Wilson's expansions

Hello

has anyone done any work with tetrachordal harmonisation using Erv Wilson's
expansion technique as described in "Divisions of the Tetrachord"? If so I'd
be interested to hear/hear about any results.

I like the resulting economy in providing musically useful harmonic
resources for the addition of only five or six extra tones.

The endogenous and duodene methods are straightforward enough but I'd be
interested in finding a relatively straightforward way of carrying out the
Wilson expansions on some of the tetrachords I use.

I started with one of my tetrachords, Ptolemy's Malakon diatonic and
(because I'm lazy and not very good at sums) found that I was using a rather
laborious trial and error method to find expansions that covered each tone
of the tetrachord. Mind you, John Chalmers does say that it is rather a
difficult problem.

Can any of the mathematicians come up with an approach that a
non-mathematician would understand?

Thanks in anticipation.

Sincerely
a.m.

--- In tuning@yahoogroups.com, Alison Monteith
<alison.monteith3@w...> wrote:
> Hello
>
> has anyone done any work with tetrachordal harmonisation using Erv
Wilson's
> expansion technique as described in "Divisions of the Tetrachord"?
If so I'd
> be interested to hear/hear about any results.
>
> I like the resulting economy in providing musically useful harmonic
> resources for the addition of only five or six extra tones.
>
> The endogenous and duodene methods are straightforward enough but
I'd be
> interested in finding a relatively straightforward way of carrying
out the
> Wilson expansions on some of the tetrachords I use.
>
> I started with one of my tetrachords, Ptolemy's Malakon diatonic and
> (because I'm lazy and not very good at sums) found that I was using
a rather
> laborious trial and error method to find expansions that covered
each tone
> of the tetrachord. Mind you, John Chalmers does say that it is
rather a
> difficult problem.
>
> Can any of the mathematicians come up with an approach that a
> non-mathematician would understand?
>
> Thanks in anticipation.
>
> Sincerely
> a.m.

i don't own a copy of john's book, so why don't you describe exactly
what you're looking for? not sure what you mean by "expansions"
and "covering" . . . personally, if limited to strict just
intonation, i'd be drawing lattices to decide which additional tones
are best for harmonizing a given set of tones . . .

on 4/6/03 9:45 pm, wallyesterpaulrus at wallyesterpaulrus@yahoo.com wrote:

> --- In tuning@yahoogroups.com, Alison Monteith
> <alison.monteith3@w...> wrote:
>> Hello
>>
>> has anyone done any work with tetrachordal harmonisation using Erv
> Wilson's
>> expansion technique as described in "Divisions of the Tetrachord"?
> If so I'd
>> be interested to hear/hear about any results.
>>
>> I like the resulting economy in providing musically useful harmonic
>> resources for the addition of only five or six extra tones.
>>
>> The endogenous and duodene methods are straightforward enough but
> I'd be
>> interested in finding a relatively straightforward way of carrying
> out the
>> Wilson expansions on some of the tetrachords I use.
>>
>> I started with one of my tetrachords, Ptolemy's Malakon diatonic and
>> (because I'm lazy and not very good at sums) found that I was using
> a rather
>> laborious trial and error method to find expansions that covered
> each tone
>> of the tetrachord. Mind you, John Chalmers does say that it is
> rather a
>> difficult problem.
>>
>> Can any of the mathematicians come up with an approach that a
>> non-mathematician would understand?
>>
>> Thanks in anticipation.
>>
>> Sincerely
>> a.m.
>
> i don't own a copy of john's book, so why don't you describe exactly
> what you're looking for? not sure what you mean by "expansions"
> and "covering" . . . personally, if limited to strict just
> intonation, i'd be drawing lattices to decide which additional tones
> are best for harmonizing a given set of tones . . .

