Tuning-Math Archive Section 1: 231

Message: 231 - Contents - Hide Contents Date: Mon, 18 Jun 2001 13:43:56 Subject: [tuning-math] unifying theory of interval "importance" (was: Re: First melodic spring results) From: monz

> ----- Original Message ----- > From: Paul Erlich <paul@s...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, June 18, 2001 11:01 AM > Subject: [tuning-math] Re: First melodic spring results > > > --- In tuning-math@y..., "John A. deLaubenfels" <jdl@a...> wrote: >> >> [I wrote:]

>>>> But then how to explain the sensation of 34-tET? It _does_ have >>>> narrow fourths (4 cents) and sixths (2 cents apiece). >> >> [Paul E:]

>>> Maybe the fourths and minor sixths are "less important" than the >>> fifths and major thirds. >>

>> Maybe, but what possible unifying theory could be behind such a >> thing? >

> The 4:3 less important than the 3:2 . . . the 8:5 less important than > the 5:4 . . . higher-number ratios less important than lower-number > ratios.

Hmmm... I see a more specific pattern. All other absolute exponent values being equal, the absolute values of the powers of 2 give an index of "importance". In vector notation where the numbers are exponents of 2, 3 and 5 respectively: 3:2 = |-1 1 0| 4:3 = | 2 -1 0| 5:4 = |-2 0 1| 8:5 = | 3 0 -1| -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: 232 - Contents - Hide Contents Date: Mon, 18 Jun 2001 21:07:12 Subject: unifying theory of interval "importance" (was: Re: First melodic spring results) From: Paul Erlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> Hmmm... I see a more specific pattern. All other absolute > exponent values being equal, the absolute values of the > powers of 2 give an index of "importance".

I don't think that's a more specific pattern, I think it's a less specific pattern. By size of the numbers I was thinking of something like n+d or n*d . . . by comparison, the absolute values of the powers of 2 would be less specific, since so many intervals would have the same value (for example, 8:3, 8:5, 8:7, 9:8, 11:8, 13:8, 15:8, 17:8, 24:11, 24:13, 24:17, 24:19, 24:23, 25:24, 29:24, 31:24, 35:24, 37:24, 41:24, 43:24, 47:24, 49:24, 5000:2401, etc.)

Message: 233 - Contents - Hide Contents Date: Mon, 18 Jun 2001 21:18:00 Subject: Re: First melodic spring results From: Paul Erlich 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?

Message: 234 - Contents - Hide Contents Date: Mon, 18 Jun 2001 14:37:59 Subject: Re: unifying theory of interval "importance" (was: Re: First melodic spring results) From: monz

> ----- Original Message ----- > From: Paul Erlich <paul@s...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, June 18, 2001 2:07 PM > Subject: [tuning-math] unifying theory of interval "importance" > > --- In tuning-math@y..., "monz" <joemonz@y...> wrote: >

>> Hmmm... I see a more specific pattern. All other absolute >> exponent values being equal, the absolute values of the >> powers of 2 give an index of "importance". >

> I don't think that's a more specific pattern, I think it's a less > specific pattern. By size of the numbers I was thinking of something > like n+d or n*d . . . by comparison, the absolute values of the > powers of 2 would be less specific, since so many intervals would > have the same value (for example, 8:3, 8:5, 8:7, 9:8, 11:8, 13:8, > 15:8, 17:8, 24:11, 24:13, 24:17, 24:19, 24:23, 25:24, 29:24, 31:24, > 35:24, 37:24, 41:24, 43:24, 47:24, 49:24, 5000:2401, etc.)

OK, got it. Of course, I meant that powers of 2 showed the "importance" together with the exponents of all the other primes. Guess I just automatically think in terms of prime-factorization when I see problems like this. -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: 235 - Contents - Hide Contents Date: Mon, 18 Jun 2001 21:48:53 Subject: unifying theory of interval "importance" (was: Re: First melodic spring results) From: Paul Erlich Hey Monz . . . at one point, you showed me a section in your book where you equated dissonance with the area of a rectangle formed by the numerator on one side and the denominator on the other side. This area, of course, equals n*d. I think Dave, Graham, and I (and Benedetti, Tenney, Pierre Lamothe, and others) have settled on n*d or a monotonic function thereof as a measure of "complexity" -- which is one of the factors determining dissonance, along with "tolerance" and "span". So it seems there was something to the insight behind your "rectangle" construction. Care to fill us in on how you came up with that construction?

