Tuning-Math Archive Section 1: 251

Message: 251 - Contents - Hide Contents Date: Tue, 19 Jun 2001 20:09:50 Subject: Re: Stretched tuning experiments (was: First melodic spring results) From: Paul Erlich --- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote:

> [Paul wrote:]

>> John, what did you think of Herman's "1/7-comma meantone with 1/7- >> comma-stretched octaves" version, as compared with, say, 55-tET? >

> Pulling them up right now! Stretched 1/7 comma meantone:

Oops -- Herman should really clarify -- this is not stretched 1/7- comma meantone, but rather a tuning where the fifths are 1/7-comma narrow and the octaves are 1/7-comma wide.

Message: 254 - Contents - Hide Contents Date: Tue, 19 Jun 2001 22:57:57 Subject: Re: Stretched tuning experiments (was: First melodic spring results) From: Herman Miller On Tue, 19 Jun 2001 14:48:00 -0600, "John A. deLaubenfels" <jdl@a...> wrote:

>[Paul:]

>> Oops -- Herman should really clarify -- this is not stretched 1/7- >> comma meantone, but rather a tuning where the fifths are 1/7-comma >> narrow and the octaves are 1/7-comma wide.

Right; this is actually one of my favorite tunings for playing traditionally notated music on the DX7II. 1/7-comma wide octaves and 1/7-comma narrow fifths. Although sometimes I just call it "1/7-comma stretched meantone" out of laziness.

>Oh, I meant that, despite fewer words. Although, I see that Herman >is saying "1/7-comma meantone with 1/7-comma tempered octaves"; perhaps >if the word "tempered" were changed to "stretched", it'd be clearer... Done.

Message: 255 - Contents - Hide Contents Date: Wed, 20 Jun 2001 03:10:27 Subject: 7/72 generator in blackjack From: jpehrson@r... Well, this isn't very advanced... but, if not math, at least it's arithmetic... I still don't understand how 7 of the 72-tET scale is a generator of blackjack. It's a great concept (spooky!) since we have been finding 7's to be very peculiar in some other instances... Would someone please go over that again, gently?? Thanks! Joseph

Message: 256 - Contents - Hide Contents Date: Tue, 19 Jun 2001 21:50:44 Subject: Re: 7/72 generator in blackjack From: monz

> ----- Original Message ----- > From: <jpehrson@r...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, June 19, 2001 8:10 PM >Subject: [tuning-math] 7/72 generator in blackjack > > > Well, this isn't very advanced... but, if not math, at least it's > arithmetic...

That's OK, Joe... this list is for math dummies like me, too, as well as guys like Paul, Dave, and Graham who understand the more esoteric stuff.

> > I still don't understand how 7 of the 72-tET scale is a generator of > blackjack. It's a great concept (spooky!) since we have been > finding 7's to be very peculiar in some other instances... > > Would someone please go over that again, gently??

Take another look at the explanation and especially the diagram below the graph at Definitions of tuning terms: MIRACLE scale, (c... * [with cont.] (Wayb.) Dave Keenan found the MIRACLE generator (~116.7 cents) by use of the "brute force" approach: he had his computer perform thousands (millions?.. billions?) of calculations and analyze the resulting scales. The ~116.7-cent generator came out on top as implying the largest number of 11-limit consonances. 2^(7/72) happens to be extremely close to the calculated MIRACLE generator (which, I should emphasize, is only *one* possible MIRACLE generator... there can be many, depending on the error method selected). The diagram on my Dictionary page shows how you cycle thru intervals of 2^(7/72) on either side of 1/1, which in this case really should be called 2^(0/72). Upon reaching the 10th note on either side, you've got Blackjack. Extending to the 15th note on either side gives you Canasta. This process is exactly analagous to constructing a meantone cycle, except that instead of a "cycle of 5ths", you're cycling thru the generator interval, whatever it may be. An interesting digression: some scales can be thought of as being constructed by more than one generator simultaneously. Naturally, our familiar old 12-EDO is one such scale. It can be thought of as a "cycle of 5ths" where each "5th" is 7 Semitones; here's an example centered on "D" (flats and sharps are of course equivalent to their enharmonic cousins): Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Or it can be thought of as a cycle of Semitones: Ab - A - Bb - B - C - C# - D - Eb - E - F - F# - G - G# -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 In either case, the generator creates a scale of 12 distinct pitches before producing a pitch which is an exact replica of one already existing. -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 257 - Contents - Hide Contents Date: Wed, 20 Jun 2001 06:10:54 Subject: Re: CS From: carl@l...