The technique is as follows : -

For harmonic chords the "unit proportion" is expressed as a string of signed
positive integers. So the major triad 4:5:6:8 is +1 +1 +2, ie the difference
between 4 and 5, 5 and 6, 6 and 8. Subharmonic chords have previxed - signs,
8:6:5:4 is therefore -2 -1 -1. The musical interest lies in the realisation
of a sequence of chords which progress from relatively discordant to a
stable consonance. For example, with Archytas Enharmonic (inverted) which is

1/1 5/4 9/7 4/3 3/2 15/8 27/14 2/1 we get

4:5:6:8 with 1/1 5/4 3/2 and 2/1

5:6:7:9 with 15/14 9/7 3/2 and 27/14

6:7:8:10 with 9/8 21/16 3/2 and 15/8

7:8:9:11 with 7/6 4/3 3/2 and 11/6

In this case the 3/2 is held constant. With subharmonic chords, the examples
(given by John Chalmers) hold the 4/3 constant.

All the tones in the tetrachord can be harmonised. Only five extra tones are
added.

My problem is to work this out for the tetrachords I use at the moment.
These are : -

Avicenna's 13 limit Diatonic - 1/1, 14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2/1.
Barbour's (re-arranged) Chromatic - 1/1, 9/8, 8/7, 4/3, 3/2, 27/16, 12/7,
2/1.
Al Farabi's Diatonic - 1/1, 49/48, 7/6, 4/3, 3/2, 49/32, 7/4, 2/1.
Ptolemy's Homalon or Equable Diatonic - 1/1, 12/11, 6/5, 4/3, 3/2, 18/11,
9/5, 2/1.
Avicenna's 11-limit Diatonic - 8/7, 40/33, 4/3, 32/21, 12/7, 20/11, 2/1,
8/7.
Ptolemy's Malakon Diatonic - 8/7, 6/5, 4/3, 32/21, 12/7, 9/5, 2/1, 8/7.

I'm wondering if there is a quicker way than trial and error, such as a
simple formula, to establish expansions that will give me a set of chords
that allow me to harmonise all the tones of the tetrachord.

Regards
a.m.

>has anyone done any work with tetrachordal harmonisation using
>Erv Wilson's expansion technique as described in "Divisions of
>the Tetrachord"?
//
>The technique is as follows : -
>
>For harmonic chords the "unit proportion" is expressed as a string
>of signed positive integers. So the major triad 4:5:6:8 is +1 +1 +2,
>ie the difference between 4 and 5, 5 and 6, 6 and 8. Subharmonic
>chords have previxed - signs, 8:6:5:4 is therefore -2 -1 -1. The
>musical interest lies in the realisation of a sequence of chords
>which progress from relatively discordant to a stable consonance.

Ok...

>For example, with Archytas Enharmonic (inverted) which is
>
>1/1 5/4 9/7 4/3 3/2 15/8 27/14 2/1 we get
>
>4:5:6:8 with 1/1 5/4 3/2 and 2/1
>
>5:6:7:9 with 15/14 9/7 3/2 and 27/14
>
>6:7:8:10 with 9/8 21/16 3/2 and 15/8
>
>7:8:9:11 with 7/6 4/3 3/2 and 11/6
>
>In this case the 3/2 is held constant. With subharmonic chords,
>the examples (given by John Chalmers) hold the 4/3 constant.
>
>All the tones in the tetrachord can be harmonised. Only five extra
>tones are added.

How does this involve the "unit proportion" thing?

(My copy of John's book is in another State. Which is a real shame,
since I haven't read it in years, and obviously should again. I
can prob. find it at my local library...)

-Carl

on 8/6/03 9:50 pm, Carl Lumma at ekin@lumma.org wrote:

>> has anyone done any work with tetrachordal harmonisation using
>> Erv Wilson's expansion technique as described in "Divisions of
>> the Tetrachord"?
> //
>> The technique is as follows : -
>>
>> For harmonic chords the "unit proportion" is expressed as a string
>> of signed positive integers. So the major triad 4:5:6:8 is +1 +1 +2,
>> ie the difference between 4 and 5, 5 and 6, 6 and 8. Subharmonic
>> chords have previxed - signs, 8:6:5:4 is therefore -2 -1 -1. The
>> musical interest lies in the realisation of a sequence of chords
>> which progress from relatively discordant to a stable consonance.
>
> Ok...
>
>> For example, with Archytas Enharmonic (inverted) which is
>>
>> 1/1 5/4 9/7 4/3 3/2 15/8 27/14 2/1 we get
>>
>> 4:5:6:8 with 1/1 5/4 3/2 and 2/1
>>
>> 5:6:7:9 with 15/14 9/7 3/2 and 27/14
>>
>> 6:7:8:10 with 9/8 21/16 3/2 and 15/8
>>
>> 7:8:9:11 with 7/6 4/3 3/2 and 11/6
>>
>> In this case the 3/2 is held constant. With subharmonic chords,
>> the examples (given by John Chalmers) hold the 4/3 constant.
>>
>> All the tones in the tetrachord can be harmonised. Only five extra
>> tones are added.
>
> How does this involve the "unit proportion" thing?

The unit proportion remains constant as explained above, each set of chords
having numbers differing by 1, 1 and 2, as in 4:5:6:8. Then you try
different chords with the same proportion and in the example given all the
tones in the tetrachord are provided with a chord which will harmonise that
tone. But it's probably best to have a read through the book as i don't want
to cut and paste a swadge of the book out of respect for copyright.

> (My copy of John's book is in another State. Which is a real shame,
> since I haven't read it in years, and obviously should again. I
> can prob. find it at my local library...)
>
> -Carl

Regards
a.m.

>>> For example, with Archytas Enharmonic (inverted) which is
>>>
>>> 1/1 5/4 9/7 4/3 3/2 15/8 27/14 2/1 we get
>>>
>>> 4:5:6:8 with 1/1 5/4 3/2 and 2/1
>>>
>>> 5:6:7:9 with 15/14 9/7 3/2 and 27/14
>>>
>>> 6:7:8:10 with 9/8 21/16 3/2 and 15/8
>>>
>>> 7:8:9:11 with 7/6 4/3 3/2 and 11/6
>>>
>>> In this case the 3/2 is held constant. With subharmonic chords,
>>> the examples (given by John Chalmers) hold the 4/3 constant.
>>>
>>> All the tones in the tetrachord can be harmonised. Only five extra
>>> tones are added.
>>
>> How does this involve the "unit proportion" thing?
>
>The unit proportion remains constant as explained above, each set of
>chords having numbers differing by 1, 1 and 2, as in 4:5:6:8. Then you
>try different chords with the same proportion and in the example given
>all the tones in the tetrachord are provided with a chord which will
>harmonise that tone.

Aha.

>But it's probably best to have a read through the book as i don't want
>to cut and paste a swadge of the book out of respect for copyright.

Swadge; nice word.

>I'm wondering if there is a quicker way than trial and error, such as
>a simple formula, to establish expansions that will give me a set of
>chords that allow me to harmonise all the tones of the tetrachord.

I dunno. Let's see...

>My problem is to work this out for the tetrachords I use at the moment.
>These are : -
>
>Avicenna's 13 limit Diatonic 1/1 14/13 7/6 4/3 3/2 21/13 7/4

So your tetrad here appears to be 1/1-7/6-3/2-7/4, or 12:14:18:21.
So the unit relation is 2,4,3. Get...

7:9:13:16
8:10:14:17
9:11:15:18
10:12:16:19
11:13:17:20

Now, see if any of them can be added. I'd probably load the raw
scale into Scala, do "show interval matrix", and look for triads
found in the above tetrads.

These look cool...

