Yahoo Tuning Groups Ultimate Backup tuning Splitting an interval in 2

Hi guys I have a minor conundrum I can't get my head around.

I have the following scale (it repeats on 3/2)

0: 1/1 0.000 unison, perfect prime
1: 9/8 203.910 major whole tone
2: 4/3 498.045 perfect fourth
3: 3/2 701.955 perfect fifth

Symmetrical scales have been on my mind recently. I would like to add another pitch to the scale, and I want it to be exactly half way between 9/8 and 4/3. Can any seasoned pro name that pitch please?

Tried to use mathematics but failed. Don't laugh at me. ;-)

Sean Archibald

🔗Tony <leopold_plumtree@...>

It's [(9:8)(4:3)]^(1:2), or [6^(1:2)]:2

--- In tuning@yahoogroups.com, "sevishmusic" <sevish@...> wrote:
>
> Hi guys I have a minor conundrum I can't get my head around.
>
> I have the following scale (it repeats on 3/2)
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 9/8 203.910 major whole tone
> 2: 4/3 498.045 perfect fourth
> 3: 3/2 701.955 perfect fifth
>
> Symmetrical scales have been on my mind recently. I would like to add another pitch to the scale, and I want it to be exactly half way between 9/8 and 4/3. Can any seasoned pro name that pitch please?
>
> Tried to use mathematics but failed. Don't laugh at me. ;-)
>
> Sean Archibald
>

🔗Mario Pizarro <piagui@...>

Sevishmusic,
((1.125)*(1.33333333333))^(1/2) = (3/2)^(1/2) = 1.22474487139
Mario
----- Original Message ----- From: "sevishmusic" <sevish@...>
To: <tuning@yahoogroups.com>
Sent: Friday, May 06, 2011 2:37 PM
Subject: [tuning] Splitting an interval in 2

> Hi guys I have a minor conundrum I can't get my head around.
>
> I have the following scale (it repeats on 3/2)
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 9/8 203.910 major whole tone
> 2: 4/3 498.045 perfect fourth
> 3: 3/2 701.955 perfect fifth
>
> Symmetrical scales have been on my mind recently. I would like to add > another pitch to the scale, and I want it to be exactly half way between > 9/8 and 4/3. Can any seasoned pro name that pitch please?
>
> Tried to use mathematics but failed. Don't laugh at me. ;-)
>
> Sean Archibald
>
>
>
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🔗john777music <jfos777@...>

498.045 - 203.91 = 294.135

294.135 / 2 = 147.0675

203.91 + 147.0675 = 350.9775

Answer = 350.9775 cents

🔗Carl Lumma <carl@...>

Hi Sean,

> I have the following scale (it repeats on 3/2)
> 0: 1/1 0.000 unison, perfect prime
> 1: 9/8 203.910 major whole tone
> 2: 4/3 498.045 perfect fourth
> 3: 3/2 701.955 perfect fifth
> Symmetrical scales have been on my mind recently. I would like
> to add another pitch to the scale, and I want it to be exactly
> half way between 9/8 and 4/3. Can any seasoned pro name that
> pitch please?

The pitch you're looking for is an irrational interval
above 1/1. That means there's no simple ratio for it.
You can find it in cents by adding the cents for 9/8 & 4/3
and then dividing by 2. Since 9/8 '+' 4/3 = 3/2, this is
the same as half a perfect fifth, or 350.978 cents.

If you want a simple ratio that is somewhere near the
middle of two JI ratios, an easy thing to do is find the
mediant. That's where you add the numerators and
denominators separately - sometimes called "freshman sums".
In this case, 9/8 '+' 4/3 gives you 13/11.

If neither of those do the job, let us know! -Carl

🔗john777music <jfos777@...>

350.9775 cents = 1.2247449.

11/9 is very close to this value, 11/9 is 347.4079 cents or 1.2222222.

John.

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> 498.045 - 203.91 = 294.135
>
> 294.135 / 2 = 147.0675
>
> 203.91 + 147.0675 = 350.9775
>
> Answer = 350.9775 cents
>
> --- In tuning@yahoogroups.com, "sevishmusic" <sevish@> wrote:
> >
> > Hi guys I have a minor conundrum I can't get my head around.
> >
> > I have the following scale (it repeats on 3/2)
> >
> > 0: 1/1 0.000 unison, perfect prime
> > 1: 9/8 203.910 major whole tone
> > 2: 4/3 498.045 perfect fourth
> > 3: 3/2 701.955 perfect fifth
> >
> > Symmetrical scales have been on my mind recently. I would like to add another pitch to the scale, and I want it to be exactly half way between 9/8 and 4/3. Can any seasoned pro name that pitch please?
> >
> > Tried to use mathematics but failed. Don't laugh at me. ;-)
> >
> > Sean Archibald
> >
>

🔗ixlramp <ixlramp@...>

🔗Valentine, Bob <bob.valentine@...>

The 'minor third' from 9/8 to 4/3 is 32/27.

Going halfway in frequency space is taking the square root (12tet tritone is square root of 2, the octave). If you want to divide by three, you need cube root, etc.

9/8 * sqrt( 32/27 ) = 1.224744871. This is 350.9775 cents which you can see is a 'neutral third (somewhere between major and minor). 11/9 is an excellent
JI neutral third. Where I am, you can go into any music store and buy keyboards with the ability to shift any note-name by a quarter tone (for Arab music)
and 350 is right there (and the neutral third and sixth are prominent in their music).

Others did the same math in the cents space. Now its something like 204 + (498 - 204 ) / 2, which gets you the same answer with addition and subtraction replacing
multiplication and division and the square root replaced by division. A cube root becomes division by 3, etc...

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🔗Mike Battaglia <battaglia01@...>

Sevish!

On Fri, May 6, 2011 at 3:37 PM, sevishmusic <sevish@...> wrote:
>
> Hi guys I have a minor conundrum I can't get my head around.
>
> I have the following scale (it repeats on 3/2)
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 9/8 203.910 major whole tone
> 2: 4/3 498.045 perfect fourth
> 3: 3/2 701.955 perfect fifth
>
> Symmetrical scales have been on my mind recently. I would like to add another pitch to the scale, and I want it to be exactly half way between 9/8 and 4/3. Can any seasoned pro name that pitch please?
>
> Tried to use mathematics but failed. Don't laugh at me. ;-)

350.978 cents is what you want. If you're lazy, 11/9 is 3 cents off
and should do the trick just fine.

-Mike

PS: any new releases coming up?

🔗sevishmusic <sevish@...>

Thanks to all who responded. I'm well on my way now!

🔗sevishmusic <sevish@...>

Hi Mike,

I'm working on something with Jacky Ligon and Tony Dubshot at the moment.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> PS: any new releases coming up?