5-Tone Mode of BP Scale

Hi,

I noticed that the mode 33331 of the chromatic Bohlen-Pierce scale has
two 3:5:7 chords and two 1/(3:5:7) chords which are all produced by
the same scale pattern. This scale is also a MOS.

Is this documented somewhere? I can't believe this hasn't been
proposed earlier.

Kalle

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
>
> Hi,
>
> I noticed that the mode 33331 of the chromatic Bohlen-Pierce scale has
> two 3:5:7 chords and two 1/(3:5:7) chords which are all produced by
> the same scale pattern. This scale is also a MOS.
>
> Is this documented somewhere? I can't believe this hasn't been
> proposed earlier.
>
> Kalle

The usual 9-note "diatonic" Bohlen-Pierce scale has a generator of 3
chromatic BP degrees (438.9 cents). So does your scale. They both can
be understood (as in part 2 of my 'Middle Path' paper) as arising from
tempering out the 'comma' 245:243 -- each would be a ring in the
corresponding horagram. As would the 4-note 3334 scale. The smallness
of the ratio 245:243 implies (in a precise way) that it doesn't take a
lot of generators to form the approximation of any consonant JI chord
formed from the primes 3, 5, and 7.

The TOP tuning of the primes would be:

cents(3)+cents(245/243)*log(3)/log(245*243) = 1903.37 cents;
cents(5)-cents(245/243)*log(5)/log(245*243) = 2784.24 cents;
cents(7)-cents(245/243)*log(7)/log(245*243) = 3366.31 cents.

Algebra shows that the approximate 9:7 serves as a generator for this
system, with the approximate 3:1 as the period. 9:7 is represented as

1903.37 + 1903.37 - 3366.31 = 440.43 cents

Here's what the horagram looks like for TOP tuning:

/tuning/files/Erlich/bphora.gif

BP pentatonic is an apt name for this scale, both to suggest how it's a
subset of the BP 'diatonic', generated in the same way, and because it
literally has five notes.

Thank you Paul, for this very lucid explanation. You'll forgive for not
being able to catch up with your article yet, won't you?

Oz.

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 01 Aral�k 2005 Per�embe 1:09
Subject: [tuning] Re: 5-Tone Mode of BP Scale

> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> >
> > Hi,
> >
> > I noticed that the mode 33331 of the chromatic Bohlen-Pierce scale has
> > two 3:5:7 chords and two 1/(3:5:7) chords which are all produced by
> > the same scale pattern. This scale is also a MOS.
> >
> > Is this documented somewhere? I can't believe this hasn't been
> > proposed earlier.
> >
> > Kalle
>
> The usual 9-note "diatonic" Bohlen-Pierce scale has a generator of 3
> chromatic BP degrees (438.9 cents). So does your scale. They both can
> be understood (as in part 2 of my 'Middle Path' paper) as arising from
> tempering out the 'comma' 245:243 -- each would be a ring in the
> corresponding horagram. As would the 4-note 3334 scale. The smallness
> of the ratio 245:243 implies (in a precise way) that it doesn't take a
> lot of generators to form the approximation of any consonant JI chord
> formed from the primes 3, 5, and 7.
>
> The TOP tuning of the primes would be:
>
> cents(3)+cents(245/243)*log(3)/log(245*243) = 1903.37 cents;
> cents(5)-cents(245/243)*log(5)/log(245*243) = 2784.24 cents;
> cents(7)-cents(245/243)*log(7)/log(245*243) = 3366.31 cents.
>
> Algebra shows that the approximate 9:7 serves as a generator for this
> system, with the approximate 3:1 as the period. 9:7 is represented as
>
> 1903.37 + 1903.37 - 3366.31 = 440.43 cents
>
> Here's what the horagram looks like for TOP tuning:
>
> /tuning/files/Erlich/bphora.gif
>
> BP pentatonic is an apt name for this scale, both to suggest how it's a
> subset of the BP 'diatonic', generated in the same way, and because it
> literally has five notes.
>
>
>
>

Of course! If you're doing so now, just remember that the interval of
equivalence in part 1 of my paper is always assumed to be an
(approximate) 2:1 or "octave", while in the BP (Bohlen-Pierce) world,
the interval of equivalence is assumed to be 3:1 instead. Also, part
1 will be improved with a wealth of suggestions (especially from
Yahya) and corrections I've received; I'd welcome yours too of course!

