is not just a succession of pure
triads. Many pieces have sequences of plain chant inbetween polyphonic
sections, and for these Pythagorean is definitely a much better guide than
meantone. Even in polyphonic sections you'll sometimes have one voice part
singing alone, e.g. beginning an imitative point. And in those cases where the
leading tone function of a note conflicts with its function as the third in a chord,
I think you have to make an artistic, interpretive judgment as to whether, in the
particular case, to emphasize one over the other, or compromise between
them.

>A good compromise of combining pure thirds and poor fifths by useful
melodic steps is: Setting equal >temperament as a reference line and tune
slightly upwards the frequencies of the root and the fifth in major chords, by
tuning downwards the thirds >- and inverse in minor chords: Tuning the
root and the fifth deeper and the minor third higher. An example (integers to
understand as Cent deviations >from >ET):C/ E/ G = +4/ -10/ +6; A/ C/ E = -6/ +
10/ -4. This tuning behavior reduces the risk of pitch drift.<

Yes, this is the sort of "rule of thumb" I was asking for. This is useful. A
somewhat similar one is given by Olivier Bettens in his fine article at virga.org/
zarlino, only he uses Pythagorean as the base, and simply raises the root and
fifth in major chords and the third in minor ones, by a full syntonic comma in
each case (chords that are major by chromatic alteration are treated like
minor chords). From the midi examples he gives this sounds far more
workable than I would have expected such a simple "adaptive" algorithm
could be, though I don't remember any of the examples including comma
shifts on held-over notes, which would be the real test. I would highly
recommend this very judicious article for any who read French (the first few
sections have an English translation).

>An additional pitfall is: A major chord in just >intonation is an unequivocal
acoustic structure, easy-to-hear with its 4 : 5 : 6 frequency ratios. But >for
minor chords there exist two different ideal >frequency ratios: The 10 : 12 : 15
frequency ratios, showing the same third intervals like a major chord >in just
intonation - and the 16 : 19 : 24 frequency ratios. The last one translated in
Cent deviations from ET: root / minor third/ fifth = 0 / -2 / >+2. This is very tight
to ET and therefore minor >chords in ET sound agreeable. Now, if the choir
would >follow naive its ears, it could happen as follows: C/ >E/ G = +4/ -10/ +
6; E/ G / B(H) = -10/ -12/ -8 (the >second E in correspondence to the first one,
the minor chord following the 16 : 19 : 24 frequency ratios). As a result of this
the pitch drifts immediately downwards. With other words: Minor chords in just
intonation have to be trained to the high minor thirds of the 10 : 12 : 15
frequency ratios, one cannot trust to a naive intonation behavior.<

I would agree that, for many reasons, intonation is more problematic for minor
chords than for major ones. In Bettens' midis, for instance, the just major
chords sound nice and pure, but the minor ones sound very rough despite
being in just ratios. The timbre used seems to accentuate the difference tones.

The 6/5 third, moreover, sounds way too high as a melodic third scale degree
in music in the classical tradition. One violin teacher gives an example from
one of the Bach partitas that begins with a minor chord followed by a
descending scale passage, and recommends that the third degree in the
scale passage be made noticeably lower than in the chord. Other approaches
may be possible, but I don't think this is necessarily "nonsense."

🔗Carl Lumma <ekin@lumma.org>

>>Existing theoretical solutions to these problems ("adaptive JI") are
>>generally too complex to be practiced in a traditional choral setting.
>>One solution -- cuing singers with synthesized tones from headphones
>>-- has been used to good effect by Toby Twining. Another might be to
>>obtain a microtonal keyboard instrument for training the choir.
>
>Twining's work is impressive, but it's extended JI, isn't it?

Yep.

>To use a similar approach for traditional choral music you would have
>to first determine the exact pitch of every note, whether by using one
>of these complex "adaptive" algorithms or otherwise.

Right.

>I for one wouldn't want to try to follow such rigid cues in actual
>performance, where intonation, like other elements, needs a certain
>interpretive leeway. For training, though, I suppose it could be useful.

Agreed.

>A question about barbershop: is it true that pitch drift is not of
>much concern there, or is it, as you say, on a piece-by-piece basis?

