Re: The Meantone CDbEFbGAbBb bbII = Diiminished Second? = Double flattened Second

Gene calls it a diminished.

I have always found this use of the term "diminished" for intervals confusing and ambiguous;
hence I avoid it; except for chords where a diminished chord contains two or more contiguous flattened seconds:

e.g. C-Eb-Gb-Bbb (Cdim7)

Does anyone else "think" that a diminished second could be interpreted as bII?

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--- In tuning-math@yahoogroups.com, Charles Lucy <lucy@...> wrote:
>
> Gene calls it a diminished.
>
> I have always found this use of the term "diminished" for intervals
> confusing and ambiguous;
> hence I avoid it; except for chords where a diminished chord contains
> two or more contiguous flattened seconds:

It's possibly confusing these days to people who are thinking solely
in terms of 12-et, where enharmonic equivalencies come in to confuse
the matter. For example, calling a diminished fifth a "tritone"
assumes you think it is three tones, which is an augmented fourth.
People do that sort of thing all the time, and it does spred confusion.

This will never happen in Lucy tuning, because it has no enharmonic
equivalencies. It comes close to one with E###### ~ Cbbbbbb', but not
quite. In other words in Lucy tuning a sextupally augmented third is
almost, but not quite (3.38 cents sharper than) a sextupally
diminished octave. (Both are around 790 cents in the unlikely event
anyone cares.)

The rule is that #s augment and bs diminish; hence C-G is a perfect
fifth, C-Gb is a diminished fifth, and C-G# is an augmented fifth.
Since a diminished major third is a minor third and a diminished major
sixth a minor sixth, augmented and diminished when aplied to thirds or
sixths is disambiguated. You can diminish a minor third, augment a
major third, diminish a minor sixth, and augment a major sixth; so
"diminished third" must mean minor third, and "augmented sixth" must
mean major sixth. You can also augment or diminish unisons, octaves,
etc. Hence, any interval of Lucy tuning corresponds to a unique
description in the vocabulary of "augmented" and "diminished" diatonic
intervals.

Not, of course, that that is necessarily the easiest way to say it.

On 6/12/06, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:
> --- In tuning-math@yahoogroups.com, Charles Lucy <lucy@...> wrote:
> >
> > Gene calls it a diminished.
> >
> > I have always found this use of the term "diminished" for intervals
> > confusing and ambiguous;
> > hence I avoid it; except for chords where a diminished chord contains
> > two or more contiguous flattened seconds:
>
> The rule is that #s augment and bs diminish; hence C-G is a perfect
> fifth, C-Gb is a diminished fifth, and C-G# is an augmented fifth.
> Since a diminished major third is a minor third and a diminished major
> sixth a minor sixth, augmented and diminished when aplied to thirds or
> sixths is disambiguated. You can diminish a minor third, augment a
> major third, diminish a minor sixth, and augment a major sixth; so
> "diminished third" must mean minor third, and "augmented sixth" must
> mean major sixth. You can also augment or diminish unisons, octaves,
> etc. Hence, any interval of Lucy tuning corresponds to a unique
> description in the vocabulary of "augmented" and "diminished" diatonic
> intervals.

Yes. There are only two ways to do it, one for perfect intervals
(unison, fourth, and fifth, plus any number of octaves), and one for
all the rest.

So there's a diminished fourth, a perfect fourth, and augmented fourth
(but no such thing as a major or minor fourth) and a diminished third,
a minor third, a major third, and an augmented third (but no such
thing as a perfect third).

The equivalence interval of 12-equal is the diminished second. E-F is
a minor second, so E-Fb is a diminished second. No ambiguity.

The equivalence interval of 19-equal is the doubly diminished second,
for example E#-Fb or G#-Ab.

Open tunings like TOP meantone or LucyTuning, of course, have no
equivalence intervals.

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...>
wrote:

> The equivalence interval of 12-equal is the diminished second. E-F is
> a minor second, so E-Fb is a diminished second. No ambiguity.
>
> The equivalence interval of 19-equal is the doubly diminished second,
> for example E#-Fb or G#-Ab.

Does anyone here program Java aps? It occurs to me that a cool online
toy would be a program which, if you fed it a 7-limit ratio, would
tell you what interval it was in this language.

You can start the ball rolling with interval classes. If we let
m7(n) = modp(n+1,7)-1, where "modp" means modulo 7 reduced to the
range 0-6, and if we set u to be <0 1 4 10| applied to the interval q,
then the pair [(u-m7(q))/7, m7(u)] tells us what interval and how much
diminished, more or less. A 6/5 goes to [-1,4] which says it is a
flattened major third, which is the "less" part. Anyway, fix that up
and add something to tell you which octave (add a seven for each
octave, so that a unison is a 1, an octave an 8, etc, and an octave up
from a fifth is 5+7 = twelvth, etc.) Also, of course, intervals are
unsigned, and q and 1/q (or a:b and b:a) go to the same interval name.

Anyway, 128/125 goes to [-2, 2] under this. [0,2] is a major second,
[-1,2] is a minor second, [-2,2] a diminished second, the equivalence
interval for 12-et. 3072/3125 goes to [

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...>
wrote:

> The equivalence interval of 19-equal is the doubly diminished second,
> for example E#-Fb or G#-Ab.

The article I was working on somehow posted itself; I suppose I hit a
key wrong. Since Yahoo is so screwed up these days I won't wait any
longer for it to appear, and just finish it here. I had gotten to the
19-et interval of equivalency, for which I used 3072/3125, smaller
than a unison, to do the calculation with. For 31, we have the
Wuerschmidt comma, 393216/390625 as an interval of equivalency. It
comes to
[-5, 4], so the interval of equivalency goes first to the minor third
[-1, 4], and then gets diminished four times to a quadruply diminished
third, [-5, 4]. 43-et has a sextuplly diminished fourth, 50-et a
septupally diminished fourth, and 81-et an eleven-times diminished
sixth as intervals of equivalence.