Just/mean-tone tuning Vs. 12-tET

Hi all,

After much musical ear-ache and testing of various
pitches, I've come to the conclusion that I'm still
not sure about the 'what the ear wants to hear' as
regards 12-tET and just/mean tuning.

Of course, what's assumed (and what many of you have
been telling me) is that pitches with pure ratios
(i.e. 3/2, 4/3, 5/4, 5/3 and 6/5 etc. etc.) have
'sweeter' sounding harmonies/chords than those done
with the 'compromising' 12-tET scale.

Well, what I've decided to do is an experiment - and have
created an mp3 with 4 tests. The tests are very simple
scale ascensions and only go up to the Major 6th, but
include the minor third in one of the 4 tests.
In a nutshell, it compares the 12t-ET pitches against
the mean/just pitches.

What I want to do is create a poll (which I'm not
completely sure how to do - to be honest) with the
following choosable options:

1: How can anyone think Just/mean tone sounds ok...
2: 12t-ET sounds 'sweeter' than Just/mean tone
3: 12t-ET sounds a fraction 'sweeter' than Just/mean tone
4: The two are roughly as good.
5: Just/mean tone sounds a fraction 'sweeter' than 12t-ET
6: Just/mean tone sounds 'sweeter' than 12t-ET
7: How can anyone think 12t-ET sounds ok...
8: I've always preferred 12t-ET, but I'm now giving
Just/mean tone second thoughts
9: I've always preferred Just/mean tone, but I'm
now giving 12t-ET second thoughts
10: I've got used to Just/mean tone and now 12t-ET
seems much better thanks to this test!
11: I've got used to 12t-ET and now Just/mean tuning
seems much better thanks to this test!

And yes... I do think option 1 or 2 is correct :P
But, I don't represent the whole world (even though I
should ;)... so lets see how it goes :D

Before I upload the mp3 (about 1meg max) to the files
section, I'd just like to check and confirm on the
/exact/ pitches I'll be using:

Just/mean-tone tuning:

C (root): 0 cents or 1.000
D (major 2nd), ~203.91 cents or 9/8 or 1.125
Eb (minor third) ~315.641 cents or 6/5 or 1.2
E (major third) ~386.313 cents or 5/4 or 1.25
F (fourth) ~498.045 cents or 4/3 or 1.333~
G (fifth) ~701.955 cents or 3/2 or 1.5
A (major sixth) ~884.358 cents or 5/3 or 1.666~

12t-ET:

C (root): 0 cents or 1.000
D (major 2nd), 200 cents or 2^(2/12) or ~1.122
Eb (minor third) 300 cents or 6/5 or 2^(3/12) ~1.189
E (major third) 400 cents or 5/4 or 2^(4/12) or ~1.2599
F (fourth) 500 cents or 4/3 or 2^(5/12) or ~1.335
G (fifth) 700 cents or 3/2 or 2^(7/12) or ~1.498
A (major sixth) 900 cents or 5/3 or 2^(9/12) or ~1.681

Also, tagged onto the mp3, is a small (~10 second)
tune I made (using only white notes to avoid
confusion). If anyone's up for converting it, I'd
love to see what it sounds like in mean-tone/pythagoras
tuning or even that 'dynamic' evolving tuning method as
researched by John deLaubenfels etc.

Cheers,

Daniel (dspwhite@email.com)

Make a difference, help support the relief efforts in the U.S.
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--- In tuning@y..., "Daniel White" <soundburst@l...> wrote:
> Hi all,
>
> After much musical ear-ache and testing of various
> pitches, I've come to the conclusion that I'm still
> not sure about the 'what the ear wants to hear' as
> regards 12-tET and just/mean tuning.
>
> Of course, what's assumed (and what many of you have
> been telling me) is that pitches with pure ratios
> (i.e. 3/2, 4/3, 5/4, 5/3 and 6/5 etc. etc.) have
> 'sweeter' sounding harmonies/chords than those done
> with the 'compromising' 12-tET scale.
>
> Well, what I've decided to do is an experiment - and have
> created an mp3 with 4 tests. The tests are very simple
> scale ascensions and only go up to the Major 6th,

Just and meantone intervals don't sound "sweeter" melodically --
only harmonically can it be claimed that they're "purer" in any
way.
>
> Just/mean-tone tuning:
>
> C (root): 0 cents or 1.000
> D (major 2nd), ~203.91 cents or 9/8 or 1.125
> Eb (minor third) ~315.641 cents or 6/5 or 1.2
> E (major third) ~386.313 cents or 5/4 or 1.25
> F (fourth) ~498.045 cents or 4/3 or 1.333~
> G (fifth) ~701.955 cents or 3/2 or 1.5
> A (major sixth) ~884.358 cents or 5/3 or 1.666~

This is not meantone tuning at all, but a rather melodically
awkward Just scale. I would never prefer it melodically to a scale
like meantone or 12-tET where the whole steps are equal
(though others might). So this is not a fair comparison.

