Yahoo Tuning Groups Ultimate Backup tuning A new diminished seventh chord

Well, the sensitivity is more towards theoretical solidness rather
than perceptual threshold. It is doubtful whether anyone can
differentiate this chord from a 12-equal diminished seventh during
live performance. But think for the sake of perfectionism!..

Oz.

✩ ✩ ✩
www.ozanyarman.com

On Nov 12, 2009, at 5:01 AM, octatonic10 wrote:

> but in a chord progression, does it really matter? The average
> tonal ear won't notice the tuning differences you describe. But of
> course for microtonal music it is interesting!
>
> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>>
>> This is an interesting and wholesome diminished seventh chord that I
>> couldn't find in the Chords database of SCALA:
>>
>> 135:160:192:225
>>
>> When the chord is unfolded, it yields:
>>
>> 5/3 x 128/75 x 5/3.
>>
>> The ratios are:
>>
>> 1/1
>> 32/27
>> 6/5
>> 75/64
>>
>> I find 75/64 to be a much more agreeable septimal minor third
>> compared
>> to 7/6.
>>
>> In Yarman-36 (a,b,c) the steps are (over the regular 12-tone layer):
>>
>> 18 8 18.
>>
>> The notation is:
>>
>> C E/b F# Ad
>>
>> Cordially,
>> Oz.
>>
>> âœ© âœ© âœ©
>> www.ozanyarman.com
>>
>
>
>
>
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🔗cameron <misterbobro@...>

--- In tuning@yahoogroups.com, "octatonic10" <octatonic10@...> wrote:
>
> but in a chord progression, does it really matter? The average >tonal ear won't notice the tuning differences you describe. But of >course for microtonal music it is interesting!

The "average tonal ear" won't notice the difference between a flat seven and a blue flat seven?

Not only is your statement wrong, it is interestingly "maximally" wrong in my experience. In introducing the theoretical concepts of JI and microtonality to people, I find that it is a blued b7 that is the instant audible eye-opener. A 5/4 often gets a puzzled, well, it's more "church" sounding..., without the comprehension that the intervals, and not just the timbre, have been changed.

>
> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@> wrote:
> >
> > This is an interesting and wholesome diminished seventh chord that I
> > couldn't find in the Chords database of SCALA:
> >
> > 135:160:192:225
> >
> > When the chord is unfolded, it yields:
> >
> > 5/3 x 128/75 x 5/3.
> >
> > The ratios are:
> >
> > 1/1
> > 32/27
> > 6/5
> > 75/64
> >
> > I find 75/64 to be a much more agreeable septimal minor third compared
> > to 7/6.
> >
> > In Yarman-36 (a,b,c) the steps are (over the regular 12-tone layer):
> >
> > 18 8 18.
> >
> > The notation is:
> >
> > C E/b F# Ad
> >
> > Cordially,
> > Oz.
> >
> > âo?=© âo?=© âo?=©
> > www.ozanyarman.com
> >
>

🔗Marcel de Velde <m.develde@...>

Hi Oz,

This is an interesting and wholesome diminished seventh chord that I
> couldn't find in the Chords database of SCALA:
>
> 135:160:192:225
>
> When the chord is unfolded, it yields:
>
> 5/3 x 128/75 x 5/3.
>
> The ratios are:
>
> 1/1
> 32/27
> 6/5
> 75/64
>
> I find 75/64 to be a much more agreeable septimal minor third compared
> to 7/6.
>
> In Yarman-36 (a,b,c) the steps are (over the regular 12-tone layer):
>
> 18 8 18.
>
> The notation is:
>
> C E/b F# Ad
>
> Cordially,
> Oz.
>

Yes this is the diminished 7th chord that I used in Beethoven's Drei Equali
No1 :)
As far as I'm personally concerned it's the only correct Just Intonation
diminished 7th.

In Tonal-JI this chord occurs as D (9/8) - F (4/3) - Ab (8/5) - B (15/8)
where the tonic is C (1/1).
I recently discussed this chord extensively on the JustIntonation Yahoo
group and gave several examples of it's use.

In the Beethoven piece it's used like this:
C (4/3) - F# (15/8) - Eb (16/5) - A (9/2)
Bb (6/5) - G (2/1) - D (3/1) - Bb (24/5)
Where the tonic is G (1/1).

Another way to use it is for instance:
F (4/3) - Ab (8/5) - B (15/8) - D (9/4) <diminished 7th>
F (4/3) - G (3/2) - B (15/8) - D (9/4) <dominant 7th>
C (1/1) - G (3/2) - C (2/1) - E (5/4) <tonic major>
In the tonic of C (1/1)

King regards,
Marcel

🔗cameron <misterbobro@...>

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>

> Yes this is the diminished 7th chord that I used in Beethoven's Drei >Equali
> No1 :)
> As far as I'm personally concerned it's the only correct Just >Intonation
> diminished 7th.
>
> In Tonal-JI this chord occurs as D (9/8) - F (4/3) - Ab (8/5) - B (15/8)
> where the tonic is C (1/1).

You're using D, F, Ab, A##. You might want to study the term "diesis" a bit.

You must understand that in doing what you doing, are not actually working with Beethoven; rather, you are doing a kind of science-fiction tuning. Groovy, go to it. If it sounds good it is good. History doesn't just go away, though.

(Ozan has taken on a heavier burden, something that has been attempted since the 19th century without complete success, though
a lot of great music has been made in the try: maqam/modal music with polyphony and functional harmony. This is a vast uncharted area, especially in the area of notation and determining what pitches actually work in real life.)

🔗Mike Battaglia <battaglia01@...>

If 1/1 is C, why would 15/8 be A##?

-Mike

On Thu, Nov 12, 2009 at 2:19 PM, cameron <misterbobro@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
> >
>
> > Yes this is the diminished 7th chord that I used in Beethoven's Drei >Equali
> > No1 :)
> > As far as I'm personally concerned it's the only correct Just >Intonation
> > diminished 7th.
> >
> > In Tonal-JI this chord occurs as D (9/8) - F (4/3) - Ab (8/5) - B (15/8)
> > where the tonic is C (1/1).
>
> You're using D, F, Ab, A##. You might want to study the term "diesis" a bit.
>
> You must understand that in doing what you doing, are not actually working with Beethoven; rather, you are doing a kind of science-fiction tuning. Groovy, go to it. If it sounds good it is good. History doesn't just go away, though.
>
> (Ozan has taken on a heavier burden, something that has been attempted since the 19th century without complete success, though
> a lot of great music has been made in the try: maqam/modal music with polyphony and functional harmony. This is a vast uncharted area, especially in the area of notation and determining what pitches actually work in real life.)
>
>

🔗Danny Wier <dawiertx@...>

I have a 7-limit chord similar to yours (it maps to the same degrees in 41, 53 and 72-edo):

* 21:25:30:35
* cumulative: 25/21 10/7 5/3
* consecutive: 25/21 6/5 7/6

Actually, I'd consider this a first inversion diminished seventh, since a root-position chord would be A-C-Eb-Gb, or 35:42:50:60 the same 7-limit system, where 6/5, 25/21 and 32/27 are the three sizes of minor third (7/6 normally is an augmented second).

~D.

----- Original Message ----- From: "Ozan Yarman":

This is an interesting and wholesome diminished seventh chord that I
couldn't find in the Chords database of SCALA:

135:160:192:225

When the chord is unfolded, it yields:

5/3 x 128/75 x 5/3.

The ratios are:

1/1
32/27
6/5
75/64

I find 75/64 to be a much more agreeable septimal minor third compared
to 7/6.

In Yarman-36 (a,b,c) the steps are (over the regular 12-tone layer):

18 8 18.

The notation is:

C E/b F# Ad

Cordially,
Oz.

🔗Danny Wier <dawiertx@...>

🔗cameron <misterbobro@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> If 1/1 is C, why would 15/8 be A##?

Oops, the "wrong note" is the Ab being a G#: D, F, G#, B. The doubly augmented sixth occurs in the Beethoven example I think, have to check again.

>
> -Mike
>
>
> On Thu, Nov 12, 2009 at 2:19 PM, cameron <misterbobro@...> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, Marcel de Velde <m.develde@> wrote:
> > >
> >
> > > Yes this is the diminished 7th chord that I used in Beethoven's Drei >Equali
> > > No1 :)
> > > As far as I'm personally concerned it's the only correct Just >Intonation
> > > diminished 7th.
> > >
> > > In Tonal-JI this chord occurs as D (9/8) - F (4/3) - Ab (8/5) - B (15/8)
> > > where the tonic is C (1/1).
> >
> > You're using D, F, Ab, A##. You might want to study the term "diesis" a bit.
> >
> > You must understand that in doing what you doing, are not actually working with Beethoven; rather, you are doing a kind of science-fiction tuning. Groovy, go to it. If it sounds good it is good. History doesn't just go away, though.
> >
> > (Ozan has taken on a heavier burden, something that has been attempted since the 19th century without complete success, though
> > a lot of great music has been made in the try: maqam/modal music with polyphony and functional harmony. This is a vast uncharted area, especially in the area of notation and determining what pitches actually work in real life.)
> >
> >
>

🔗cameron <misterbobro@...>

...and the B in Marcel's example there is a major diesis flat, so I don't know what to call it. :-)

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > If 1/1 is C, why would 15/8 be A##?
>
> Oops, the "wrong note" is the Ab being a G#: D, F, G#, B. The doubly augmented sixth occurs in the Beethoven example I think, have to check again.
>
>
> >
> > -Mike
> >
> >
> > On Thu, Nov 12, 2009 at 2:19 PM, cameron <misterbobro@> wrote:
> > >
> > >
> > >
> > > --- In tuning@yahoogroups.com, Marcel de Velde <m.develde@> wrote:
> > > >
> > >
> > > > Yes this is the diminished 7th chord that I used in Beethoven's Drei >Equali
> > > > No1 :)
> > > > As far as I'm personally concerned it's the only correct Just >Intonation
> > > > diminished 7th.
> > > >
> > > > In Tonal-JI this chord occurs as D (9/8) - F (4/3) - Ab (8/5) - B (15/8)
> > > > where the tonic is C (1/1).
> > >
> > > You're using D, F, Ab, A##. You might want to study the term "diesis" a bit.
> > >
> > > You must understand that in doing what you doing, are not actually working with Beethoven; rather, you are doing a kind of science-fiction tuning. Groovy, go to it. If it sounds good it is good. History doesn't just go away, though.
> > >
> > > (Ozan has taken on a heavier burden, something that has been attempted since the 19th century without complete success, though
> > > a lot of great music has been made in the try: maqam/modal music with polyphony and functional harmony. This is a vast uncharted area, especially in the area of notation and determining what pitches actually work in real life.)
> > >
> > >
> >
>

🔗Charles Lucy <lucy@...>

It seems to me that the diminished seventh is three steps of bIII (L+s) ; producing I, bIII, bV, and bbVII e.g. C Eb Gb Bbb

Obviously splitting the octave into four equal intervals in 12 edo, and hence may be called and treated by any of four names.

On 12 Nov 2009, at 19:19, cameron wrote:

>
>
> --- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
> >
>
> > Yes this is the diminished 7th chord that I used in Beethoven's Drei >Equali
> > No1 :)
> > As far as I'm personally concerned it's the only correct Just >Intonation
> > diminished 7th.
> >
> > In Tonal-JI this chord occurs as D (9/8) - F (4/3) - Ab (8/5) - B (15/8)
> > where the tonic is C (1/1).
>
> You're using D, F, Ab, A##. You might want to study the term "diesis" a bit.
>
> You must understand that in doing what you doing, are not actually working with Beethoven; rather, you are doing a kind of science-fiction tuning. Groovy, go to it. If it sounds good it is good. History doesn't just go away, though.
>
> (Ozan has taken on a heavier burden, something that has been attempted since the 19th century without complete success, though
> a lot of great music has been made in the try: maqam/modal music with polyphony and functional harmony. This is a vast uncharted area, especially in the area of notation and determining what pitches actually work in real life.)
>
>

Charles Lucy
lucy@...

-- Promoting global harmony through LucyTuning --

For more information on LucyTuning go to:

http://www.lucytune.com

LucyTuned Lullabies (from around the world) can found at:

http://www.lullabies.co.uk

🔗Marcel de Velde <m.develde@...>

Hello Cameron,

You're using D, F, Ab, A##. You might want to study the term "diesis" a bit.
>
>

I just recently learned enharmonic spelling etc.
And I can now say with confidence that I do mean a B :)

Also please see my 2 examples of use of the diminished 7th, they are also
spelled correctly I believe.