Piece by piece. For pieces that have a strong tonic, the problem
is generally well-controlled in Barbershop at the professional level.

-Carl

🔗Kurt Bigler <kkb@breathsense.com>

🔗Werner Mohrlok <wmohrlok@hermode.com>

Actually only Logic Pro. Sorry.

One alternate possibility is: The Kontakt2 of Native
Instruments now will content Hermode Tuning.
But Native Instruments in the first step has implemented
it very problematical. The Hermode Tuning function will be
managed by every instrument seperately. In this way
Hermode Tuning will only run perfectly for a certain
instrument if and when it will get all Note-on and Note-off
messages of the played music.

Maybe for hear education this restriction will be less
important, as for such aims you can play all notes with
the same instrument.

I hope, Native Instruments will change soon this nonsense.

The third alternative solution is: Content Organs
www.content-organs.com
produces electronic church organs with Hermode Tuning
and soon will export them (again) to the USA.
This is the most expensive solution, but for hear education
the sounds of church organs are best qualified.

The forth alternative solution is a church organ expander
with an additional windows programme of ours. With this
programme it is possible to perform as well Hermode Tuning as
also every fixed tuning model, even non-octave-repeating
tuning models, for instance partial tone tuning models.
It costs 1,705.00 Euro, additionally tax and freight.

Best

Werner

> -----Ursprï¿½ngliche Nachricht-----
> Von: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com]Im Auftrag
von Kurt Bigler
> Gesendet: Dienstag, 10. Mai 2005 09:16
> An: tuning@yahoogroups.com
> Betreff: [tuning] to Werner: hermode & logic 7
>

> on 5/1/05 10:41 PM, Werner Mohrlok <wmohrlok@hermode.com> wrote:
>
>> Do you possess a MAC and LOCIC7? Sorry for making PR -
>> but there you will find Hermode Tuning, which is a living,
>> self-correcting tuning model so that you could listen to the
>> acoustic result of pure thirds and fifths, following the line
>> of ET as best as possible.
>
> Is this part of Logic Express or only Logic Pro?
>
> -Kurt

🔗Ozan Yarman <ozanyarman@superonline.com>

Dear Werner, this is directed to the second article of your old message dated 2 May.

You might be pleased to know that the leading tones in Maqam Music are actually performed with just intonation in mind, such that F# is an apotome (rather than a limma) away from G.

The nonsense which you speak of is also well ingrained in my brain, though I am not educated as a string instrument player. I (and others like me, especially Yalçın Tura) consider leading tones to be only as distant to the fundamental tone as the smallest possible semitone, while this certainly is not the case with Maqam Music. Take for instance Maqam Segah given below where the third degree is the tonic:

0: 1/1 0.000 unison, perfect prime
1: 13/11 289.210 tridecimal minor third
2: 5/4 386.314 major third
3: 4/3 498.045 perfect fourth
4: 3/2 701.955 perfect fifth
5: 5/3 884.359 major sixth, BP sixth
6: 15/8 1088.269 classic major seventh
7: 2/1 1200.000 octave

Still, I think there is much more to beatless harmonics to what makes a chord consonant.

Cordially,
Ozan

----- Original Message -----
From: Werner Mohrlok
To: tuning@yahoogroups.com
Sent: 02 Mayıs 2005 Pazartesi 8:41
Subject: AW: [tuning] choral tuning

To your questions:

1. As to my opinion one should distinguish between the tuning models
of keyboards and the living tuning behavior of human musicians.
The idea of pure thirds has been invented by singers, probably by singers
in the England of the 14th century. Singers as well as human musicians with
string or wind instruments can tune in pure thirds and(!) fifths. Key instruments
against this have to be tuned to meantone tuning models as soon as they follow
the idea of pure thirds. The narrow meantone fifths are no ideal, they only
are a inevitable compromise.

