14 tone adaptive just intonation system.

0 702 386 969 204
702 204 1088 471 906
386 1088 772 155 590
969 471 155 738 1173
204 906 590 1173 408

0, 155, 204, 386, 408, 471, 590, 702, 738, 772, 906, 969, 1088, 1173, (1200)

The goal was to have an adaptive just intonation system based on pure odd limit intervals with a low number of tones. It consist of the 3, 5, 7, 9, odd limits
which means there's five different keys to modulate to 0, 702, 386, 969, and 204. There are a few unique tones that are not shared with other keys which some of them are 772, 738 and 408.

Hi Andrew - I'm not sure I follow. When you say "adaptive" you mean
that the notes will change in pitch on the fly to match certain
ratios, right?

Can you write your 14-note scale out in order, and then say which
ratios are adaptively covered by which scale degrees?

Thanks,
Mike

On Mon, Jan 21, 2013 at 7:22 PM, andrewstevenmolina
<andrewstevenmolina@yahoo.com> wrote:
>
> 0 702 386 969 204
> 702 204 1088 471 906
> 386 1088 772 155 590
> 969 471 155 738 1173
> 204 906 590 1173 408
>
> 0, 155, 204, 386, 408, 471, 590, 702, 738, 772, 906, 969, 1088, 1173, (1200)
>
> The goal was to have an adaptive just intonation system based on pure odd limit intervals with a low number of tones. It consist of the 3, 5, 7, 9, odd limits
> which means there's five different keys to modulate to 0, 702, 386, 969, and 204. There are a few unique tones that are not shared with other keys which some of them are 772, 738 and 408.

--- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
>
> Hi Andrew - I'm not sure I follow. When you say "adaptive" you mean
> that the notes will change in pitch on the fly to match certain
> ratios, right?
>
> Can you write your 14-note scale out in order, and then say which
> ratios are adaptively covered by which scale degrees?
>
> Thanks,
> Mike
>
>
> On Mon, Jan 21, 2013 at 7:22 PM, andrewstevenmolina
> wrote:
> >
> > 0 702 386 969 204
> > 702 204 1088 471 906
> > 386 1088 772 155 590
> > 969 471 155 738 1173
> > 204 906 590 1173 408
> >
> > 0, 155, 204, 386, 408, 471, 590, 702, 738, 772, 906, 969, 1088, 1173, (1200)
> >
> > The goal was to have an adaptive just intonation system based on pure odd limit intervals with a low number of tones. It consist of the 3, 5, 7, 9, odd limits
> > which means there's five different keys to modulate to 0, 702, 386, 969, and 204. There are a few unique tones that are not shared with other keys which some of them are 772, 738 and 408.
>
For example If I play a just intonated major triad in the one of the five keys based off the harmonic series, lets say in the key '702' then my major triad would consist of 1088 the major third and 204 the perfect fifth then if modulate the key '386' then the major triad would consist of 1088 the perfect fifth and 772 the major third. You can play up to five notes in each of the five keys all based on the harmonic series.

--- In tuning-math@yahoogroups.com, "andrewstevenmolina" wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
> >
> > Hi Andrew - I'm not sure I follow. When you say "adaptive" you mean
> > that the notes will change in pitch on the fly to match certain
> > ratios, right?
> >
> > Can you write your 14-note scale out in order, and then say which
> > ratios are adaptively covered by which scale degrees?
> >
> > Thanks,
> > Mike
> >
> >
> > On Mon, Jan 21, 2013 at 7:22 PM, andrewstevenmolina
> > wrote:
> > >
> > > 0 702 386 969 204
> > > 702 204 1088 471 906
> > > 386 1088 772 155 590
> > > 969 471 155 738 1173
> > > 204 906 590 1173 408
> > >
> > > 0, 155, 204, 386, 408, 471, 590, 702, 738, 772, 906, 969, 1088, 1173, (1200)
> > >
> > > The goal was to have an adaptive just intonation system based on pure odd limit intervals with a low number of tones. It consist of the 3, 5, 7, 9, odd limits
> > > which means there's five different keys to modulate to 0, 702, 386, 969, and 204. There are a few unique tones that are not shared with other keys which some of them are 772, 738 and 408.
> >
> For example If I play a just intonated major triad in the one of the five keys based off the harmonic series, lets say in the key '702' then my major triad would consist of 1088 the major third and 204 the perfect fifth then if modulate the key '386' then the major triad would consist of 1088 the perfect fifth and 772 the major third. You can play up to five notes in each of the five keys all based on the harmonic series.
> Technically I guess the '702' major triad would look more like this 702, 1088, 1404.

