Talk:Chords of diaschismic/WikispacesArchive
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Seeking Clarification
This page has info I want in a form I don't yet understand well enough. I'd appreciate some help. I'm familiar with diaschismic temperament as found in, say, 58edo. I want to be able to read, write and converse about its chords, in a manner consistent with the xenharmonic efforts toward a generalized or plenitonal music theory. What I do understand are the transversals and some, but not all, of the normal mapping. I don't understand much more of what's here.
Let's begin here: "The normal mapping for diaschismic is dia = [<2 0 11 31 45|, <0 1 -2 -8 -12|]." The second part of this mapping is clear enough. It's a list of the generator steps required to approximate the prime harmonics in order. Getting to 2/1 requires no steps. Getting to 3/2 requires +1 step. Getting to 5/4 requires -2 steps. Getting to 7/4 requires -8 steps. Getting to 11/8 requires -12 steps. But what is meant by <2 0 11 31 45|, the first part of this mapping? I suspect the initial 2 is the number of periods per octave, but what do the other numbers mean? The Mappings page implies they also refer to the period chain, but whence then the 0 that follows the 2?
Let's proceed to the next sentence: "From this we may derive a val v = dia[1] - 100 dia[2] = <2 -100 211 831 1245| which we may use to sort and normalize the chords of diaschismic." This operation isn't hard to follow. Two minus the product 0*100 is +2. Zero minus the product 1*100 is -100. Eleven minus the product -2*100 is +211. Thirty-one minus the product -8*100 is +831. And 45 minus the product -12*100 is +1245. However, the meaning of the operation, and the reason for choosing 100 as the multiplier for dia[2], both baffle me, especially given my confusion and ignorance on the first part of the mapping.
The following sentence is crucial: "Under 'Chord' is listed the chord, normalized to start from zero, in the mapping by v."
Looking down to the table of triads, the first is 0-104-208, with transversal 1-4/3-16/9, a chord that a simpleton like me might label a 9:12:16. So in the 104 that represents 4/3, the 100 represents 1*(1/3) and the 4 represents 2*2. The 208 is twice this, just as 16/9 is a stack of two 4/3 intervals. Am I right so far?
Suppose the 4:5:6 major triad were listed among the chords of diaschismic, with transversal 1-5/4-3/2. It can be played in diaschismic; in 58edo it sounds fresher and sweeter than in 12edo. The normalized chord value would start with 0-207, with 207 for 5/4 because the 5/ would yield +211 and the /4 would yield -4. But what number would represent 3/2? The 3/ would yield -100 and the /2 would yield +2, with -98 as the sum. Yet none of the normalized chord values in the tables on this page include negative numbers, do they? (The hyphens are delimiters, not subtraction or negation signs.) Then again, the tables don't include the 3/2 transversal anywhere, do they? The Euler-Fokker genus tetrad 8:10:12:15 can certainly be played in diaschismic, with transversal 1-5/4-3/2-15/8, but it isn't listed, is it? At least, not in that form.
I'm betting that vital pieces of this puzzle are present, either on this very page, or elsewhere in the wiki, but that gaps in my comprehension have prevented me from spotting them. Could somebody, or plural somebodies, please fill me in? I'd appreciate it.
- manuphonic December 17, 2015, 08:28:51 AM UTC-0800
The first val in the mapping [<2 0 11 31 45|, <0 1 -2 -8 -12|] represents the half-octave period P = 45/32, and the second val represents the generator g = 3/1. It can be read as following:
2/1 = 2P + 0g
3/2 = 0P + 1g
5/4 = 11P - 2g
7/4 = 31P - 8g
11/8 = 45P - 12g
If we ignore octaves, we can simply ignore even values in the first val, and add P if it is odd:
2/1 = 0
3/2 = 1g
5/4 = P - 2g
7/4 = P - 8g
11/8 = P - 12g
You can also use the mapping [<2 3 5 7 9|, <0 1 -2 -8 -12|], with period P = 45/32 and generator g = 16/15, s.th. P + g = 3/2 (in a logarithmic sense). This is the mapping used by Graham Breed's temperament finder, btw.:
http://x31eq.com/cgi-bin/rt.cgi?ets=12_46&limit=11&tuning=po
- Gedankenwelt December 17, 2015, 12:15:17 PM UTC-0800
...sorry, I messed that up, the first listing should read:
2/1 = 2P + 0g
3/1 = 0P + 1g
5/1 = 11P - 2g
7/1 = 31P - 8g
11/1 = 45P - 12g
- Gedankenwelt December 17, 2015, 12:16:50 PM UTC-0800
Thanks for helping me work through the mapping for diaschismic. Using 16/15 instead of 3/2 as generator makes a sensible basis; having the generator smaller than the period makes the mapping easier to follow. This gives us
2/1 = 2P + 0g
3/1 = 3P + 1g
5/1 = 5P - 2g
7/1 = 7P - 8g
11/1 = 9P - 12g
So how does this affect the val for chord sorting and normalization, v = dia[1] - 100 dia[2]? IIUC it yields <2 -97 205 807 1209|, not greatly different from what Gene Ward Smith put on the page above. So having the 4/3 transversal in a chord would be normalized +101, whereas 3/2 would still be negative at -93.
