Tuning-Math Archive Section 1: 277

Message: 277 - Contents - Hide Contents Date: Thu, 21 Jun 2001 19:18 +0 Subject: Re: recap of decimal notation From: graham@m... In-Reply-To: <9gsrt9+o9t0@e...> Joseph Pehrson wrote:

> 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.

What's sophisticated about it? You understand staff notation don't you? There are 7 notes (A, B, C, D, E, F and G) and 2 things you can do to them. I decimal notation, there are 10 notes (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) and still 2 things you can do to them (^ and v or > and < depending on which page you look at). The 10 notes are this scale in 72-equal: 0 1 3 4 5 6 7 8 9 0 7 7 7 7 7 7 7 7 9 The ^,v,>,< or whatever tell you to move by 2 steps of 72-equal. Graham

Message: 278 - Contents - Hide Contents Date: Thu, 21 Jun 2001 19:18 +0 Subject: Re: 7/72 generator in blackjack From: graham@m... In-Reply-To: <9gpf0f+i66g@e...>

> By what figure-of-demerit and at what odd-limits can we claim that the > MIRACLE generator is the best?

11-limit by most figures of demerit, although that is ignoring some obviously too complex scales. 7-limit it's second by the default FOD, whatever that was. 9-limit it doesn't score so well.

> 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?

13/41, consistent with 19 and 22 note scales. Schismic is second. Miracle does really badly in the 7-limit with this measure. See <5 12 19 22 26 27 29 31 41 46 50 53 58 60 68 70... * [with cont.] (Wayb.)> etc.

> I'm sure some folks would be interested in the 13-limit result too.

Right. I've added, but not checked, that. <Automatically generated temperaments * [with cont.] (Wayb.)> It should be easily hackable now. Although there's a lot of code in writetemper.py, you should be able to work out how to supply different limits, or figures of demerit. Obviously, you don't need the bit that uploads to my website. Graham

Message: 279 - Contents - Hide Contents Date: Thu, 21 Jun 2001 18:24:16 Subject: Re: Sonance degree (DEFINITION) From: Paul Erlich --- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> Paul (or Dave or...), > > Please explain in simplified terms what Pierre wrote here.

I wish I could!

Message: 280 - Contents - Hide Contents Date: Fri, 22 Jun 2001 00:40:43 Subject: Re: 7/72 generator in blackjack From: Dave Keenan 31-EDO is "MIRACLE" in exactly the same way that 12-tET is meantone. i.e. Not really. At best borderline, with large deviations from JI. We have the following approximate analogy Meantone MIRACLE ---------------- 12-EDO 31-EDO 31-EDO 72-EDO 19-EDO 41-EDO

Message: 281 - Contents - Hide Contents Date: Fri, 22 Jun 2001 01:26:27 Subject: Re: 7/72 generator in blackjack From: Dave Keenan

> <Automatically generated temperaments * [with cont.] (Wayb.)>

Your page says "11 limit" where it should be "13 limit".

Message: 282 - Contents - Hide Contents Date: Fri, 22 Jun 2001 01:42:55 Subject: Re: Sonance degree (DEFINITION) From: Dave Keenan --- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:

> Hi Joe, > > <<Paul (or Dave or...), Please explain in simplified terms what Pierre > wrote here.>> > > Pierre's posts are really the only ones where I consistently find > myself in need of a 'math to English' translator! Robert Walker has > been very helpful in the past in this regard, you might want to ask > him. > > --Dan Stearns

My apologies Dan, Monz, and Pierre. Here's where it becomes obvious that I'm not a "real" mathematician. It's such a shame because the math should be the universal language that overcomes the French/English barrier. But I don't know all the math that Pierre knows. Maybe I could get a handle on it if I spent a long time, but I just can't justify the time. Reminds me of one time when I _did_ spend the time to "translate" some stuff of yours Dan. :-) You've come a long way since then. Dan, you asked once before if I could explain some stuff Pierre wrote, generalising the golden mean to two dimensions. I didn't respond because I had already explained as much of it as I understood, but I didn't understand how to relate it to scales. Regards, -- Dave Keenan

Message: 293 - Contents - Hide Contents Date: Fri, 22 Jun 2001 19:00:31 Subject: Hypothesis revisited From: Paul Erlich Progress seems to have halted on the paper that was to introduce MIRACLE . . . I suggest the title _The Relationship Between Just Intonation and Well-Formed Scales_ and some sort of "proof" of the hypothesis (I know, it doesn't always work). If we can do the following math problem, we'll be fine: Given a k-by-k matrix, containing k-1 commatic unison vectors and 1 chromatic unison vector, delimiting a periodicity block, find: (a) the generator of the resulting WF (MOS) scale; (b) the integer N such that the interval of repetition is 1/N octaves. If we can derive a general formula of this nature, the status of the pathological cases (e.g., Monz' shruti block) should become clear (hopefully). Then we can give a few examples, including the diatonic and MIRACLE scales. So, who's going to be our hero?

Message: 294 - Contents - Hide Contents Date: Fri, 22 Jun 2001 19:04:50 Subject: Re: 7/72 generator in blackjack From: Paul Erlich --- In tuning-math@y..., jpehrson@r... wrote:

> But isn't it a bit strange that a generator that is 7 units of 72- > tET should create FIVES... and that 5+2, the small interval = 7. > > I'm not entirely understanding why that is... > > ??

