AKJ's tuning of Neidhardt I

Hi Aaron!

I didn't quite understand your procedure here. So I've interjected
a few questions.

Aaron Krister Johnson wrote:
> Subject: The great Bach temperament debate (was: Re: Bach WTC)
>
> Johnny, can you please put quote levels on in your response, it's hard to
> parse who said what, at least for an outsider.....

Amen to that! :-)

<much snipt>

> ... I came up with my own interpretation of Neidhardt I which I quite
like:
> Starting at C, do pure C-F-Bb-Eb-A; ...

That would be Ab, wouldn't it?

> ... you have a resulting ascending C-Ab sixth.
> We'll divide that up into two equal beating thirds. So, make C-E beat
equal
> to E-G#. Now we have a ascending minor sixth E-C that we will repeat the
> procedure to, to soften the Ab-C Pythagorean third a bit: so bring G# up a
> bit ...

G# being the same as Ab, of course?

> ... so that E-G# now beats equal to G#-C. (of course, it will no longer
beat
> equal with C-E). This layout ultimately favors slightly the flat keys in
> terms of consonance, because of the equal beating step.
>
> Neidhardt I has C-F-Bb as pure fifths/fourths, so we'll keep that from the
> beginning. These are followed by two slightly impure fifths.
>
> So next we retune Eb-Bb ascending fifth ...

Does that mean the Bb changes, while the Eb remains the same?

> ... to be the same size (a la meantone) as
> the ascending fifth Ab-Eb.

Does "a la meantone" mean so that the two fifths beat equally?

> The E-G# is divided analogously to Ab-C; two pure, then two impure fifths.
>
> So then we have and G#-C# pure descending fifth
> Then C#-F# pure ascending fourth
> F#-B descending fifth same size as B-E descending fifth.
>
> Finally, we then distribute C-E a la meantone between C-G-D-A-E.

Same question here ...

> A nice, subtle well temperament, the distant key signatures not too harsh,
and
> it's easily done. And it has a large number of subtly, differently-sized
> intervals, if you like maximully distinct counterpoint! ;)

Final question: Why do you call this your _interpretation_ of Neidhardt I?
I had supposed - perhaps wrongly - from what you wrote about it earlier
that there wasn't a lot of room for interpretation of the temperament ...

Regards,
Yahya

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On Tuesday 16 August 2005 9:30 pm, Yahya Abdal-Aziz wrote:
> Hi Aaron!
>
> I didn't quite understand your procedure here. So I've interjected
> a few questions.
>
> > ... I came up with my own interpretation of Neidhardt I which I quite
>
> like:
> > Starting at C, do pure C-F-Bb-Eb-A; ...
>
> That would be Ab, wouldn't it?

oops, yes, pardon my hastiness....Ab indeed!

>
> > ... you have a resulting ascending C-Ab sixth.
> > We'll divide that up into two equal beating thirds. So, make C-E beat
>
> equal
>
> > to E-G#. Now we have a ascending minor sixth E-C that we will repeat the
> > procedure to, to soften the Ab-C Pythagorean third a bit: so bring G# up
> > a bit ...
>
> G# being the same as Ab, of course?

yup.

>
> > ... so that E-G# now beats equal to G#-C. (of course, it will no longer
>
> beat
>
> > equal with C-E). This layout ultimately favors slightly the flat keys in
> > terms of consonance, because of the equal beating step.
> >
> > Neidhardt I has C-F-Bb as pure fifths/fourths, so we'll keep that from
> > the beginning. These are followed by two slightly impure fifths.
> >
> > So next we retune Eb-Bb ascending fifth ...
>
> Does that mean the Bb changes, while the Eb remains the same?

no. I meant 'we'll keep that' to mean 'keep C-F-Bb what they were: pure'

> > ... to be the same size (a la meantone) as
> > the ascending fifth Ab-Eb.
>
> Does "a la meantone" mean so that the two fifths beat equally?

no, the higher fifth beats ~1.5 times the lower, to make it closer to
mathematically correct.