>>> >he Consonant-CS thing makes sense, but for the more general >>> case I think you can leave out the computational complexity of >>> Harmonic Entropy and just say that a scale is "CS to a tolerance >>> of X cents". >>

>> Well, I suppose it would be the same difference as rating an ET >> by max cents error, or something like Erlich's accuracy. >

> Not really. In the case of a CS tolerance, more cents is better.

Yes, but a scale with many (bad errors, near collisions) may have the same (max error, min tolerance) as a scale with only one (error, near collision).

> The probability of the average listener perceiving as the same, two > pitches differing by some amount. It's from that dude's experiment > that Paul quotes for HE. 1% change in frequency ~= 17 cents.

My goodness! I don't think we want Goldstein's 1% bit involved here.

> i.e. ... of the scale's smallest step. Yes, that would be useful to > know, as well as the absolute tolerance. But as a figure of merit > in comparing improper MOS, the absolute tolerance seems more > important to me. Can you explain further. Maybe with an example.

Well, I'd argue that a near-collision of 10 cents would not be as important in an otherwise-even pentatonic as it would in a decatonic. I'll note here that non-CS may be desirable. As I once said, 'ambiguous intervals [collisions] are the common tones of melodic modulation'. -Carl

Message: 258 - Contents - Hide Contents Date: Wed, 20 Jun 2001 06:14:39 Subject: Re: 7/72 generator in blackjack From: Dave Keenan --- In tuning-math@y..., "monz" <joemonz@y...> wrote: Joseph, As a complement to Monz's explanation, see what I just added to http://uq.net.au/~zzdkeena/Music/MiraclePitchChart.gif - Type Ok * [with cont.] (Wayb.) It shows the octave as a circle divided into 72 parts. Start at D> and follow the (new) straight line segments clockwise. You'll see each one jumps 7/72 of an octave. When you've done that 20 times and wound up at D<, you've generated Blackjack.

> Dave Keenan found the MIRACLE generator (~116.7 cents) by use > of the "brute force" approach: he had his computer perform > thousands (millions?.. billions?) of calculations and analyze > the resulting scales. > > The ~116.7-cent generator came out on top as implying the > largest number of 11-limit consonances. ...

Er, no. That all came _after_ the discovery, and merely confirmed its "miraculous" nature (as a 7-limit or 11-limit generator, but not necc. 9-limit). I was afraid there might have been some holes in my search strategy, but since then Graham Breed has performed a search using a completely different method to mine, and (I think?) further confirmed it. Strictly speaking, the MIRACLE generator was discovered by Paul Erlich, who extracted it from a scale that I posted, the 31-noter that we now call Canasta. Paul then recognised that there was a more manageable (although improper) MOS with 21 notes using the same generator. So historically it went: Canasta - MIRACLE generator - Blackjack - Decimal scale (although, apart from "Blackjack" we didn't call them that immediately). But logically it goes: MIRACLE generator - Decimal scale - Blackjack - Canasta. So Graham, By what figure-of-demerit and at what odd-limits can we claim that the MIRACLE generator is the best? Does cardinality_of_smallest_MOS_containing_a_complete_otonality divided by exp(-(minimax_error/17c)^2) do it at 7 and 11 limits? What's the best 9-limit generator by this FoD? I'm sure some folks would be interested in the 13-limit result too. Regards, -- Dave Keenan

Message: 259 - Contents - Hide Contents Date: Wed, 20 Jun 2001 02:43:52 Subject: Decatonic (in 72-tET) From: Pierre Lamothe I wish I would have prepared lattice drawings and full comments to publish with these decatonic structures ib1183 and ib1215 <decatonic * [with cont.] (Wayb.)> but I have to quit for few weeks or months. Since these structures are closed related to temperament 72, I think someone could be interested, even without any comment. (I used a scale with 36 divisions but it is easy to detect points on divisions and points in-between). The ib1183 structure is the minimal one corresponding to the Paul Erlich's decatonic scales, as explained in "For all strict-JI fans . . ." at Yahoo groups: /tuning/message/20746 * [with cont.] and the ib1215 is the minimal decatonic in non-degenerated 11-limit. ----- Since Monzo has opened a door about complexity notion, I would add soon a new definition about complexity in a next message, before to quit. Pierre