You must understand that in doing what you doing, are not actually working
> with Beethoven; rather, you are doing a kind of science-fiction tuning.
> Groovy, go to it. If it sounds good it is good. History doesn't just go
> away, though.
>

Hm I hope myself that I'm not doing sciene fiction but real Just Intonation
according to what the music says. But I can understand opinions can differ
on this.
But even if the opinion on this differs, I don't think I'm going against
history or against Beethoven if I may guess roughly what you ment by this.

> (Ozan has taken on a heavier burden, something that has been attempted
> since the 19th century without complete success, though
> a lot of great music has been made in the try: maqam/modal music with
> polyphony and functional harmony. This is a vast uncharted area, especially
> in the area of notation and determining what pitches actually work in real
> life.)
>

This "burden" :) I'm taking on aswell.
Especially if one extends Tonal-JI to the 7th harmonic limit.
Tonal-JI would describe Maqam music and how to modulate with it etc.
But I'm not going into harmonic 7-limit and Maqam music myself yet untill I
have a better understanding first of the way common practice western
classical music works in 6-limit Tonal-JI.

Marcel

🔗Ozan Yarman <ozanyarman@...>

Now that I think of it, the first inversion of the chord, which is
160:192:225:270 or 316 + 275 + 316 cents is more pleasing to my ears
actually:

0: 1/1 C Dbb unison, perfect prime
1: 6/5 minor third
2: 45/32 F# Gb diatonic tritone
3: 27/16 A Bbb Pythagorean major sixth

1st inversion of this chord unfolded yields:

27/16 x 5/3 x 128/75

Very neat, since we are in the 5-limit. And already, the 75/64 minor third resonates superior to 7/6.

In Yarman-36 notation, this would be:

Cd Eb F|<| A

Correction on the steps!

10 + 8 + 10 (~320 + ~280 + ~320 cents).

Cordially,
Oz.

✩ ✩ ✩
www.ozanyarman.com

On Nov 12, 2009, at 6:02 PM, Marcel de Velde wrote:

>
>
> Hi Oz,
>
> This is an interesting and wholesome diminished seventh chord that I
> couldn't find in the Chords database of SCALA:
>
> 135:160:192:225
>
> When the chord is unfolded, it yields:
>
> 5/3 x 128/75 x 5/3.
>
> The ratios are:
>
> 1/1
> 32/27
> 6/5
> 75/64
>
> I find 75/64 to be a much more agreeable septimal minor third compared
> to 7/6.
>
> In Yarman-36 (a,b,c) the steps are (over the regular 12-tone layer):
>
> 18 8 18.
>
> The notation is:
>
> C E/b F# Ad
>
> Cordially,
> Oz.
>
>
> Yes this is the diminished 7th chord that I used in Beethoven's Drei
> Equali No1 :)
> As far as I'm personally concerned it's the only correct Just
> Intonation diminished 7th.
>
> In Tonal-JI this chord occurs as D (9/8) - F (4/3) - Ab (8/5) - B
> (15/8) where the tonic is C (1/1).
> I recently discussed this chord extensively on the JustIntonation
> Yahoo group and gave several examples of it's use.
>
> In the Beethoven piece it's used like this:
> C (4/3) - F# (15/8) - Eb (16/5) - A (9/2)
> Bb (6/5) - G (2/1) - D (3/1) - Bb (24/5)
> Where the tonic is G (1/1).
>
> Another way to use it is for instance:
> F (4/3) - Ab (8/5) - B (15/8) - D (9/4) <diminished 7th>
> F (4/3) - G (3/2) - B (15/8) - D (9/4) <dominant 7th>
> C (1/1) - G (3/2) - C (2/1) - E (5/4) <tonic major>
> In the tonic of C (1/1)
>
> King regards,
> Marcel
>
>

🔗Ozan Yarman <ozanyarman@...>

Dear brother Danny,

While 7-limit has a cool flavour all its own, I would like to obtain a
5-limit diminished seventh for my own selfish purposes.

Cordially,
Oz.

✩ ✩ ✩
www.ozanyarman.com

On Nov 12, 2009, at 10:05 PM, Danny Wier wrote:

> Turns out that chord is listed in Scala as a third inversion... ~D.
>
> ----- Original Message -----
> From: "Danny Wier"
>
>
>> I have a 7-limit chord similar to yours (it maps to the same
>> degrees in
>> 41, 53 and 72-edo):
>>
>> * 21:25:30:35
>> * cumulative: 25/21 10/7 5/3
>> * consecutive: 25/21 6/5 7/6
>>
>> Actually, I'd consider this a first inversion diminished seventh,
>> since a root-position chord would be A-C-Eb-Gb, or 35:42:50:60 the
>> same 7-limit system, where 6/5, 25/21 and 32/27 are the three sizes
>> of
>> minor third (7/6 normally is an augmented second).
>>
>> ~D.
>
>
>
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>
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🔗duckfeetbilly <billygard@...>

🔗Marcel de Velde <m.develde@...>

🔗Daniel Forro <dan.for@...>

This kind of diminished chord is on VIIth step of the major/minor
scale, so your chords except the last one are not in their root
position. Have you written it intentionally just for a better
comparison how spelling is changed?

BTW the same chord can have more enharmonic versions, for example:
H# - D# - F# - A (resolves into C# maj/min)
Eb - Gb - Bbb - Dbb (into Fb)
Gx - H# - D# - F# (into A#)

Daniel Forro

On 13 Nov 2009, at 7:33 AM, Marcel de Velde wrote:

>
>
> Hello Charles,
>
> It seems to me that the diminished seventh is three steps of bIII (L
> +s) ; producing I, bIII, bV, and bbVII e.g. C Eb Gb Bbb
> I don't understand what bIII (L+s) etc means.

Minor third (as a combination from L-arge second + s-mall second). I
don't like this confusing way of writing, because not always the
higher note of this minor third has "b" accidental (like for example
C# - E, or D# - F#).
And concerning Ls system, that's maybe useful for scales, but not for
description of intervals. If we want to count somehow size of bigger
intervals, then using small second as a measure is enough good (minor
third = 3, fifth = 7 etc.).

> But as far as spelling the diminished 7th chord C Eb Gb Bbb yes you
> can!
>
> The 4 different way to spell the diminished 7th and all it's
> inversion are:
>
> In the tonic of Bb (16/9): C (1/1) - Eb (32/27) - Gb (64/45) - A (5/3)
>
> In the tonic of G (3/2): C (1/1) - Eb (6/5) - F# (45/32) - A (27/16)
> And may I add I agree with Oz that this is the nicest sounding one
> (due to 9/4 beeing higher from tonic than 9/8).
>
> In the tonic of E (5/4): C (1/1) - D# (75/64) - F# (45/32) - A (5/3)
>
> In the tonic of Db (16/15): C (1/1) - Eb (6/5) - Gb (64/45) - Bbb
> (128/75)
> C Eb Gb Bbb indeed :)
>
> Kind regards,
> Marcel
>
>
>>

🔗Marcel de Velde <m.develde@...>

🔗monz <joemonz@...>

Back around 1999, when i "justified" my piece
_3 Plus 4_, i wrote some posts here about my
empirical experiments where i ended up deciding
to tune a particular "minor-3rd" to 75/64.
I summarized them a couple of years ago here:

/tuning/topicId_72699.html#72769

-monz
http://tonalsoft.com/tonescape.aspx
Tonescape microtonal music software

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> Hi Oz,
>
> Very neat, since we are in the 5-limit. And already, the 75/64 minor
> > third resonates superior to 7/6.
> >
> I don't think 75/64 resonates superior to 7/6 by itself.
>
> I personally don't see 75/64 as a basic "harmonic 6-limit building block",
> but as an interval forming between 2 very simple intervals from the tonic,
> namely 8/5 and 15/8.
> As an interval directly from the tonic 75/64 does not occur (atleast not
> untill very high harmonic interval permutations).
> 7/6 does occur directly from the tonic in harmonic 7-limit though, and I
> personally see 7/6 as much more consonant than 75/64 by itself.
> Only the structure 9/8 4/3 8/5 15/8 I see as more simple than 7-limit. It is
> a 6-limit structure even though it has a 75/64 interval between 8/5 and
> 15/8.
>
> Marcel
>

🔗octatonic10 <octatonic10@...>

that's a good point. Thanks. I don't know much about microtonal music or tuning. (Curiously, I believe it is, to some extent, irrelant in tonal music, which is more interested in the gross interval category, for purposed of counterpoint and chord construction. I notice there are different musical psychology paradigms between tuning theorists and psychoacoustic researchers who try to integrate acoustics with tonality. Peace!

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "octatonic10" <octatonic10@> wrote:
> >
> > but in a chord progression, does it really matter? The average >tonal ear won't notice the tuning differences you describe. But of >course for microtonal music it is interesting!
>
> The "average tonal ear" won't notice the difference between a flat seven and a blue flat seven?
>
> Not only is your statement wrong, it is interestingly "maximally" wrong in my experience. In introducing the theoretical concepts of JI and microtonality to people, I find that it is a blued b7 that is the instant audible eye-opener. A 5/4 often gets a puzzled, well, it's more "church" sounding..., without the comprehension that the intervals, and not just the timbre, have been changed.
>
>
>
>
>
>
> >
> > --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@> wrote:
> > >
> > > This is an interesting and wholesome diminished seventh chord that I
> > > couldn't find in the Chords database of SCALA:
> > >
> > > 135:160:192:225
> > >
> > > When the chord is unfolded, it yields:
> > >
> > > 5/3 x 128/75 x 5/3.
> > >
> > > The ratios are:
> > >
> > > 1/1
> > > 32/27
> > > 6/5
> > > 75/64
> > >
> > > I find 75/64 to be a much more agreeable septimal minor third compared
> > > to 7/6.
> > >
> > > In Yarman-36 (a,b,c) the steps are (over the regular 12-tone layer):
> > >
> > > 18 8 18.
> > >
> > > The notation is:
> > >
> > > C E/b F# Ad
> > >
> > > Cordially,
> > > Oz.
> > >
> > > âo?=© âo?=© â�©
> > > www.ozanyarman.com
> > >
> >
>

🔗Jacques Dudon <fotosonix@...>

Lovely theme Monz !

May I disgress here by adding why 75/64 is a very special minor third for me ?
(besides being in 5-limit !)
On the contrary to 7/6, 75/64 's differential tone is a neutral fourth (11/64) which transposed to the same octave,
introduces a 88/75 interval that repeats pretty much the same interval (they differ by a schisma of 5632/5625).
This is a fractal property that is illustrated in a infinity of recurrent sequences such as 35:41:48:56, 99:116:136:160, 169:198:232:272, ...
and so on where x^2 = 8(x - 1) - that is called "Isrep" in my terminology, and has 4 - 2^(1/2) = 1,171572875254 as solution.

One of those sequences is 64:75:88:104:128:192 (differentials 11 > 13 > 16 > 24) and if it may be seen more rapidly degenerating than other series, it does it in unique way, since the spiral crosses the octave (128) of the first tone.
128 - 104 = 24 then introduces a new tone, but extends luckily the diminished seventh chord (if ever you still want to hear it that way) to a 64:75:88:96:104 (n-1) class -c (differentially coherent) pentatonic scale :

isrep_75

1/1
75/64
11/8
3/2
13/8
2/1

No idea if the 13/8 and 11/8 quartertones here, in combination to 75/64, would be of any interest in maqam music (or extrapolations), may be Ozan knows ?
However it has some ethiopian accents, and other charms.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Jacques

🔗a_sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "monz" <joemonz@...> wrote:

> to tune a particular "minor-3rd" to 75/64.

> > I don't think 75/64 resonates superior to 7/6 by itself.....

> > 7/6 does occur directly from the tonic in harmonic 7-limit...

in deed, because 75/64 can be considered as crude
5-limit approximation of the "septimal-minor-3rd" 7/6

(75/64) := (7/6)*(225/224) or in logarithmic Cent units
~274.6 C := ~266.9 C + ~7.7 C

with the ingredients

7/6 http://en.wikipedia.org/wiki/Septimal_minor_third
~266.9 Cents

* or +

225/224 http://en.wikipedia.org/wiki/Septimal_kleisma
~7.7 Cents
=
75/64 classic augmented second := (9/8)*(25/24)
http://www.huygens-fokker.org/docs/intervals.html
~274.6 Cents

instead of

7/6 := (9/8)*(28/27) diatonic

with 'Archytas's 1/3 tone (28/27) ~63 Cents
that appears at first in all his 3 septimal tetrachords

http://de.wikipedia.org/wiki/Archytas_von_Tarent
"
enharmonisches Tetrachord: (28:27)(36:35)(5:4)
chromatisches Tetrachord: (28:27)(15:14)(6:5)
diatonisches Tetrachord: (28:27)(8:7)(9:8)
"

bye
A.S.