Barbour's (re-arranged) Chromatic
1/1 9/8 8/7 4/3 3/2 27/16 12/7
Ptolemy's Homalon/Equable Diatonic
1/1 12/11 6/5 4/3 3/2 18/11 9/5
Ptolemy's Malakon Diatonic
1/1 21/20 7/6 4/3 3/2 63/40 7/4

-Carl

on 9/6/03 9:32 pm, Carl Lumma at ekin@lumma.org wrote:

>>>> For example, with Archytas Enharmonic (inverted) which is
>>>>
>>>> 1/1 5/4 9/7 4/3 3/2 15/8 27/14 2/1 we get
>>>>
>>>> 4:5:6:8 with 1/1 5/4 3/2 and 2/1
>>>>
>>>> 5:6:7:9 with 15/14 9/7 3/2 and 27/14
>>>>
>>>> 6:7:8:10 with 9/8 21/16 3/2 and 15/8
>>>>
>>>> 7:8:9:11 with 7/6 4/3 3/2 and 11/6
>>>>
>>>> In this case the 3/2 is held constant. With subharmonic chords,
>>>> the examples (given by John Chalmers) hold the 4/3 constant.
>>>>
>>>> All the tones in the tetrachord can be harmonised. Only five extra
>>>> tones are added.
>>>
>>> How does this involve the "unit proportion" thing?
>>
>> The unit proportion remains constant as explained above, each set of
>> chords having numbers differing by 1, 1 and 2, as in 4:5:6:8. Then you
>> try different chords with the same proportion and in the example given
>> all the tones in the tetrachord are provided with a chord which will
>> harmonise that tone.
>
> Aha.
>
>> But it's probably best to have a read through the book as i don't want
>> to cut and paste a swadge of the book out of respect for copyright.
>
> Swadge; nice word.
>
>> I'm wondering if there is a quicker way than trial and error, such as
>> a simple formula, to establish expansions that will give me a set of
>> chords that allow me to harmonise all the tones of the tetrachord.
>
> I dunno. Let's see...
>
>> My problem is to work this out for the tetrachords I use at the moment.
>> These are : -
>>
>> Avicenna's 13 limit Diatonic 1/1 14/13 7/6 4/3 3/2 21/13 7/4
>
> So your tetrad here appears to be 1/1-7/6-3/2-7/4, or 12:14:18:21.
> So the unit relation is 2,4,3. Get...
>
> 7:9:13:16
> 8:10:14:17
> 9:11:15:18
> 10:12:16:19
> 11:13:17:20
>
> Now, see if any of them can be added. I'd probably load the raw
> scale into Scala, do "show interval matrix", and look for triads
> found in the above tetrads.

Right - I'll get Scala and try it. It's the arithmetic that I can't be
bothered with.
>
> These look cool...
>
> Barbour's (re-arranged) Chromatic
> 1/1 9/8 8/7 4/3 3/2 27/16 12/7
> Ptolemy's Homalon/Equable Diatonic
> 1/1 12/11 6/5 4/3 3/2 18/11 9/5
> Ptolemy's Malakon Diatonic
> 1/1 21/20 7/6 4/3 3/2 63/40 7/4

They are.

Best
a.m.

>

Hello Alison!
Looking at the pages again from Chalmers book (135-137), it appears that this methods really harmonizes the above tetrachord with the harmonic and the lower with the subharmonic. It seems you have to start with these ratios against your static tone and find a common unit proportion that works for them all. In example 6 there is a printer's mistake for the ratios
reading down are 5/4 9/7 and 27/22 and the order should be 9/7 5/4 27/22. The 9/5 should be over to the right also.

