Re Decimals, etc.

I don't know of anyone still using Savarts or rationalized Savarts. Yasser used centitones (600 to the octave) in A Theory of Evolving Tonality, but I've not seen them used elsewhere. A.G. Piklers article Logarithmic Frequency Systems, Journal of the Acoustic Society of America 39(6): 1102-1120, 1966 is the most comprehensive I know of, but it's limited to the literature pre-1965.

The late composer Paul Beaver suggested "Harmos," the cycle of 1728 to the octave (12 cubed) in duodecimal notation to me. This system divides the semitone into 144 parts. For Bohlen-Pierce fans, 1300 to the 3/1 might be useful.

--John

I also like a system that divides the octave purely dozenally/duodecimally. The (minor) problem I have with cents is that it's a multiple-radix system (dozenal and decimal hybrid), not the fact it has anything to do with twelve. A single base unit and single number base, be it ten or twelve, is all that's needed.

Beyond intervals, binary logs in dozenal or decimal notation can be used for specifying pitch. Something of an absolute interval compared against the second.

> The late composer Paul Beaver suggested "Harmos," the cycle of 1728 to
> the octave (12 cubed) in duodecimal notation to me. This system divides
> the semitone into 144 parts.

--- In tuning@yahoogroups.com, "John H. Chalmers" <JHCHALMERS@...> wrote:

> The late composer Paul Beaver suggested "Harmos," the cycle of 1728 to
> the octave (12 cubed) in duodecimal notation to me. This system divides
> the semitone into 144 parts.

I hope you turned him down.

If someone really wants to indulge a taste for powers of a small number for such a system, may I suggest 3125? It's aces for the 7-limit and is five to the fifth power, which anyone adept at such things could easily endow with mystical significance. If that's too many notes, 441 is there, it's square, and I've actually used it for tuning purposes. Neither could be accused of having a bias in favor of 12 equal.

Of course, you need a whole new way of writing numbers to go along with this. For 3125, I suggest digits +1 and +2 are N and R, -1 and -2 are the Cyrilic letters I and Ya, which look like backwards N and bsckwards R. That should be exotic enough for the people looking for such things.

Would that be a reverse notation?  You don't really need new digits for quinary numeration, which would be another benefit over dozenal.

--- On Wed, 7/7/10, genewardsmith <genewardsmith@...> wrote:

From: genewardsmith <genewardsmith@...>
Subject: [tuning] Re: Re Decimals, etc.
To: tuning@yahoogroups.com
Date: Wednesday, July 7, 2010, 1:51 PM

--- In tuning@yahoogroups.com, "John H. Chalmers" <JHCHALMERS@...> wrote:

> The late composer Paul Beaver suggested "Harmos," the cycle of 1728 to

> the octave (12 cubed) in duodecimal notation to me. This system divides

> the semitone into 144 parts.

I hope you turned him down.

If someone really wants to indulge a taste for powers of a small number for such a system, may I suggest 3125? It's aces for the 7-limit and is five to the fifth power, which anyone adept at such things could easily endow with mystical significance. If that's too many notes, 441 is there, it's square, and I've actually used it for tuning purposes. Neither could be accused of having a bias in favor of 12 equal.

Of course, you need a whole new way of writing numbers to go along with this. For 3125, I suggest digits +1 and +2 are N and R, -1 and -2 are the Cyrilic letters I and Ya, which look like backwards N and bsckwards R. That should be exotic enough for the people looking for such things.

Gene>"If that's too many notes, 441 is there, it's square, and I've actually
used it for tuning purposes. Neither could be accused of having a bias in favor
of 12 equal."
True enough.

--- In tuning@yahoogroups.com, Tony Taylor <leopold_plumtree@...> wrote:
>
> Would that be a reverse notation?  You don't really need new digits for quinary numeration, which would be another benefit over dozenal.

Yes, but why use the boring digits {0,1,2,3,4} when if you use {-2,-1,0,1,2} you don't need special notation for negative numbers? Plus, everything gets adjusted from your 1/1 symmetrically, the way we'd like it to be.