Message: 236 - Contents - Hide Contents Date: Mon, 18 Jun 2001 23:26:47 Subject: Re: CS From: Dave Keenan --- In tuning-math@y..., carl@l... wrote:

>> Example: Generator/Period = 6/19 (e.g. narrow major third/octave), >> 10 note improper MOS. This has steps of 3/19 and also has 3 >> consecutive steps of 1/19. >

> I see the 3rd of 189 cents but not the 2nd. According to Scala > this is CS.

Oops. Sorry. You have to take it out to 13 notes before the 3/19 oct seconds appear. I wonder how much "slop" Scala allows in determining whether products of intervals are equal for this purpose?

> Please note that while my experience tells me Chalmers' formula > is correct, I do not have a proof. I've been thinking about one, > but haven't come up with anything yet. Probably came to a stop > sign at the key instant. :) Hee hee. > [Note: Myhill's property means there are only two kinds of 2nds, > and that they are distributed in a certain way, so that there will > be exactly two kinds of the other scalar intervals. Perhaps this > is where the 2 in Chalmers' formula comes from...]

Clearly if L/s = n, an integer, and n consecutive "s"s appear in the scale then it is not CS. That is the case with the 13 of 6/19 oct generator above, where n = 3. But not all scales of L/s = 2 have 2 consecutive "s"s, so yes a proof looks tricky. But I'm not much interested in a property that is defined so that it dissapears with an inaudible, nay immeasurable!, change to the scale. Regards, -- Dave Keenan

Message: 237 - Contents - Hide Contents Date: Mon, 18 Jun 2001 23:59:13 Subject: unifying theory of interval "importance" (was: Re: First melodic spring results) From: Dave Keenan --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> I think Dave, Graham, and I (and > Benedetti, Tenney, Pierre Lamothe, and others) have settled on n*d or > a monotonic function thereof as a measure of "complexity" -- which is > one of the factors determining dissonance, along with "tolerance" > and "span".

What about "rootedness", i.e. having the smaller number be a power of 2? Is 8:11 more consonant than 7:11?

Message: 238 - Contents - Hide Contents Date: Tue, 19 Jun 2001 00:10:22 Subject: unifying theory of interval "importance" (was: Re: First melodic spring results) From: Paul Erlich --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> > What about "rootedness", i.e. having the smaller number be a power > of 2? Is 8:11 more consonant than 7:11?

That's a tough call . . . on the other hand, it's pretty hard not to conclude that 16:1 is more consonant than 15:1 or 17:1 . . . so clearly octave-equivalence (and thus powers of 2) plays some role.

Message: 244 - Contents - Hide Contents Date: Tue, 19 Jun 2001 04:36:44 Subject: Re: CS From: carl@l...

>> > see the 3rd of 189 cents but not the 2nd. According to Scala >> this is CS. >

>Oops. Sorry. You have to take it out to 13 notes before the 3/19 >oct seconds appear.

Wow. "Show data" on this scale crashes my copy of Scala 1.7. This reminds me of a bug I thought was fixed. Manuel? "Show intervals" works, and the scale is clearly not CS. Chalmers isn't wrong, it's my fault. If a scale is proper, and isn't CS, then L:s = 2:1. But if a scale is improper, L:s doesn't seem to apply.

> I wonder how much "slop" Scala allows in determining whether > products of intervals are equal for this purpose?

My guess is none, within the precision setting. Manuel?

> But I'm not much interested in a property that is defined so > that it dissapears with an inaudible, nay immeasurable!, change > to the scale.

The property isn't designed to blindly measure scales in any kind of search. It's just an idea that Wilson wants you to think about. In search of a measure, what did you think of my smooth-CS suggestions? -Carl

Message: 245 - Contents - Hide Contents Date: Tue, 19 Jun 2001 00:41:29 Subject: [tuning-math] Monzo theory of sonance (was: unifying theory of interval "importance") From: monz

> ----- Original Message ----- > From: Paul Erlich <paul@s...> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, June 18, 2001 2:48 PM > Subject: [tuning-math] unifying theory of interval "importance" > > Hey Monz . . . at one point, you showed me a section in your book > where you equated dissonance with the area of a rectangle formed by > the numerator on one side and the denominator on the other side. This > area, of course, equals n*d. I think Dave, Graham, and I (and > Benedetti, Tenney, Pierre Lamothe, and others) have settled on n*d or > a monotonic function thereof as a measure of "complexity" -- which is > one of the factors determining dissonance, along with "tolerance" > and "span". So it seems there was something to the insight behind > your "rectangle" construction. Care to fill us in on how you came up > with that construction?