>
> From: Alison Monteith <alison.monteith3@which.net>
> Subject: Re: Re: Wilson's expansions
>
> on 4/6/03 9:45 pm, wallyesterpaulrus at wallyesterpaulrus@yahoo.com wrote:
>
> > --- In tuning@yahoogroups.com, Alison Monteith
> > <alison.monteith3@w...> wrote:
> >> Hello
> >>
> >> has anyone done any work with tetrachordal harmonisation using Erv
> > Wilson's
> >> expansion technique as described in "Divisions of the Tetrachord"?
> > If so I'd
> >> be interested to hear/hear about any results.
> >>
> >> I like the resulting economy in providing musically useful harmonic
> >> resources for the addition of only five or six extra tones.
> >>
> >> The endogenous and duodene methods are straightforward enough but
> > I'd be
> >> interested in finding a relatively straightforward way of carrying
> > out the
> >> Wilson expansions on some of the tetrachords I use.
> >>
> >> I started with one of my tetrachords, Ptolemy's Malakon diatonic and
> >> (because I'm lazy and not very good at sums) found that I was using
> > a rather
> >> laborious trial and error method to find expansions that covered
> > each tone
> >> of the tetrachord. Mind you, John Chalmers does say that it is
> > rather a
> >> difficult problem.
> >>
> >> Can any of the mathematicians come up with an approach that a
> >> non-mathematician would understand?
> >>
> >> Thanks in anticipation.
> >>
> >> Sincerely
> >> a.m.
> >
> > i don't own a copy of john's book, so why don't you describe exactly
> > what you're looking for? not sure what you mean by "expansions"
> > and "covering" . . . personally, if limited to strict just
> > intonation, i'd be drawing lattices to decide which additional tones
> > are best for harmonizing a given set of tones . . .
>
> The technique is as follows : -
>
> For harmonic chords the "unit proportion" is expressed as a string of signed
> positive integers. So the major triad 4:5:6:8 is +1 +1 +2, ie the difference
> between 4 and 5, 5 and 6, 6 and 8. Subharmonic chords have previxed - signs,
> 8:6:5:4 is therefore -2 -1 -1. The musical interest lies in the realisation
> of a sequence of chords which progress from relatively discordant to a
> stable consonance. For example, with Archytas Enharmonic (inverted) which is
>
> 1/1 5/4 9/7 4/3 3/2 15/8 27/14 2/1 we get
>
> 4:5:6:8 with 1/1 5/4 3/2 and 2/1
>
> 5:6:7:9 with 15/14 9/7 3/2 and 27/14
>
> 6:7:8:10 with 9/8 21/16 3/2 and 15/8
>
> 7:8:9:11 with 7/6 4/3 3/2 and 11/6
>
> In this case the 3/2 is held constant. With subharmonic chords, the examples
> (given by John Chalmers) hold the 4/3 constant.
>
> All the tones in the tetrachord can be harmonised. Only five extra tones are
> added.
>
> My problem is to work this out for the tetrachords I use at the moment.
> These are : -
>
> Avicenna's 13 limit Diatonic - 1/1, 14/13, 7/6, 4/3, 3/2, 21/13, 7/4, 2/1.
> Barbour's (re-arranged) Chromatic - 1/1, 9/8, 8/7, 4/3, 3/2, 27/16, 12/7,
> 2/1.
> Al Farabi's Diatonic - 1/1, 49/48, 7/6, 4/3, 3/2, 49/32, 7/4, 2/1.
> Ptolemy's Homalon or Equable Diatonic - 1/1, 12/11, 6/5, 4/3, 3/2, 18/11,
> 9/5, 2/1.
> Avicenna's 11-limit Diatonic - 8/7, 40/33, 4/3, 32/21, 12/7, 20/11, 2/1,
> 8/7.
> Ptolemy's Malakon Diatonic - 8/7, 6/5, 4/3, 32/21, 12/7, 9/5, 2/1, 8/7.
>
> I'm wondering if there is a quicker way than trial and error, such as a
> simple formula, to establish expansions that will give me a set of chords
> that allow me to harmonise all the tones of the tetrachord.
>
> Regards
> a.m.
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

on 10/6/03 8:15 pm, Kraig Grady at kraiggrady@anaphoria.com wrote:

>>
>
> Hello Alison!
> Looking at the pages again from Chalmers book (135-137), it appears that this
> methods really harmonizes the above tetrachord with the harmonic and the lower
> with the subharmonic. It seems you have to start with these ratios against
> your static tone and find a common unit proportion that works for them all. In
> example 6 there is a printer's mistake for the ratios
> reading down are 5/4 9/7 and 27/22 and the order should be 9/7 5/4 27/22. The
> 9/5 should be over to the right also.
>
>

Noted and many thanks Kraig.

Sincerely
a.m.