JI tuning question

One of my goals is to try to create a JI tuning that doesn't resort to a chain of 5ths or something like that in one direction and also doesn't worry about octaves.

To try to do this I started from a just C major chord and did all of the math in cents

C 1/1 0c
E 5/4 386c
G 3/2 702c

from here I extended to other notes

E to B is 3/2 = 386 + 702 = 1088

and I can also say that Eb - G is
702 - 386 = 315c = Eb

Then Eb to Bb is 702c so 315 + 702 = 1017

I continued on in this manner and found a (surprisingly) 17 tone tuning though I have a problem with Ab being anomalously low.

My reality check says I'm re-inventing someone's wheel so are there any comments, resources for what I'm trying to do?

My incomplete excel spread sheet is here:

http://micro.soonlabel.com/various/JI-chain.xls

--- In tuning@yahoogroups.com, "christopherv" <chrisvaisvil@...> wrote:

> My incomplete excel spread sheet is here:
>
> http://micro.soonlabel.com/various/JI-chain.xls

Why not a Scala file instead, or as well?

because I can't calculate in a scala file.

I will be posting a text version of this in a second.

On Wed, May 11, 2011 at 11:18 AM, genewardsmith <genewardsmith@...
> wrote:

>
>
>
> --- In tuning@yahoogroups.com, "christopherv" <chrisvaisvil@...> wrote:
>
> > My incomplete excel spread sheet is here:
> >
> > http://micro.soonlabel.com/various/JI-chain.xls
>
> Why not a Scala file instead, or as well?
>
>
>

lets see if this works

column 1 is the meantone note name,
column 2 is the cents
column 3 is how the value was derived.

C cents 1/1
C#
Db 32.62406 F-5/4
D 103.29648 F#-5/4
D# 244.96858 Gb-5/4
Eb 315.641 G-5/4
E 386.3137125 5/4
E#
Fb
F 418.93777 A-5/4
F# 560.28258 A#-5/4
Gb 631.28229 Bb-5/4
G 701.9549984 3/2
G# 772.62742 E+5/4
Ab 734.57906 Db+3/2
A 805.25148 D+3/2
A# 946.92358 D#+3/2
Bb 1017.596 Eb+3/2
B 1088.26871 E+3/2
B# 1158.94113 G#+5/4
Cb
C 1262.56487 2/1

On Wed, May 11, 2011 at 11:18 AM, genewardsmith <genewardsmith@...
> wrote:

>
>
>
> --- In tuning@yahoogroups.com, "christopherv" <chrisvaisvil@...> wrote:
>
> > My incomplete excel spread sheet is here:
> >
> > http://micro.soonlabel.com/various/JI-chain.xls
>
> Why not a Scala file instead, or as well?
>
>
>

the 2nd octave C is from A# plus 3/2

On Wed, May 11, 2011 at 11:23 AM, Chris Vaisvil <chrisvaisvil@...>wrote:

> lets see if this works
>
> column 1 is the meantone note name,
> column 2 is the cents
> column 3 is how the value was derived.
>
> C cents 1/1
> C#
> Db 32.62406 F-5/4
> D 103.29648 F#-5/4
> D# 244.96858 Gb-5/4
> Eb 315.641 G-5/4
> E 386.3137125 5/4
> E#
> Fb
> F 418.93777 A-5/4
> F# 560.28258 A#-5/4
> Gb 631.28229 Bb-5/4
> G 701.9549984 3/2
> G# 772.62742 E+5/4
> Ab 734.57906 Db+3/2
> A 805.25148 D+3/2
> A# 946.92358 D#+3/2
> Bb 1017.596 Eb+3/2
> B 1088.26871 E+3/2
> B# 1158.94113 G#+5/4
> Cb
> C 1262.56487 2/1
>
>
> On Wed, May 11, 2011 at 11:18 AM, genewardsmith <
> genewardsmith@...> wrote:
>
>>
>>
>>
>> --- In tuning@yahoogroups.com, "christopherv" <chrisvaisvil@...> wrote:
>>
>> > My incomplete excel spread sheet is here:
>> >
>> > http://micro.soonlabel.com/various/JI-chain.xls
>>
>> Why not a Scala file instead, or as well?
>>
>>
>>
>
>

--- "christopherv" <chrisvaisvil@...> wrote:

> One of my goals is to try to create a JI tuning that doesn't
> resort to a chain of 5ths or something like that in one
> direction [snip]
> from here I extended to other notes
> E to B is 3/2 = 386 + 702 = 1088

If you connect notes by JI intervals (as you're doing here)
you'll eventually get the complete JI lattice, which is the
starting point of the "regular mapping paradigm" we keep
talking about. Of course you can never have the whole
lattice, but you can have sections of it, like this (fixed
width font)

5/3-----------5/4----------15/8
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
4/3-----------1/1-----------3/2-----------9/8

Your scale will have more consonances if it corresponds to
a lattice section that's roundish. The above is oblong but
still pretty round compared to what it could be.

