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"Because unsaturated subgroups of Z^n are problematic…"
It seems like the article took an unexpected turn as soon as I read this sentence. Can somebody explain to me why this is problematic?
- Sarzadoce October 27, 2012, 11:52:11 AM UTC-0700
The previous two paragraphs consist of an explanation for this claim.
- genewardsmith October 27, 2012, 08:06:19 PM UTC-0700
I assume you are talking about this:
"… if C isn't saturated it may be regarded as pathological, as it has notes which cannot be reached from the unison by tempered rational intervals."
So I'll reiterate my question: Why is this problematic?
- Sarzadoce October 27, 2012, 11:59:00 PM UTC-0700
This doesn't seem correct:
"If C represents the commas (nullspace or kernel) of a supposed regular temperament, ie the intervals it tempers out, then if C isn't saturated it may be regarded as pathological, as it has notes which cannot be reached from the unison by tempered rational intervals. Similarly, if V is the subgroup of vals of the temperament, and is not saturated, then we obtain a temperament of sorts in which all of the notes cannot be reached by tempered intervals."
The whole concept of having notes which cannot be reached from the unison by tempered rational intervals would seem to apply only if V is unsaturated, not C. If C isn't saturated, then if G is the group of JI intervals, the problem is that G/C has torsion, not that it has unmapped intervals.
- mbattaglia1 October 28, 2012, 12:09:30 AM UTC-0700
Yes, there is one example, but it is rarely applicable for ordinary musicians, composers or mikrotonalists who have no mathematical background. Can you show us how this concept can be used in microtonal / xenharmonic music?
- ) Thanks in advance
- xenwolf June 16, 2011, 11:59:40 PM UTC-0700