Spend a good deal of time listening to Renaissance music
recorded in meantone tuning. Go back to 12-tET and it will sound
truly horrible. But you'll get used to it again if you use it or listen to
music in it . . .

Hi Paul,

> > Well, what I've decided to do is an experiment - and have
> > created an mp3 with 4 tests. The tests are very simple
> > scale ascensions and only go up to the Major 6th,
>
> Just and meantone intervals don't sound "sweeter" melodically --
> only harmonically can it be claimed that they're "purer" in any
> way.

Oh, so you mean 'harmonically' as a chord?
Surely if this was the case though, you couldn't ever
change chord, because this would be in effect a moving
line of simultaneous melodies...

> >
> > Just/mean-tone tuning:
> >
> > C (root): 0 cents or 1.000
> > D (major 2nd), ~203.91 cents or 9/8 or 1.125
> > Eb (minor third) ~315.641 cents or 6/5 or 1.2
> > E (major third) ~386.313 cents or 5/4 or 1.25
> > F (fourth) ~498.045 cents or 4/3 or 1.333~
> > G (fifth) ~701.955 cents or 3/2 or 1.5
> > A (major sixth) ~884.358 cents or 5/3 or 1.666~
>
> This is not meantone tuning at all, but a rather melodically
> awkward Just scale. I would never prefer it melodically to a scale

Oh... How would I alter the test to make it a better poll?

> like meantone or 12-tET where the whole steps are equal

What are the pitches I should use for meantone then?
I need C, D, Eb, E, F G and A as accurate floatings.
You said before how 1.25 is preferable over 1.259 for
the major third. But it seems one can only use this
just 1.25 chord if the tune consists of nothing except
this one chord (otherwise it's changing (i.e. melody) :)

> (though others might). So this is not a fair comparison.

I see. I must admit, I thought it seemed a bit strange that
the 12-note version seemed to 'win' a bit too easily.

> Spend a good deal of time listening to Renaissance music
> recorded in meantone tuning. Go back to 12-tET and it will sound
> truly horrible.
> But you'll get used to it again if you use it or listen to
> music in it . . .

Well, if I were going to hear Renaissance music, maybe I'd
rather hear it in 12-tET, and maybe it should sound better
in 12t-ET anyway.

Well, I'm glad I didn't post the poll. I thought I might've
got a couple of things wrong... :)

Cheers,

Daniel (dspwhite@email.com)

Daniel wrote:

> What are the pitches I should use for meantone then?

C 0.0 cents
D# 76.0 cents
D 193.2 cents
Eb 310.3 cents
E 386.3 cents
F 503.4 cents
F# 579.5 cents
G 696.6 cents
G# 772.6 cents
A 889.7 cents
Bb 1006.8 cents
B 1082.9 cents

> I need C, D, Eb, E, F G and A as accurate floatings.
> You said before how 1.25 is preferable over 1.259 for
> the major third. But it seems one can only use this
> just 1.25 chord if the tune consists of nothing except
> this one chord (otherwise it's changing (i.e. melody) :)

I don't understand your problem here. Because a tuning's good for static
harmony doesn't mean you can't also use it for melody.

Graham

--- In tuning@y..., "Daniel White" <soundburst@l...> wrote:
>
> Oh, so you mean 'harmonically' as a chord?
> Surely if this was the case though, you couldn't ever
> change chord, because this would be in effect a moving
> line of simultaneous melodies...

I don't know what you mean!

> >
> > This is not meantone tuning at all, but a rather melodically
> > awkward Just scale. I would never prefer it melodically to a
scale
>
> Oh... How would I alter the test to make it a better poll?
>
> > like meantone or 12-tET where the whole steps are equal
>
> What are the pitches I should use for meantone then?
> I need C, D, Eb, E, F G and A as accurate floatings.

Well, one really would want to use a different test -- one that shows
off the purer harmonies of meantone. But here goes:

C 1.0000
D 1.1180
Eb 1.1963 (note -- this is not D#)
E 1.2500
F 1.3375
G 1.4953
A 1.6719

> You said before how 1.25 is preferable over 1.259 for
> the major third. But it seems one can only use this
> just 1.25 chord if the tune consists of nothing except
> this one chord (otherwise it's changing (i.e. melody) :)

In meantone tuning, one can change chords freely (though one needs
separate pitches for Eb and D#, etc.)

