The Meantone Five

There are five weakly proper 7-note 12-et scales (meaning proper, but
not strictly proper.) If spelled correctly, these five scales are all
proper in any meantone system with a fifth flatter than 700 cents;
hence, five of the six strictly proper 7-note scales in 19-et are
these same scales in a different tuning. I think it would be
reasonable to regard these as the only legitimate meantone strictly
proper 7-note scales, though the question of how to quantify that is
interesting.

In any case, here are the Meantone Five:

Diatonic: C D E F G A B = [0,2,-3,-1,3,5]

Ascending minor: C D Eb F G A B = [0,2,-3,-1,3,5]

Harmonic minor: C D Eb F G Ab B = [0,2,-3,-1,1,-4,5]

Harmonic major: C D E F G Ab B = [0,2,4,-1,1,-4,5]

Locrian major: C D E F Gb Ab Bb = [0,2,4,-1,-6,-4,-2]

Here the numbers on the right are generator steps, ie, meantone fifth
steps. Of course, each of these scales has transpositions.

On 6/9/06, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:
> Diatonic: C D E F G A B = [0,2,-3,-1,3,5]

Of course this should be [0,2,4,-1,1,3,5].

> Ascending minor: C D Eb F G A B = [0,2,-3,-1,3,5]

And this should be [0,2,-3,-1,1,3,5]. The rest are all right.

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...>
wrote:
>
> On 6/9/06, Gene Ward Smith <genewardsmith@...> wrote:
> > Diatonic: C D E F G A B = [0,2,-3,-1,3,5]
>
> Of course this should be [0,2,4,-1,1,3,5].
>
> > Ascending minor: C D Eb F G A B = [0,2,-3,-1,3,5]
>
> And this should be [0,2,-3,-1,1,3,5]. The rest are all right.

Thanks, Keenan. It seems it is really the Meantone Six, not five:
we also have the following exotic entry, which does not exist in
12-et, but which has a complete circle of major and minor thirds, and
is a proper scale in meantone tunings under 700 cents:

C Db E Fb G Ab Bb = [0,-5,4,-8,1,-4,-2]

It doesn't exist in 12-et for the simple reason that E and Fb are the
same note. Here it is in 31-et:

! prop31strange.scl
Strange diatonic-like strictly proper scale
7
!
116.129032
387.096774
425.806452
696.774194
812.903226
1006.451613
1200.000000
! [0,-5,4,-8,1,-4,-2]

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

> It doesn't exist in 12-et for the simple reason that E and Fb are the
> same note. Here it is in 31-et:
>
> ! prop31strange.scl
> Strange diatonic-like strictly proper scale
> 7
> !
> 116.129032
> 387.096774
> 425.806452
> 696.774194
> 812.903226
> 1006.451613
> 1200.000000
> ! [0,-5,4,-8,1,-4,-2]

Now of course the question is, if we stick to 31-et and play Pachelbel
in each scale, what kind of warpage results?