If you try to make the scale even, not leaving any huge gaps
between notes, then you'll tend to wind up with familiar
numbers of notes/octave, like 12, 17, 19, 22 etc.

> My incomplete excel spread sheet is here:
> http://micro.soonlabel.com/various/JI-chain.xls

If your Ab were C -5/4, it'd be 814 cents above C.

It seems your A may also be too low. You might try
C -6/5, which would be 884 cents.

-Carl

I think what I am doing is as you say - generating the lattice. I was most
surprised 17 notes came popping out.

What I want to do is to generate a scala file for all 88 keys of my piano
that doesn't worry about octaves being pure or commas.
Sort of a "just let it all hang out" and see if I can make music with it. So
eventually this will start from.. A0 (I think it is ).

What I don't want to do is to start involving other intervals in the
*generation* of the tuning.

I don't want to temper anything. It may sound like crap. Its one of those
things I need to do for myself.

I'll look at the A and A flat - thanks for pointing that out.

Chris

On Wed, May 11, 2011 at 12:47 PM, Carl Lumma <carl@...> wrote:

>
>
> --- "christopherv" <chrisvaisvil@...> wrote:
>
> > One of my goals is to try to create a JI tuning that doesn't
> > resort to a chain of 5ths or something like that in one
> > direction [snip]
>
> > from here I extended to other notes
> > E to B is 3/2 = 386 + 702 = 1088
>
> If you connect notes by JI intervals (as you're doing here)
> you'll eventually get the complete JI lattice, which is the
> starting point of the "regular mapping paradigm" we keep
> talking about. Of course you can never have the whole
> lattice, but you can have sections of it, like this (fixed
> width font)
>
> 5/3-----------5/4----------15/8
> / \ / \ / \
> / \ / \ / \
> / \ / \ / \
> / \ / \ / \
> / \ / \ / \
> / \ / \ / \
> 4/3-----------1/1-----------3/2-----------9/8
>
> Your scale will have more consonances if it corresponds to
> a lattice section that's roundish. The above is oblong but
> still pretty round compared to what it could be.
>
> If you try to make the scale even, not leaving any huge gaps
> between notes, then you'll tend to wind up with familiar
> numbers of notes/octave, like 12, 17, 19, 22 etc.
>
>
> > My incomplete excel spread sheet is here:
> > http://micro.soonlabel.com/various/JI-chain.xls
>
> If your Ab were C -5/4, it'd be 814 cents above C.
>
> It seems your A may also be too low. You might try
> C -6/5, which would be 884 cents.
>
> -Carl
>
>
>

Hi Chris,
I can't make heads or tails of your spreadsheet (maybe because I use OpenOffice and it's reading the formatting wrong or something) so I don't know what the scale is that you came up with. You say you're not worrying about octaves but you also say you came up with a 17 note scale...does that scale have a repeat interval other than the octave? Or does it not repeat into other registers?

As you created that web of intervals that emanate from C, did you reduce intervals by octaves? For instance If you went C-E-G and then G-B-D, would that D be 204 cents or 1404 cents in your scale? Let me put it another way in case that doesn't make sense: if you are generating your scale by going up and down by 5/4's and 3/2's and you don't reduce to an octave, C-Eb-E-G-Bb-B-D-F-F#-A-C' (for example) will be:
0
315
386
702
1017
1088
1404
1719
1790
2106
2421
(approximately)

If you do reduce to the octave, it will look like:
0
21
204
315
386
519
590
702
906
1017
1088
1200

If you are doing the latter (which I suspect you are) you will eventually end up with the 5-limit lattice, which can also be looked at as an infinite number of chains of 3/2's offset from each other by 81/80's. If you're stopping at 17 notes you probably already have some notes that are only about 21 cents apart (that's 81/80 doing its thing). I wonder why you decided to stop there?

If you keep going with this process past 17 notes, this leads to some wacky stuff because the Syntonic comma differs from the Pythagorean comma by the Schisma, which is just a little over a cent. For instance if you go down in 3/2's from C to go C-F-Bb-Eb-Ab-Db-Gb-Cb-Fb, that Fb will be a Schisma away from your E that you got by going up a 5/4 from C. In truth, you could easily get a scale that sounds indistinguishable from 5-limit JI with a single chain of 3/2's. Just FYI.

