Re: optimal tuning for diatonic scale; Euler

Dave Keenan wrote,

>I know we're talking diatonic here but it should be noted that once you
>allow ratios of 7, 1/4 comma looks a lot better than 3/14 comma.

The dominant seventh, tuned 4:5:6:7, is better in Pythagorean than in 3/14-comma
meantone, let alone 1/4-comma meantone. We're talking diatonic so augmented sixths don't
count. You must have been using the augmented sixth, right?

Joe Monzo wrote,

>OK, re-reading that has me seeing it from your point
>of view. It sounds like Euler *was* interested in a
>tuning where these septimal commas vanish.

>Perhaps he decided 7 was being used as a factor
>in the music of his time *because* of 12-Eq or
>meantone temperaments in use at the time which have
>these commas vanish.

In meantone temperament, the septimal comma only vanishes if you derive your 7 from the
dominant seventh chord. If you derive it from the augmented sixth, you get a far purer
seventh and the septimal comma doesn't vanish. Huygens understood meantone in this
latter way, since augmented sixths (and meantone) were in use in the music of his time.
The septimal comma is then the meantone "wolf", as is the 36/35.

[Me, Dave Keenan]
>>I know we're talking diatonic here but it should be noted that once you
>>allow ratios of 7, 1/4 comma looks a lot better than 3/14 comma.

[Brett Barbaro]
>The dominant seventh, tuned 4:5:6:7, is better in Pythagorean than in
3/14-comma
>meantone, let alone 1/4-comma meantone. We're talking diatonic so
augmented sixths don't
>count. You must have been using the augmented sixth, right?

Yes.

I was *not* talking diatonic when I referred to ratios of 7. That's why I
used "but" instead of "so" above. Sorry I wasn't clearer.

I don't consider the dominant seventh (defined as a root and steps of 4, 1
and -2 fifths from this root on a chain fifths) to *ever* be an
approximation of 4:5:6:7 (at least not when the fifths in the chain are all
the same size). If one allows the 4:5 to be approximated by -8 fifths
instead of +4, the closest approach is indeed very close to Pythagorean but
the maximum error is still 27.9 cents (in the 5:7 and 6:7 where it really
matters).

[Brett Barbaro in response to Joe Monzo]
>In meantone temperament, the septimal comma only vanishes if you derive
your 7 from the
>dominant seventh chord. If you derive it from the augmented sixth, you get
a far purer
>seventh and the septimal comma doesn't vanish. Huygens understood meantone
in this
>latter way, since augmented sixths (and meantone) were in use in the music
of his time.

I don't think there is any such thing as *the* septimal comma. I think you
guys are talking about two different "septimal commas" here. When talking
meantone the relevant septimal comma relates to the augmented sixth and so
*does* dissappear.

>The septimal comma is then the meantone "wolf", as is the 36/35.

I think this is an abuse of the term "wolf". I noticed the same mistake in
Edward Dunne's article
http://www.math.okstate.edu/~dunne/students/Temperament.html

I understand a "wolf" (without qualification) to be a fifth where the error
is considered intolerable. It's the meantone G# to Eb that is the wolf, not
the G# to Ab (if it existed). Of course one can also speak of a "wolf
third" or other badly-tuned normally-consonant interval.

Regards,
-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote:

>I don't consider the dominant seventh (defined as a root and steps of 4, 1
>and -2 fifths from this root on a chain fifths) to *ever* be an
>approximation of 4:5:6:7 (at least not when the fifths in the chain are all
>the same size). If one allows the 4:5 to be approximated by -8 fifths
>instead of +4, the closest approach is indeed very close to Pythagorean but
>the maximum error is still 27.9 cents (in the 5:7 and 6:7 where it really
>matters).

Barbershop quartets can hide 28-cent errors and do use 4:5:6:7 dominant sevenths.

> [Brett Barbaro in response to Joe Monzo]
> >In meantone temperament, the septimal comma only vanishes if you derive
> your 7 from the
> >dominant seventh chord. If you derive it from the augmented sixth, you get
> a far purer
> >seventh and the septimal comma doesn't vanish. Huygens understood meantone
> in this
> >latter way, since augmented sixths (and meantone) were in use in the music
> of his time.
>
> I don't think there is any such thing as *the* septimal comma. I think you
> guys are talking about two different "septimal commas" here. When talking
> meantone the relevant septimal comma relates to the augmented sixth and so
> *does* dissappear.

Dave, the septimal comma is defined as 64:63, much as the syntonic comma is defined as
81:80. Now go back and read my statement above. Joe Monzo was referring to the case
where the 7 is derived from the dominant seventh chord.

> >The septimal comma is then the meantone "wolf", as is the 36/35.
>
> I think this is an abuse of the term "wolf". I noticed the same mistake in
> Edward Dunne's article
> http://www.math.okstate.edu/~dunne/students/Temperament.html
>
> I understand a "wolf" (without qualification) to be a fifth where the error
> is considered intolerable. It's the meantone G# to Eb that is the wolf, not
> the G# to Ab (if it existed). Of course one can also speak of a "wolf
> third" or other badly-tuned normally-consonant interval.

You are correct. G# to Ab did exist on many Renaissance and Baroque keyboards --
perhaps we should call it the meantone "subsemitone".