Message: 262 - Contents - Hide Contents Date: Wed, 20 Jun 2001 17:09:16 Subject: Challenging problem about sonance From: Pierre Lamothe Since my abstract thoughts seems not to be well appreciated, here is a "concrete" challenging problem having goal only to introduce my definition of _Sonance degree_ in my next post. If I have no feedback, I could so have illusion that it's not only by lack of interest :-) Could you calculate an approximative "sonance" value for any interval represented by a coordinate vector (x0 x1 .. xN)* in a primal basis <p0 p1 .. pN> where only p0 is known, say as the prime 2? As such that question would have no sense, but with a given coherent set of unison vectors defining a system, this problem has a deep sense : the first coordinate (the power of 2) retains IN A COHERENT SYSTEM sufficient information to obtain this well-approximated value for the "sonance". More concretely. Let U = {(-4 4 -1)*,(-3 -1 2)*} a such set of unison vectors defined by its coordinates. Without using the fact that these vectors would represent 81/80 and 25/24 in <2 3 5>, it is possible to easily calculate a numerical value which is the _sonance degree_ in this system for any interval represented by (x y z)*, say (-1 1 0)* and (-3 1 1)* which would be 3/2 and 10/9 in <2 3 5>. My definition of _Sonance degree_ simply generalises that. Pierre

Message: 263 - Contents - Hide Contents Date: Thu, 21 Jun 2001 01:54:56 Subject: Re: Stretched tuning experiments (was: First melodic spring results) From: Dave Keenan --- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:

> On Tue, 19 Jun 2001 14:48:00 -0600, "John A. deLaubenfels"

>> Oh, I meant that, despite fewer words. Although, I see that Herman >> is saying "1/7-comma meantone with 1/7-comma tempered octaves"; perhaps >> if the word "tempered" were changed to "stretched", it'd be clearer... > > Done.

No. I think maybe John still doesn't get it. It's definitely "with octaves tempered 1/7-comma wide" and _not_ "with a stretch of 1/7 comma per octave". These are very different things. A "stretch" is in fact applied to _all_ intervals, not merely the octave. However it is usually specified as so many cents per octave. Its purpose is usually to compensate for a stretched inharmonic timbre or a property of human pitch perception. A meantone with a tempered octave means that for a given note we not only need to know where it is on the chain of fifths, but how many octaves one has to reduce its stacked fifths by, to bring it back to the "home" octave. Herman's scale gives the optimum distributiion of the syntonic comma for the intervals 1:2, 2:3 and 4:5. It essentially doesn't give a damn about other 5-limit intervals like 3:4, 5:6, 3:5, 5:8. Dan Stearns, are you reading? This is an excellent example of a tuning that optimises only the rooted intervals (in this case including the octave), giving them all equal weight. Here are the three scales to the nearest cent 1. Ordinary 1/7 comma meantone 2. 1/7 comma meantone with a stretch of 1/7 comma per octave 3. 1/7 comma meantone with octaves tempered 1/7 comma wide. Name Octaves Fifths 1/7 with with comma stretch tempered octaves ----------------------------------------------- C 0 0 0 0 0 C# -4 7 92 92 80 D -1 2 198 198 195 Eb 2 -3 303 304 309 E -2 4 396 397 389 F 1 -1 501 502 504 F# -3 6 593 595 584 G 0 1 699 701 699 G# -4 8 791 793 779 A -1 3 897 899 894 Bb 2 -2 1002 1005 1008 B -2 5 1094 1097 1088 C 1 0 1200 1203 1203 Look at E. Notice how stretch makes the major thirds (4:5 = 386c) slightly worse, but wide-tempered-octaves makes them significantly better. Regards, -- Dave Keenan

Message: 264 - Contents - Hide Contents Date: Thu, 21 Jun 2001 02:04:13 Subject: Re: 7/72 generator in blackjack From: jpehrson@r... --- In tuning-math@y..., "monz" <joemonz@y...> wrote: Yahoo groups: /tuning-math/message/256 * [with cont.]