🔗Ozan Yarman <ozanyarman@...>

I don't see why (9/8 x 28/27) should be superior to (9/8 x 25/24). I
don't think 76/64 is a crude interval or an approximation. It is a
legitimate, nicely resonating augmented second/minor third. It's just
the right flavour I need in my formulations for maqam scales. 7/6 is
rather inappropriate for my purposes in fact.

Oz.

✩ ✩ ✩
www.ozanyarman.com

On Nov 16, 2009, at 6:20 PM, a_sparschuh wrote:

> --- In tuning@yahoogroups.com, "monz" <joemonz@...> wrote:
>
>> to tune a particular "minor-3rd" to 75/64.
>
>>> I don't think 75/64 resonates superior to 7/6 by itself.....
>
>>> 7/6 does occur directly from the tonic in harmonic 7-limit...
>
> in deed, because 75/64 can be considered as crude
> 5-limit approximation of the "septimal-minor-3rd" 7/6
>
> (75/64) := (7/6)*(225/224) or in logarithmic Cent units
> ~274.6 C := ~266.9 C + ~7.7 C
>
> with the ingredients
>
> 7/6 http://en.wikipedia.org/wiki/Septimal_minor_third
> ~266.9 Cents
>
> * or +
>
> 225/224 http://en.wikipedia.org/wiki/Septimal_kleisma
> ~7.7 Cents
> =
> 75/64 classic augmented second := (9/8)*(25/24)
> http://www.huygens-fokker.org/docs/intervals.html
> ~274.6 Cents
>
> instead of
>
> 7/6 := (9/8)*(28/27) diatonic
>
> with 'Archytas's 1/3 tone (28/27) ~63 Cents
> that appears at first in all his 3 septimal tetrachords
>
> http://de.wikipedia.org/wiki/Archytas_von_Tarent
> "
> enharmonisches Tetrachord: (28:27)(36:35)(5:4)
> chromatisches Tetrachord: (28:27)(15:14)(6:5)
> diatonisches Tetrachord: (28:27)(8:7)(9:8)
> "
>
> bye
> A.S.
>
>
>
> ------------------------------------
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
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🔗martinsj013 <martinsj@...>

--- In tuning@yahoogroups.com, Daniel Forro <dan.for@...> wrote:
> Minor third (as a combination from L-arge second + s-mall second). I
> don't like this confusing way of writing, because not always the
> higher note of this minor third has "b" accidental (like for example
> C# - E, or D# - F#).

The L, s method is most appropriate in the context of a meantone system. (L is equivalent to +2 steps in the chain of 5ths, s is equivalent to -5 steps; therefore L+s is equivalent to -3 steps.) In the example F#-A-C-Eb, if there are three identical steps of L+s then to complete the octave we need a different step of 2L-s: the octave is 5L+2s not 4L+4s (these are equal if and only if L=2s). This makes sense because Eb-F# is an augmented 2nd not a minor 3rd.

However, Marcel and Oz are not talking about meantone but a 5-limit JI system, so I visualise a lattice with C-(G-D-)A on one line, Eb on the line below (between C-G), F# on the line above (between D-A), as below: (this will only look right with Courier font or similar):

F#
\
C -(G)-(D)- A
\
Eb

Two of the intervals are the same size but the other two are distinct from them and each other; the diagram captures this fact. Each of Marcel's four inversions starts from a different point on this diagram (though he uses different note names). His note names are correct; the 75/64 interval corresponds to the augmented second.

>> C (1/1) - Eb (32/27) - Gb (64/45) - A (5/3)
>> C (1/1) - Eb (6/5) - F# (45/32) - A (27/16)
>> C (1/1) - D# (75/64) - F# (45/32) - A (5/3)
>> C (1/1) - Eb (6/5) - Gb (64/45) - Bbb (128/75)

Daniel's examples are transpositions of Marcel's last one:
> H# - D# - F# - A (into C# maj/min)
> Eb - Gb - Bbb - Dbb (into Fb)
> Gx - H# - D# - F# (into A#)

Steve M.

🔗a_sparschuh <a_sparschuh@...>

🔗Ozan Yarman <ozanyarman@...>

Dear Jacques,

I find the proximity between certain lower prime limit and higher
prime limit intervals always interesting: Take for instance 35/32 versus 12/11, or 55/49 versus 9/8. My experience with Maqam musicians
have shown that they find 9/8 too high for a whole tone. Does this
have something to do with the fact that equal temperament has so much
penetrative power nowadays? Or is there something else going on with
the higher primes? Evidence also suggests that 35/32 is sometimes
preferred over 12/11.

What do you think of the sequence 64 75 90 105 128, which is limited
to prime 7?

0: 1/1 0.000 unison, perfect prime
1: 75/64 274.582 classic augmented second
2: 45/32 590.224 diatonic tritone
3: 105/64 857.095 septimal neutral sixth
4: 2/1 1200.000 octave

This yields a nice diminished seventh chord. Differentials are 11 > 15
> 15 > 23.

With the addition of the perfect fifth, the sequence becomes:

64 75 90 96 105 128

And the scale is:

0: 1/1 0.000 unison, perfect prime
1: 75/64 274.582 classic augmented second
2: 45/32 590.224 diatonic tritone
3: 3/2 701.955 perfect fifth
4: 105/64 857.095 septimal neutral sixth
5: 2/1 1200.000 octave

Cordially,
Oz.

✩ ✩ ✩
www.ozanyarman.com

On Nov 16, 2009, at 2:25 AM, Jacques Dudon wrote:

>
>
> Lovely theme Monz !
>
> May I disgress here by adding why 75/64 is a very special minor
> third for me ?
> (besides being in 5-limit !)
> On the contrary to 7/6, 75/64 's differential tone is a neutral
> fourth (11/64) which transposed to the same octave,
> introduces a 88/75 interval that repeats pretty much the same
> interval (they differ by a schisma of 5632/5625).
> This is a fractal property that is illustrated in a infinity of
> recurrent sequences such as 35:41:48:56, 99:116:136:160,
> 169:198:232:272, ...
> and so on where x^2 = 8(x - 1) - that is called "Isrep" in my
> terminology, and has 4 - 2^(1/2) = 1,171572875254 as solution.
>
> One of those sequences is 64:75:88:104:128:192 (differentials 11 >
> 13 > 16 > 24) and if it may be seen more rapidly degenerating than
> other series, it does it in unique way, since the spiral crosses the
> octave (128) of the first tone.
> 128 - 104 = 24 then introduces a new tone, but extends luckily the
> diminished seventh chord (if ever you still want to hear it that
> way) to a 64:75:88:96:104 (n-1) class -c (differentially coherent)
> pentatonic scale :
>
> isrep_75
>
> 1/1
> 75/64
> 11/8
> 3/2
> 13/8
> 2/1
>
> No idea if the 13/8 and 11/8 quartertones here, in combination to
> 75/64, would be of any interest in maqam music (or extrapolations),
> may be Ozan knows ?
> However it has some ethiopian accents, and other charms.
>
> - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
> Jacques
>

🔗Ozan Yarman <ozanyarman@...>

An interesting scale from the differentials of 75/64:

Original > Differential > Normalized
75/64 > 11/64 > 88/64 (11/8)
88/64 > 13/64 > 13/8
13/8 > 5/8 > 5/4
1/4 > 2/1

Ascending order:

75/64
5/4
11/8
13/8
2/1

Oz.

✩ ✩ ✩
www.ozanyarman.com

On Nov 16, 2009, at 2:25 AM, Jacques Dudon wrote:

🔗Ozan Yarman <ozanyarman@...>

Umm, I am not exactly concerned with the overtone series after 5/4
failed to be the correct equivalent of perde segah under many
circumstances. Just as I find 550/441 to be the "right segah", I find
75/64 to be the "right kürdi" in the proper melodic/harmonic setting.

Check for instance this Rast-Segah combo scale:

0: 1/1 0.000 unison, perfect prime
1: 55/49 199.980 quasi-equal major second
2: 75/64 274.582 classic augmented second
3: 550/441 382.384
4: 147/110 501.975
5: 3/2 701.955 perfect fifth
6: 165/98 901.935
7: 21609/12100 1003.950
8: 275/147 1084.339
9: 441/220 1203.930 Werckmeister's undecimal
septenarian schisma +1 octave

Where perde segah is the 3rd degree. It is excellent both melodically
and harmonically.

Oz.

✩ ✩ ✩
www.ozanyarman.com

On Nov 18, 2009, at 5:57 PM, a_sparschuh wrote:

> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>>
>> I don't see why (9/8 x 28/27) should be superior to (9/8 x 25/24).
>> I don't think 76/64 is a crude interval or an approximation.
>> It is a legitimate,
>> nicely resonating augmented second/minor third.
>> It's just the right flavour I need in my formulations for maqam
>> scales. 7/6 is rather inappropriate for my purposes in fact.
>>
> Agreed Oz,
> that for 5-limit in maquam 75/64 turns out to be the better choice,
> but how about the lower embedding of 7/6 within the overtone-series:
>
> 1 : 2 : 4 : 5 : (6 : 7) : 8 :9....63 : (64) : 65...74 : (75) : 76....?
>
> bye
> A.S.
>
>
>
> ------------------------------------
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
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🔗Marcel de Velde <m.develde@...>

Hi Oz,

These are some very high ratios.
How do you come by these? By ear?

Umm, I am not exactly concerned with the overtone series after 5/4
> failed to be the correct equivalent of perde segah under many
> circumstances. Just as I find 550/441 to be the "right segah", I find
> 75/64 to be the "right kürdi" in the proper melodic/harmonic setting.
>
> Check for instance this Rast-Segah combo scale:
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 55/49 199.980 quasi-equal major second
> 2: 75/64 274.582 classic augmented second
> 3: 550/441 382.384
> 4: 147/110 501.975
> 5: 3/2 701.955 perfect fifth
> 6: 165/98 901.935
> 7: 21609/12100 1003.950
> 8: 275/147 1084.339
> 9: 441/220 1203.930 Werckmeister's undecimal
> septenarian schisma +1 octave
>
> Where perde segah is the 3rd degree. It is excellent both melodically
> and harmonically.
>

I wouldn't call 550/441 excellent harmonically though :)
If you accept 7-limit there's a much more consonant 7-limit interval fairly
close that to me makes much more sense harmonically (and melodically)

56/45 378.602 cents
5/4 386.314 cents major third

The 56/45 occurs between 9/8 and 7/5, 5/4 and 14/9, 3/2 and 28/15, 15/8 and
7/3.

The closest I could get to your scale in 7-limit tonal-ji in a single tonic
is the following scale:
0: 1/1 0.000 unison, perfect prime
1: 28/25 196.198 middle second
2: 7/6 266.871 septimal minor third
3: 56/45 378.602
4: 4/3 498.045 perfect fourth
5: 3/2 701.955 perfect fifth
6: 42/25 898.153 quasi-tempered major sixth
7: 9/5 1017.596 just minor seventh, BP seventh
8: 28/15 1080.557 grave major seventh
9: 2/1 1200.000 octave

The tonic of this scale is on 8/5 (which is not in your scale)

Seen from the tonic this scale is:
5/4 7/5 35/24 14/9 5/3 15/8 21/10 9/4 7/3

Seems like a great 7-limit scale, but I have very little experience with
7-limit and maqam music in general so I'll let you be the judge if you like
it or not :)

Also this seems like the best solution for the scale if it is in 1 tonic,
but there are other possiblities in 1 tonic in 7-limit though not as close
to the pitches you gave, and there are many many more possiblities for the
scale if the scale results from a modulation and is in 2 tonics or even
more.

Marcel

🔗Ozan Yarman <ozanyarman@...>

O Marcel,

16/9 should be in place of 9/5, because we want a "4:3" between that
and 4/3. Your scale seems rigid enough, except:

1. There is an ugly 56:75 (506 cent) fourth between 28/25 and 3/2.
This, when flipped, yields a most unsavoury 694 cent fifth.

2. 28/25 is too low a whole tone.

3. 56/45 is not more consonant than 550/441. The latter produces
proportional beating chords.

4. It is boring to keep the octave "pure" when there arepsychoacoustical reasons to stretch it, even without the effect of
inharmonicity.

Therefore, your scale seems to be a eyeful 7-limit approximation to
the better and more consonant Werckmeisterian flavour I stick to.