It's very simple. 72 divided by 7 is 10, with a remainder of 2. So after 10 generators in the cycle, you're 2 units short of where you started. Another generator later, you're 7-2 = 5 units beyond where you started, and 2 units short of the second note in the chain. Similarly, you'll divide each of the 7-unit steps of the first cycle into a large (5-unit) step and a small (2-unit) step as you go around the second cycle. When you've completed the second cycle, you've completed the blackjack scale. When you've completed the third cycle, you've completed the canasta scale. When you've completed the fourth cycle, you've completed the MIRACLE- 41 scale.

Message: 295 - Contents - Hide Contents Date: Fri, 22 Jun 2001 21:39 +0 Subject: Re: Hypothesis revisited From: graham@m... Paul wrote:

> Given a k-by-k matrix, containing k-1 commatic unison vectors and 1 > chromatic unison vector, delimiting a periodicity block, find: > > (a) the generator of the resulting WF (MOS) scale;

That's the bit I'm not sure about

> (b) the integer N such that the interval of repetition is 1/N octaves.

Easy. It'll usually be the determinant of the matrix. You can always get it by solving the matrix equation. Say you have Where H is the logs of the primes, H' is the approximation, a1...ak are the unison vectors, where ak is chromatic, and a0 is the octave (1 0 0 ... 0). You solve it to get (a0)-1 (a0) (a1) (a1) H' = (a2) ( 0)H (..) (..) (ak) ( 0) From which you know the first column of (a0) (a0)-1 (a0) (a1) (a1) (a1) det(a2) (a2) ( 0) (..) (..) (..) (ak) (ak) ( 0) will be a vector of integers specifying the number of steps to each prime interval. You then reduce them by any common factor, and the one on top will be the number of steps to an octave. Or say that it's pathological if there is a common factor.

> If we can derive a general formula of this nature, the status of the > pathological cases (e.g., Monz' shruti block) should become clear > (hopefully). Then we can give a few examples, including the diatonic > and MIRACLE scales.

If you supplied two different chromatic unison vectors, that would give two equal temperaments that could be plugged into my Python script to yield everything else we need to know. Ideally, we could do without chromatic unison vectors altogether, but I don't see how to do that bit. You could do a brute force search over all consistent ETs, like my program does, but that's not the elegant way of solving this problem. So are we aiming for musicians or mathematicians? Graham

Message: 296 - Contents - Hide Contents Date: Fri, 22 Jun 2001 20:57:33 Subject: Re: Hypothesis revisited From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:

>> (b) the integer N such that the interval of repetition is 1/N octaves. >

> Easy. It'll usually be the determinant of the matrix.

Huh? The determinant of the matrix is usually the number of notes, not the number of repetitions per octave (which is usually just 1). You can always get

> it by solving the matrix equation. Say you have > > > Where H is the logs of the primes,

Looks like you left something out here, yes? Let's leave out the octave, octave-equivalence will be assumed (yes, in a more general case it won't be, but let's not bite off more than we can chew). It's fine if the paper is a bit mathematical if that helps it obtain a more powerful result. Music theory can get very mathematical these days.

Message: 297 - Contents - Hide Contents Date: Fri, 22 Jun 2001 21:04:34 Subject: Re: 7/72 generator in blackjack From: jpehrson@r... --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: Yahoo groups: /tuning-math/message/294 * [with cont.]

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

>> But isn't it a bit strange that a generator that is 7 units of 72- >> tET should create FIVES... and that 5+2, the small interval = 7. >> >> I'm not entirely understanding why that is... >> >> ?? >

> It's very simple. > > 72 divided by 7 is 10, with a remainder of 2. > > So after 10 generators in the cycle, you're 2 units short of where > you started. > > Another generator later, you're 7-2 = 5 units beyond where you > started, and 2 units short of the second note in the chain. > > Similarly, you'll divide each of the 7-unit steps of the first

cycle into a large (5-unit) step and a small (2-unit) step as you go around the second cycle.

> > When you've completed the second cycle, you've completed the > blackjack scale.

Of course... thanks Paul... that is an easy concept. I wonder if I'd been able to figure it out if I'd continued to puzzle over it... Well, anyway, you saved me some time... thanks! _________ ______ ______ Joseph Pehrson

Message: 299 - Contents - Hide Contents Date: Fri, 22 Jun 2001 20:36:07 Subject: Re: 7/72 generator in blackjack From: Herman Miller On Fri, 22 Jun 2001 03:04:34 -0000, "Dave Keenan" <D.KEENAN@U...> wrote:

>Could you describe this keyboard mapping in more detail? The main >problem I'm having here is that there are 11 "unnaturals" in Blackjack >but only 10 black keys in 2 octaves (although Manuel argued for 11 >naturals and 10 unnaturals in Blackjack).

0^ 1^ 2^ 3^ 4^ 5^ 6^ 7^ 8^ 9^ 0 1 2 2 3 4 5 5 6 7 7 8 9 0 0v C# Eb F# G# Bb C# Eb F# G# Bb C D E F G A B C D E F G A B C I also have a 24-note mapping with the duplicate notes mapped to more useful ones: 0^ 1^ 2^ 3^ 4^ 5^ 6^ 7^ 8^ 9^ 0 1 2 3 4 5 6 7 8 9 0 2v 5v 7v 0v It's not a contiguous MIRACLE scale, but it just happens to work out nicely, since 2v and 5v complete the 4:5:6:7:9:11 chord over 0. -- see my music page ---> ---<The Music Page * [with cont.] (Wayb.)>-- hmiller (Herman Miller) "If all Printers were determin'd not to print any @io.com email password: thing till they were sure it would offend no body, \ "Subject: teamouse" / there would be very little printed." -Ben Franklin