> > Take another look at the explanation and especially the diagram > below the graph at > Definitions of tuning terms: MIRACLE scale, (c... * [with cont.] (Wayb.) > > > Dave Keenan found the MIRACLE generator (~116.7 cents) by use > of the "brute force" approach: he had his computer perform > thousands (millions?.. billions?) of calculations and analyze > the resulting scales. > > The ~116.7-cent generator came out on top as implying the > largest number of 11-limit consonances. 2^(7/72) happens > to be extremely close to the calculated MIRACLE generator > (which, I should emphasize, is only *one* possible MIRACLE > generator... there can be many, depending on the error method > selected). > Hello Monz!

Thank you so much for this interesting response. Well, I read ahead, and I guess Dave Keenan credits Paul Erlich with the discovery of the MIRACLE generator. I guess that's how I remember it, too, looking back at the chain (literally!) of events... Well, from what you are saying, then, 31-EDO at with a 116.1 generator is also a MIRACLE scale, yes... of a sort?? Of course, I had always heard of the "special" properties of 31-EDO... Then, it is not too coincidental that a 31-tone NON-EDO using 116.7 would be "miraculous" as well... I mean it's not so miraculous that both scales would have 31 notes, correct?? I found your "MIRACLE GENERATOR" page fascinating. I guess I really hadn't read that one as carefully as the page that pertained PARTICULARLY to Blackjack... I remember when Paul Erlich did the calculus to figure out the "RMS" or root mean square method to find the errors. Although I can't specifically follow this in the entire, it seems rather related to a discussion that I had with Graham Breed and John deLaubenfels about finding "errors" by squaring things and then taking the square root... Is that correct? It looks as though Paul, in his calculations, is trying to find the least errors for all the various intervals he is considering, in squaring them and so forth, and then puts it all nicely back in a pie (not a pie chart!) to determine the MIRACLE generator at 116.7 cents. Am I getting that at all...?? ANYWAY, I also very much appreciated the 72-EDO chart at the bottom of the MIRACLE generator page... That REALLY did a good job of breaking down how the various scales are related to the generator... Well, the chart with the colors for the various scales did too... It seems I'm gradually coming to a greater appreciation of this process, finally... although I have to admit I was rather "left in the dust" when it happened. (In fact, I became so confused, that Paul thought I had forgotten the entire discussion!) Well, anyway, thanks for the help! _________ _______ _________ Joseph Pehrson

Message: 265 - Contents - Hide Contents Date: Thu, 21 Jun 2001 02:09:00 Subject: Re: 7/72 generator in blackjack From: jpehrson@r... --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote: Yahoo groups: /tuning-math/message/258 * [with cont.]

> --- In tuning-math@y..., "monz" <joemonz@y...> wrote: > Joseph, > > As a complement to Monz's explanation, see what I just added to > http://uq.net.au/~zzdkeena/Music/MiraclePitchChart.gif - Type Ok * [with cont.] (Wayb.) > > It shows the octave as a circle divided into 72 parts. Start at D>

and follow the (new) straight line segments clockwise. You'll see each one jumps 7/72 of an octave. When you've done that 20 times and wound up at D<, you've generated Blackjack.

> Hello Dave!

Thanks for this "enhancement" of your "Miracle Wheel." I understand it better than ever, now. I did have one question, though... It doesn't really look as though one goes around 20 times with those lines to get blackjack... I see a going around 10 times and then I get to D< but then there is not a connecting line going from that to the D> which begins a second "chain" of 10. And besides, between those two notes there are only 4 degrees of 72, not 7. What am I doing wrong?? Thanks! ________ _______ _______ Joseph Pehrson

Message: 266 - Contents - Hide Contents Date: Thu, 21 Jun 2001 02:29:28 Subject: Re: 7/72 generator in blackjack From: Paul Erlich --- In tuning-math@y..., jpehrson@r... wrote:

> > Is that correct? It looks as though Paul, in his calculations, is > trying to find the least errors for all the various intervals he is > considering, in squaring them and so forth, and then puts it all > nicely back in a pie (not a pie chart!) to determine the MIRACLE > generator at 116.7 cents. > > Am I getting that at all...??

Yup . . . it's identical in form to the derivation Woolhouse did (in 1835?) of the optimal meantone temperament, which turned out to have a fifth (the generator) tempered by 7/26 of a syntonic comma. Monz has a webpage on that . . .

