Pisot-Vijayaraghavan numbers

http://en.wikipedia.org/wiki/Pisot-Vijayaraghavan_number

While following links from the Padovan number page, I came across these whose magnitudes
which are between the Padovan and Golden Ratio numbers. I haven't noticed them before on
this list or Tuning Math. They might define interesting 4ths, tritones, fifths and sixths for
generating scales or temperaments.

--John

--- In MakeMicroMusic@yahoogroups.com, "John H.
Chalmers" <JHCHALMERS@...> wrote:
>
> http://en.wikipedia.org/wiki/Pisot-Vijayaraghavan_number
>
> While following links from the Padovan number page, I came across
these whose magnitudes
> which are between the Padovan and Golden Ratio numbers. I haven't
noticed them before on
> this list or Tuning Math. They might define interesting 4ths,
tritones, fifths and sixths for
> generating scales or temperaments.
>
> --John

0: 1/1 0.000 unison, perfect prime
1: 750797/662359 216.971
2: 269.851 cents 269.851
3: 662359/500000 486.822
4: 750797/500000 703.794
5: 920.765 cents 920.765
6: 973.645 cents 973.645
7: 1190.616 cents 1190.616

The "plastic number" was already familiar to me from my search for
"shadows" and points of useful (and, frankly, pretty) "otherness" in
general.

The above tuning (see #1 and #5 on the chart at Wikipedia) has some
interesting properties, to my ears- the detuned octave seems to ride
in one of those little spots that are curiously less dissonant than
they "should" be, and triadic tall chords, ie playing every other
note in the scale, are just stacks of 7/6s and 9/7s about 2-3 cents
sharp of Just. There is a also a distinctive "look Ma, no wrong
notes!" sound to it.

This tuning is part of a personal study of "comma pump? who cares!".
For we may consider the result of a simple repetitive Just or
otherwise coherent intervals a thing in it's own right and not, say,
a tonic now "off" by a comma. A consideration justified by the ear,
in my opinion.

-Cameron Bobro

Have you made one of the Tritriadic scales John has mapped out as a possibility, maybe?

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Cameron Bobro wrote:
>
> --- In MakeMicroMusic@yahoogroups.com > <mailto:MakeMicroMusic%40yahoogroups.com>, "John H.
> Chalmers" <JHCHALMERS@...> wrote:
> >
> > http://en.wikipedia.org/wiki/Pisot-Vijayaraghavan_number > <http://en.wikipedia.org/wiki/Pisot-Vijayaraghavan_number>
> >
> > While following links from the Padovan number page, I came across
> these whose magnitudes
> > which are between the Padovan and Golden Ratio numbers. I haven't
> noticed them before on
> > this list or Tuning Math. They might define interesting 4ths,
> tritones, fifths and sixths for
> > generating scales or temperaments.
> >
> > --John
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 750797/662359 216.971
> 2: 269.851 cents 269.851
> 3: 662359/500000 486.822
> 4: 750797/500000 703.794
> 5: 920.765 cents 920.765
> 6: 973.645 cents 973.645
> 7: 1190.616 cents 1190.616
>
> The "plastic number" was already familiar to me from my search for
> "shadows" and points of useful (and, frankly, pretty) "otherness" in
> general.
>
> The above tuning (see #1 and #5 on the chart at Wikipedia) has some
> interesting properties, to my ears- the detuned octave seems to ride
> in one of those little spots that are curiously less dissonant than
> they "should" be, and triadic tall chords, ie playing every other
> note in the scale, are just stacks of 7/6s and 9/7s about 2-3 cents
> sharp of Just. There is a also a distinctive "look Ma, no wrong
> notes!" sound to it.
>
> This tuning is part of a personal study of "comma pump? who cares!".
> For we may consider the result of a simple repetitive Just or
> otherwise coherent intervals a thing in it's own right and not, say,
> a tonic now "off" by a comma. A consideration justified by the ear,
> in my opinion.
>
> -Cameron Bobro
>
>

--- In MakeMicroMusic@yahoogroups.com, Kraig Grady <kraiggrady@...>
wrote:
>
> Have you made one of the Tritriadic scales John has mapped out as a
> possibility, maybe?

Not on purpose- I have never read the Xenharmonikon articles about
Tritriadic scales and only know of them by accident sort of, as
they've been used in computer/generative music.

Your intervals are all your own so not implying that.
basically he outline the continuum of having three triads in a row which i believe inspired possibly by the most well known triads such as

A E B
F C G D

as the archetype. It is a popular scale building tool of my wife BTW.
what computer/generative music might have missed our radar?