-Igs

--- In tuning@yahoogroups.com, "christopherv" <chrisvaisvil@...> wrote:
>
> One of my goals is to try to create a JI tuning that doesn't resort to a chain of 5ths or
> something like that in one direction and also doesn't worry about octaves.
>
> To try to do this I started from a just C major chord and did all of the math in cents
>
> C 1/1 0c
> E 5/4 386c
> G 3/2 702c
>
> from here I extended to other notes
>
> E to B is 3/2 = 386 + 702 = 1088
>
> and I can also say that Eb - G is
> 702 - 386 = 315c = Eb
>
> Then Eb to Bb is 702c so 315 + 702 = 1017
>
> I continued on in this manner and found a (surprisingly) 17 tone tuning though I have a problem with Ab being anomalously low.
>
> My reality check says I'm re-inventing someone's wheel so are there any comments, resources for what I'm trying to do?
>
> My incomplete excel spread sheet is here:
>
> http://micro.soonlabel.com/various/JI-chain.xls
>

--- Chris Vaisvil <chrisvaisvil@...> wrote:

> What I want to do is to generate a scala file for all
> 88 keys of my piano that doesn't worry about octaves
> being pure or commas.

If you

1. choose arbitrary generating intervals and
2. force the scale to be ascending and
3. force it to have 88 tones over roughly the same total
pitch compass as a piano

then it seems to me you will wind up with a scale on the
lattice of your chosen generators (obviously) where the
chromatic steps are intervals of roughly 100 cents found
near to the origin of that lattice.

The more distinct generating intervals you chose, the
greater the number of different chromatic step sizes you
will have. But "distinct" here means some sort of unique
factors of your chosen intervals. For example, if I
chose 316, 386 and 702 cents, that will be the same as
just choosing 316 and 386, since 702 is their sum and will
come out on the lattice anyway. This is easier to deal
with by using ratios and multiplication rather than cents
and addition, since then you can prime-factor your chosen
generators.

Yeah, so an easy way to do this is to find all intervals
between 50 & 150 cents within some radius of the origin of
the lattice, and then just stack those to get the scale.

I know, I just took all the fun out of your project. Boo!

-Carl

It will take a bit to answer this

I arbitrarily set up the spreadsheet with a 21 note meantone - so every 12
equal step has a sharp and also has a distinct flat. This was just a
framework - a place to start.

There is a row. for me row 6, that describes how I generated the values in
row 7.
Row 7 is in cents from C - but this is an arbitrary starting point. It could
be cents from A for instance.
The rows below 7 contain the cent differences between pitches. Yellow high
light indicates Just Major chord members.

When I say 17 notes I am stating that I cannot generate any more notes
within the compass of the pseudo - octave I am working with. (I think)
The cents value for the pseudo - octave is 1262 and change. All it reall
indicates is that I am back to "C" in my arbitrary 21 note meantone naming.

I am purposely NOT reducing by octaves - that would negate the purpose of
the exercise.

In fact

"If you keep going with this process past 17 notes, this leads to some wacky
stuff because the Syntonic comma differs from the Pythagorean comma by the
Schisma, which is just a little over a cent."

I am searching for the wacky stuff. :-)

Did the guide to the meaning of the rows help your interpretation of the
spreadsheet?

Thanks for your time and interest!

Chris

On Wed, May 11, 2011 at 3:19 PM, cityoftheasleep <igliashon@...>wrote:

>
>
> Hi Chris,
> I can't make heads or tails of your spreadsheet (maybe because I use
> OpenOffice and it's reading the formatting wrong or something) so I don't
> know what the scale is that you came up with. You say you're not worrying
> about octaves but you also say you came up with a 17 note scale...does that
> scale have a repeat interval other than the octave? Or does it not repeat
> into other registers?
>
>

Carl,

I think you are assuming too many constraints. For one, if you look at row
6, assuming you can now interpet it after I explained it, you'll see I
applied 5/4 and 3/2 down as well as up.

The assumption I'm making for instance is that C to G is 3/2 and then G to B
is 5/4 then G *down* to Eb is 5/4 the Eb *up* to Bb is Eb + 3/2
You can do that math more easily and directly if you work in cents.

It does seem though that no one else has done what I'm doing, more or less,
so that is heartening. If it will amount to anything is another story. I
will at least have learned something.

let me look into the A and A flat issue. I've been busy the whole time since
you mentioned that.

Chris

On Wed, May 11, 2011 at 4:27 PM, Carl Lumma <carl@...> wrote:

>
>
>
> If you
>
> 1. choose arbitrary generating intervals and
> 2. force the scale to be ascending and
> 3. force it to have 88 tones over roughly the same total
> pitch compass as a piano
>
> then it seems to me you will wind up with a scale on the
> lattice of your chosen generators (obviously) where the
> chromatic steps are intervals of roughly 100 cents found
> near to the origin of that lattice.
>
>
> I know, I just took all the fun out of your project. Boo!
>
> -Carl
>
>
>
>

--- Chris Vaisvil <chrisvaisvil@...> wrote:

> Carl,
> I think you are assuming too many constraints.

I thought you said you wanted 88 tones and that you want to
pick a list of generating intervals.
The only other assumptions are that the scale is ascending,
and that it covers roughly the same pitch compass as a piano.

Addition and subtraction are both covered by the lattice
abstraction, as they are simply inverses. Working in cents
introduces rounding errors and sum-free sets are a bit more
obscure than relatively-prime ones
http://en.wikipedia.org/wiki/Sum-free_set
but sure, you can do it.