Message: 267 - Contents - Hide Contents Date: Thu, 21 Jun 2001 02:32:42 Subject: Re: 7/72 generator in blackjack From: Paul Erlich --- In tuning-math@y..., jpehrson@r... wrote:

> It doesn't really look as though one goes around 20 times with those > lines to get blackjack... > > I see a going around 10 times and then I get to D< but then there is > not a connecting line going from that to the D> which begins a > second "chain" of 10. And besides, between those two notes there are > only 4 degrees of 72, not 7. > > What am I doing wrong?? >

There is only one chain. It _starts_ at D>. It goes around the circle once, producing 9 more notes. Then D pops up. Then 9 more notes. Then D< is last.

Message: 268 - Contents - Hide Contents Date: Wed, 20 Jun 2001 22:29:35 Subject: Re: Stretched tuning experiments (was: First melodic spring results) From: Herman Miller On Thu, 21 Jun 2001 01:54:56 -0000, "Dave Keenan" <D.KEENAN@U...> wrote:

>A "stretch" is in fact applied to _all_ intervals, not merely the >octave. However it is usually specified as so many cents per octave. >Its purpose is usually to compensate for a stretched inharmonic timbre >or a property of human pitch perception.

Hmm, good point. I guess I'll call it a "widened" octave rather than a "stretched" one.

Message: 269 - Contents - Hide Contents Date: Thu, 21 Jun 2001 02:46:11 Subject: Re: 7/72 generator in blackjack From: jpehrson@r... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: Yahoo groups: /tuning-math/message/267 * [with cont.]

> --- In tuning-math@y..., jpehrson@r... wrote: >

>> It doesn't really look as though one goes around 20 times with those >> lines to get blackjack... >> >> I see a going around 10 times and then I get to D< but then there is >> not a connecting line going from that to the D> which begins a >> second "chain" of 10. And besides, between those two notes there are >> only 4 degrees of 72, not 7. >> >> What am I doing wrong?? >>

> There is only one chain. It _starts_ at D>. It goes around the circle > once, producing 9 more notes. Then D pops up. Then 9 more notes. Then > D< is last.

Thanks, Paul... I just started on the wrong note... :) ________ _______ ______ Joseph Pehrson

Message: 271 - Contents - Hide Contents Date: Wed, 20 Jun 2001 23:20:51 Subject: Re: 7/72 generator in blackjack From: monz

> ----- Original Message ----- > From: <jpehrson@r...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, June 20, 2001 7:04 PM > Subject: [tuning-math] Re: 7/72 generator in blackjack > > > Well, from what you are saying, then, 31-EDO at with a 116.1 > generator is also a MIRACLE scale, yes... of a sort?? Of course, I > had always heard of the "special" properties of 31-EDO... > > Then, it is not too coincidental that a 31-tone NON-EDO using 116.7 > would be "miraculous" as well... I mean it's not so miraculous that > both scales would have 31 notes, correct??

Yes, Joe, you're on the right track. If I understand him correctly, Graham considers 31-EDO to be a MIRACLE temperament "of a sort". But that ~0.7-cent error accumulates further down the chain to produce intervals that are rather farther away from JI than the 72-EDO-based MIRACLE scales. The MIRACLE *generator* is the magical element in these tunings. You can generate a number of different-sized scales from it and they will all have essentially the same harmonic properties, the differences being simply a matter of the gaps in the smaller scales. So the 10-tone version *is* a very useable scale for a composer who wants many JI implications with a tiny pitch-set. You've become convinced that Blackjack (21-tone MIRACLE) will do the trick for you, and I've decided to focus on Canasta (31-tone MIRACLE). The 41-tone MIRACLE scale is yet another scale offering even *more* JI implications, and Graham has recommended one a bit larger than that (was it 46?). And of course, the full 72-EDO set closes the possibilities of the 2^(7/72) generator.

> ANYWAY, I also very much appreciated the 72-EDO chart at the bottom > of the MIRACLE generator page... That REALLY did a good job of > breaking down how the various scales are related to the generator...

Yes, quite a bit of the MIRACLE discussion was elaborated via Graham's decimal notation, which is (I think) a lot harder to grasp for a "regular" performer/composer microtonalist who's used to using deviations from 12-EDO, altho from a theoretical perspective decimal *is* more elegant to describe these tunings. So even for me, 72-EDO notation makes it easier to understand. (I'm just sorry that I'm so devoted to my own 72-EDO notation that now it goes up against the version everyone else has decided to use.)