Cordially,
Oz.

✩ ✩ ✩
www.ozanyarman.com

On Nov 19, 2009, at 2:00 AM, Marcel de Velde wrote:

>
>
> Hi Oz,
>
> These are some very high ratios.
> How do you come by these? By ear?
>
> Umm, I am not exactly concerned with the overtone series after 5/4
> failed to be the correct equivalent of perde segah under many
> circumstances. Just as I find 550/441 to be the "right segah", I find
> 75/64 to be the "right kürdi" in the proper melodic/harmonic setting.
>
> Check for instance this Rast-Segah combo scale:
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 55/49 199.980 quasi-equal major second
> 2: 75/64 274.582 classic augmented second
> 3: 550/441 382.384
> 4: 147/110 501.975
> 5: 3/2 701.955 perfect fifth
> 6: 165/98 901.935
> 7: 21609/12100 1003.950
> 8: 275/147 1084.339
> 9: 441/220 1203.930 Werckmeister's undecimal
> septenarian schisma +1 octave
>
> Where perde segah is the 3rd degree. It is excellent both melodically
> and harmonically.
>
>
>
> I wouldn't call 550/441 excellent harmonically though :)
> If you accept 7-limit there's a much more consonant 7-limit interval
> fairly close that to me makes much more sense harmonically (and
> melodically)
>
> 56/45 378.602 cents
> 5/4 386.314 cents major third
>
> The 56/45 occurs between 9/8 and 7/5, 5/4 and 14/9, 3/2 and 28/15,
> 15/8 and 7/3.
>
>
> The closest I could get to your scale in 7-limit tonal-ji in a
> single tonic is the following scale:
> 0: 1/1 0.000 unison, perfect prime
> 1: 28/25 196.198 middle second
> 2: 7/6 266.871 septimal minor third
> 3: 56/45 378.602
> 4: 4/3 498.045 perfect fourth
> 5: 3/2 701.955 perfect fifth
> 6: 42/25 898.153 quasi-tempered major sixth
> 7: 9/5 1017.596 just minor seventh, BP seventh
> 8: 28/15 1080.557 grave major seventh
> 9: 2/1 1200.000 octave
>
> The tonic of this scale is on 8/5 (which is not in your scale)
>
> Seen from the tonic this scale is:
> 5/4 7/5 35/24 14/9 5/3 15/8 21/10 9/4 7/3
>
> Seems like a great 7-limit scale, but I have very little experience
> with 7-limit and maqam music in general so I'll let you be the judge
> if you like it or not :)
>
> Also this seems like the best solution for the scale if it is in 1
> tonic, but there are other possiblities in 1 tonic in 7-limit though
> not as close to the pitches you gave, and there are many many more
> possiblities for the scale if the scale results from a modulation
> and is in 2 tonics or even more.
>
> Marcel

🔗Marcel de Velde <m.develde@...>

Hi Oz,

O Marcel,
>
> 16/9 should be in place of 9/5, because we want a "4:3" between that
> and 4/3. Your scale seems rigid enough, except:
>
> 1. There is an ugly 56:75 (506 cent) fourth between 28/25 and 3/2.
> This, when flipped, yields a most unsavoury 694 cent fifth.
>
Well if you're looking to use all fifths as pure fifths (and fourths as pure
fourths) in your scale then you must be modulating.
If you're modulating then the scale I gave is of no use.
And if you're modulating then all fixed scales will be temperaments.

>
> 2. 28/25 is too low a whole tone.
>
And 9/8 is too high??
There isn't anything inbetween 28/25 and 9/8 in low ratio JI.

>
> 3. 56/45 is not more consonant than 550/441. The latter produces
> proportional beating chords.
>
I personally don't call proportionally beating consonant.

>
> 4. It is boring to keep the octave "pure" when there are
> psychoacoustical reasons to stretch it, even without the effect of
> inharmonicity.
>

I personally can't stand a streched octave except in a jazz piano or
something like that.

>
> Therefore, your scale seems to be a eyeful 7-limit approximation to
> the better and more consonant Werckmeisterian flavour I stick to.
>

I agree if you're modulating within this scale a temperament like
werckmeister is much better.
As I said my proposed 7-limi ji scale was only for a sinlge tonic and
without my understanding of how maqam music works.
I guess I guessed wrong and maqam scales are scales arising from
modulations.

>
> Cordially,
> Oz.
>
Kind regards,
Marcel

🔗Marcel de Velde <m.develde@...>

Hi Joe,

> Back around 1999, when i "justified" my piece
> _3 Plus 4_, i wrote some posts here about my
> empirical experiments where i ended up deciding
> to tune a particular "minor-3rd" to 75/64.
> I summarized them a couple of years ago here:
>

Very nice piece :)

If I understand correctly you use the 75/64 in a minor triad aswell as 1/1
75/64 3/2?
I like that minor triad too, as 8/5 15/8 6/5 (from tonic 1/1)
For instance:
Ab (8/5) - B (15/8) - Eb (12/5)
G (3/2) - C (2/1) - Eb (12/5)

or going to the diminished chord that started the 75/64 talk this time:
F (4/3) - Ab (8/5) - B (15/8) - Eb (12/5)
F (4/3) - Ab (8/5) - B (15/8) - D (9/4) <-
C (1/1) - G (3/2) - C (2/1) - Eb (12/5)

But there's another thing that puzzles me.
For a long time (before my tonal-ji theory) I though the German sixth was
1/1 5/4 3/2 225/128. (75/64 between 3/2 and 225/128)
But now my tonal-ji theory (that works in everything so far) says this is
not a good chord.
And after carefull listening I must agree that 1/1 5/4 3/2 225/128 doesn't
sound very good. It sounds out of tune to me.
1/1 5/4 3/2 7/4 sounds good, 1/1 5/4 3/2 16/9 sounds good, 1/1 5/4 3/2 9/5
sounds good to me.
Which one is the German sixth and how does the German sixth work in
functional harmony? I don't know.
It could be 1/1 5/4 3/2 7/4 and that there's no modulation when it resolves
to V.
Could be that it's 1/1 5/4 3/2 9/5 and that there's a modulation when it
"resolves" to V.
Still experimenting.

Marcel

🔗Mike Battaglia <battaglia01@...>

Marcel: your theory includes lots of extremely dissonant chords with
comma-adjusted intervals, correct? Why do you not make use instead of
comma-adjusted melodic jumps, while leaving the chords consonant?

What I'm saying is, why do you prioritize melodic consonance over
harmonic consonance?

-Mike

On Wed, Nov 18, 2009 at 10:11 PM, Marcel de Velde <m.develde@...> wrote:
>
>
>
> Hi Joe,
>>
>> Back around 1999, when i "justified" my piece
>> _3 Plus 4_, i wrote some posts here about my
>> empirical experiments where i ended up deciding
>> to tune a particular "minor-3rd" to 75/64.
>> I summarized them a couple of years ago here:
>
> Very nice piece :)
>
> If I understand correctly you use the 75/64 in a minor triad aswell as 1/1 75/64 3/2?
> I like that minor triad too, as 8/5 15/8 6/5 (from tonic 1/1)
> For instance:
> Ab (8/5) - B (15/8) - Eb (12/5)
> G (3/2) - C (2/1) - Eb (12/5)
>
> or going to the diminished chord that started the 75/64 talk this time:
> F (4/3) - Ab (8/5) - B (15/8) - Eb (12/5)
> F (4/3) - Ab (8/5) - B (15/8) - D (9/4) <-
> C (1/1) - G (3/2) - C (2/1) - Eb (12/5)
>
>
> But there's another thing that puzzles me.
> For a long time (before my tonal-ji theory) I though the German sixth was 1/1 5/4 3/2 225/128. (75/64 between 3/2 and 225/128)
> But now my tonal-ji theory (that works in everything so far) says this is not a good chord.
> And after carefull listening I must agree that 1/1 5/4 3/2 225/128 doesn't sound very good. It sounds out of tune to me.
> 1/1 5/4 3/2 7/4 sounds good, 1/1 5/4 3/2 16/9 sounds good, 1/1 5/4 3/2 9/5 sounds good to me.
> Which one is the German sixth and how does the German sixth work in functional harmony? I don't know.
> It could be 1/1 5/4 3/2 7/4 and that there's no modulation when it resolves to V.
> Could be that it's 1/1 5/4 3/2 9/5 and that there's a modulation when it "resolves" to V.
> Still experimenting.
>
> Marcel
>
>
>

🔗Ozan Yarman <ozanyarman@...>

✩ ✩ ✩
www.ozanyarman.com

On Nov 19, 2009, at 5:49 AM, Marcel de Velde wrote:

Modulating to other keys or transposing to different ahenks (diapasons).

>
> 2. 28/25 is too low a whole tone.
>
> And 9/8 is too high??

Yes. We are looking for something in between.

> There isn't anything inbetween 28/25 and 9/8 in low ratio JI.
>

Yes there is: 55/49. That is, if you accept 11 to be low enough for a
prime. It's just the next one after 7.

>
> 3. 56/45 is not more consonant than 550/441. The latter produces
> proportional beating chords.
>
> I personally don't call proportionally beating consonant.
>

You have to hear it to believe it.

>
> 4. It is boring to keep the octave "pure" when there are
> psychoacoustical reasons to stretch it, even without the effect of
> inharmonicity.
>
>
> I personally can't stand a streched octave except in a jazz piano or
> something like that.
>

Jazz pianos are console or grand pianos that are encumbered with the
same inharmonicity as all other pianos. With console pianos, octave
stretching is more towards the extreme ends of the register.

Are you talking about additional octave stretching next to that due to
inharmonicity?

The fifth has been tempered as much as 7 cents in history. Why
shouldn't the octave be tempered by 4 cents in retrospect?

>
> Therefore, your scale seems to be a eyeful 7-limit approximation to
> the better and more consonant Werckmeisterian flavour I stick to.
>
>
> I agree if you're modulating within this scale a temperament like
> werckmeister is much better.

You misunderstand. I call the 11-limit scale Werckmeisterian due to
the undeciamal septenarian schisma (441/440) attributed to Andreas
Werckmeister. You'll notice that 550/441 is this schisma smaller than
5/4 and 147/110 larger than 4/3 by the same amount. So is 9/8 compared
to 55/49. The octave is likewise stretched by 441/440.

> As I said my proposed 7-limi ji scale was only for a sinlge tonic
> and without my understanding of how maqam music works.
> I guess I guessed wrong and maqam scales are scales arising from
> modulations.
>

There is limited modulation - so far - in traditional renditions of
Maqam music. However, the cycle of fifths is an indispensible quality.

>
> Cordially,
> Oz.
>
> Kind regards,
> Marcel
>
>

🔗Ozan Yarman <ozanyarman@...>

Have you tried the chord

882:1100:1323:2205

In Yarman-36c? The ratios are:

1/1
550/441
3/2
5/2

Tell me if what you hear is consonant or not. Now compare it with an
instance where you replace 550/441 with your 7-limit equivalent 56/45.
Which one is more consonant?

Oz.

✩ ✩ ✩
www.ozanyarman.com

On Nov 19, 2009, at 5:49 AM, Marcel de Velde wrote:

>
>
> Hi Oz,
>
> O Marcel,
>
> 16/9 should be in place of 9/5, because we want a "4:3" between that
> and 4/3. Your scale seems rigid enough, except:
>
> 1. There is an ugly 56:75 (506 cent) fourth between 28/25 and 3/2.
> This, when flipped, yields a most unsavoury 694 cent fifth.
>
> Well if you're looking to use all fifths as pure fifths (and fourths
> as pure fourths) in your scale then you must be modulating.
> If you're modulating then the scale I gave is of no use.
> And if you're modulating then all fixed scales will be temperaments.
>
>
> 2. 28/25 is too low a whole tone.
>
> And 9/8 is too high??
> There isn't anything inbetween 28/25 and 9/8 in low ratio JI.
>
>
> 3. 56/45 is not more consonant than 550/441. The latter produces
> proportional beating chords.
>
> I personally don't call proportionally beating consonant.
>
>
> 4. It is boring to keep the octave "pure" when there are
> psychoacoustical reasons to stretch it, even without the effect of
> inharmonicity.
>
>
> I personally can't stand a streched octave except in a jazz piano or
> something like that.
>
>
> Therefore, your scale seems to be a eyeful 7-limit approximation to
> the better and more consonant Werckmeisterian flavour I stick to.
>
>
> I agree if you're modulating within this scale a temperament like
> werckmeister is much better.
> As I said my proposed 7-limi ji scale was only for a sinlge tonic
> and without my understanding of how maqam music works.
> I guess I guessed wrong and maqam scales are scales arising from
> modulations.
>
>
> Cordially,
> Oz.
>
> Kind regards,
> Marcel
>
>
>

🔗Marcel de Velde <m.develde@...>

Hi Mike,

Marcel: your theory includes lots of extremely dissonant chords with
> comma-adjusted intervals, correct? Why do you not make use instead of
> comma-adjusted melodic jumps, while leaving the chords consonant?
>
> What I'm saying is, why do you prioritize melodic consonance over
> harmonic consonance?
>
> -Mike
>

I'm not prioritizing melodic consonance over harmonic consonance.
I am however not accepting a comma jump in a held note.
Further I belief my theory gives the most consonant structure possible from
a certain tonic.
6-limit tonal-ji structure (in one octave) is 1/1 (tonic) 9/8 6/5 5/4 4/3
3/2 8/5 5/3 9/5 15/8 2/1
I belief my theory gives the harmonically most consonant way to handle
certain chord progressions.
Also for instance:
4/3 8/5 15/8 9/4
4/3 3/2 15/8 9/4
1/1 3/2 2/1 5/4
How could this possibly be played in a more consonant way harmonically or
melodically?
I don't see how.