> let me look into the A and A flat issue. I've been busy the
> whole time since you mentioned that.

You mentioned that you had a problem with Ab being too low!
I was just responding to that.

-Carl

Chris, I made a trace of the continuous (non-octave-equivalent) zig-zaggy
path you described through a 5-limit lattice of note names (that match those
you use): http://img96.imageshack.us/img96/4729/lattpath.jpg

Well, I tried re-doing the excercise but only a set of 12 pitches. So I obtained 9 steps of 70 cents starting from A and rising chromatically. BUT 3 pitches, C, Eb, G were 174 (or so) higher than the previous chromatic pitch. I did look up just semitone on wikipedia and found some information about asymetric 5-limit semitones.

So it seems I've repeated that though right now I don't understand why it happens. And I find it quite interesting that the natural semitone seems to be about 70 cents which within an octave gives you about 17 equal.

I got a graphic of what I did.

-----Original Message-----
From: "Carl Lumma" <carl@lumma.org>
Sender: tuning@yahoogroups.com
Date: Wed, 11 May 2011 21:52:51
To: <tuning@yahoogroups.com>
Reply-To: tuning@yahoogroups.com
Subject: [tuning] Re: JI tuning question

--- Chris Vaisvil <chrisvaisvil@...> wrote:

> Carl,
> I think you are assuming too many constraints.

I thought you said you wanted 88 tones and that you want to
pick a list of generating intervals.
The only other assumptions are that the scale is ascending,
and that it covers roughly the same pitch compass as a piano.

Addition and subtraction are both covered by the lattice
abstraction, as they are simply inverses. Working in cents
introduces rounding errors and sum-free sets are a bit more
obscure than relatively-prime ones
http://en.wikipedia.org/wiki/Sum-free_set
but sure, you can do it.

> let me look into the A and A flat issue. I've been busy the
> whole time since you mentioned that.

You mentioned that you had a problem with Ab being too low!
I was just responding to that.

-Carl

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:

> And I find it quite interesting that the natural semitone seems
> to be about 70 cents which within an octave gives you about
> 17 equal.

Though you're trying to avoid temperament and known theory,
you're retracing the steps that lead to it. The semitone
you're encountering is 25/24, which comes out tops in 5-limit
lattice searches for small intervals.

-Carl

Yeah, I can't avoid it.

Here is my motivation.

I want to set up my 88 key controller (or the AXiS) in harmonic series
tuning - but with more than one root. I was looking for a way to make 12
notes (12 only because of the keyboard design) Just to each other with out
resorting to octave reduction or an arbitrary interval (like 100 cents).

I need to think about this more - unless you (or someone else) has an idea.

Chris

On Wed, May 11, 2011 at 7:33 PM, Carl Lumma <carl@...> wrote:

>
>
> --- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> > And I find it quite interesting that the natural semitone seems
> > to be about 70 cents which within an octave gives you about
> > 17 equal.
>
> Though you're trying to avoid temperament and known theory,
> you're retracing the steps that lead to it. The semitone
> you're encountering is 25/24, which comes out tops in 5-limit
> lattice searches for small intervals.
>
> -Carl
>
>
>

On Wed, May 11, 2011 at 8:31 PM, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> Yeah, I can't avoid it.
>
> Here is my motivation.
>
> I want to set up my 88 key controller (or the AXiS) in harmonic series tuning - but with more than one root. I was looking for a way to make 12 notes (12 only because of the keyboard design) Just to each other with out resorting to octave reduction or an arbitrary interval (like 100 cents).
>
> I need to think about this more - unless you (or someone else) has an idea.

So your base intervals are 3/2 and 5/4, and you want to ignore 2/1,
and just generate all other intervals by compounding those two like
lego blocks and seeing what comes out? Are you trying to work 7/4 or
11/8 in there or anything like that?

-Mike

On Wed, May 11, 2011 at 7:24 PM, <chrisvaisvil@...> wrote:
>
> Well, I tried re-doing the excercise but only a set of 12 pitches. So I obtained 9 steps of 70 cents starting from A and rising chromatically. BUT 3 pitches, C, Eb, G were 174 (or so) higher than the previous chromatic pitch. I did look up just semitone on wikipedia and found some information about asymetric 5-limit semitones.
>
> So it seems I've repeated that though right now I don't understand why it happens. And I find it quite interesting that the natural semitone seems to be about 70 cents which within an octave gives you about 17 equal.

I see you've worked the theory out after all. However, if you do this,
you'll find that the major and minor thirds are a bit far apart from
just another. Dividing the 17-equal step in half gives you 34-equal,
which has nearly-just thirds, fourths that are only 4 steps bright and
pleasantly so, still contains the 70 cent interval you want, and has
an awesome color in general. When people talk about 53-equal as being
this masterful nearly-just tuning, I wish they'd skip it and look at
34-equal instead. I know that you don't want to temper, so maybe just
file this away in a footnote for now.