Sure thing. I'll give a brief overview first, then simply quote an exchange between myself and Dave Keenan from 2 years ago (at the height of the discussion on harmonic complexity) which explains it in good detail. Basically, I was trying to illustrate a foundation of my theory of sonance, which I first found clearly explained by Ben Johnston, but which actually AFAIK dates back to Euler. This is: that absolute (or maximum) consonance is expressible, in prime-factor terms, as n^0 (the 1:1 ratio) and dissonance increases directly as *both* the number and size of the prime-factors and the absolute value of the exponents get larger. The larger the area covered by the graph - in all dimensions - the greater the dissonance of the interval. If I had a good understanding of trigonometry, I'd probably modify this now to reflect my (or any other) lattice formula. Any ideas on that, Paul? -monz ------- first explanatory email to Dave Keenan --------- From: monz@j... To: d.keenan@u... Date: Thu, 8 Apr 1999 12:52:23 -0400 Subject: my sonance theory Hey Dave, I was just thinking about my sonance theory and how it may be another type of complexity measure you haven't considered yet. You probably have - I just haven't yet grasped (or spent the time trying to grasp) all the info that's on your spreadsheet. (Plus I have to play with the charts in some graphics program to reduce their size so I can print them.) This is it, in case you haven't covered it yet: On plain square graph-paper, I drew a line representing the cents value of the ratio, then a box the size of 1 cell for each exponent in the numerator and denominator on either side of the line respectively. Boxes would be arranged according to size of prime-bases and how exponents were stacked, or how prime-factors combined in composites. For example, 15:8 factors to 2^-3 * 3^1 * 5^1. 5 is the highest prime factor, so the length of the line representing the note's pitch will be 5 units long along the horizontal (x-) axis. The numerator factors into 3 times 5, so there 3 groups of 5 boxes, stacked 3 high, above the line. Since the denominator factors ultimately to 1 for octave-equivalent music-theory purposes, the denominator is represented by only 1 box under the line. The total number of boxes, and the length of both the line horizontal and the height of the stacked boxes vertically, describes the relative sonance (consonance/dissonance) of the dyad. Does it sound like something already discussed? You're welcome to respond to this publicly on the List if it merits further discussion. -monz ------- Dave's response to my first explanatory email ---- From: Dave Keenan <d.keenan@u...> To: monz@j... Date: Fri, 09 Apr 1999 10:10:35 +1000 Subject: Re: my sonance theory No, but your above description is really too ambiguous/contradictory and vague for me to be sure. It sounds a bit like an octave-reduced n+d which I would think would be useless. It's octave equivalent so I assume we can cast out any 2's before we start and there would always be at least one box on each side of the line. I'm confused about what happens with odd-prime exponents greater than 1. What does 9/8 look like, or 45/32? How do "The total number of boxes, and the length of both the line horizontal and the height of the stacked boxes vertically" go together to produce a single complexity (= dissonance for simple ratios) number. At first you say the (horiz?) line represents the cents value (how?) but in the example the line length is given by the highest prime factor. Yours confusedly, -- Dave Keenan ------- second explanatory email to Dave Keenan --------- From: monz@j... To: d.keenan@u... Date: Thu, 8 Apr 1999 22:21:48 -0400 Subject: Re: my sonance theory Sorry about not being clear. The cents value is represented along the y-axis (vertical), but in truth, it's more or less irrelevant to the sonance issue as being described. I used the cents value only to have a value along which to plot the different ratios and know how to recognize them (by the pitch-height). I have no idea how to tell you to get one complexity number out of the two or three different measurements I described. That's why I use a graph :) and asked you to do the numbers :) On the graph, no matter how complex the formula is, you can see at a glance exactly how much 'space' the ratio takes up (on paper), and how it's arranged, which I think also carries important harmonic or sonance information. One thing I forgot to mention: with this kind of plotting, all complementary intervals have an equal 'value'. It's a graph made of all squares, so compared to some of the ASCII lattices I've drawn, it's a piece of cake. Here are some examples, we'll just disregard the whole issue of cents value: _ |_| 1:1 and 2:1 |_| _ _ _ |_|_|_| 3:2 (invert for 4:3) |_| _ _ _ _ _ |_|_|_|_|_| 5:4 (invert for 8:5) |_| _ _ _ |_|_|_|_ _ 6:5 (invert for 5:3) |_|_|_|_|_| _ _ _ _ _ _ _ |_|_|_|_|_|_|_| 7:4 (inv. 8:7) |_| _ _ _ _ _ _ _ |_|_|_|_|_|_|_| 7:6 (inv. 12:7) |_|_|_| _ _ _ |_|_|_| |_|_|_| |_|_|_| 9:8 (inv. 16:9) |_| _ _ _ _ _ |_|_|_|_|_| 10:9 (inv. 9:5) |_|_|_| |_|_|_| |_|_|_| I think this visual model represents well what I've read of the most commonly-accepted sonance theories (Helmholtz, Ellis, Ben Johnston, Partch, lots of people), and what my own ears inform me of. I was trying to find a way to model how both increasing size of prime-base *and* increasing size of exponents leads to increasing dissonance, but still take account of both the uniqueness of the primes themselves, and the multi-dimensionality of their combination. So now we're up to a point where several different ratios can have the same 'box count' value, thus the same sonance (= level of relative consonance/dissonance). For example, 7:6 and 9:8 both have a total of 10 boxes. So I would say that they have approximately the same consonance or dissonance, but with two different *qualities*, described by the distibution of primes and exponents in the ratio, and by the different shapes on the graph (even tho they take up the same area or volume). Now, ideally, this would be a 3-or-more-dimensional graph, so that complicated multiples could be portrayed simply by using a different dimension for each prime. This would probably be kind of confusing for this type of graph, but it's exactly what worked for me in my lattice diagrams. Even plotting them 2-dimensionally, using unique angles and vector-lengths for the different primes gives the lattice a multi-dimensional aspect. So anyway, to graph 45:32, it would be best to stack a horizontal layer of 5 both 3 high and 3 deep. To me, this is the best way to portray 5-limit composite factoring. I'll try my best to show you in ASCII what it would look like, in a view from the top front: _ _ _ _ _ /_/_/_/_/_/| /_/_/_/_/_/|/ 45:32 /_/_/_/_/_/|// |_|_|_|_|_|/// <-- just use your imagination |_|_|_|_|_|// for this part |_|_|_|_|_|/ |_|/ <-- that's just one tiny little block on the bottom You can easily see it as 15*3 or 9*5. I think this 'double meaning' is what makes the composite ratios so interesting, and probably why they're important enough for some people to like odd-limit. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)

