Interval names (was: Thanks to Dave Keenan)

[Gerald Eskelin, TD 520.11]
>> 5:7 augmented fourth
>> 7:10 diminished fifth
>
>Aren't these last two backward?

I just knew you were gonna say that. :-)

>The 5:7 tritone is smaller than the 7:10 tritone,

Yes.

>thus the 5:7 "contracts" (E-Bb) in resolution (diminished fifth)
>and the 7:10 (E-A#) expands in resolution (augmented fourth).

I'd say the 7:10 (E-Bb) contracts in resolution (diminished fifth)
and the 5:7 (E-A#) expands in resolution (augmented fourth).

Or putting them both in C major, B:F (7:10) contracts in resolution to C:E (4:5) and F:B (5:7) expands in resolution to E:C (5:8), not because of their initial sizes but just because of where the nearest _diatonic_ notes are.

Of course whether B:F is closer to 5:7 or 7:10 depends on your tuning. I have assumed something tending toward meantone above (since these are optimum for diatonic). For tunings tending toward (or beyond) the Pythagorean, B:F would be closer to 5:7 which one might then call a "Pythagorean diminished fifth", or (as I'd prefer) a "subdiminished fifth".

>Remember, Paul and Joe hold that "context is everything."

It's certainly very important, and the system, as explained on my web page, http://dkeenan.com/Music/IntervalNaming.htm allows for alternative names for the same ratio in different contexts, as with 5:7 = aug 4th = subdim 5th above.

>(You don't suppose Mr. Fokker would reconsider these, do you?

No.

> Too late, you say?

As in "the late Mr Fokker", yes.

>Okay. Do you suppose he'd mind if we messed with it a bit?)

Yes. The whole system would be screwed if we swapped these two. To be fair to Mr Fokker, strictly speaking, he gave the names to the intervals of 31-tET (a meantone), not to the JI ratios which they approximate. The latter seems like an obvious move to many people, but I understand that Paul Erlich is still dubious about it because of the apparent disagreement, in the case of 22-tET (and other super-Pythagorean tunings), between the A-G, #, b, /, \ names for the notes, based on chains of approximate 2:3's, and the Fokker-style names for the intervals, based on the 11-limit ratios most nearly approximated.

22-tET intervals from C to:

C perfect unison
C/ super unison
Db/ minor second
D\ neutral second
D major second, supermajor second
Eb subminor third
Eb/ minor third, neutral third
E\ major third
E supermajor third
F perfect fourth
F/ super fourth
F#\, Gb/ augmented fourth, diminished fifth
G\ sub fifth
G perfect fifth
Ab subminor sixth
Ab/ minor sixth
A\ neutral sixth, major sixth
A supermajor sixth
Bb subminor seventh, minor seventh
Bb/ neutral seventh
B\ major seventh
C\ sub octave
C perfect octave

Personally, I don't have a problem with that. But one does need to know (at least) whether the tuning being used has G# < Ab or G# > Ab.

Regards,

-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote:

> The whole system would be screwed if we swapped these two. To be fair to Mr
> Fokker, strictly speaking, he gave the names to the intervals of 31-tET
> (a meantone), not to the JI ratios which they approximate.

He did give names to JI ratios too, in _Rekenkundige bespiegeling der
muziek_ (Mathematical reflection on music), albeit without any accompanying
explanation. This book hardly goes into 31-tone equal temperament.
But Dave is right. He even named 6:7 an augmented second (harmonische vergrote
sekunde) and 7:8 a diminished third (harmonische verkleinde terts).

Manuel Op de Coul coul@ezh.nl

>The latter seems like an obvious move to many people, but I understand that
Paul >Erlich is still dubious about it because of the apparent disagreement,
in the >case of 22-tET (and other super-Pythagorean tunings), between the
A-G, #, b, /, >\ names for the notes, based on chains of approximate 2:3's,
and the Fokker->style names for the intervals, based on the 11-limit ratios
most nearly >approximated.

>22-tET intervals from C to:

>C perfect unison
>C/ super unison
>Db/ minor second
>D\ neutral second
>D major second, supermajor second
>Eb subminor third
>Eb/ minor third, neutral third
>E\ major third
>E supermajor third
>F perfect fourth
>F/ super fourth
>F#\, Gb/ augmented fourth, diminished fifth
>G\ sub fifth
>G perfect fifth
>Ab subminor sixth
>Ab/ minor sixth
>A\ neutral sixth, major sixth
>A supermajor sixth
>Bb subminor seventh, minor seventh
>Bb/ neutral seventh
>B\ major seventh
>C\ sub octave
>C perfect octave

The only sensible solutions for 22-tET seem to be a consistent application
of the former, in which it is understood that all ratios involving 5 will
involve the sub or super modifiers; or a version of decatonic notation
described in my paper. Although 22-tET is consistent with the set of
11-limit ratios, it is not consistent with conventional diatonic
nomenclature, because the syntonic comma does not vanish. In your scheme,
the interval between the "major second" and the "major sixth" is not a
"perfect fifth" but a "sub fifth". I find that screwy. If you don't, at
least you're in good company -- Ben Johnston's JI notation of C major is
similar.