As another note, in 17-equal this 70 cent interval is mapped to the
diatonic semitone (e.g. the difference between E and F), not the
chromatic semitone (e.g. the difference between Eb and E). If you want
it to be the chromatic semitone, that gives you about 19 equal.

-Mike

Thank you for this. How did you make the graphic?

Chris

On Wed, May 11, 2011 at 6:38 PM, Daniel Nielsen <nielsed@...> wrote:

>
>
> Chris, I made a trace of the continuous (non-octave-equivalent) zig-zaggy
> path you described through a 5-limit lattice of note names (that match those
> you use): http://img96.imageshack.us/img96/4729/lattpath.jpg
>
>
>

Well, Just 7th (or 9th) chords sound like a great option. I'm just not sure
then what would be the relationship between them.

Since these chords would be stacking just major 3rds, lets say for argument
we'll stick with 7ths, then I have 3 of them in 12 (unless I ignore keys,
which is an option.)

Hmmm ignoring keys - perhaps not a bad idea - but then I'm crunch so many
octaves into 88 or 98 I may go beyond hearign range.

Any suggestions Mike or Carl?

Chris

On Wed, May 11, 2011 at 9:28 PM, Mike Battaglia <battaglia01@...>wrote:

>
>
>
>
> So your base intervals are 3/2 and 5/4, and you want to ignore 2/1,
> and just generate all other intervals by compounding those two like
> lego blocks and seeing what comes out? Are you trying to work 7/4 or
> 11/8 in there or anything like that?
>
> -Mike
>
>
>

Yep, I guess I had to prove it to myself. This is what I came up with

http://micro.soonlabel.com/various/in1oct-A.jpg

I'll file away 34 - it does sound interesting - lets see how my 22 edo stick
guitar goes.

Now, I'm a bit puzzled on what you said about diatonic and chromatic
semitones.
First off 1200 / 17 = 70 cents and 1200 / 19 = 63 cents

It sounds as if you are equating them yet coming out with different edos.

On Wed, May 11, 2011 at 9:27 PM, Mike Battaglia <battaglia01@...>wrote:

>
>
>
>
> I see you've worked the theory out after all. However, if you do this,
> you'll find that the major and minor thirds are a bit far apart from
> just another. Dividing the 17-equal step in half gives you 34-equal,
> which has nearly-just thirds, fourths that are only 4 steps bright and
> pleasantly so, still contains the 70 cent interval you want, and has
> an awesome color in general. When people talk about 53-equal as being
> this masterful nearly-just tuning, I wish they'd skip it and look at
> 34-equal instead. I know that you don't want to temper, so maybe just
> file this away in a footnote for now.
>
> As another note, in 17-equal this 70 cent interval is mapped to the
> diatonic semitone (e.g. the difference between E and F), not the
> chromatic semitone (e.g. the difference between Eb and E). If you want
> it to be the chromatic semitone, that gives you about 19 equal.
>
> -Mike
>
>
>

--- Chris Vaisvil <chrisvaisvil@...> wrote:

> I want to set up my 88 key controller (or the AXiS) in harmonic
> series tuning - but with more than one root. I was looking for
> a way to make 12 notes (12 only because of the keyboard design)
> Just to each other with out resorting to octave reduction or an
> arbitrary interval (like 100 cents).

Well, you can take harmonic series segments and overlay them.
That's been done. For instance

!
Two harmonic series segments (cap 16) rooted a 3:2 apart.
12
!
13/12
9/8
7/6
5/4
4/3
11/8
3/2
13/8
5/3
7/4
15/8
2/1
!
! Carl Lumma, after David Canright.

!
Denny Genovese, harmonics 8-16 and subharmonics 6-12.
12
!
12/11
9/8
6/5
5/4
4/3
11/8
3/2
13/8
12/7
7/4
15/8
2/1
!

!
16 harmonics on 1/1 and 16 subharmonics on 15/8.
12
!
15/14
9/8
15/13
5/4
15/11
11/8
3/2
13/8
5/3
7/4
15/8
2/1
!
! George Secor, TL 60761 (2005.09.27)

Magnus Johnson even made a notation for this kind of thing,
and suggested stuff like

!
[1 3 5] x [1 3 5] x u[1 3 5] cross set, Magnus Jonsson 2005.
12
!
25/24
9/8
6/5
5/4
4/3
3/2
25/16
8/5
5/3
9/5
15/8
2/1
!
! Also John Chalmers' "Major Wing".

Lots more if you want something other than 12 tones/octave
or if you want to repeat at something other than an octave.

Denny used to tune an entire acoustic piano to an overlay
of harmonics and subharmonics with no repetition -- just
straight up and down the keyboard until he ran out of keys.
He never told me exactly how, but rumor has it that he
did it to John Starrett's piano once.