The chord listings on this page and elsewhere in the wiki imply that negative numbers are not allowed in chord normalizations but, if they were, a 4:5:6 major triad with transversal 1-5/4-3/2 would be normalized 0-201-(-93), and a generic 8:10:12:15 with transversal 1-5/4-3/2-15/8 would be normalized how? The 15/8 has a numerator 15/ that would yield 205-97=108 and a denominator /8 that would yield -3*2=-6 for a sum of 102? Which would give us 0-201-(-93)-102? Jeez, no wonder negative numbers are deprecated in the tables!
So how then do we talk about 4:5:6 and 8:10:12:15 and other chords with 3/2 and/or 9/8, et cetera, in them in diaschismic? Again, I'm betting the answer is obvious to people who comprehend musical math better than I. If I had to guess, I'd speculate that inverse intervals are treated as equivalents in this system, such that 1-16/15-5/4-4/3 or even 1-5/4-4/3-5/3 would substitute for 8:10:12:15. But that guess must be wrong, since neither of those transversals are listed among the diaschismic chords on this page. There isn't even a 1-5/4-4/3 transversal to substitute for the major triad.
Lost, I still am. Again I appeal for help.
- manuphonic December 18, 2015, 04:19:13 AM UTC-0800
Triad number 2 has transversal 1-4/3-5/3. Triad number 3 has transversal 1-5/4-5/3. So why is there no tetrad with transversal 1-5/4-4/3-5/3?
- manuphonic December 18, 2015, 04:24:15 AM UTC-0800
More on diaschismic from Graham Breed's temperament finder:
http://x31eq.com/cgi-bin/rt.cgi?ets=12f_46&limit=17
- manuphonic December 18, 2015, 04:39:04 AM UTC-0800
Sorry for the delayed response - I wasn't completely familiar with the subject either, and used the incentive to catch up on it.
I think the general idea here is to generate dyadic chords (here: octaves and inversion
don't matter) over the consonance set of 11-odd-limit intervals in diaschismic
temperament, characterize them by type (o-/u-/ambitonal or tempered), and sort them by
Graham complexity.
First of all, here's a list of all 11-(odd-)limit consonances that are generated by
stacking the generator downward, and applying some periods. The first value is the value calculated from v, which you see under "Chord", followed by the actual interval. The () contain the number of generators g required to generate the interval:
0: 1/1 (0)
104: 4/3 (-1)
207: 5/4 (-2)
208: 16/9 (-2)
311: 5/3 (-3)
413: 10/9 (-4)
414: 11/7 (-4)
620: 7/5 (-6)
827: 7/4 (-8)
929: 7/6 (-9)
1032: 11/10 (-10)
1033: 14/9 (-10)
1239: 11/8 (-12)
1343: 11/6 (-13)
1445: 11/9 (-14)
The Graham complexity for each interval is the number of generators (absolute value) required to generate the interval, multiplied by the number of octaves. In other words, it's the absolute value of the number in () times 2. For example, for 5/4, the Graham complexity is |-2| * 2 = 2 * 2 = 4. As the page states, we get the same value by dividing the values to the left by 100, rounding, and multiplying 2.
The chords are generated by combining the above intervals, starting with 1/1, and sorted by complexity. This is why the 4th triad is listed as 1-4/3-10/9, and not as 1-10/9-4/3. Generating chords by only using the above list of intervals helps us avoid redundancy, and we will get 1-4/3-5/3, but not its inversion 1-5/4-3/2. Also, in this representation, the Graham complexity of the chord is simply the Graham complexity of the most complex interval (i.e. the rightmost interval in the sequence).
Since only dyadic chords are considered, chords that feature intervals that are not in the consonance set have to be removed. For example, 1/1-5/4-4/3 is not a dyadic chord over the set of 11-odd-limit consonances, because it contains a 16/15 between 5/4 and 4/3.
I assume 100 is multiplied because it's large enough s.th. periods can be ignored (they're not relevant in the interval composition for Graham complexity), and because it's a power of 10, which allows to easily derive the Graham complexity w/o using a calculator.
(can someone confirm this?)
What I don't completely understand is how the normal mapping is defined, and how it is different from [<2 3 5 7 9|, <0 1 -2 -8 -12|]. Or are both considered a normal mapping, and the wording - which suggests there's a unique normal mapping - is just misleading?
- Gedankenwelt December 20, 2015, 12:30:49 AM UTC-0800
"The Graham complexity for each interval is the number of generators (absolute value) required to generate the interval, multiplied by the number of octaves."
Sorry, that should be "number of periods per octave". Also, the formatting at the beginning got messed up somehow.
- Gedankenwelt December 20, 2015, 12:36:02 AM UTC-0800
Oh, I had been thinking "11-limit" meant prime limit rather than odd limit. Thank you, you've clarified things a lot.
- manuphonic December 21, 2015, 12:46:36 PM UTC-0800