As for the example of the 8/5 15/8 6/5 minor triad.
Yes this is a much more dissonant triad than 1/1 6/5 3/2.
But it's nice to be able to play such a dissonant triad, and tonal-ji theory
sais when and how one can play such a triad and what it's function is
(getting there anyway, i'm sure there's much more indepth exploration
warranted in tonal-ji).

Marcel

🔗Marcel de Velde <m.develde@...>

Hi Oz,

Have you tried the chord
>
> 882:1100:1323:2205
>
> In Yarman-36c? The ratios are:
>
> 1/1
> 550/441
> 3/2
> 5/2
>
> Tell me if what you hear is consonant or not. Now compare it with an
> instance where you replace 550/441 with your 7-limit equivalent 56/45. Which
> one is more consonant?
>
> Oz.
>

Well I'd prefer 1/1 5/4 3/2 5/2 by far :)
550/441 sounds like an out of tune 5/4 to me in the chord you just gave.

But if I actually want an interval lower than 5/4 I'd use 56/45 myself.
For instance 1/1 5/4 14/9 15/8 (56/45 between 5/4 and 15/8) though I'd
prefer 1/1 5/4 25/16 15/8 (tonic 5/4, 5/3 or 15/8) for any common practice
music, but I though since you so clearly gave a lower pitch than 5/4 that
maybe 56/45 is a grave interval that should indeed be played in this maqam.
But as I said allready, maqam is not my specialty :)
Though when we're talking about consonance.
Perhaps 550/441 is percieved as more consonant than 56/45 in certain
circumstances, but probably simply because it's closer to 5/4 and harmonic
enthropy etc. Still this makes it an "out of tune" or "tempered" interval in
my book.

But I just wanted to chime in to see if maybe the ratios I gave you were of
any help as I saw all those high ratios.
But I see it wasn't :)
So I'm not interested in discussing which ratios is better or more consonant
etc, as I allready said when it comes to maqam etc I'm just guessing.
Good luck with your tuning.

Kind regards,
Marcel

🔗Jacques Dudon <fotosonix@...>

Dear Ozan,
I am totally convinced of the complementarity of higher and lower primes, just like God is, who did not create high primes for nothing ! ; )
You mention 35/32 and 12/11, but we could also point that 35/32 and 128/117 differ only by 4096/4095,
thus making 35 quasi-equivalent to a 13-limit interval, ETC...
And I also use 55/49, if I remember in connection with certain arabian neutral thirds series such as 40:49:60.

I would not think that middle-east musicians would be necessarily "equal"-temperament influenced whenever they show preferences for tempered intervals, which is frequent, even if we can't exclude that some deviant modern luthery practices could raise the question. I think there are numerous cases where temperaments, and not necessarily equal nor irrational, would be relevant in oriental music contexts, and you would certainly be the kind of person to know that !

That 35/32 may be prefered over 12/11 could be explained by differential coherence (in the case of 35/32 it is pretty evident), but not only. As always it is just a question of context : 12/11 can be more pertinent under the tonic, such as in an arabian Rast, and 35/32 rather after the tonic ; then, for a differential coherence addict as I am, when a 13-limit interval such as 128/117 under the tonic generates a 11-limit differential interval it is quite precious.

What do I think of 64 75 90 105 128 ? 7-limit is fine and it sounds perfectly nice on the spectral level !
It is much more "diminished seventh-correct" than my bizarre fractal 64:75:88:104:128.
On the -c level, 11 > 15 > 15 > 23 lacks differential coherence,
but it is very easy to improve, by lowering your 1/1 to 255/256 :
255:300:360:420:510 makes it more -c since 300-255 = 255-210 = 45 = octave of 360

Then I would be tempted to complete it with 240 in order to get a fully class(1+2) -c scale :
240:255:300:360:420:480:510
(n-1) diffs = 15 > 45 > 60 > 60> 60 > 30
(n-2) diffs = 60 > 105 > 120 > 120> 90
note that it can well be simplified by 15 :
16:17:20:24:28:32 : a nice harmonic scale ! - kind of defective Basant Mukhari
( but I would rather keep your 256 along with 255, because 16/15 and 17/16 are both meaningful and I enjoy these types of commas myself ! )

BTW, someting is interesting here : 17/16 is quasi-exactly the remnant semitone of four isrep minor thirds to the octave :
2 / (1.171572875254 ^4)
and therefore 17/16 suggests this "heavy-Isrep" pentatonic :
204 239 280 328 384 408
diffs = 35 > 41 > 48 > 56 > 24
(sorry for the primes...)
- - - - - -
Jacques

On nov 18, 2009, Ozan Yarman wrote :

> Dear Jacques,
>
> I find the proximity between certain lower prime limit and higher
> prime limit intervals always interesting: Take for instance 35/32
> versus 12/11, or 55/49 versus 9/8. My experience with Maqam musicians
> have shown that they find 9/8 too high for a whole tone. Does this
> have something to do with the fact that equal temperament has so much
> penetrative power nowadays? Or is there something else going on with
> the higher primes? Evidence also suggests that 35/32 is sometimes
> preferred over 12/11.
>
> What do you think of the sequence 64 75 90 105 128, which is limited
> to prime 7?
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 75/64 274.582 classic augmented second
> 2: 45/32 590.224 diatonic tritone
> 3: 105/64 857.095 septimal neutral sixth
> 4: 2/1 1200.000 octave
>
> This yields a nice diminished seventh chord. Differentials are 11 > 15
> > 15 > 23.
>
> With the addition of the perfect fifth, the sequence becomes:
>
> 64 75 90 96 105 128
>
> And the scale is:
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 75/64 274.582 classic augmented second
> 2: 45/32 590.224 diatonic tritone
> 3: 3/2 701.955 perfect fifth
> 4: 105/64 857.095 septimal neutral sixth
> 5: 2/1 1200.000 octave
>
> Cordially,
> Oz.

🔗Jacques Dudon <fotosonix@...>

🔗duckfeetbilly <billygard@...>

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
> But there's another thing that puzzles me.
> For a long time (before my tonal-ji theory) I though the German sixth was
> 1/1 5/4 3/2 225/128. (75/64 between 3/2 and 225/128)
> But now my tonal-ji theory (that works in everything so far) says this is
> not a good chord.
> And after carefull listening I must agree that 1/1 5/4 3/2 225/128 doesn't
> sound very good. It sounds out of tune to me.
> 1/1 5/4 3/2 7/4 sounds good, 1/1 5/4 3/2 16/9 sounds good, 1/1 5/4 3/2 9/5
> sounds good to me.
> Which one is the German sixth and how does the German sixth work in
> functional harmony? I don't know.
> It could be 1/1 5/4 3/2 7/4 and that there's no modulation when it resolves
> to V.
> Could be that it's 1/1 5/4 3/2 9/5 and that there's a modulation when it
> "resolves" to V.
> Still experimenting.

Going by the JI scale alone, I find that the choice between 16/9 and 9/5 for the 7th depends on what note in the scale the chord is rooted on, since some of the whole tones in the JI scale are 9/8's and some are 10/9's. For instance, I would expect the II7 and V7 to have the 16/9 and the VI7 and III7 to have the 9/5.

But if you want to bend scale tones to tune the chord, I'd rather just use the 4:5:6:7 chord all the way.

Billy

🔗Marcel de Velde <m.develde@...>

Hi Billy,

Going by the JI scale alone, I find that the choice between 16/9 and 9/5 for
> the 7th depends on what note in the scale the chord is rooted on, since some
> of the whole tones in the JI scale are 9/8's and some are 10/9's. For
> instance, I would expect the II7 and V7 to have the 16/9 and the VI7 and
> III7 to have the 9/5.
>

Sorry but I'm still studying normal harmonic theory.
I never went to music school and taught myself JI.
I know a lot about music and harmony but the names used in normal harmonic
theory I don't all know so communication to others is sometimes difficult
for me.
I don't know harmonic names well enough yet (but will soon)

II7 means D F A C in the tonic of C?
I would tune this as 9/8 4/3 5/3 2/1 so indeed a minor 7th of 16/9 (but this
can offcourse not be a German sixth)

V7 as G (3/2) - B (15/8) - D (9/4) - F (8/3) in the tonic of C (1/1)
Indeed a minor 7th of 16/9

VI7 as F (4/3) - A (5/3) - C (2/1) - Eb (12/5) in the tonic of C (1/1)
Indeed a minor 7th of 9/5

III7 as E (5/4) - G (3/2) - B (15/8) - D (9/4) in the tonic of C (1/1) (but
this can also not be a German sixth)
Indeed a minor 7th of 9/5

So I'm not sure if I interpreted II7 and III7 correctly if you ment them as
German sixths.
If II7 means D F# A C then it's not a possible chord in my 6-limit tonal-ji
theory as there's no F# from the tonic in 6-limit tonal-ji.
As I currently still see it as most likely that just about all common
practice classical music doesn't use the 7th harmonic.
If the 7th harmonic is used then D F# A C is possible in the tonic of C, and
the F# would be 7/5 (or even 35/24).

If III7 starts on Eb instead of E the it's again not possible with a minor
7th since the minor 7th on Eb is Db and there's no Db in 6-limit tonal JI
(there is a 21/20 Db and a 35/32 Db in 7-limit tonal-ji.

> But if you want to bend scale tones to tune the chord, I'd rather just use
> the 4:5:6:7 chord all the way.
>
> Billy
>

No never bending anything, music is perfect :)
But if the German sixth and it's resolving to V is all in one tonic then
according to my theory it must be the 7th harmonic for the minor 7th.
In the tonic of C (1/1):
Ab (8/5) - C (2/1) - Eb (12/5) - Gb (7/5) German sixth?
G (3/2) - C (2/1) - D (9/4) - F (8/3)
G (3/2) - B (15/8) - D (9/4) - F (8/3) V7
C (1/1) - C (2/1) - E (12/5) - G (3/1) tonic minor
Or something like that.

But the 7th harmonic in common practice classical music I dunno.. and can
possibly think up many examples where this 7th harmonic is strange
melodically.
Isn't it far more likely that there's a modulation from the German sixth to
V?
Normal music theory sais that there isn't but if I find that if there's
anything where I don't trust normal music theory it's in where it places the
tonic and how it sees modulations etc.

So with common modulations of 3/2, 4/3, 5/4, 8/5, 6/5, 5/3 and with the
German sixth as either 1/1 5/4 3/2 16/9 or 1/1 5/4 3/2 9/5 (excluding 1/1
32/25 3/2 9/5 and 1/1 5/4 40/27 16/9 for now).
The 1/1 5/4 3/2 16/9 only occurs as 3/2 15/8 9/4 16/9 in the tonic 1/1, and
1/1 5/4 3/2 9/5 occurs both as this in tonic 1/1 and as 4/3 5/3 2/1 12/5 in
tonic 1/1.
What is it then that defines the German sixth?
I don't know enough about music theory to define this yet and make an
educated guess at how it should most often be tuned.
I can only tune it now in a real music retuning (because then I'll know the
tonic preceding the German sixth) but haven't yet retuned anything with a
German sixth in it.

