Diminished - why #??? - an alternative way of looking at your scales and intervals.

Your scale/chord is a D# diminished 7 as the intervals between the notes are three flattened thirds (bIIIrd) (L+s), so it is not surprising that it has a diminished quality.

for D#-F#-A-C

C-G-D-A-E-B-F#-C#-G#-D#
1 2 3 4 5 6 7 8 9 10
gives us;
9/235689/10

Looking at it starting from C you get:

Cdim7 contains C-Eb-Gb-Bbb

Bbb-Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C
1 2 3 4 5 6 7 8 9 10
gives us
9/235689/10

The two examples above are transpositions of the same chord or scale by a sharp second (#IInd) (2L-s),
or nine steps along the circle/spiral of fourths/fifths.

If you were to substitute an Eb for the D# in your example and get Eb-F#-A-C you will produce a very similar chord
as the first interval will become a #II instead of a bIII, yet if you were to arranges the named notes as F#-A-C-Eb in the diminished seventh pattern with

Eb-Bb-F-C-G-D-A-E-B-F#
1 2 3 4 5 6 7 8 9 10
gives us;
9/235689/1

(same chain length and missing notes, yet different "tonic")

In 12edo this is much easy to analyse as it splits and octave into four equal intervals of three semitones each
in a similar matter to the augmented which splits the octave into three equal intervals of four semitones each.

In meantone you have to consider the intervals as (L+s) or (2L) which by multiple addition do not hit the octave at (5L+2s).

4 intervals of L+s = 4L+4s which is (5L+2s)-(4L+4s) = (L-2s) i.e. a bbIInd above or below the octave
3 intervals of 2L = 6L which is (5L+2s)-(6L) = (-L+2s) i.e. a bbIInd above or below the octave.

Whether above or below the octave depends upon whether the Large interval is > or < 200 cents. (i.e. a negative or positive meantone)

I hope this clarifies what is happening in this example, and that you can figure out which is neg and which is pos,
and why it works in opposite ways for Pythagorean and negative meantone tunings ;-).

On 7 Jun 2005, at 05:29, tuning@yahoogroups.com wrote:

> Message: 5
> Date: Mon, 6 Jun 2005 11:04:20 -0400
> From: pgreenhaw@nypl.org
> Subject: Re: The _other_ minor scale
>
> Here's my guess --
>
> Lydian (as it is used contemporarily) has the raised 4th scale > degree --
> as does the scale in question -- and a diminished triad is formed > if you
> use the D# as an enharmonic Eb. D#-F#-A-C...... there is a diminished
> quality in the set.
>
> P
>
>
> ____________________
>
>
>
>
> "Dave Keenan" <d.keenan@bigpond.net.au>
> Sent by: tuning@yahoogroups.com
> 06/06/2005 12:53 AM
> Please respond to tuning
>
>
> To: tuning@yahoogroups.com
> cc:
> Subject: [tuning] The _other_ minor scale
>
>
>
> Scala describes this _other_ minor 12-tone mode (e.g. A B C D#E F# G#
> A) as:
>
> 2131221 Lydian Diminished, Mela Dharmavati, Raga Arunajualita,
> Dumyaraga, Madhuvanti, Ambika
>
> I don't understand why the name "Lydian diminished" is appropriate at
> all, since the diminished scale is the 8-note scale 21212121 and the
> Greek Lydian mode is just the natural major 2212221.
>
>

Charles Lucy - lucy@harmonics.com
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