Message: 246 - Contents - Hide Contents Date: Tue, 19 Jun 2001 07:48:59 Subject: Re: CS From: Dave Keenan --- In tuning-math@y..., carl@l... wrote:

> In search of a measure, what did you think of my smooth-CS > suggestions?

You mean those in Yahoo groups: /tuning-math/message/118 * [with cont.] The 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". Which means that, so long as you don't identify (i.e. consider as equal) interval sizes that differ by more than X cents (X cents or more?), there is no interval size that can be arrived at by different numbers of steps. So "CS to 0 cents" means not CS at all. I'd think CS to 17 cents is somewhat borderline (one standard deviation). CS to 63 cents is obviously CS. Blackjack (with 7/72 oct generator) is only CS to 17 cents. It has 67c in 2 steps and 83c in 1 step, 183c in 4 steps and 200c in 3 steps, 300c in 6 steps and 317c in 5 steps, 417c in 8 steps and 433c in 7 steps, 533c in 10 steps and 550c in 9 steps, etc. So I don't think it's meaningful to say simply that Blackjack is CS. It's borderline CS. With a 3/31 oct generator it is not CS at all (but it's proper with L/s = 2). With a 4/41 oct generator it is improper (L/s = 3) but at its maximum CS-ness, having a 29c (one step of 41-EDO) tolerance. But these extremes are no fun harmnonically. -- Dave Keenan

Message: 248 - Contents - Hide Contents Date: Tue, 19 Jun 2001 15:27:00 Subject: Re: CS From: carl@l...

>> >n search of a measure, what did you think of my smooth-CS >> suggestions? >

> You mean those in > Yahoo groups: /tuning-math/message/118 * [with cont.] Yep. > The 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.

> Which means that, so long as you don't identify (i.e. consider as > equal) interval sizes that differ by more than X cents (X cents or > more?), there is no interval size that can be arrived at by > different numbers of steps. So "CS to 0 cents" means not CS at > all. I'd think CS to 17 cents is somewhat borderline (one standard > deviation). CS to 63 cents is obviously CS.

Of what is 17 cents one standard deviation? My feeling is that this ought to be expressed as a fraction of the scale's smallest 2nd. -Carl