But here are the possibilities in 6-limit tonal-ji:
(again, excluding 1/1 32/25 3/2 9/5 and 1/1 5/4 40/27 16/9)

For 1/1 5/4 3/2 16/9 German sixth:
Ab (8/5) - C (2/1) - Eb (12/5) - Gb (64/45) German sixth, Tonic 16/15
G (3/2) - C (2/1) - D (9/4) - F (8/3) Tonic 1/1 (modulation of 16/15)
G (3/2) - B (15/8) - D (9/4) - F (8/3) V7
C (1/1) - C (2/1) - E (12/5) - G (3/1) tonic minor
Now this one gives a modulation of 16/15. That's not a consonant modulation.
I don't think this one can be correct at all.
So either this "German sixth" of 1/1 5/4 3/2 16/9 doesn't go to V, or it's
wrong alltogether.
Probably wrong alltogether, I'd even rather use a 7th harmonic here than
such a crazy modulation or making some crazy chord out of the V instead of
the V.

For 1/1 5/4 3/2 9/5 German sixth:
Ab (8/5) - C (2/1) - Eb (12/5) - Gb (36/25) German sixth, Tonic 8/5 or 6/5.
G (3/2) - C (2/1) - D (9/4) - F (8/3) Tonic 1/1 (modulation of 5/4 / 8/5 or
6/5 / 5/3)
G (3/2) - B (15/8) - D (9/4) - F (8/3) V7
C (1/1) - C (2/1) - E (12/5) - G (3/1) tonic minor
Now this one works very well.
There are 2 possibilities for the tonic, many many musical possibilities.
It also sounds great to me, very clear.

So I'd much much prefer 9/5 for minor 7th in the German sixth than 16/9.
But about the 7/4.
That one sounds good too.
It's way different tuning than 9/5 but still sounds good.
I just can't get rid of the idea that the German sixth may also be an
example of the 7th harmonic in common practice music perhaps hopefull
thinking..
I hate using the 7th harmonic because in the past it allways turned out in
the end I was wrong to use the 7th harmonic and it should have been a normal
prime-5-limit interval afterall, but somewhere I've allways kept hoping for
a strong musical example which shows a the actual use of the 7th harmonic is
common practice music somewhere.
My examples above doesn't give any definitive outcome.
Hope to retune a piece soon that has many German sixths :)

Marcel

🔗Ozan Yarman <ozanyarman@...>

Dear Jacques,

I skipped 32/25 as a substitute for 9/7. 49/60 is another interval I
neglected to mention that emulates 11/9. You have pointed out the
latter.

The differential coherence you mention appears to depend on your
formula x^2 = 8(x - 1), which requires that 4 - 2^(1/2) or
1.171572875254 is the multiple between the numbers of a given series.
It is a special property I'll admit, but is it the only one for
differential coherence?

255:300:360:420:510 is 17-limit. I am restricting myself to 11-limit
nowadays. However, your scales are charming.

Cordially,
Oz.

✩ ✩ ✩
www.ozanyarman.com

On Nov 19, 2009, at 5:57 PM, Jacques Dudon wrote:

>
>
> Dear Ozan,
> I am totally convinced of the complementarity of higher and lower
> primes, just like God is, who did not create high primes for
> nothing ! ; )
> You mention 35/32 and 12/11, but we could also point that 35/32 and
> 128/117 differ only by 4096/4095,
> thus making 35 quasi-equivalent to a 13-limit interval, ETC...
> And I also use 55/49, if I remember in connection with certain
> arabian neutral thirds series such as 40:49:60.
>
> I would not think that middle-east musicians would be necessarily
> "equal"-temperament influenced whenever they show preferences for
> tempered intervals, which is frequent, even if we can't exclude that
> some deviant modern luthery practices could raise the question. I
> think there are numerous cases where temperaments, and not
> necessarily equal nor irrational, would be relevant in oriental
> music contexts, and you would certainly be the kind of person to
> know that !
>
> That 35/32 may be prefered over 12/11 could be explained by
> differential coherence (in the case of 35/32 it is pretty evident),
> but not only. As always it is just a question of context : 12/11 can
> be more pertinent under the tonic, such as in an arabian Rast, and
> 35/32 rather after the tonic ; then, for a differential coherence
> addict as I am, when a 13-limit interval such as 128/117 under the
> tonic generates a 11-limit differential interval it is quite precious.
>
> What do I think of 64 75 90 105 128 ? 7-limit is fine and it
> sounds perfectly nice on the spectral level !
> It is much more "diminished seventh-correct" than my bizarre fractal
> 64:75:88:104:128.
> On the -c level, 11 > 15 > 15 > 23 lacks differential coherence,
> but it is very easy to improve, by lowering your 1/1 to 255/256 :
> 255:300:360:420:510 makes it more -c since 300-255 = 255-210 =
> 45 = octave of 360
>
> Then I would be tempted to complete it with 240 in order to get a
> fully class(1+2) -c scale :
> 240:255:300:360:420:480:510
> (n-1) diffs = 15 > 45 > 60 > 60> 60 > 30
> (n-2) diffs = 60 > 105 > 120 > 120> 90
> note that it can well be simplified by 15 :
> 16:17:20:24:28:32 : a nice harmonic scale ! - kind of defective
> Basant Mukhari
> ( but I would rather keep your 256 along with 255, because 16/15 and
> 17/16 are both meaningful and I enjoy these types of commas myself ! )
>
> BTW, someting is interesting here : 17/16 is quasi-exactly the
> remnant semitone of four isrep minor thirds to the octave :
> 2 / (1.171572875254 ^4)
> and therefore 17/16 suggests this "heavy-Isrep" pentatonic :
> 204 239 280 328 384 408
> diffs = 35 > 41 > 48 > 56 > 24
> (sorry for the primes...)
> - - - - - -
> Jacques
>
>
> On nov 18, 2009, Ozan Yarman wrote :
>
>> Dear Jacques,
>>
>> I find the proximity between certain lower prime limit and higher
>> prime limit intervals always interesting: Take for instance 35/32
>> versus 12/11, or 55/49 versus 9/8. My experience with Maqam musicians
>> have shown that they find 9/8 too high for a whole tone. Does this
>> have something to do with the fact that equal temperament has so much
>> penetrative power nowadays? Or is there something else going on with
>> the higher primes? Evidence also suggests that 35/32 is sometimes
>> preferred over 12/11.
>>
>> What do you think of the sequence 64 75 90 105 128, which is limited
>> to prime 7?
>>
>> 0: 1/1 0.000 unison, perfect prime
>> 1: 75/64 274.582 classic augmented second
>> 2: 45/32 590.224 diatonic tritone
>> 3: 105/64 857.095 septimal neutral sixth
>> 4: 2/1 1200.000 octave
>>
>> This yields a nice diminished seventh chord. Differentials are 11 >
>> 15
>> > 15 > 23.
>>
>> With the addition of the perfect fifth, the sequence becomes:
>>
>> 64 75 90 96 105 128
>>
>> And the scale is:
>>
>> 0: 1/1 0.000 unison, perfect prime
>> 1: 75/64 274.582 classic augmented second
>> 2: 45/32 590.224 diatonic tritone
>> 3: 3/2 701.955 perfect fifth
>> 4: 105/64 857.095 septimal neutral sixth
>> 5: 2/1 1200.000 octave
>>
>> Cordially,
>> Oz.
>
>
>

🔗Ozan Yarman <ozanyarman@...>

🔗duckfeetbilly <billygard@...>

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> ...So I'm not sure if I interpreted II7 and III7 correctly if you ment them as
> German sixths.
> If II7 means D F# A C then it's not a possible chord in my 6-limit tonal-ji
> theory as there's no F# from the tonic in 6-limit tonal-ji.
> As I currently still see it as most likely that just about all common
> practice classical music doesn't use the 7th harmonic.
> If the 7th harmonic is used then D F# A C is possible in the tonic of C, and
> the F# would be 7/5 (or even 35/24).....
>

To fill in some specifics, I did mean the major-minor 7th in the case of the II7, III7 and VI7. I was using a convention that assumes the major-minor 7th when 7 is added to an upper-case roman numeral. And I meant them as examples of secondary dominant 7ths rather than German 6ths, as the major-minor dominant 7th wants to resolve down a fifth.

I would think that easiest thing to spot about a German sixth chord is the resolution of the "seventh" upward, rather than down like in a dominant. A German sixth chord wants to resolve down a half step, but more commonly is resolved up a major third.

Billy

🔗Marcel de Velde <m.develde@...>

Hi Billy,

Thanks for your message.

To fill in some specifics, I did mean the major-minor 7th in the case of the
> II7, III7 and VI7. I was using a convention that assumes the major-minor 7th
> when 7 is added to an upper-case roman numeral. And I meant them as examples
> of secondary dominant 7ths rather than German 6ths, as the major-minor
> dominant 7th wants to resolve down a fifth.
>

Ah ok, my tonal-ji theory sais there is no II7 chord (unless one wants a
56/45 or 35/27 major third, never that seems to me).
Strange and impossible as this may sound at first, it does work out in the
end as my theory sees it as either a modulation to for instance (most
likely) V, or that the tonic wasn't I to begin with but actually the tonic
was what was thought to be IV, and the II7 is actually V7 for instance.

>
> I would think that easiest thing to spot about a German sixth chord is the
> resolution of the "seventh" upward, rather than down like in a dominant.
>

Could make sense with the 9/5 as upwards is 25/24 and downward is a 27/25
step. But I know too about this now, especially within a modulation.

> A German sixth chord wants to resolve down a half step, but more commonly
> is resolved up a major third.
>

My example does the half step down to V and then to I, a major third up
would be going directly to I.
But my example with the 9/5 does this with a modulation.
This is the reason why I think it may be the 7th harmonic after all.
As so far it looks like chords generally want to resolve in thesame tonic.
In the tonic of C the Ab - C - Eb - F# german sixth can't be with 9/5 7th,
the 7th must be 7/4 (7/5 from C).
But perhaps in other ways the German sixth is made clear to want to do a
modulation and resolve with a modulation, the the 7th can be 9/5.
This is my main doubt.

Marcel

🔗Marcel de Velde <m.develde@...>

Ok I'll add one more possibility for the German sixth as it's starting to
look more and more like the right solution in many cases.
It doesn't work in my example a few messages back because of the clear V7.
But what if the German sixth does indeed stay in thesame tonic when it
resolves yet isn't a prime-7-limit chord.
Then the only solution I see is that the German sixth is the III7 chord.

E (5/4) - Ab (8/5) - B (15/8) - D (9/4) German sixth. Defenately needs
resolving!
Eb (6/5) - G (3/2) - Bb (9/5) - Eb (12/5) Resolving half a step down.

E (5/4) - Ab (8/5) - B (15/8) - D (9/4) German sixth.
Eb (6/5) - Ab (8/5) - C (2/1) - Eb (12/5) Resolving major third up.

E (5/4) - Ab (8/5) - B (15/8) - D (9/4) German sixth.
Eb (6/5) - G (3/2) - C (2/1) - Eb (12/5) Resolving to tonic.

D (9/8) - E (5/4) - Ab (8/5) - B (15/8) German sixth.
C (1/1) - E (5/4) - G (3/2) - C (2/1) Resolving to tonic.

E (5/4) - Ab (8/5) - B (15/8) - D (9/4) German sixth.
F (4/3) - Ab (8/5) - B (15/8) - D (9/4) Going to diminished 7th.

It seems very likely to me now afte more thinking that this is the German
sixth that is most used.
And the 9/5 "German sixth" with a modulation beeing the other one used
sometimes perhaps?

Marcel

2009/11/22 Marcel de Velde <m.develde@...>

> Hi Billy,
>
> Thanks for your message.
>
> To fill in some specifics, I did mean the major-minor 7th in the case of
>> the II7, III7 and VI7. I was using a convention that assumes the major-minor
>> 7th when 7 is added to an upper-case roman numeral. And I meant them as
>> examples of secondary dominant 7ths rather than German 6ths, as the
>> major-minor dominant 7th wants to resolve down a fifth.
>>
>
> Ah ok, my tonal-ji theory sais there is no II7 chord (unless one wants a
> 56/45 or 35/27 major third, never that seems to me).
> Strange and impossible as this may sound at first, it does work out in the
> end as my theory sees it as either a modulation to for instance (most
> likely) V, or that the tonic wasn't I to begin with but actually the tonic
> was what was thought to be IV, and the II7 is actually V7 for instance.
>
>
>
>>
>> I would think that easiest thing to spot about a German sixth chord is the
>> resolution of the "seventh" upward, rather than down like in a dominant.
>>
>
> Could make sense with the 9/5 as upwards is 25/24 and downward is a 27/25
> step. But I know too about this now, especially within a modulation.
>
>
>
>> A German sixth chord wants to resolve down a half step, but more commonly
>> is resolved up a major third.
>>
>
> My example does the half step down to V and then to I, a major third up
> would be going directly to I.
> But my example with the 9/5 does this with a modulation.
> This is the reason why I think it may be the 7th harmonic after all.
> As so far it looks like chords generally want to resolve in thesame tonic.
> In the tonic of C the Ab - C - Eb - F# german sixth can't be with 9/5 7th,
> the 7th must be 7/4 (7/5 from C).
> But perhaps in other ways the German sixth is made clear to want to do a
> modulation and resolve with a modulation, the the 7th can be 9/5.
> This is my main doubt.
>

🔗Mike Battaglia <battaglia01@...>

> Ah ok, my tonal-ji theory sais there is no II7 chord (unless one wants a 56/45 or 35/27 major third, never that seems to me).
> Strange and impossible as this may sound at first, it does work out in the end as my theory sees it as either a modulation to for instance (most likely) V, or that the tonic wasn't I to begin with but actually the tonic was what was thought to be IV, and the II7 is actually V7 for instance.