3. A good compromise of combining pure thirds and poor fifths by useful
melodic steps is: Setting equal temperament as a reference line and
tune slightly upwards the frequencies of the root and the fifth in major chords,
by tuning downwards the thirds - and inverse in minor chords: Tuning the
root and the fifth deeper and the minor third higher.
An example (integers to understand as Cent deviations from ET):
C/ E/ G = +4/ -10/ +6; A/ C/ E = -6/ +10/ -4.
This tuning behavior reduces the risk of pitch drift.
See our website:
www.hermode.com
You will find at "historical" on the end, at "software driven tuning",
a musical example with diagrams of 4 different ideas of tuning models
in just intonation, each of these 4 ideas with different pitch results.

4. An additional pitfall is: A major chord in just intonation is an unequivocal
acoustic structure, easy-to-hear with its 4 : 5 : 6 frequency ratios.
But for minor chords there exist two different ideal frequency ratios:
The 10 : 12 : 15 frequency ratios, showing the same third intervals
like a major chord in just intonation - and the 16 : 19 : 24 frequency
ratios. The last one translated in Cent deviations from ET:
root / minor third/ fifth = 0 / -2 / +2. This is very tight to ET and
therefore minor chords in ET sound agreeable.
Now, if the choir would follow naive its ears, it could happen as follows:
C/ E/ G = +4/ -10/ +6; E/ G / B(H) = -10/ -12/ -8 (the second E in
correspondence to the first one, the minor chord following
the 16 : 19 : 24 frequency ratios). As a result of this the pitch
drifts immediately downwards.
With other words: Minor chords in just intonation have to be
trained to the high minor thirds of the 10 : 12 : 15 frequency ratios,
one cannot trust to a naive intonation behavior.

Werner

🔗Werner Mohrlok <wmohrlok@hermode.com>

Dear Ozan,

This is perhaps a misunderstanding. I didn't speak of "all music",
I only spoke of "multivoiced music in western tradition, written
with fifth and third intervals and equipped with major and minor
chords".
In such music the upwards leading tone is almost a
major third. An example: C-major, dominant to tonica =
G-B-D-(F) to C-E-G.
As soon as you will tune the thirds to just intonation you will
get a very distinct audible bass forth (or fifth) jump, G - C.
Tuning the B as leading tone upwards will destroy this
acoustic effect. This is why I prefer and suggest to tune thirds
in such music to just intonation.

In music with only one voice or in other music cultures everyone
may handle it according to his personal preferences or to the
ideas of these cultures.

Besides, I don't like the term "beatfree", as every multivoiced
music creates beats. Even in just intonation. But in just intonation
these beats are almost in resonance with the original frequencies.

And last not least, please don't ask me how to handle the "F" in
the example, shown above...

Best

Werner

> -----Ursprï¿½ngliche Nachricht-----
> Von: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com]
> Im Auftrag von Ozan Yarman
> Gesendet: Mittwoch, 11. Mai 2005 17:32
> An: tuning@yahoogroups.com
> Betreff: Re: [tuning] choral tuning
>
>
> Dear Werner, this is directed to the second article of your old message
> dated 2 May.
>
> You might be pleased to know that the leading tones in Maqam Music are
> actually performed with just intonation in mind, such that F# is an
> apotome (rather than a limma) away from G.
>
> The nonsense which you speak of is also well ingrained in my brain,
> though I am not educated as a string instrument player. I (
> and others like me, especially Yalï¿½ï¿½n Tura) consider leading tones
> to be only as distant to the fundamental tone as the smallest possible
> semitone, while this certainly is not the case with Maqam Music.
> Take for instance Maqam Segah given below where the third degree is the
tonic:
>
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 13/11 289.210 tridecimal minor third
> 2: 5/4 386.314 major third
> 3: 4/3 498.045 perfect fourth
> 4: 3/2 701.955 perfect fifth
> 5: 5/3 884.359 major sixth, BP sixth
> 6: 15/8 1088.269 classic major seventh
> 7: 2/1 1200.000 octave
>
> Still, I think there is much more to beatless harmonics to what makes a
> chord consonant.
>
> Cordially,
> Ozan

----- Original Message -----
From: Werner Mohrlok
To: tuning@yahoogroups.com
Sent: 02 Mayï¿½s 2005 Pazartesi 8:41
Subject: AW: [tuning] choral tuning

...