--- In tuning@yahoogroups.com, <oyarman@o...> wrote:
>
> Thank you Paul, for this very lucid explanation. You'll forgive for
not
> being able to catch up with your article yet, won't you?
>
> Oz.
>
> ----- Original Message -----
> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> To: <tuning@yahoogroups.com>
> Sent: 01 Aralýk 2005 Perþembe 1:09
> Subject: [tuning] Re: 5-Tone Mode of BP Scale
>
>
> > --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> > >
> > > Hi,
> > >
> > > I noticed that the mode 33331 of the chromatic Bohlen-Pierce
scale has
> > > two 3:5:7 chords and two 1/(3:5:7) chords which are all
produced by
> > > the same scale pattern. This scale is also a MOS.
> > >
> > > Is this documented somewhere? I can't believe this hasn't been
> > > proposed earlier.
> > >
> > > Kalle
> >
> > The usual 9-note "diatonic" Bohlen-Pierce scale has a generator
of 3
> > chromatic BP degrees (438.9 cents). So does your scale. They both
can
> > be understood (as in part 2 of my 'Middle Path' paper) as arising
from
> > tempering out the 'comma' 245:243 -- each would be a ring in the
> > corresponding horagram. As would the 4-note 3334 scale. The
smallness
> > of the ratio 245:243 implies (in a precise way) that it doesn't
take a
> > lot of generators to form the approximation of any consonant JI
chord
> > formed from the primes 3, 5, and 7.
> >
> > The TOP tuning of the primes would be:
> >
> > cents(3)+cents(245/243)*log(3)/log(245*243) = 1903.37 cents;
> > cents(5)-cents(245/243)*log(5)/log(245*243) = 2784.24 cents;
> > cents(7)-cents(245/243)*log(7)/log(245*243) = 3366.31 cents.
> >
> > Algebra shows that the approximate 9:7 serves as a generator for
this
> > system, with the approximate 3:1 as the period. 9:7 is
represented as
> >
> > 1903.37 + 1903.37 - 3366.31 = 440.43 cents
> >
> > Here's what the horagram looks like for TOP tuning:
> >
> > /tuning/files/Erlich/bphora.gif
> >
> > BP pentatonic is an apt name for this scale, both to suggest how
it's a
> > subset of the BP 'diatonic', generated in the same way, and
because it
> > literally has five notes.
> >
> >
> >
> >
>

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> >
> > Hi,
> >
> > I noticed that the mode 33331 of the chromatic Bohlen-Pierce scale has
> > two 3:5:7 chords and two 1/(3:5:7) chords which are all produced by
> > the same scale pattern. This scale is also a MOS.
> >
> > Is this documented somewhere? I can't believe this hasn't been
> > proposed earlier.
> >
> > Kalle
>
> The usual 9-note "diatonic" Bohlen-Pierce scale has a generator of 3
> chromatic BP degrees (438.9 cents). So does your scale. They both can
> be understood (as in part 2 of my 'Middle Path' paper) as arising from
> tempering out the 'comma' 245:243 -- each would be a ring in the
> corresponding horagram. As would the 4-note 3334 scale. The smallness
> of the ratio 245:243 implies (in a precise way) that it doesn't take a
> lot of generators to form the approximation of any consonant JI chord
> formed from the primes 3, 5, and 7.

Yes, it is amazing that there can be so many consonant *triads* in a
5-tone scale! Did Bohlen/Pierce consider those 1/(3:5:7) chords as
valid harmonies? I have a hunch that they didn't.