Uh, II7 definitely exists. In common practice harmony you might see it
labeled more often as "V7/V", but it's used -all- the time.

> Ok I'll add one more possibility for the German sixth as it's starting to look more and more like the right solution in many cases.
> It doesn't work in my example a few messages back because of the clear V7.
> But what if the German sixth does indeed stay in thesame tonic when it resolves yet isn't a prime-7-limit chord.
> Then the only solution I see is that the German sixth is the III7 chord.

The most common use for a German sixth is over bVI.

As in, in C major, it would be Ab C Eb F#.

So using your "theory", we would see 4/5 1/1 6/5 45/32, resolving
downward to 3/4 15/16 9/8 3/2. If you want to be anal about it, you'll
note that this creates a parallel fifth between the Ab-Eb and the G-D,
which is usually dealt with by resolving to 3/4 1/1 5/4 3/2 first, and
then to 3/4 15/16 9/8 3/2.

The augmented sixth of Ab and F# here is a 225/128 interval, which is
976.537 cents wide. On the other hand, a subminor seventh of 7/4 is
968.826 cents. The difference between them is 225/224, which you will
no doubt remember as also being the difference between 7/6 and 75/64.
So the discussion about whether the augmented sixth should be 7/4 or
225/128 is really the same thing as the discussion about whether the
augmented second should be 7/6 or 75/64.

-Mike

🔗Marcel de Velde <m.develde@...>

Hello Mike,

Uh, II7 definitely exists. In common practice harmony you might see it
> labeled more often as "V7/V", but it's used -all- the time.
>

My theory produces only the intervals 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/4 9/5
15/8 2/1 from the 1/1 tonic.
In (harmonic interval permutation) 6-limit tonal-ji that is.
There is no Db / C# or F# / Gb from the tonic C in 6-limit tonal-ji. One
would have to go to 7-limit tonal-ji to find those intervals from the tonic.
In 7-limit tonal-ji there are no less than 20 intervals from the tonic.
But it is fairly clear to me that these intervals are seldomly or more
likely never used in common practice classical music.

I started with a blank page when creating my theory, focussing on tuning and
logic behind tuning only.
I took the position beforehand that normal harmonic theory must be wrong
since it can't do JI properly.
If others have failed the past centuries / millenia in decribing working JI,
then I'm not going to succeed with thesame line of thinking used up till
now.
Since I beleif tuning to be the basis of music and that music is perfect,
then normal harmonic theory must have errors, is my line of reasoning.
I think these kinds of things we're talking about now are the errors of
normal harmonic theory in naming these chords and putting them in correct
location relevant to the tonic.
This is what I belief is the main error of normal harmonic theory, the often
wrong placement of the tonic and the resulting wrong determining of
modulations etc.

When I say there is no II7 chord, it's mainly a name thing.
In my theory one can play exactly thesame thing as in normal harmonic
theory, only the tonics and modulations become different.
And as a result one can play the thing in tonal-ji in perfect tune.
Something impossible with normal harmonic theory as it produces out of tune
music.
Normal harmonic theory isn't mathematically consistent. Tonal-ji so far
seems to be mathematically consistent (yet much remains to be worked out).

So yes in my theory the II7 chord becomes for instance a V7 (other things
are possible too depending on what's played)
This in itself doesn't say my theory is wrong at all.

> Ok I'll add one more possibility for the German sixth as it's starting to
> look more and more like the right solution in many cases.
> > It doesn't work in my example a few messages back because of the clear
> V7.
> > But what if the German sixth does indeed stay in thesame tonic when it
> resolves yet isn't a prime-7-limit chord.
> > Then the only solution I see is that the German sixth is the III7 chord.
>
> The most common use for a German sixth is over bVI.
>
> As in, in C major, it would be Ab C Eb F#.
>
> So using your "theory", we would see 4/5 1/1 6/5 45/32, resolving
> downward to 3/4 15/16 9/8 3/2.
>

As you can see, there is no 45/32 (or any other F# in the tonic of C) in my
tonal-ji.
So no this is not a possibility in my theory in the tonic of 1/1.
Same story as with the II7 chord, my theory disagrees with normal harmonic
theory about the tonic here.

> If you want to be anal about it, you'll
> note that this creates a parallel fifth between the Ab-Eb and the G-D,
> which is usually dealt with by resolving to 3/4 1/1 5/4 3/2 first, and
> then to 3/4 15/16 9/8 3/2.
>
> The augmented sixth of Ab and F# here is a 225/128 interval, which is
> 976.537 cents wide.
>

My 6-limit tonal-ji theory has as the most dissonant interval the 75/64
interval.
After this there comes 7-limit intervals in 7-limit tonal-ji which I don't
think are used in common practice music.
The 225/128 interval is not a possible interval in any way in 6-limit
tonal-ji.

> On the other hand, a subminor seventh of 7/4 is
> 968.826 cents. The difference between them is 225/224, which you will
> no doubt remember as also being the difference between 7/6 and 75/64.
> So the discussion about whether the augmented sixth should be 7/4 or
> 225/128 is really the same thing as the discussion about whether the
> augmented second should be 7/6 or 75/64.
>

No it is not thesame discussion at all.
75/64 does occur in 6-limit tonal-ji between 8/5 and 15/8. (it is the
largest ratio interval that occurs)
225/128 does not ever occur is 6-limit tonal-ji.

As you may notice that by seeing the German sixth as a III7 chord that it
indeed does have a 75/64 interval, but not a 225/128.
Seen from the tonic 1/1 my German sixth chord is 5/4 8/5 15/8 9/4
Seen from 5/4 this makes 1/1 32/25 3/2 9/5
Inversion seen from 8/5 makes 1/1 75/64 45/32 25/16
Notice the 45/32 from 32/25 to 9/5
This is thesame as the 45/32 from 5/4 to 225/128 in your German sixth tuning
example.
Yet mine doesn't have a 225/128 interval.

Kind regards,
Marcel

🔗Mike Battaglia <battaglia01@...>

🔗Marcel de Velde <m.develde@...>

Hi Mike,

> Then it seems like your theory doesn't cut it. All of these issues are
> addressed in my new "Battaglia-JI" system. It contains the following
> intervals:
>
> 1/1 16/15 10/9 9/8 6/5 5/4 4/3 45/32 3/2 25/16 8/5 5/3 27/16 16/9 9/5 15/8
> 2/1
>
> Battaglia-JI solves all of the problems caused by the outdated
> DeVelde-JI system, as well as problems thought insurmountable for
> centuries by the world's finest musical minds.
>
> I strongly recommend everyone update to the new system... while it's still
> free.

I'm sorry to see that you're seeing my message as some sort of self
promotion or promotion of tonal-ji.
I did not mean my messages regarding 75/64 or the German sixth as such.
My messages come from a genuine interest to find out how music and ji works.
I the context of my messages in this thread I wish I wasn't the person who
invented (or perhaps better put "found") tonal-ji.
I would have made the exact same messages if that were the case yet my
motives wouldn't be questioned.
The reason I aproach the German sixth etc with tonal-ji is because it's the
only thing that currently makes sense to me and that produces a to me
correct sounding result.
I've thrown a lot of problems towards tonal-ji allready and it manages to
work these out and make sense of them and make them sound good.
The things we're discussing now is one of the few examples where I'm not
completely sure how to see it.
But the fact that tonal-ji solves so many problems for me that I wasn't able
to solve previously makes that I'm using it to approach for instance the
German sixth.
As for tonal-ji not cutting it. Please show me where it would fail as if
what you're saying is true I most certainately want to know.
I do not wish to live in a fantasy world where I fool myself with a wrong
theory. If there's truly something wrong with tonal-ji please show me what
it is.
A musical example that cannot be tuned to satisfaction in tonal-ji.
Something similar to the many comma shift problems that plagued classic
5-limit JI.
Again, I'm not saying this because I have personal interest in tonal-ji
because I discovered it. I'm saying this because I have personal interest in
finding out how music and JI works.
Btw as for your remark "while it's still free", I'm not crazy enough to
think that I'll ever make a single cent with any of my present or possible
future just intonation discoveries :)
Perhaps one day as a composer. Though if I'll ever go after money again I'll
start up an internet company like I have done in the past.
I'm nt in music for money. Perhaps I used to be in it partly because of
achieving something and somewhere wished recognition / fame in this, but no
longer so. I do wish to achieve something in my life that I hold important
and that has become JI. Further my main other hope is simply that music
theory and JI progresses, wether this is due to my research or to anybody
elses research is irrelevant to me.

Marcel

🔗Jacques Dudon <fotosonix@...>

(this thread has gone another direction but I reply here to Ozan's
message on19th of november) -

Dear Ozan,
I am not certain to understand your question, so please ask again if
I missed the point !
Anyway I realize I have done some shortcuts in my initial message
(below), that I will try to develop now - in my limited english.

When I say, in my comment on 64 : 75 : 88 : 104 : 128 : 192 as a
differentially-coherent recurrent sequence :
> "and so on where x^2 = 8(x - 1) that has 4 - 2^(1/2) =
1,171572875254 as solution"
That's a langage shortcut.
75 - 64 = 11,
88 - 75 = 13
104 - 88 = 16
128 - 104 = 24
and the same is found for many differentially-coherent similar series
(such as 35 : 41 : 48 : 56 : 64 etc.), that verify
Hn = 8H(n-1) - 8 H(n-2)
where H(n-2), H(n-1), Hn , etc. are successive terms of one series
(ex: 88 = 600 - 512, then 104 = 704 -600, 128 = 832 - 704, etc.)
Now if many whole numbers series verify this precise algorithm, there
is only one solution x = Hn / H(n-1) that does it for all terms
together, which means that in a serie such as : 1 : x : x^2 :
x^3 ... : x^n,
x satisfies the equation x2 = 8x - 8
and this is x = 4 - 2^(1/2) = 1,171572875254

But the various Hn / H(n-1) ratios do not need to equal x to be
coherent intervals in the series, nor even to converge towards x :
75/64, 88/75, 104/88 ... actually diverge rapidly away from x.

Now, scales differential coherence properties certainly do not need
that a scale follows a recurrent sequence,
(as 64 : 75 : 88 : 104 : 128 : 192 does).
However many traditional scales, more than we think, can be found in
various differentially-coherent recurrent sequences.
Let's take one simple example with a basic harmonic scale :
1/1
9/8
5/4
3/2
7/4
2/1
This pentatonic scale can be resumed as 5 : 6 : 7 : 8 : 9 : 10
and is a highly -c scale, but can be qualified as trivial, and evenboring ! - because all differentials mostly focus on one term and its
octaves, (here 8, and more rarely 6) :
class (n-1) = 1
class (n-2) = 2
class (n-3) = 3 (better)
class (n-4) = 4

Despite its class (n-1) intervals (2nds) being very unequal, here is
one recurrent series that includes the scale :
32 : 36 : 40 : 48 : 56 : 64 : 73 : 83 : 99 : 113 : 129 (the 6 first
6 terms resume to 8 : 9 : 10 : 12 : 14 : 16)
Hn = H(n-1) + 1/4 H(n-5)
(= Slendra, or x^5 = x^4 + 1/4, solution 1.1452994122)

Its class (n-2) intervals are more even and here is another recurrent
series that includes the scale,
and two possible algorithms :
3 : 4 : 5 : 7 : 9 : 12 : 16 : 21 : 28 : 37 : 49 ... (the first 7
terms resume to the same scale)
Hn = H(n-2) + H(n-3)
Hn = H(n-1) + H(n-5)
(= Natté, Meta-slendro, Psi, number of the architects, etc., or x^3 =
x + 1, solution 1.3247179572447)

Same with class (n-3) intervals of the same scale :
7 : 10 : 16 : 24 : 36 : 56 : 84 : 128 : 196 ...(the first 6 terms
resume to the same scale)
Hn = H(n-2) + 2H(n-3)
(= Myriadjerm or x^3 = x + 2, solution 1.5213797068)

What the interest of that ? knowing the fractal aspects of a scale,
even if already coherent, lets you extend it in many ways, along the
same series or others series, of one or several algorithms, unfolding
an infinity of versions, each one with its own flavor.
Depending on their recurrent algorithms and the number of them, they
will loose or gain differential coherence properties, but will always
gain to be less trivial, showing a higher diversity of coherent
differentials.
- - - - - -
Jacques

On Nov 19, 2009, Ozan Yarman wrote:

> The differential coherence you mention appears to depend on your
> formula x^2 = 8(x - 1), which requires that 4 - 2^(1/2) or
> 1.171572875254 is the multiple between the numbers of a given series.
> It is a special property I'll admit, but is it the only one for
> differential coherence?