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Cameron Bobro wrote:
>
> --- In MakeMicroMusic@yahoogroups.com > <mailto:MakeMicroMusic%40yahoogroups.com>, Kraig Grady <kraiggrady@...>
> wrote:
> >
> > Have you made one of the Tritriadic scales John has mapped out as a
> > possibility, maybe?
>
> Not on purpose- I have never read the Xenharmonikon articles about
> Tritriadic scales and only know of them by accident sort of, as
> they've been used in computer/generative music.
>
>

--- In MakeMicroMusic@yahoogroups.com, Kraig Grady <kraiggrady@...>
wrote:
>
> Your intervals are all your own so not implying that.

Oh I didn't think you were implying that, and anyway I'm always quite
relieved if something I do has been done before, which happens all
the time, because then I don't feel like a nut.

> basically he outline the continuum of having three triads in a row
> which i believe inspired possibly by the most well known triads
such as
>
>
> A E B
> F C G D
>
> as the archetype. It is a popular scale building tool of my wife
BTW.

Is that using higher integer ratios?

> what computer/generative music might have missed our radar?

Hmmm...my own approach to "generative/interactive" music, which I
have to do more and more but still isn't ready for prime time yet,
is based on generating the actual materials first rather than
triggering pre-determined scales and such but I'm always surfing
around to see what others are doing. Let's see...

"Tritriadic Chimes: bells in just intonation | Terumi Narushima
"Tritriadic Chimes: bells in just intonation" is a set of 12 short
pieces that were originally designed to function as chimes that ring
on the hour from a bell tower. The work explores experimental tunings
called tritriadic scales which have been developed by the American
tuning theorist John Chalmers. The pieces are realised using bell
timbres synthesised in Csound."

>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> Mesotonal Music from:
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://
anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
>
>
>
> Cameron Bobro wrote:
> >
> > --- In MakeMicroMusic@yahoogroups.com
> > <mailto:MakeMicroMusic%40yahoogroups.com>, Kraig Grady
<kraiggrady@>
> > wrote:
> > >
> > > Have you made one of the Tritriadic scales John has mapped out
as a
> > > possibility, maybe?
> >
> > Not on purpose- I have never read the Xenharmonikon articles about
> > Tritriadic scales and only know of them by accident sort of, as
> > they've been used in computer/generative music.
> >
> >
>

I think John was more concerned with the 'structure' and the different families possible which he has charted out.
/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Cameron Bobro wrote:
>
> --- In MakeMicroMusic@yahoogroups.com > <mailto:MakeMicroMusic%40yahoogroups.com>, Kraig Grady <kraiggrady@...>
> wrote:
> >
> > Your intervals are all your own so not implying that.
>
> Oh I didn't think you were implying that, and anyway I'm always quite
> relieved if something I do has been done before, which happens all
> the time, because then I don't feel like a nut.
>
> > basically he outline the continuum of having three triads in a row
> > which i believe inspired possibly by the most well known triads
> such as
> >
> >
> > A E B
> > F C G D
> >
> > as the archetype. It is a popular scale building tool of my wife
> BTW.
>
> Is that using higher integer ratios?
>
> > what computer/generative music might have missed our radar?
>
> Hmmm...my own approach to "generative/interactive" music, which I
> have to do more and more but still isn't ready for prime time yet,
> is based on generating the actual materials first rather than
> triggering pre-determined scales and such but I'm always surfing
> around to see what others are doing. Let's see...
>
> "Tritriadic Chimes: bells in just intonation | Terumi Narushima
> "Tritriadic Chimes: bells in just intonation" is a set of 12 short
> pieces that were originally designed to function as chimes that ring
> on the hour from a bell tower. The work explores experimental tunings
> called tritriadic scales which have been developed by the American
> tuning theorist John Chalmers. The pieces are realised using bell
> timbres synthesised in Csound."
>
> >
> > /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> > Mesotonal Music from:
> > _'''''''_ ^North/Western Hemisphere:
> > North American Embassy of Anaphoria Island <http://anaphoria.com/ > <http://anaphoria.com/>>
> >
> > _'''''''_ ^South/Eastern Hemisphere:
> > Austronesian Outpost of Anaphoria <http://
> anaphoriasouth.blogspot.com/>
> >
> > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> >
> >
> >
> >
> > Cameron Bobro wrote:
> > >
> > > --- In MakeMicroMusic@yahoogroups.com > <mailto:MakeMicroMusic%40yahoogroups.com>
> > > <mailto:MakeMicroMusic%40yahoogroups.com>, Kraig Grady
> <kraiggrady@>
> > > wrote:
> > > >
> > > > Have you made one of the Tritriadic scales John has mapped out
> as a
> > > > possibility, maybe?
> > >
> > > Not on purpose- I have never read the Xenharmonikon articles about
> > > Tritriadic scales and only know of them by accident sort of, as
> > > they've been used in computer/generative music.
> > >
> > >
> >
>
>

