ARCHIVED WIKISPACES DISCUSSION BELOW
Please do not add any new discussion to this archive page.
All new discussion should go on Talk:Hobbits.
Can we explain these things in English as well?
It would be a nice thing to not immediately assault the reader with PhD mathematical treatise-speak when they learn about scale classification....
Granted are two points:
1) Much scale research goes on on the (various) Yahoo tuning lists. The research is valuable and discoveries are made, etc.
2) Potentially, many of these discoveries are of musical value.
BUT---if we care/want musicians to even tinker with these concepts, we will scare them away with talks of "vals" and "monzos"...to the outsider, these are simply neologisms.
Just a thought. The xenwiki is setup as a resource, one would hope, for musicians, and not simply for PhD mathematicians.
- akjmicro March 07, 2011, 01:01:50 PM UTC-0800
I agree with you. I tried to force a definition of val, monzo by adding links to (future) wiki pages, but this will not improve the readability.
As I've learned, some notations are very efficient, but some of them I cannot decode at all. Maybe some examples for these techniques will suffice for the non-mathematician of us.
- xenwolf March 07, 2011, 11:53:22 PM UTC-0800
Here's an outline for how I think the page can be improved. Unfortunately the details will need to be filled in by Gene.
In 2001 it was discovered that [MOS] scales correspond to rank 2 regular temperamnts. For a regular temperament of arbitrary rank, there is a corroseponding hobbit scale (named for their relationship to [dwarf scales]). Examples may be found on the [Scalesmith] page.
Given a regular tempearment, we first find an equal temperament [val] v which supports it. Next, we define an [octave equivalent seminorm] for the temperament. This seminorm applies to monzos and sends any comma of the temperament, and also the octave, to zero. It measures the complexity of the octave-equivalent pitch class to which a monzo belongs. For example, in meantone temperament, the OE seminorm of 3/2 is _ and of 3/1 is _ and of 25/16 is ___. Roughly speaking, the hobbit is the scale consisting of the lowest OE complexity for each scale step mapped to the integer i by v (see [epimorphic] for details).
In the rank 2 case, the hobbit for v is identical to the MOS for v because...
Consider the 7-note hobbit for meantone temperament ... yields the familiar diatonic scale (MOS)...
Next, consider the 22-note hobbit for minerva temperament...
Denoting the OE seminorm...
- clumma March 08, 2011, 10:16:42 PM UTC-0800
Wow, I cannot edit this. Useless.
- clumma March 08, 2011, 10:17:22 PM UTC-0800