-Carl

>
> Thank you for this. How did you make the graphic?
>
> Chris
>

Just a screen capture from Graham's site (http://x31eq.com/lattice.htm) and
MS Paint, unfortunately nothing fancier.

--- Chris Vaisvil <chrisvaisvil@...> wrote:

> Well, Just 7th (or 9th) chords sound like a great option.
> I'm just not sure then what would be the relationship
> between them.

To get 7-limit dominant 7th chords, make your interval
set 386, 316, and 267. For the 9-limit, include 435 also
(these are 5/4, 6/5, 7/6, and 9/7 respectively).
You should already be getting 5-limit major 7ths as
386 + 316 + 386.

> Yep, I guess I had to prove it to myself. This is what
> I came up with
> http://micro.soonlabel.com/various/in1oct-A.jpg

As I say here
/tuning/topicId_98996.html#99012
you've ended up with a scale containing a few unique
chromatic steps -- in this case 25/24 and 3456/3125.

> Now, I'm a bit puzzled on what you said about diatonic and
> chromatic semitones.
> First off 1200 / 17 = 70 cents and 1200 / 19 = 63 cents

I'm not sure why you got 17 and 12. But if you stack
JI intervals until you get a scale with no big gaps, you
will tend to wind up with 12, 17 etc tones. That's
what periodicity blocks are all about (MOS when you have
only a single chain).

-Carl

I appreciate these and I will try them. I think I'm after what Denny did to
those pianos.

Michael Harrison has this section
http://www.michaelharrison.com/web/pure_tunings.htm but it has not been
populated.

This description (and the CD Revelations I bought)
"* By combining carefully selected pitch relationships with various
performance techniques, this tuning creates undulating waves of shimmering
and pulsating sounds, with what sound like  phase shifting and  note
bending effects and other acoustical phenomena. Sometimes the overtones are
so audible that it sounds as if many different instruments are resonating
from the piano. The tuning has so many beautiful and exotic sounds latent
within it, that for the first few months, every time I played it, I
discovered new harmonic regions and felt like an explorer in unknown and
distant realms. "*

makes me want to try it. He further describes:
http://www.michaelharrison.com/web/pure_intonation.htm

* Revelation takes this concept to another level by incorporating three sets
of adjacently tuned  celestial commas (a 64:63 ratio, or approximately an
 eighth tone) into the harmonic fabric of every octave of the tuning. When
these commas sound simultaneously or in rapid succession, they produce
never-before-heard combinations of modes, harmonies, and acoustical
phenomena. The comma is thus freed from its restricted status as an
"out-of-tune" dissonance that, until recently, was disguised, avoided or
obliterated by tempered tunings, compositional styles, performance practices
and instrument designs. *

oh wait - I've not read this page before

*The tuning's unique qualities exist in the relationships between the black
and white keys, which reveal a wide variety of exotic and colorful
intervals. The  revelation tuning has a practical symmetry whereby all of
the white keys form a series of Pythagorean fifths (a 3:2 ratio), and all of
the black keys form another series of Pythagorean fifths, with each black
key tuned to the seventh overtone (a 7:4 ratio) above each corresponding
white key. The 7:4 ratio is the naturally occurring minor seventh that
exists in the overtone series (approximately 31% of a semi-tone flat from
the equal tempered minor seventh). As a result, three black keys are tuned
to the  celestial comma (a 64:63 ratio) below three adjacent white keys.
This creates ample opportunity to use what I refer to as the  pulsating
comma effect in a variety of different harmonic contexts, where the
adjacent commas sound simultaneously. This symmetrical layout of the white
and black keys allows for a very intuitive approach to playing the piano.
For example, the white keys are purely diatonic; by adding any black keys
into the mix you will get either  septimal minor intervals or the 
pulsating comma effect. *

*The twelve pitches are tuned to the following twelve harmonics or 
overtones of the fundamental note F. For white keys: F=1, C=3, G=9, D=27,
A=81, E=243, and B=729 (all of which are multiples of the prime number 3);
and for black keys: E-flat=7, B-flat=21, F#=63, C#=189, and G#=567 (all of
which are multiples of the primes 3 and 7).

That would seem to be it. I wonder if pianoteq is up to the task...
*
On Wed, May 11, 2011 at 10:28 PM, Carl Lumma <carl@...> wrote:

>
>
>
> Denny used to tune an entire acoustic piano to an overlay
> of harmonics and subharmonics with no repetition -- just
> straight up and down the keyboard until he ran out of keys.
> He never told me exactly how, but rumor has it that he
> did it to John Starrett's piano once.
>
> -Carl
>

On Wed, May 11, 2011 at 10:13 PM, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> Well, Just 7th (or 9th) chords sound like a great option. I'm just not sure then what would be the relationship between them.
>
> Since these chords would be stacking just major 3rds, lets say for argument we'll stick with 7ths, then I have 3 of them in 12  (unless I ignore keys, which is an option.)
>
> Hmmm ignoring keys - perhaps not a bad idea - but then I'm crunch so many octaves into 88  or 98 I may go beyond hearign range.
>
> Any suggestions Mike or Carl?