> > The E-G# is divided analogously to Ab-C; two pure, then two impure
> > fifths.
> >
> > So then we have and G#-C# pure descending fifth
> > Then C#-F# pure ascending fourth
> > F#-B descending fifth same size as B-E descending fifth.
> >
> > Finally, we then distribute C-E a la meantone between C-G-D-A-E.
>
> Same question here ...

Same answer: proportional beating. The higher fifth G-D is 1.5 times faster
than the lower fifth C-G....

> > A nice, subtle well temperament, the distant key signatures not too
> > harsh,
>
> and
>
> > it's easily done. And it has a large number of subtly, differently-sized
> > intervals, if you like maximully distinct counterpoint! ;)
>
> Final question: Why do you call this your _interpretation_ of Neidhardt I?
> I had supposed - perhaps wrongly - from what you wrote about it earlier
> that there wasn't a lot of room for interpretation of the temperament ...

The 'interpretation' comes from the initial 'equal beating' layout of the
'skeleton' thirds C-E-G#-C. In the original, The thirds are to beat such that
G#-C is a bit faster than E-G#.

My result is close to the original, but a tiny bit easier for a beginner, and
perhaps adds a satisfying layer of sonic goodness with the two equal beating
thirds. Let your taste decide!

One *could* do a variant where all the thirds in the 'skeleton' beat equally,
but in order to more preserve traditional key color and tension, put the C-E
on top, so we have E-G#-C-E. The reason is that any interval that beat at the
same rate decrease in size as we ascend in pitch, and we want C-major for
instance to have the smallest third. So we see that (G#)Ab-C will be closer
to pure than E-G# in this case, and my proposed original. This makes flat
keys a bit mellower than sharp keys, which I personally don't see as a
problem.

Hope this helps.

Best,
Aaron.

--- In tuning@yahoogroups.com, Aaron Krister Johnson <aaron@a...> wrote:

> any interval that beat at the
> same rate decrease in size as we ascend in pitch

A minor correction: this is only true if the interval is on the large
side of just to begin with. If it's on the small side of just, the
opposite is true. And that's very relevant if you're talking about,
say, the perfect fifths in these tunings.

Aaron Krister Johnson wrote:

[much clipped]

> > > ... to be the same size (a la meantone) as
> > > the ascending fifth Ab-Eb.
> >
> > Does "a la meantone" mean so that the two fifths beat equally?
>
> no, the higher fifth beats ~1.5 times the lower, to make it closer to
> mathematically correct.

OK!

[clipped]

> > Final question: Why do you call this your _interpretation_ of Neidhardt
I?
> > I had supposed - perhaps wrongly - from what you wrote about it earlier
> > that there wasn't a lot of room for interpretation of the temperament
...
>
> The 'interpretation' comes from the initial 'equal beating' layout of the
> 'skeleton' thirds C-E-G#-C. In the original, The thirds are to beat such
that
> G#-C is a bit faster than E-G#.

Maybe in about the ratio 5:4, ie ~1.25 times? And the same ratio between
the two lower thirds?

> My result is close to the original, but a tiny bit easier for a beginner,
and
> perhaps adds a satisfying layer of sonic goodness with the two equal
beating
> thirds. Let your taste decide!
>
> One *could* do a variant where all the thirds in the 'skeleton' beat
equally,
> but in order to more preserve traditional key color and tension, put the
C-E
> on top, so we have E-G#-C-E. The reason is that any interval that beat at
the
> same rate decrease in size as we ascend in pitch, and we want C-major for
> instance to have the smallest third. So we see that (G#)Ab-C will be
closer
> to pure than E-G# in this case, and my proposed original. This makes flat
> keys a bit mellower than sharp keys, which I personally don't see as a
> problem.
>
> Hope this helps.

Yes, indeedy! :-)
Yahya

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