About the major 2nd.

Hello!

I have some doubts about the the major 2nd of a major scale. It seems clear to me, that a chord of "sixte ajout�e" (in C Major: F,A,C,D) should have the following intervals:

A/F=5/4
C/A=6/5
D/C=10/9
---> C/F=3/2, D/F=5/3, D/A=4/3

It also seems clear that a Dominant chord (G-B-D) should be:

B/G=5/4
D/B=6/5
---> D/G=3/2

The chord of "sixte ajout�e" is often followed by the Dominant, so that, if we assume that G/C=3/2, then we get two different D's! The first one is D1/C=10/9, and the second one D2/C=9/8.

So, what would be the best solution? Sacrifying one chord by using only one of the D's? Sacrifying both chords by using a D3 that is in the middle of the other two? Sacrifying a melodic line by using both D1 and D2? Any other solution?

Please, help me. Give me/us your oppinion.

Thanks.

> David Mezquita wrote:
>
> Hello!
>
> I have some doubts about the the major 2nd of a major scale. It seems clear to me, that a chord of "sixte ajout�e" (in C Major: F,A,C,D) should have the following intervals:
>
> A/F=5/4
> C/A=6/5
> D/C=10/9
> ---> C/F=3/2, D/F=5/3, D/A=4/3

I believe there is more to this chord than meets the eye! it tends to sound more consonant than a major chord in a way!

>
>
> It also seems clear that a Dominant chord (G-B-D) should be:
>
> B/G=5/4
> D/B=6/5
> ---> D/G=3/2
>
> The chord of "sixte ajout�e" is often followed by the Dominant, so that, if we assume that G/C=3/2, then we get two different D's! The first one is D1/C=10/9, and the second one D2/C=9/8.
> Sacrifying a melodic line by using both D1 and D2?

Commatic shifts if any thing can enhance a melodic line. Boomsliter and creel showed how it occurred all the time!

>
>
> Please, help me. Give me/us your oppinion.
>
> Thanks.
>

-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

> David Mezquita wrote:
>
> I have some doubts about the the major 2nd of a major scale.
> It seems clear to > me, that a chord of "sixte ajout�e"
> (in C Major: F,A,C,D) should have the following intervals:
>
> A/F=5/4
> C/A=6/5
> D/C=10/9
> ---> C/F=3/2, D/F=5/3, D/A=4/3

note f : a : c : d
common subharmonic 6 : : 9 : 10
4 : 5 : 6
common harmonic 1/5 : 1/4 : : 1/3
otonal 12 : 15 : 18 : 20
utonal 1/15 : 1/12 : 1/10 : 1/9
Frequency 176Hz: 220Hz: 264Hz: 293.3Hz

> It also seems clear that a Dominant chord (G-B-D) should be:
>
> B/G=5/4
> D/B=6/5
> ---> D/G=3/2

note g : b : (c) : d
common subharmonic 4 : 5 : : 6
6 : : 8 : 9
otonal 12 : 15 : 16 : 18
Frequency 198Hz: 247.5Hz: 264Hz: 297Hz

> The chord of "sixte ajout�e" is often followed by the Dominant,
> so that, if we assume that G/C=3/2, then we get two different D's!
> The first one is D1/C=10/9, and the second one D2/C=9/8.

> So, what would be the best solution?
> Sacrifying one chord by using only one of the D's?
> Sacrifying both chords by using a D3 that is in the middle of
> the other two?
> Sacrifying a melodic line by using both D1 and D2?
> Any other solution?

Sacrifying a melodic line by using both D1 and D2.
Maybe you can slide between D1 and D2 regarding the context
to keep an optimal JI at each very moment.
Yes. Retuning on the fly.

Peter Mulkers

David Mezquita wrote,

>I have some doubts about the the major 2nd of a major scale. It seems clear
to me, that a >chord of "sixte ajout�e" (in C Major: F,A,C,D) should have
the following intervals:

>A/F=5/4
>C/A=6/5
>D/C=10/9
>---> C/F=3/2, D/F=5/3, D/A=4/3

>It also seems clear that a Dominant chord (G-B-D) should be:

>B/G=5/4
>D/B=6/5
>---> D/G=3/2

>The chord of "sixte ajout�e" is often followed by the Dominant, so that, if
we assume that G/C=>3/2, then we get two different D's! The first one is
D1/C=10/9, and the second one D2/C=>9/8.

>So, what would be the best solution? Sacrifying one chord by using only one
of the D's? >Sacrifying both chords by using a D3 that is in the middle of
the other two? Sacrifying a >melodic line by using both D1 and D2? Any other
solution?

Actually, David, you have found the "problem of the syntonic comma" which
explains why 5-limit just intonation never gained much importance in Western
music. The syntonic comma is 81/80, about 21.5 cents, the difference (ratio)
between 9/8 and 10/9. Historically, the solution to your dilemma, from about
1500-1700 (and to some extent, through 1850), was meantone temperament.
Meantone takes each perfect fifth in the chain of fifths and flattens it by
a small fraction (typically a quarter) of a syntonic comma. The result is
that the thirds are nearly pure; a major third, constructed from four
perfect fifths, will typically be a full comma flat relative to a
Pythagorean major third 81/64, thus having ratio 80/64 = 5/4. The fifths and
minor thirds are only 5 or 6 cents flat, which is not too different from
just intonation unless the music goes _very_ slowly. In meantone diatonic
music, every notated pitch has only one actual frequency, but all thirds and
sixths are consonant as well as fouths and fifths. Every diatonic key has
three consonant major triads and three consonant minor triads, which is not
possible in just intonation (unless you allow two alternative tunings for
the second degree). The whole tones (major seconds), which are not consonant
in common practice music and so do not themselves need to approximate simple
ratios, are a compromise between 9/8 and 10/9 in meantone temperament (thus
the name "mean-tone"). Modern 12-tone equal temperament is a variant of
meantone temperament (each fifth is flattened by about one eleventh of a
syntonic comma) which allows for unlimited modulation, but the thirds are
closer to their dissonant Pythagorean proportions than to the consonant ones
of older meantone temperaments. 19- and particularly 31-tone equal
temperament are well within the range of typical meantone temperaments, and
also allow unlimited modulation, but 12-tone won out in the 1700s for
reasons of convenience.