> It seems I'm gradually coming to a greater appreciation of this > process, finally... although I have to admit I was rather "left in > the dust" when it happened. > > (In fact, I became so confused, that Paul thought I had forgotten the > entire discussion!)

You're not alone, Joe. I too was quite mystified during the first week or so of the MIRACLE discussion, and it was only when I realized the theoretical *AND* practical importance of it (i.e, how easily it can be mapped to a Ztar) that I devoted some serious study to it and began to understand. The MIRACLE tunings, especially Canasta for me, solve a lot of the riddles and problems I have been facing in dealing with large extended JI systems - specifically, how to map so many damn pitches to a playable instrument. MIRACLE solves the problem by distributing the small errors so well that a very small pitch-set can represent a huge number of JI structures. I've begun making a lattice of the implications of the Canasta scale (to go on my Canasta page), but it's got so many ratios on it that I fear it will be another "spaghetti lattice"... and I'm only up to 7-limit, haven't even plotted 11 yet! If you think of using the 2^(7/72) generator to create extended (i.e., >12) scales in the same way that you can use a tempered (i.e., narrowed) meantone "5th" or the wider-than-12-EDO Pythagorean "5th" to create >12 "extended" cycles, I think that will help make the whole process clearer. The meantone and Pythagorean "5ths" don't close the cycle at the 13th note, because the 1st and 13th are separated by a small interval which is exactly or approximately one of the commas. Similarly, if you call your origin 0 and create an 11-tone MIRACLE scale, the notes at either end (-5 and +5 generators away from the origin) will be separated by 2^(2/72), or 33&1/3 cents. This is a bit bigger than a "comma", but the process is the same. By continuing to extend the cycle beyond these 11 notes, you get pairs of pitches all separated by 2^(2/72) - this is exactly why the Blackjack scale has L=5 s=2 (in terms of 72-EDO degrees). By the time you reach a cycle bounded by -15 and +15 generators, you've filled out the "octave" pitch-space pretty evenly, hence the Canasta scale. Extending it to one more note on either side would give a pitch separated from the one on the other end by only 2^(1/72), or 16&2/3 cents. So now all MIRACLE scales above cardinality 31 will have pairs of pitches separated by *that* interval. Then finally the 72nd generator closes the cycle... in other words, at that point the separation of pitches on either end finally reduces to 2^(0/72), which is a unison. -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 272 - Contents - Hide Contents Date: Thu, 21 Jun 2001 03:46:51 Subject: Sonance degree (DEFINITION) From: Pierre Lamothe -------------- Sonance degree -------------- Complexity (n*d) and _Sonance_ (log n + log d) are microtonal concepts on which it seems there exist now (with varied terminology) a large agreement. I want to introduce here the _Sonance degree_ concept, a macrotonal one. While the _Sonance_ corresponds to a universal rational function applicable to any isolated irreducible ratio n/d, the _Sonance degree_ is defined only inside a discrete Z-module (lattice) with a coherent set of unison vectors. (A coherent set implies a CS structure, in other words a periodicity block). For an isolated ratio n/d, the _Sonance degree_ has no sense like the usual degree (tone rank in a scale). Dividing the usual degree of an interval by the degree number of the octave, we obtain the _Width degree_ of this interval which is the counterpart of the _Sonance degree_ at macrotonal level. To help the understanding of that concept the microtonal dyad (Width, Sonance) is compared here with the macrotonal dyad (Width_degree, Sonance_degree) at definition level. The microtonal definitions of _Width_ and _Sonance_ in the context of a lattice might be written _Width_ (X) = log(B) X / log(2) _Sonance_ (X) = log(B) |X| / log(2) where X = (X0 X1 .. XN)* is the coordinate vector (* indicate a column vector) of any interval in the basis B = <B0 B1 .. BN> where B0 = 2 (octaviant system) and the other independant components are normally simple primes (primal basis). The log operator applied to B gives log(B) = log <2 B1 .. BN> = [log(2) log(B1) .. log(BN)] and the absolute operator || applied to X gives |X| = (|X0| |X1| .. |XN|)* so these definitions might also be written _Width_ (X) = Sum(log(Bi)* Xi ) / log(2) _Sonance_ (X) = Sum(log(Bi)*|Xi|) / log(2) ----- Now, the main property of a coherent musical system G is the existence of an epimorphism D applying G on the relative integers Z. That implies D(xy) = D(x) + D(y) and considering the octave modularity D(xy mod <2>) = D(x) + D(y) mod [d] where d is the number of degree in the octave. The quotient G/D defines the congruence classes of interval in the system. ----- I already shown how to explicit a such epimorphism in the form of the degree function D(X) using the unison vectors to calculate it. See <Yahoo groups: /tuning/message/18625 * [with cont.] > <Yahoo groups: /tuning/message/20746 * [with cont.] > I had calculated the D(X) function for 3 systems using two distinct methods. D(X) = 5x + 8y + 14z (Slendro) D(X) = 7x + 11y + 16z (Zarlino) D(X) = 10x + 16y + 23z + 28t (Erlich decatonic) Using the formalism inside the precedent definitions we have D(X) = [5 8 14] X D(X) = [7 11 16] X D(X) = [10 16 23 28] X So D might be represented by a matrix operator [D0 D1 .. DN] applied to X. We have to understand here that the multiple ratios D0:D1:..:DN of the components in a such epimorphism is a rational approximation of this irrational one log(B0):log(B1):..:log(BN) depending of the unison vectors used. Here we have 5:8:14 ~ log(2):log(3):log(7) 7:11:16 ~ log(2):log(3):log(5) 10:16:23:28 ~ log(2):log(3):log(5):log(7) ----- Recalling _Width_ (X) = log(B) X / log(2) _Sonance_ (X) = log(B) |X| / log(2) ---------------------------------- BY DEFINITION _Width degree_ (X) = D X / d _Sonance degree_ (X) = D |X| / d ---------------------------------- where the matrix D = [d D1 .. DN] is the degree operator expliciting the system epimorphism in which the first component D0 = d is the degree of the octave and where X = (X0 X1 X2 .. XN)* is the coordinate vector (* indicate a column vector) of any interval in a basis <2 B1 B2 .. BN> where only the first component B0 = 2 (octaviant system) has to be known. The absolute operator || applied to X gives |X| = (|X0| |X1| .. |XN|)* so these definitions might also be written _Width degree_ (X) = Sum(Di* Xi ) / d _Sonance degree_ (X) = Sum(Di*|Xi|) / d ----- In the challenging problem we had two unison vectors U = {(-4 4 -1)*,(-3 -1 2)*} Using the determinant method to calcultate D (x -4 -3) D(X) = det (y 4 -1) = 7x + 11y + 16z (z -1 2) then D = [7 11 16]. Then SD(X) the _Sonance degree_ of the two intervals (-1 1 0)* (-3 1 1)* is simply [7 11 16](1 1 0)* / 7 = 18/7 [7 11 16](3 1 1)* / 7 = 48/7 ----- Comparing values rounded to three decimals for the sonance in <2 3 5> with the sonance_degree using U we have sonance_degree (-1 1 0) = 2,571 sonance (3/2) = 2,585 sonance_degree (3 1 1) = 6,857 sonance (15/8) = 6,907 Pierre