> The TOP tuning of the primes would be:
>
> cents(3)+cents(245/243)*log(3)/log(245*243) = 1903.37 cents;
> cents(5)-cents(245/243)*log(5)/log(245*243) = 2784.24 cents;
> cents(7)-cents(245/243)*log(7)/log(245*243) = 3366.31 cents.

And for equal temperament with the mapping [13 19 23] for 3, 5 and 7 I
get 1904.19, 2783.04 and 3368.95 cents.

> Algebra shows that the approximate 9:7 serves as a generator for this
> system, with the approximate 3:1 as the period. 9:7 is represented as
>
> 1903.37 + 1903.37 - 3366.31 = 440.43 cents
>
> Here's what the horagram looks like for TOP tuning:
>
> /tuning/files/Erlich/bphora.gif

Yep, very nice!

> BP pentatonic is an apt name for this scale, both to suggest how it's a
> subset of the BP 'diatonic', generated in the same way, and because it
> literally has five notes.

I instantly thought about that name too! It is certainly the most
natural and self-evident name for it.

For me the 9-tone diatonic and 13-tone chromatic BP scales are way too
complex to be heard as coherent melodic and harmonic gestalts
especially because there are no octaves and the scale repeats at the
"tritave". I've been playing with the BP pentatonic scale a bit in FM7
and I think it sounds very interesting, sort of mindwarping.

Sometimes I can even hear the tritave equivalence. :)

Kalle

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
> > >
> > > Hi,
> > >
> > > I noticed that the mode 33331 of the chromatic Bohlen-Pierce
scale has
> > > two 3:5:7 chords and two 1/(3:5:7) chords which are all
produced by
> > > the same scale pattern. This scale is also a MOS.
> > >
> > > Is this documented somewhere? I can't believe this hasn't been
> > > proposed earlier.
> > >
> > > Kalle
> >
> > The usual 9-note "diatonic" Bohlen-Pierce scale has a generator
of 3
> > chromatic BP degrees (438.9 cents). So does your scale. They both
can
> > be understood (as in part 2 of my 'Middle Path' paper) as arising
from
> > tempering out the 'comma' 245:243 -- each would be a ring in the
> > corresponding horagram. As would the 4-note 3334 scale. The
smallness
> > of the ratio 245:243 implies (in a precise way) that it doesn't
take a
> > lot of generators to form the approximation of any consonant JI
chord
> > formed from the primes 3, 5, and 7.
>
>
> Yes, it is amazing that there can be so many consonant *triads* in a
> 5-tone scale! Did Bohlen/Pierce consider those 1/(3:5:7) chords as
> valid harmonies? I have a hunch that they didn't.

They both considered them, yes, but neither found them to be one of
the most consonant triads in the scale. You can read Bohlen's website
here:

http://members.aol.com/bpsite/index.html

And you should read both Pierce's book _The Science of Musical Sound_
and the article on his scale in _Harmony and Tonality_ edited by J.
Sundberg.

> > The TOP tuning of the primes would be:
> >
> > cents(3)+cents(245/243)*log(3)/log(245*243) = 1903.37 cents;
> > cents(5)-cents(245/243)*log(5)/log(245*243) = 2784.24 cents;
> > cents(7)-cents(245/243)*log(7)/log(245*243) = 3366.31 cents.
>
>
> And for equal temperament with the mapping [13 19 23] for 3, 5 and
7 I
> get 1904.19, 2783.04 and 3368.95 cents.

Yup!

> > Algebra shows that the approximate 9:7 serves as a generator for
this
> > system, with the approximate 3:1 as the period. 9:7 is
represented as
> >
> > 1903.37 + 1903.37 - 3366.31 = 440.43 cents
> >
> > Here's what the horagram looks like for TOP tuning:
> >
> > /tuning/files/Erlich/bphora.gif
>
>
> Yep, very nice!

Thanks. I'd love to hear your FM explorations in this 'pentatonic'
scale.