Alright, let's start with the 5-limit. I think you'll get what you
want with an AXiS, which will enable you to easily retune things to
whatever you want.

If we're sticking to the 5-limit, then it seems like what you want is
to have 3/2 and 5/4 as your base intervals. So that's exactly how I'd
set the AXiS up - make one of the directions be 3/2, and another
direction be 5/4. This is basically what the AXiS's "harmonic table"
layout already is - one direction is a major third, and another is a
perfect fifth. All of the 5-limit intervals in the system that you
mentioned above will be reachable by this layout, and the whole thing
will be really intuitive because you'll have a visual diagram that
tells you exactly what the logic of the system is.

OK, but now you want the 7-limit as well. Well, you actually said you
want the 9-limit, but since you have 3/2's already, then 9/4 is
already covered. How do we throw the 7-limit in there? Well, the
solution, of course, is for you to buy an infinite number of AXiS
controllers and stack them vertically on top of one another, so that
this new vertical direction represents 7/4.

If that seems unsatisfactory, then there is one small compromise you
can make which will probably be good enough for what you want, and
that's to microtemper. Microtemperaments are temperaments that are so
close to just that you can't tell the difference. If you can find a
way to work 7/4 in as a combination of 3/2 and 5/4, then you'll be
able to map the entire thing out on a 2d harmonic lattice. Since it
seems like what you want is to learn how JI "logic" works, but have to
avoid buying infinite AXiSes, this will probably be good enough.

One good option is to take the 5/4 axis and divide it by two, so each
step in that direction moves you up by half a major third. If you just
ignore every other step, you end up with 5-limit JI. But, by doing
this, then two and a half major thirds puts you to 7/4, so you can get
the 7-limit that way. If you use this tuning, then I'd set the major
thirds equal to 387 cents, meaning that they're less than one cent
sharp of just. Then the 7/4's will be 967.5 cents, which is 1.3 cents
flat of just. You'd never hear the difference. So the end result is
that you can pretty much think of it as being JI, but doing something
like this lets you multiplex the whole thing onto a two-dimensional
keyboard like the AXiS.

There are other options than this as well, some of which are even
closer to just, but the closer you get, the harder it'll be to fit the
whole thing manageably onto an AXiS.

-Mike

On 12 May 2011 05:02, Mike Battaglia <battaglia01@...> wrote:

> One good option is to take the 5/4 axis and divide it by two, so each
> step in that direction moves you up by half a major third. If you just
> ignore every other step, you end up with 5-limit JI. But, by doing
> this, then two and a half major thirds puts you to 7/4, so you can get
> the 7-limit that way. If you use this tuning, then I'd set the major
> thirds equal to 387 cents, meaning that they're less than one cent
> sharp of just. Then the 7/4's will be 967.5 cents, which is 1.3 cents
> flat of just. You'd never hear the difference. So the end result is
> that you can pretty much think of it as being JI, but doing something
> like this lets you multiplex the whole thing onto a two-dimensional
> keyboard like the AXiS.

You seem to be talking about Parahemwuer here:

http://x31eq.com/cgi-bin/rt.cgi?ets=99_31_19&limit=7

It's interesting, but not the obvious planar temperament to be
applying. Dividing the third messes up 5-limit fingerings, for
example.

> There are other options than this as well, some of which are even
> closer to just, but the closer you get, the harder it'll be to fit the
> whole thing manageably onto an AXiS.

And the obvious option is Marvel, where two 16:15 semitones approximate an 8:7.

http://x31eq.com/cgi-bin/rt.cgi?ets=22_31_19&limit=7

Simpler then Parahemwuer and I think it qualifies as a microtemperament.

For higher accuracy (and about the same complexity as Parahemwuer) you
can divide the fifth into neutral thirds, and give them a 7-limit
association. This amounts to tempering out 2401:2400, and tends to
work with a square lattice, rather than the AXiS keys. Still, here it
is:

http://x31eq.com/cgi-bin/rt.cgi?ets=72_31_99&limit=7

Here's a long list of 7-limit rank 3 temperaments that might work:

http://x31eq.com/cgi-bin/more.cgi?r=3&limit=7&error=2

In all cases you need to work out how to fit the rank 3 lattice onto a
rank 2 keyboard. Either you optimize for one inversion of chords, or
break the regularity somewhere.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> You seem to be talking about Parahemwuer here:
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=99_31_19&limit=7

Or just hemimean:

http://xenharmonic.wikispaces.com/Hemimean+family

Either way, tempering out 3136/3125. I should have a comma pump example up soon.

> For higher accuracy (and about the same complexity as Parahemwuer) you
> can divide the fifth into neutral thirds, and give them a 7-limit
> association.

And no question that this is a microtemperament.