I think John was more concerned with the 'structure' and it potential and from there, the different qualities the different proportion would produce. He has charted these out

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Cameron Bobro wrote:
>
> --- In MakeMicroMusic@yahoogroups.com > <mailto:MakeMicroMusic%40yahoogroups.com>, Kraig Grady <kraiggrady@...>
> wrote:
> >
> > Your intervals are all your own so not implying that.
>
> Oh I didn't think you were implying that, and anyway I'm always quite
> relieved if something I do has been done before, which happens all
> the time, because then I don't feel like a nut.
>
> > basically he outline the continuum of having three triads in a row
> > which i believe inspired possibly by the most well known triads
> such as
> >
> >
> > A E B
> > F C G D
> >
> > as the archetype. It is a popular scale building tool of my wife
> BTW.
>
> Is that using higher integer ratios?
>
> > what computer/generative music might have missed our radar?
>
> Hmmm...my own approach to "generative/interactive" music, which I
> have to do more and more but still isn't ready for prime time yet,
> is based on generating the actual materials first rather than
> triggering pre-determined scales and such but I'm always surfing
> around to see what others are doing. Let's see...
>
> "Tritriadic Chimes: bells in just intonation | Terumi Narushima
> "Tritriadic Chimes: bells in just intonation" is a set of 12 short
> pieces that were originally designed to function as chimes that ring
> on the hour from a bell tower. The work explores experimental tunings
> called tritriadic scales which have been developed by the American
> tuning theorist John Chalmers. The pieces are realised using bell
> timbres synthesised in Csound."
>
> >
> > /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> > Mesotonal Music from:
> > _'''''''_ ^North/Western Hemisphere:
> > North American Embassy of Anaphoria Island <http://anaphoria.com/ > <http://anaphoria.com/>>
> >
> > _'''''''_ ^South/Eastern Hemisphere:
> > Austronesian Outpost of Anaphoria <http://
> anaphoriasouth.blogspot.com/>
> >
> > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> >
> >
> >
> >
> > Cameron Bobro wrote:
> > >
> > > --- In MakeMicroMusic@yahoogroups.com > <mailto:MakeMicroMusic%40yahoogroups.com>
> > > <mailto:MakeMicroMusic%40yahoogroups.com>, Kraig Grady
> <kraiggrady@>
> > > wrote:
> > > >
> > > > Have you made one of the Tritriadic scales John has mapped out
> as a
> > > > possibility, maybe?
> > >
> > > Not on purpose- I have never read the Xenharmonikon articles about
> > > Tritriadic scales and only know of them by accident sort of, as
> > > they've been used in computer/generative music.
> > >
> > >
> >
>
>

Cameron Bobro wrote:

> 0: 1/1 0.000 unison, perfect prime
> 1: 750797/662359 216.971
> 2: 269.851 cents 269.851
> 3: 662359/500000 486.822
> 4: 750797/500000 703.794
> 5: 920.765 cents 920.765
> 6: 973.645 cents 973.645
> 7: 1190.616 cents 1190.616
> > The "plastic number" was already familiar to me from my search for > "shadows" and points of useful (and, frankly, pretty) "otherness" in > general.
> > The above tuning (see #1 and #5 on the chart at Wikipedia) has some > interesting properties, to my ears- the detuned octave seems to ride > in one of those little spots that are curiously less dissonant than > they "should" be, and triadic tall chords, ie playing every other > note in the scale, are just stacks of 7/6s and 9/7s about 2-3 cents > sharp of Just. There is a also a distinctive "look Ma, no wrong > notes!" sound to it.
> > This tuning is part of a personal study of "comma pump? who cares!".
> For we may consider the result of a simple repetitive Just or > otherwise coherent intervals a thing in it's own right and not, say, > a tonic now "off" by a comma. A consideration justified by the ear, > in my opinion.

My first thought on hearing this scale is that it would make a great tuning for a set of wind chimes!