Myhill's property

Rick Sanford and Carl Lumma recently mentioned Myhill's property.
Gerald Myerson and I defined Myhill's property (MP) for an artbitrary
set of numbers mod n: iff every generic interval appears in
exactly two sizes, the set is said to have MP. A generic interval
is what we think of musically as a 2nd, 3rd, etc. (but excluding the
prime or modular interval itself). For example,
the set {0, 2, 4, 5, 7, 9, 11} in mod 12 qualifies as MP.

Our paper "Musical Scales and the Generalized Circle of Fifths" with the
definition of MP appeared in Amer. Mathematical Monthly,
93/9 (Nov. 1986).

John Myhill BTW was a very famous logician who was involved in
computer music at SUNY-Buffalo in his later years.

John Clough

John Clough wrote,

>Gerald Myerson and I defined Myhill's property (MP) for an artbitrary
>set of numbers mod n: iff every generic interval appears in
>exactly two sizes, the set is said to have MP. A generic interval
>is what we think of musically as a 2nd, 3rd, etc. (but excluding the
>prime or modular interval itself). For example,
>the set {0, 2, 4, 5, 7, 9, 11} in mod 12 qualifies as MP.

John -- Is it true that Myhill's property implies distributional
evenness, but the converse is not necessarily true?

To John Clough;
I was wondering your thoughts on the 2nd layer MOS or what your
might call the Myhill of Myhill as seen in the TANABE CYCLE . These
common scales taken from the very series you mentioned (0,2,4,5,7,9,11)
seem to have the property that the generator is always the same size
subtended by the same number of steps even though some of the thirds are
smaller than some of the seconds.
-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

Message text written by INTERNET:tuning@onelist.com
>John Myhill BTW was a very famous logician who was involved in
computer music at SUNY-Buffalo in his later years.

John Clough<

Myhill was one of the leaders in constructive mathematics (a successor to
Brouwer's intuitionism) as well as a pioneer in algorithmic composition.
The relationship between the two activities is of interest.

Right, Paul, MP implies DE, but DE does not imply MP.

John Clough

On Fri, 12 Feb 1999, Paul H. Erlich wrote:

> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
> John Clough wrote,
>
> >Gerald Myerson and I defined Myhill's property (MP) for an artbitrary
> >set of numbers mod n: iff every generic interval appears in
> >exactly two sizes, the set is said to have MP. A generic interval
> >is what we think of musically as a 2nd, 3rd, etc. (but excluding the
> >prime or modular interval itself). For example,
> >the set {0, 2, 4, 5, 7, 9, 11} in mod 12 qualifies as MP.
>
> John -- Is it true that Myhill's property implies distributional
> evenness, but the converse is not necessarily true?
>
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