On Nov 16, 2009, at 2:25 AM, Jacques Dudon wrote:

>> > On the contrary to 7/6, 75/64 's differential tone is a neutral
>> > fourth (11/64) which transposed to the same octave,
>> > introduces a 88/75 interval that repeats pretty much the same
>> > interval (they differ by a schisma of 5632/5625).
>> > This is a fractal property that is illustrated in a infinity of
>> > recurrent sequences such as 35:41:48:56, 99:116:136:160,
>> > 169:198:232:272, ...
>> > and so on where x^2 = 8(x - 1) - that is called "Isrep" in my
>> > terminology, and has 4 - 2^(1/2) = 1.171572875254 as solution.
>> >
>> > One of those sequences is 64:75:88:104:128:192 (differentials 11 >
>> > 13 > 16 > 24) and if it may be seen more rapidly degenerating than
>> > other series, it does it in unique way, since the spiral crosses
>> the
>> > octave (128) of the first tone.
>> > 128 - 104 = 24 then introduces a new tone, but extends luckily the
>> > diminished seventh chord (if ever you still want to hear it that
>> > way) to a 64:75:88:96:104 (n-1) class -c (differentially coherent)
>> > pentatonic scale :
>> >
>> > isrep_75
>> >
>> > 1/1
>> > 75/64
>> > 11/8
>> > 3/2
>> > 13/8
>> > 2/1

🔗Marcel de Velde <m.develde@...>

With a fresh ear, the "German sixth" tuning I posted before doesn't sound
very good at all.
For my ear now it's either 1/1 5/4 3/2 7/4 or 1/1 5/4 3/2 225/128.
Yes 1/1 5/4 3/2 225/128, the one my 6-limit tonal-ji doesn't allow.

However, I've recently found a different harmonic interval permutation
structure with even a lower harmonic interval limit.
My new 5-limit harmonic permutation structure gives the following intervals
(when reduced to one octave)
1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8 2/1
Looks familiar doesn't it :)
I'll make a fresh thread about it soon as harmonic permutation structure is
very interesting musically.
Anyhow, this one does indeed allow a 1/1 5/4 3/2 225/128 German sixth.
Like for instance this with tonic of 1/1:
3/2 - 9/4 - 3/1 - 15/4
8/5 - 12/5 - 3/1 - 4/1
8/5 - 12/5 - 45/16 - 4/1 (German Sixth)
3/2 - 9/4 - 3/1 - 15/4
1/1 - 5/2 - 3/1 - 4/1
Though I like the sound of 8/3 there much better than 45/16

Another place where it would occur in tonic 1/1 would be here:
1/1 - 3/2 - 2/1 - 5/2
16/15 - 8/5 - 2/1 - 8/3
16/15 - 8/5 - 15/8 - 8/3 (German sixth)
1/1 - 3/2 - 2/1 - 5/2

Marcel

Ok I'll add one more possibility for the German sixth as it's starting to
> look more and more like the right solution in many cases.
> It doesn't work in my example a few messages back because of the clear V7.
> But what if the German sixth does indeed stay in thesame tonic when it
> resolves yet isn't a prime-7-limit chord.
> Then the only solution I see is that the German sixth is the III7 chord.
>
> E (5/4) - Ab (8/5) - B (15/8) - D (9/4) German sixth. Defenately needs
> resolving!
> Eb (6/5) - G (3/2) - Bb (9/5) - Eb (12/5) Resolving half a step down.
>
> E (5/4) - Ab (8/5) - B (15/8) - D (9/4) German sixth.
> Eb (6/5) - Ab (8/5) - C (2/1) - Eb (12/5) Resolving major third up.
>
> E (5/4) - Ab (8/5) - B (15/8) - D (9/4) German sixth.
> Eb (6/5) - G (3/2) - C (2/1) - Eb (12/5) Resolving to tonic.
>
> D (9/8) - E (5/4) - Ab (8/5) - B (15/8) German sixth.
> C (1/1) - E (5/4) - G (3/2) - C (2/1) Resolving to tonic.
>
> E (5/4) - Ab (8/5) - B (15/8) - D (9/4) German sixth.
> F (4/3) - Ab (8/5) - B (15/8) - D (9/4) Going to diminished 7th.
>
> It seems very likely to me now afte more thinking that this is the German
> sixth that is most used.
> And the 9/5 "German sixth" with a modulation beeing the other one used
> sometimes perhaps?
>
> Marcel
>

🔗duckfeetbilly <billygard@...>

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
> >
> > E (5/4) - Ab (8/5) - B (15/8) - D (9/4) German sixth. Defenately needs
> > resolving!
> > Eb (6/5) - G (3/2) - Bb (9/5) - Eb (12/5) Resolving half a step down.

The fullest resolution, which causes the outward resolution of all three augmented intervals in the chord, An aug 6th, aug 4th, and aug 2nd.

> >
> > E (5/4) - Ab (8/5) - B (15/8) - D (9/4) German sixth.
> > Eb (6/5) - Ab (8/5) - C (2/1) - Eb (12/5) Resolving major third up.
> >

A common resolution to avoid a parallel fifth, but is often used as a true resolution in the romantic period and barbershop.

> >
> > E (5/4) - Ab (8/5) - B (15/8) - D (9/4) German sixth.
> > Eb (6/5) - G (3/2) - C (2/1) - Eb (12/5) Resolving to tonic.
> >

This actually sounds more like the resolution of an "English 6th chord", which is still another enharmonic spelling apart from the dominant 7th or German 6th.

AbCEbGb = dominant; AbCEbF# = German; AbCD#F# = English

Note the different location in the chord of the augmented second, and that what sounds like a perfect fifth is really a doubly-augmented fourth. Kind of a far-fetched enharmonic when you think about it.

When I hear a major-minor 7th chord in the key of C, I hear a secondary dominant 7th chord rather than a German 6th. It sound like it wants to cadence to an A chord.

Billy

🔗Marcel de Velde <m.develde@...>

🔗Charles Lucy <lucy@...>

🔗Marcel de Velde <m.develde@...>

Hello Billy,

Thanks for your reply.
Your knowledge of these chords is very helpfull.

> > E (5/4) - Ab (8/5) - B (15/8) - D (9/4) German sixth. Defenately needs
> > > resolving!
> > > Eb (6/5) - G (3/2) - Bb (9/5) - Eb (12/5) Resolving half a step down.
>
> The fullest resolution, which causes the outward resolution of all three
> augmented intervals in the chord, An aug 6th, aug 4th, and aug 2nd.

Sorry but I've since come back from this tuning and spelling.
As you can see I've spelled the major third in this example as a diminished
fourth, which it actually is in this tuning.

I've come to a few new realizations concerning my JI theory and now see the
German sixth tuning as follows:
Db (16/15) - Ab (8/5) - B (15/8) - F (8/3) German sixth
C (1/1) - G (3/2) - C (2/1) - E (5/2) Tonic 1/1
True resolution

Or
Ab (8/5) - Eb (12/5) - F# (45/16) - C (4/1) German sixth
G (3/2) - D (9/4) - G (3/1) - B (15/4) V
To dominant.

> >
> > > E (5/4) - Ab (8/5) - B (15/8) - D (9/4) German sixth.
> > > Eb (6/5) - Ab (8/5) - C (2/1) - Eb (12/5) Resolving major third up.
> > >
>
> A common resolution to avoid a parallel fifth, but is often used as a true
> resolution in the romantic period and barbershop.

This I would now tune:
Db (16/15) - Ab (8/5) - B (15/8) - F (8/3) German sixth
C (1/1) - A (5/3) - C (2/1) - F (8/3) IV
To subdominant

or
Ab (8/5) - Eb (12/5) - F# (45/16) - C (4/1) German sixth
G (3/2) - E (5/2) - G (3/1) - C (4/1) Tonic 1/1
True resoluton (in inversion)

> >
> > > E (5/4) - Ab (8/5) - B (15/8) - D (9/4) German sixth.
> > > Eb (6/5) - G (3/2) - C (2/1) - Eb (12/5) Resolving to tonic.
> > >
>
> This actually sounds more like the resolution of an "English 6th chord",
> which is still another enharmonic spelling apart from the dominant 7th or
> German 6th.

I think this one is possible and don't think it's an English 6th, nor German
6th, nor dominant 7th, I don't know what it's called.
One could also play it like this:
B (15/16) - F (4/3) - Ab (8/5) - B (15/8) - D (9/4)
B (15/16) - E (5/4) - Ab (8/5) - B (15/8) - D (9/4) Unknown chordname.
C (1/1) - E (5/4) - G (3/2) - C (2/1) - E (5/4) Tonic

To spell the unknown chord from C would be C-Fb-G-Bb

>
> AbCEbGb = dominant; AbCEbF# = German; AbCD#F# = English
>

Ok so dominant 7th from C would be:
C (1/1) - E (5/4) - G (3/2) - Bb (16/9)
German 6th from C would be:
C (1/1) - E (5/4) - G (3/2) - A# (225/128)
English from C could be:
C (1/1) - E (5/4) - Fx (40/27) - A# (16/9) ?
Unknown chord name from C:
C (1/1) - Fb (32/25) - G (3/2) - Bb (9/5)
Another unknown chord name from C:
C (1/1) - E (5/4) - G (3/2) - Bb (9/5)
This chord goes to C (1/1) - F (4/3) - Ab (8/5) in drei equali nr1.

> Note the different location in the chord of the augmented second, and that
> what sounds like a perfect fifth is really a doubly-augmented fourth. Kind
> of a far-fetched enharmonic when you think about it.
>
> When I hear a major-minor 7th chord in the key of C, I hear a secondary
> dominant 7th chord rather than a German 6th. It sound like it wants to
> cadence to an A chord.
>
> Billy

What's the difference between a secondary dominant 7th and a dominant 7th?
In C I'd most likely tune a major minor 7th as 1/1 5/4 3/2 9/5
While a dominant 7th on V is clearly 1/1 5/4 3/2 16/9 (3/2 15/8 9/4 8/3)

Kind regards,
Marcel

🔗duckfeetbilly <billygard@...>

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> > I think this one is possible and don't think it's an English 6th, nor
> > German 6th, nor dominant 7th, I don't know what it's called.
> > One could also play it like this:
> > B (15/16) - F (4/3) - Ab (8/5) - B (15/8) - D (9/4)
> > B (15/16) - E (5/4) - Ab (8/5) - B (15/8) - D (9/4) Unknown chordname.
> > C (1/1) - E (5/4) - G (3/2) - C (2/1) - E (5/4) Tonic
> >
> > To spell the unknown chord from C would be C-Fb-G-Bb
> >
>
> Just wanted to add that should this chord not have a name yet (which
> probably is the case as it doesn't sound very good), I hereby name it ugly
> chord :)
>

While it is definitely not spelled as an English 6th chord, you resolved it like one in a previous example. In your spelling you have a dim 4, aug 2 and m3. I don't think this spelling is documented. The spelling for the English would be C-E-Fx=A#, and based on the principle of resolution in the direction of alteration, it should resolve to BD#G#B. I actually think this spelling of the chord may have been forced into existence by the desire to always spell the chord notes as notes altered in the direction they will be resolving. How else would they come up with a 4th that's doubly-augmented to sound like a 5th?

By the way the source that referenced an English 6th chord also called it a Swiss 6th. Check

http://www.dolmetsch.com/musictheory17.htm

"the English augmented 6th differs from the German augmented 6th in its 'spelling'. This is why the English augmented 6th is sometimes known as the misspelled German, Swiss, Alsatian or doubly augmented fourth"

Billy