>
> > Spend a good deal of time listening to Renaissance music
> > recorded in meantone tuning. Go back to 12-tET and it will sound
> > truly horrible.
> > But you'll get used to it again if you use it or listen to
> > music in it . . .
>
> Well, if I were going to hear Renaissance music, maybe I'd
> rather hear it in 12-tET, and maybe it should sound better
> in 12t-ET anyway.

The standard tuning in the Renaissance was meantone tuning. It
definitely "should" sound better in meantone, and in fact it does,
once you've given it a fair shot at "unbrainwashing" you, and of
course attended to other performance considerations. The early music
community is now fairly unanimous that Renaissance music, at least
the keyboard repertoire, be performed in meantone.

Hi Paul,

> > Oh, so you mean 'harmonically' as a chord?
> > Surely if this was the case though, you couldn't ever
> > change chord, because this would be in effect a moving
> > line of simultaneous melodies...
>
> I don't know what you mean!

Sorry - didn't make myself clear. What I meant was
that if mean/just-tone can play a major third sweeter
than the 12-note system, then one would assume this
would mean /melodically/ (in a note sequence) as well
as harmonically (in a chord). So my point was that if
you changed from another triad to the major triad
in the just mean/tone system, this is equivalent to
being a /melodic/ adjustment even though all 3 notes
are changing......... thus making it 'imperfect'.

It seems as though it's a contradiction in terms if
a certain scale is perfect for harmony but not for
melody... or vice versa.

If I play the notes C, then E, then G as seperate
notes (1 second intervals instead of a chord), is
this counted as a melody?

It sounds stupid to say that for perfection:

0 cents (root), 400 cents and 700 cents is needed for
a melody where C, E and G is played with 1 second
intervals....
....compared to...
0 cents (root), 5/4 and 3/2 needed for the the same
notes but played as a /chord/.

Fixed cent amounts seems to make much more sense
for both melody and harmony.

> > > This is not meantone tuning at all, but a rather melodically
> > > awkward Just scale. I would never prefer it melodically to a
> scale
> >
> > Oh... How would I alter the test to make it a better poll?
> >
> > > like meantone or 12-tET where the whole steps are equal
> >
> > What are the pitches I should use for meantone then?
> > I need C, D, Eb, E, F G and A as accurate floatings.
>
> Well, one really would want to use a different test -- one that shows
> off the purer harmonies of meantone. But here goes:

OK, what other test should I try? Maybe this could
go into two polls if you think that would be a good
idea.

> C 1.0000
> D 1.1180
> Eb 1.1963 (note -- this is not D#)
> E 1.2500
> F 1.3375
> G 1.4953
> A 1.6719

Thanks for these (Graham - also thanks :)

> > You said before how 1.25 is preferable over 1.259 for
> > the major third. But it seems one can only use this
> > just 1.25 chord if the tune consists of nothing except
> > this one chord (otherwise it's changing (i.e. melody) :)
>
> In meantone tuning, one can change chords freely (though one needs
> separate pitches for Eb and D#, etc.)

hmmm... so I guess D# is 'needed' when changing key to
say... C# or G# major/minor.....? Or is it a note to
be used in standard C major/minor?

> > > Spend a good deal of time listening to Renaissance music
> > > recorded in meantone tuning. Go back to 12-tET and it will sound
> > > truly horrible.
> > > But you'll get used to it again if you use it or listen to
> > > music in it . . .
> >
> > Well, if I were going to hear Renaissance music, maybe I'd
> > rather hear it in 12-tET, and maybe it should sound better
> > in 12t-ET anyway.
>
> The standard tuning in the Renaissance was meantone tuning. It
> definitely "should" sound better in meantone, and in fact it does,
> once you've given it a fair shot at "unbrainwashing" you, and of

Is there any (mathematical) proof that it's better? :)
I know this might sound a bit naive on my part, but
I could argue you've got used to the 5/4 pitch, just like
so many other people get used to pitches even 'stranger'
than this. For example, as you pointed out earlier, the
medievalists preferred a sharper Major third (fractionally
/above/ 2^(5/12)) (this is over 20 cents out from the
beat-less third!!).
I realise there's the evidence of the mean/just third
being beat-less, but this is not good enough reason (IMHO)
to mean that it's actually 'sweeter' than the 12t-ET M.third.

I'm wondering if it's just been accepted for historical reasons
rather than being selected for its pure merits (i.e. basing
it on how it should sound)... and people have got used to it.