On Mon, Jan 21, 2013 at 8:53 PM, andrewstevenmolina
<andrewstevenmolina@yahoo.com> wrote:
>
> For example If I play a just intonated major triad in the one of the five
> keys based off the harmonic series, lets say in the key '702' then my major
> triad would consist of 1088 the major third and 204 the perfect fifth then
> if modulate the key '386' then the major triad would consist of 1088 the
> perfect fifth and 772 the major third. You can play up to five notes in each
> of the five keys all based on the harmonic series.

OK, but in what sense is this adaptive? Isn't this just a 14-note JI scale?

-Mike

> OK, but in what sense is this adaptive? Isn't this just a 14-note JI scale?
>
> -Mike
>
I guess adaptive in sense that it allows for some modulation unlike regular just intonation.

--- In tuning-math@yahoogroups.com, "andrewstevenmolina" wrote:
>
> > OK, but in what sense is this adaptive? Isn't this just a 14-note JI scale?
> >
> > -Mike
> >
> I guess adaptive in sense that it allows for some modulation unlike regular just intonation.
>
Maybe I'm just confused about just intonaion but I'v always believed JI was based on a fixed pitch or to quote "There is a widespread belief that Just Intonation has to be based in a given key. This is not exactly the case, although there are many text books that talk about such things as "the Just Intonation Scale in the key of G", or something like that. This is misleading, as there is no single JI scale. JI can be used to generate an infinite variety of scales. Nor do they have to be in a specific key.' http://www.patmissin.com/tunings/tun1.html

--- In tuning-math@yahoogroups.com, "andrewstevenmolina" wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "andrewstevenmolina" wrote:
> >
> > > OK, but in what sense is this adaptive? Isn't this just a 14-note JI scale?
> > >
> > > -Mike
> > >
> > I guess adaptive in sense that it allows for some modulation unlike regular just intonation.
> >
> Maybe I'm just confused about just intonaion but I'v always believed JI was based on a fixed pitch or to quote "There is a widespread belief that Just Intonation has to be based in a given key. This is not exactly the case, although there are many text books that talk about such things as "the Just Intonation Scale in the key of G", or something like that. This is misleading, as there is no single JI scale. JI can be used to generate an infinite variety of scales. Nor do they have to be in a specific key.' http://www.patmissin.com/tunings/tun1.html
>
essentially we derive JI intervals from a single pitch but that doesn't mean a violinist has play in one key only in order to maintain pure harmony.

On Mon, Jan 21, 2013 at 9:49 PM, andrewstevenmolina
<andrewstevenmolina@yahoo.com> wrote:
>
> Maybe I'm just confused about just intonaion but I'v always believed JI
> was based on a fixed pitch or to quote "There is a widespread belief that
> Just Intonation has to be based in a given key. This is not exactly the
> case, although there are many text books that talk about such things as "the
> Just Intonation Scale in the key of G", or something like that. This is
> misleading, as there is no single JI scale. JI can be used to generate an
> infinite variety of scales. Nor do they have to be in a specific key.'
> http://www.patmissin.com/tunings/tun1.html

The term has sometimes historically been used differently, but these
days JI most commonly refers to a tuning system in which every
interval is, when viewed as a ratio of frequencies, a perfectly tuned
rational number. We identify rational numbers with intervals in this
tuning system, so that 5/4 refers to the ~386 cent interval most
typically perceived as a type of major third, and 3/2 refers to the
~702 cent interval typically called the perfect fifth, and 81/80
refers to the ~20 cent interval which is tempered out in meantone
temperament.

So for instance, 5-limit JI refers to the set of all intervals with a
maximum prime factor of 5, 7-limit JI refers to the set of all
intervals with a maximum prime factor of 7, etc. Note that both of
these tuning systems have an -infinite- number of pitches in them, and
aren't any sort of finite scale. You then use this as a framework to
construct finite scales, such as the 7-note 1/1 9/8 5/4 4/3 3/2 5/3
15/8 2/1 JI major scale, or the 14-note 7-limit scale you outline
above.

So in modern parlance, there is no single "Just Intonation scale." JI
is an infinite lattice of pitches all related by rational harmonic
relationships, from which you can create any number of scales.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
>
> On Mon, Jan 21, 2013 at 8:53 PM, andrewstevenmolina
> wrote:
> >
> > For example If I play a just intonated major triad in the one of the five
> > keys based off the harmonic series, lets say in the key '702' then my major
> > triad would consist of 1088 the major third and 204 the perfect fifth then
> > if modulate the key '386' then the major triad would consist of 1088 the
> > perfect fifth and 772 the major third. You can play up to five notes in each
> > of the five keys all based on the harmonic series.
>
> OK, but in what sense is this adaptive? "Isn't this just a 14-note JI scale?"
>
> -Mike
>
I suppose if this was based on a fixed pitch it would be based on these harmonic limits 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27. Which differs from the previous system that only goes up to the 9-limit.