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more general definition
I added the slightly wrong example of the major scale in 12edo, according to the given definition, it should be the Locrian mode.
What about the more general formula
with i and disp in the range [0,steps-1] ?
- xenwolf January 10, 2014, 01:23:16 AM UTC-0800
what is quasi-equal?
I've noticed that folks tend to use the phrase "quasi-equal" in a less strict sense than "maximally even". This might be a good way to go, and I wonder how people feel about it. We could use QE to describe:
1. Scales that are "close" to equal, leaving "close" subjective and context-sensitive.
2. Scales where L:s is within a specific range.
In my MOS investigations page, I noted a few different ranges of L:s and the consequences this has on chromatic alterations of the scale. In particular, there is:
Case A: L/s<2. The chroma is smaller than the s step (c<s).
Case B: L/s=2, ie. L=2s. The chroma and the small step are the same (c=s).
Case C: L/s>2. The chroma is larger than the s step (c>s).
Case A1: 2<L/s<3/2. The diminished step (s-c) is smaller than the chroma (d<c).
Case A2: L/s=3/2. The diminished step and the chroma are the same (d=c).
Case A3: 3/2<L/s<1. The diminished step is larger than the chroma (d>c).
So I wonder if we want to reserve the term "quasi-equal" for Case A3, where 3/2<L/s<1. Is this a useful range to have a name for, and does it make sense to use this name for that?
Another possibility is that QE could also include Case A2, where L/s = 3/2. What say ye?
- Andrew_Heathwaite June 20, 2012, 07:46:18 AM UTC-0700
As I understand Mandelbaum's definition, it just says two intervals of adjacent size - I.e. L-s=1, nothing else.
For "maximally even", there is an additional condition: that the intervals are distributed as evenly as possible. This means that, e.g., the melodic minor scale in 12edo is quasi-equal but not maximally even.
- hstraub December 25, 2013, 08:56:08 AM UTC-0800
These are exactly the same thing as maximally even scales, right?
- keenanpepper November 26, 2011, 02:03:45 PM UTC-0800
Hm, looking now at the Wikipedia page for "Maximally Even," I think that is the case. I initially got the concept and phrase from Joel Mandelbaum's dissertation on 19-one equal temperament (which I think I read on the web, but I can't see to find now). I wasn't aware there was another phrase for this property.
- Andrew_Heathwaite November 26, 2011, 07:01:11 PM UTC-0800
That sounds as if the claim in Wikipedia that Clough and Douthett were the first with the idea is false.
- genewardsmith November 26, 2011, 07:04:31 PM UTC-0800
I would very much love to hear that Mandelbaum preceded Clough and Douthett with maximal evenness. Is that true?
- mbattaglia1 November 26, 2011, 09:04:16 PM UTC-0800
I am absolutely not making that claim! As far as I know, Mandelbaum only preceded Clough and Douthett in my own experience. I'll be the first to tell you that there's a lot I don't know about. As I said above, I'm pretty sure I found Mandelbaum's dissertation on the web somewhere, and if you can find it that might clear up the confusion a bit. I'm not attached to the phrase "quasi-equal," it's just the only one I knew about. If "maximal evenness" is more standard, it's no problem for me to start using it.
- Andrew_Heathwaite November 27, 2011, 07:44:48 AM UTC-0800
Ok, here it is -- from 1961.
- Andrew_Heathwaite November 27, 2011, 07:46:34 AM UTC-0800
Hmm, in which chapter of Mandelbaum's dissertation did you see the definition analogous to maximal evenness?
- hstraub December 25, 2013, 02:55:07 AM UTC-0800
Chapter 14, page 4-5.
- Andrew_Heathwaite December 25, 2013, 07:40:28 AM UTC-0800
"The systems considered here will all have in common that they are based on scales containing intervals as equal as is possible within a 19-tone system. ...as the diatonic scale in 12-tone equal temperament is composed only of the intervals 1/12 and 2/12 ... so all of the scales to be considered here contain a range of two intervals of adjacent size; the size of the intervals will depend on the number of tones in the scale. This is what is meant by the term 'quasi-equal' in the title of this supplement."
- Andrew_Heathwaite December 25, 2013, 07:44:20 AM UTC-0800
Thanks. Indeed, what Mandelbaum calls "quasi-equal-interval-symmetrical" (QEIS) is, as far as I can see, exactly the same as maximally even. Quasi-equal alone is a little less.
- hstraub December 25, 2013, 08:38:26 AM UTC-0800
Ah yes. Looking more closely at the text, I see that you are correct. Thanks for the clarification!
- Andrew_Heathwaite December 25, 2013, 10:50:59 AM UTC-0800