2. Equal temperament and Pythagorean tuning are pleasant tuning models
for
melodic lines, but not for multivoiced music with third and fifth
intervals.
Raising leading tones is contrary to the idea of pure thirds. I don't
know
who invented this nonsense, but I know that this idea is still in
the brain of many string instrument players. ..

🔗Ozan Yarman <ozanyarman@superonline.com>

I was referring to the tendency to achieve the best harmonic consonance in chords the way you explained, but in an obscure manner in the heterophonic practice of Turkish Maqam Music. A hidden desire for homophonic unity may be involved beneath this tradition. Don't you find it interesting that the Ney's third natural key is normally 5/4 away from Rast, while violin virtuosos almost always interpret the third degree of the diatonical scale in a pythagorean manner?

The multi-voiced music performed by Turkish orchestras nowadays is mostly out of tune for my taste, that being probably a result of the neglect in alignment with your theoretical observation for strictly harmonic textures.

Thus, I agree with you overall, save for the fact that I find 81/64 just as pleasing (if not more) as 5/4, and would think that B could just as well be a pythagorean major third away from G, and E from C in the chordal modulation example you gave.

Therefore:

G-B-D-F

1/1
81/64
3/2
16/9

F-A-C

4/3
27/16
2/1

Which is quite pleasant to me.

Cordially,
Ozan
----- Original Message -----
From: Werner Mohrlok
To: tuning@yahoogroups.com
Sent: 11 Mayıs 2005 Çarşamba 21:15
Subject: AW: [tuning] choral tuning

Dear Ozan,

In music with only one voice or in other music cultures everyone
may handle it according to his personal preferences or to the
ideas of these cultures.

Besides, I don't like the term "beatfree", as every multivoiced
music creates beats. Even in just intonation. But in just intonation
these beats are almost in resonance with the original frequencies.

And last not least, please don't ask me how to handle the "F" in
the example, shown above...

Best

Werner

🔗Werner Mohrlok <wmohrlok@hermode.com>

Regarding violin virtuosos and opera singers: I believe
they follow the sentence "Better too high than false".

But - and more seriously-
One cannot argue about musical (or other) taste.
I like forceful sounding harmonies and you
follow a certain idea of melodic distances.

Best

Werner

-----Ursprï¿½ngliche Nachricht-----
Von: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com]Im Auftrag von
Ozan Yarman
Gesendet: Mittwoch, 11. Mai 2005 21:01
An: tuning@yahoogroups.com
Betreff: Re: [tuning] choral tuning

I was referring to the tendency to achieve the best harmonic consonance in
chords the way you explained, but in an obscure manner in the heterophonic
practice of Turkish Maqam Music. A hidden desire for homophonic unity may be
involved beneath this tradition. Don't you find it interesting that the
Ney's third natural key is normally 5/4 away from Rast, while violin
virtuosos almost always interpret the third degree of the diatonical scale
in a pythagorean manner?

The multi-voiced music performed by Turkish orchestras nowadays is mostly
out of tune for my taste, that being probably a result of the neglect in
alignment with your theoretical observation for strictly harmonic textures.

Thus, I agree with you overall, save for the fact that I find 81/64 just
as pleasing (if not more) as 5/4, and would think that B could just as well
be a pythagorean major third away from G, and E from C in the chordal
modulation example you gave.

Therefore:

G-B-D-F

1/1
81/64
3/2
16/9

F-A-C

4/3
27/16
2/1

Which is quite pleasant to me.

Cordially,
Ozan
----- Original Message -----
From: Werner Mohrlok
To: tuning@yahoogroups.com
Sent: 11 Mayï¿½s 2005 ï¿½arï¿½amba 21:15
Subject: AW: [tuning] choral tuning

Dear Ozan,

In music with only one voice or in other music cultures everyone
may handle it according to his personal preferences or to the
ideas of these cultures.

Besides, I don't like the term "beatfree", as every multivoiced
music creates beats. Even in just intonation. But in just intonation
these beats are almost in resonance with the original frequencies.

And last not least, please don't ask me how to handle the "F" in
the example, shown above...

Best

Werner