Message: 273 - Contents - Hide Contents Date: Thu, 21 Jun 2001 00:59:10 Subject: Re: Sonance degree (DEFINITION) From: monz ----- Original Message ----- From: Pierre Lamothe <plamothe@a...> To: <tuning-math@xxxxxxxxxxx.xxx> Sent: Thursday, June 21, 2001 12:46 AM Subject: [tuning-math] Sonance degree (DEFINITION)

> > -------------- > Sonance degree > --------------

Paul (or Dave or...), Please explain in simplified terms what Pierre wrote here. It seems from what I gleaned from it that he's talking about something which relates to my "finity" concept. I'm very interested, but having a hard time following all the math. -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 274 - Contents - Hide Contents Date: Thu, 21 Jun 2001 13:13:13 Subject: recap of decimal notation (posted to "biggie" also) From: jpehrson@r... Monz... this addition to the MIRACLE page is just terrific. It relates the whole "generator" process to our "traditional" Pythagorean comma in 12-tET, etc... It *really* put things together for me... Now, could I please ask you to elaborate a bit on Graham Breed's decimal notation, so I can understand it?? And *yes*, I have been on that page of Graham's SEVERAL times, and I *never* get it. It's all very sophisticated, but I get lost in the presentation. Would you mind recapping that in another way, so I can understand *that* notation. I would really appreciate it! Thanks! Joe ________ ________ ________ Joseph Pehrson