On Thu, May 12, 2011 at 2:53 AM, Graham Breed <gbreed@...> wrote:
>
> On 12 May 2011 05:02, Mike Battaglia <battaglia01@...> wrote:
>
> > One good option is to take the 5/4 axis and divide it by two, so each
> > step in that direction moves you up by half a major third. If you just
> > ignore every other step, you end up with 5-limit JI. But, by doing
> > this, then two and a half major thirds puts you to 7/4, so you can get
> > the 7-limit that way. If you use this tuning, then I'd set the major
> > thirds equal to 387 cents, meaning that they're less than one cent
> > sharp of just. Then the 7/4's will be 967.5 cents, which is 1.3 cents
> > flat of just. You'd never hear the difference. So the end result is
> > that you can pretty much think of it as being JI, but doing something
> > like this lets you multiplex the whole thing onto a two-dimensional
> > keyboard like the AXiS.
>
> You seem to be talking about Parahemwuer here:
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=99_31_19&limit=7

He doesn't want 2/1, but he does want 3/2 and 5/4, so I'm not sure
what I'm talking about really...

> > There are other options than this as well, some of which are even
> > closer to just, but the closer you get, the harder it'll be to fit the
> > whole thing manageably onto an AXiS.
>
> And the obvious option is Marvel, where two 16:15 semitones approximate an 8:7.
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=22_31_19&limit=7

Yeah, but how will this work without 2/1?

> For higher accuracy (and about the same complexity as Parahemwuer) you
> can divide the fifth into neutral thirds, and give them a 7-limit
> association. This amounts to tempering out 2401:2400, and tends to
> work with a square lattice, rather than the AXiS keys. Still, here it
> is:
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=72_31_99&limit=7
>
> Here's a long list of 7-limit rank 3 temperaments that might work:
>
> http://x31eq.com/cgi-bin/more.cgi?r=3&limit=7&error=2
>
> In all cases you need to work out how to fit the rank 3 lattice onto a
> rank 2 keyboard. Either you optimize for one inversion of chords, or
> break the regularity somewhere.

Since he doesn't want 2/1, this won't be a problem, but it will be a
problem if the chosen microtemperament lands you at 7/2 instead of
7/4, which is what he wants. I just typed in stuff here

http://x31eq.com/cgi-bin/pregular.cgi?limit=3%2F2.5%2F4.7%2F4&error=5.0

and got this as the winner

http://x31eq.com/cgi-bin/rt.cgi?ets=19p_31p&limit=3%2F2_5%2F4_7%2F4

Second on the list seems to be keemun, which I thought might be a bit
too out there for what he wants.

-Mike

--- In tuning@yahoogroups.com, "christopherv" <chrisvaisvil@...> wrote:
>
> One of my goals is to try to create a JI tuning that doesn't resort to a chain of 5ths or something like that in one direction and also doesn't worry about octaves.
>
> To try to do this I started from a just C major chord and did all of the math in cents
>
> C 1/1 0c
> E 5/4 386c
> G 3/2 702c
>
> from here I extended to other notes
>
> E to B is 3/2 = 386 + 702 = 1088
>
> and I can also say that Eb - G is
> 702 - 386 = 315c = Eb
>
> Then Eb to Bb is 702c so 315 + 702 = 1017
>
> I continued on in this manner and found a (surprisingly) 17 tone tuning though I have a problem with Ab being anomalously low.
>
> My reality check says I'm re-inventing someone's wheel so are there any comments, resources for what I'm trying to do?
>
> My incomplete excel spread sheet is here:
>
> http://micro.soonlabel.com/various/JI-chain.xls
>

Mark:
Here are some scales I've created based on my understanding of what you're trying to do. I apologise if I am repeating the suggestions of others who have replied to this thread. It was taking a long time to read through all the posts.

With 1296/625 as an octave (about 1262.6 cents):

1, 25/24, 625/576, 6/5, 5/4, 125/96, 36/25, 3/2, 25/16, 216/125, 9/5, 15/8, 1296/625

With 125/64 as an octave (about 1158.9 cents):

1, 25/24, 144/125, 6/5, 5/4, 125/96, 36/25, 3/2, 25/64, 625/384, 9/5, 15/8, 125/64

With 9375/4608 as an octave (about 1229.6 cents):

1, 25/24, 625/576, 6/5, 5/4, 125/96, 3125/2304, 3/2, 25/16, 125/64, 15/8, 375/192, 9375/4608

I have tried to maximise the instances of the intervals 3/2, 5/4 and 6/5 (and maximise instances of pure major (1, 5/4, 3/2) and pure minor (1, 6/5, 3/2) triads), while limiting myself to 12 notes in the approximate octave and making sure that all the notes are linked together using only 3/2 and 5/4. I have tried to make the transition from one approximate octave to the next smooth, with a lot of notes in one approximate octave linked to notes in the next approximate octave by 3/2 and 5/4.

Unfortunately I have not been able to solve the problem with the minor sixth being too flat.

I may produce diagrams showing these tunings soon.