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:

> > BP pentatonic is an apt name for this scale, both to suggest how
it's a
> > subset of the BP 'diatonic', generated in the same way, and because
it
> > literally has five notes.
>
>
> I instantly thought about that name too! It is certainly the most
> natural and self-evident name for it.

Manuel Op de Coul should add some rotation of it to his "List of
Musical Modes" document, since there's already a BP section there.
(Also I noticed that although Herman Miller's 7-note Porcupine scale is
listed under 22-equal, the Igliashon Jones 8-note Porcupine isn't -- I
think 33133333 was his favorite rotation. Manuel?)

i thought i would remind others that the BP scale can be considered a diamond with a unique interval of equivalence
http://www.anaphoria.com/images/BPdiamond.gif
also of interest is
http://www.anaphoria.com/images/BPdia2.gif

> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
>
> i thought i would remind others that the BP scale can be considered a
> diamond with a unique interval of equivalence
> http://www.anaphoria.com/images/BPdiamond.gif

I don't understand this, since there are only 7 distinct pitch classes
in a 3-by-3 diamond. The vertical spine consists of three identical
pitches, which you seem to label as '3' or equivalently as '1' (the
other factors cancel out). But the BP diatonic scale has *9* pitch
classes. You could get to it by adding 25/21 and 25/9 to your diamond.

> also of interest is
> http://www.anaphoria.com/images/BPdia2.gif

These seems to merely add more 'tritave equivalents' of the 7 pitch
classes above, but it's still 7 and not 9.

since the interval of equivalence is a 3/1 they fall on the same spot on the diamond matrix.
>
>Message: 4 > Date: Tue, 06 Dec 2005 19:34:31 -0000
> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
>Subject: Re: 5-Tone Mode of BP Scale
>
>--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> >
>>i thought i would remind others that the BP scale can be considered a >>diamond with a unique interval of equivalence
>>http://www.anaphoria.com/images/BPdiamond.gif
>> >>
>
>I don't understand this, since there are only 7 distinct pitch classes >in a 3-by-3 diamond. The vertical spine consists of three identical >pitches, which you seem to label as '3' or equivalently as '1' (the >other factors cancel out). But the BP diatonic scale has *9* pitch >classes. You could get to it by adding 25/21 and 25/9 to your diamond.
>
> >
>>also of interest is
>>http://www.anaphoria.com/images/BPdia2.gif
>> >>
>
>These seems to merely add more 'tritave equivalents' of the 7 pitch >classes above, but it's still 7 and not 9.
>
>
>
>
>
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

Hi Kraig,

I know the interval of equivalence is 3/1 and said so myself below.
I still don't see how 25/21 and 25/9 fall on the diamond matrix, let
alone the same spot. You can't get from one to the other by
multiplying or dividing by 3.

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
> since the interval of equivalence is a 3/1 they fall on the same
spot on
> the diamond matrix.
>
>
> >
> >Message: 4
> > Date: Tue, 06 Dec 2005 19:34:31 -0000
> > From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> >Subject: Re: 5-Tone Mode of BP Scale
> >
> >--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> >
> >
> >>i thought i would remind others that the BP scale can be
considered a
> >>diamond with a unique interval of equivalence
> >>http://www.anaphoria.com/images/BPdiamond.gif
> >>
> >>
> >
> >I don't understand this, since there are only 7 distinct pitch
classes
> >in a 3-by-3 diamond. The vertical spine consists of three
identical
> >pitches, which you seem to label as '3' or equivalently as '1'
(the
> >other factors cancel out). But the BP diatonic scale has *9* pitch
> >classes. You could get to it by adding 25/21 and 25/9 to your
diamond.
> >
> >
> >
> >>also of interest is
> >>http://www.anaphoria.com/images/BPdia2.gif
> >>
> >>
> >
> >These seems to merely add more 'tritave equivalents' of the 7
pitch
> >classes above, but it's still 7 and not 9.
> >
> >
> >
> >
> >
> >
> >
>
> --
> Kraig Grady
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
> The Wandering Medicine Show
> KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles
>