> course attended to other performance considerations. The early music
> community is now fairly unanimous that Renaissance music, at least
> the keyboard repertoire, be performed in meantone.

Well, it's by no means a 'definitive' test, but I'd like
to see how well the poll goes :)

Cheers,

Daniel (dspwhite@email.com)

--- In tuning@y..., "Daniel White" <soundburst@l...> wrote:
> Hi Paul,
>
> > > Oh, so you mean 'harmonically' as a chord?
> > > Surely if this was the case though, you couldn't ever
> > > change chord, because this would be in effect a moving
> > > line of simultaneous melodies...
> >
> > I don't know what you mean!
>
> Sorry - didn't make myself clear. What I meant was
> that if mean/just-tone can play a major third sweeter
> than the 12-note system, then one would assume this
> would mean /melodically/ (in a note sequence) as well
> as harmonically (in a chord).

You see just about every conceivable melodic interval in world
musical cultures based on melody. Harmony is much more restrictive.

> So my point was that if
> you changed from another triad to the major triad
> in the just mean/tone system, this is equivalent to
> being a /melodic/ adjustment even though all 3 notes
> are changing......... thus making it 'imperfect'.

Not 'imperfect' -- just "different" if you grew up hearing 12-tET
melodic intervals all the time.

> It seems as though it's a contradiction in terms if
> a certain scale is perfect for harmony but not for
> melody... or vice versa.

Not at all. Ivor Darreg wrote articles and articles about how the
best tunings for melody were not the best tunings for harmony, and
vice versa.
>
> If I play the notes C, then E, then G as seperate
> notes (1 second intervals instead of a chord), is
> this counted as a melody?
>
> It sounds stupid to say that for perfection:
>
> 0 cents (root), 400 cents and 700 cents is needed for
> a melody where C, E and G is played with 1 second
> intervals....
> ....compared to...
> 0 cents (root), 5/4 and 3/2 needed for the the same
> notes but played as a /chord/.
>
> Fixed cent amounts seems to make much more sense
> for both melody and harmony.

What do you mean, "fixed cent amounts"? 1/1, 5/4, and 3/2 (0, 386,
702) are just as "fixed" as 0, 400, 700.

> OK, what other test should I try? Maybe this could
> go into two polls if you think that would be a good
> idea.

A piece of music with a harmonic progression -- preferable something
written before Beethoven.

>
> > C 1.0000
> > D 1.1180
> > Eb 1.1963 (note -- this is not D#)
> > E 1.2500
> > F 1.3375
> > G 1.4953
> > A 1.6719
>
> Thanks for these (Graham - also thanks :)
>
> > > You said before how 1.25 is preferable over 1.259 for
> > > the major third. But it seems one can only use this
> > > just 1.25 chord if the tune consists of nothing except
> > > this one chord (otherwise it's changing (i.e. melody) :)
> >
> > In meantone tuning, one can change chords freely (though one
needs
> > separate pitches for Eb and D#, etc.)
>
> hmmm... so I guess D# is 'needed' when changing key to
> say... C# or G# major/minor.....?

Yes.

> Or is it a note to
> be used in standard C major/minor?

Only if you're tonicizing E minor or something like that.

> Is there any (mathematical) proof that it's better? :)

You're right to smile -- this is not a mathematical question.

> I know this might sound a bit naive on my part, but
> I could argue you've got used to the 5/4 pitch, just like
> so many other people get used to pitches even 'stranger'
> than this. For example, as you pointed out earlier, the
> medievalists preferred a sharper Major third (fractionally
> /above/ 2^(5/12)) (this is over 20 cents out from the
> beat-less third!!).

Yes, but in Medieval music, the major third was a _dissonance_ that
had to resolve! In Renaissance music, the major third was a _stable
consonance_ -- very different!

> I realise there's the evidence of the mean/just third
> being beat-less, but this is not good enough reason (IMHO)
> to mean that it's actually 'sweeter' than the 12t-ET M.third.

There are other psychoacoustical phenomena we could go into besides
beating -- such as roughness, virtual pitch, and combinational tones -
- but I suspect such arguments won't convince you anyway.
>
> I'm wondering if it's just been accepted for historical reasons
> rather than being selected for its pure merits (i.e. basing
> it on how it should sound)... and people have got used to it.

Well, all I can say is spend a few years playing with different
tuning systems, until you can accept them _all_ for their unique
qualities. _Then_ you can view all types of major thirds without
bias, and _then_ you can decide if one is truly "sweeter" or
something in a unique way.