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Message: 50 - Contents - Hide Contents

Date: Mon, 28 May 2001 09:14:28

Subject: Re: Fwd: optimizing octaves in MIRACLE scale..

From: Paul Erlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > > Yahoo groups: /tuning-math/message/31 * [with cont.] > >> I wrote: >>
>>> Oh, Monz . . . you're not expecting the result to be a >>> stretched or squashed 72-tET, are you? 'Cause if you are, >>> then it's a one-parameter optimization -- much easier. >>
>> And if it is, the answer is 71.959552-tET, or 72-tET with >> the octave stretched to 1200.6745¢. > >
> With a step-size of 16.67603472 cents. > > Thanks, Paul. Uh... I don't think "expecting" is the way > I'd say it, but yes, I *was* *guessing* that it would be > a stretched 72-EDO. > > But I'm unclear on why my expectation would have any effect > on the type of optimization. ...?
Well, because it's easier to solve the problem of how best to stretch or squashed 72-tET for the 12-integer-limit (a univariate optimization), than to solve the problem of what the best size of generator _and_ the best size of octave are for MIRACLE in the 12- integer limit (a multivariate optimization). If you _want_ a stretched or squashed 72-tET, then there you have it, I'm done. If not, I'm hitting the Matlab Optimization Toolbox manual.
> > Also, on the asking of what are probably elementary questions > like this to the rest of you on this list: is it OK for me to > ask questions like this here?
You better believe it!
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Message: 51 - Contents - Hide Contents

Date: Mon, 28 May 2001 09:23:22

Subject: Re: Temperament program issues

From: Paul Erlich

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> I had a hard time recognising Paultone among your top-10 7-limit > generators.
Holy $#!% -- totally missed that. And how did MIRACLE end up on the _bottom_ for 9-limit?
> > I think there's a bug in your MA optimiser.
MA optimization is hard. Go with RMS.
> In the case where one or > more intervals is purely a multiple of the period (zero generators) > you need to either give a _range_ for the optimum generator, or > preferably eliminate the zero-generator intervals from the > optimisation. > > If you do that you should get 109.36 cents (not 111.04) for Paultone. Yes. > Not by me. How about giving us the top-tens ranked according to RMS > error times (n*d)^2, generators in cents. > Good idea.
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Message: 52 - Contents - Hide Contents

Date: Mon, 28 May 2001 09:24:45

Subject: Re: Temperament program issues

From: Paul Erlich

--- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> Doesn't the fact that meantone, (the single most popular 5-limit > temperament of all time), doesn't even make the top-ten, mean that > there is something very wrong with our figure of demerit?
Very wrong. Very, very wrong.
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Message: 53 - Contents - Hide Contents

Date: Mon, 28 May 2001 09:26:51

Subject: Re: Temperament program issues

From: Paul Erlich

--- In tuning-math@y..., graham@m... wrote:
> D.KEENAN@U... (Dave Keenan) wrote: >
>> Doesn't the fact that meantone, (the single most popular 5-limit >> temperament of all time), doesn't even make the top-ten, mean that >> there is something very wrong with our figure of demerit? >> >
> You're out of date.
Oh, OK! So where is it in the 5-limit ranking now?
> It's this one: > > > 66/157 >
How is this approximation decided upon? What makes it 66/157 and not some more or less complex approximation?
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Message: 54 - Contents - Hide Contents

Date: Mon, 28 May 2001 13:54 +0

Subject: Re: Temperament program issues

From: graham@m...

Paul Erlich wrote:

> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote: >
>> I had a hard time recognising Paultone among your top-10 7-limit >> generators. >
> Holy $#!% -- totally missed that. And how did MIRACLE end up on the > _bottom_ for 9-limit?
It's moved up now.
>> I think there's a bug in your MA optimiser. >
> MA optimization is hard. Go with RMS.
I used that algorithm *because* it's easy. If you have another one, write the code and plug it in.
>> In the case where one or >> more intervals is purely a multiple of the period (zero generators) >> you need to either give a _range_ for the optimum generator, or >> preferably eliminate the zero-generator intervals from the >> optimisation. >> >> If you do that you should get 109.36 cents (not 111.04) for > Paultone. > > Yes.
AIUI, this wouldn't affect the minimax. So does it matter?
>> Not by me. How about giving us the top-tens ranked according to RMS >> error times (n*d)^2, generators in cents. >> > Good idea.
Well, go on and implement it. Sorry if this sounds dismissive, but I would like to get some music written today. Graham
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Message: 55 - Contents - Hide Contents

Date: Mon, 28 May 2001 13:54 +0

Subject: Re: Temperament program issues

From: graham@m...

Paul wrote:


>> You're out of date. >
> Oh, OK! So where is it in the 5-limit ranking now?
<counts> 5. The files should be on the website, and not have changed since Saturday night (my time).
>
>> It's this one: >> >> >> 66/157 >>
> How is this approximation decided upon? What makes it 66/157 and not > some more or less complex approximation?
The temperaments are chosen from pairs of consistent ETs. To prevent duplicates, more complex temperaments always overwrite simpler ones with the same mapping. The program's searching ETs from 1 to 99, and so the most complex meantone pair is 88 and 69. This is something that should be fixed, to give a simpler default and all the versions in a detailed list, but I haven't done so yet. Graham
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Message: 56 - Contents - Hide Contents

Date: Mon, 28 May 2001 17:04:29

Subject: Re: Temperament program issues

From: Paul Erlich

--- In tuning-math@y..., graham@m... wrote:
> > AIUI, ?? >this wouldn't affect the minimax. So does it matter?
Yup -- the 6:5 and 5:3 are kinda hairy at your value.
> Sorry if this sounds dismissive, but I would like to get some music > written today.
Yay! By the way, your Blackjack chord progression totally blew away and inspired Joseph Pehrson. You should take this as a sign that you're on the right track!
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Message: 57 - Contents - Hide Contents

Date: Mon, 28 May 2001 20:55 +0

Subject: Re: Temperament program issues

From: graham@m...

Paul wrote:

> There's only one interval that sounds like a repetition of the same > pitch to humans and other mammals. It's pretty close to 2:1.
That isn't what I found by playing the Bohlen-Pierce and O'Connell scales with the matching timbres. I'll keep looking for alternative periods until my ears tell me otherwise. Graham
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Message: 58 - Contents - Hide Contents

Date: Mon, 28 May 2001 20:55 +0

Subject: Re: Temperament program issues

From: graham@m...

paul@s... (Paul Erlich) wrote:


>> AIUI, >
As I Understand It
>> this wouldn't affect the minimax. So does it matter? >
> Yup -- the 6:5 and 5:3 are kinda hairy at your value.
But I'm only doing the optimization in order to compare temperaments. The way I'm doing it is good enough for that. The real tuning would only become important if you actually wanted to play one of the things, which would require sufficient effort that you could work out the optimum then. I can do my own optimization by ear when it comes to that. One think I don't think I've mentioned yet. Where the period is an octave, your "miraculous" property (more complete chords than notes) is present whenever there's a tonality diamond, isn't it? It's 2n+1 notes required each time. So we have a theorem: Any MOS containing a tonality diamond will have at least as many complete chords (otonal and utonal) as notes.
>> Sorry if this sounds dismissive, but I would like to get some music >> written today. > > Yay!
Actually, that was more desperation than resolution. I went out for a walk after that, dropped in at the office to listen to some MP3s, haven't got anything written on the holiday, although I do still like the stuff I recorded yesterday. The things you mention can be done. When I next look at the program I'll implement them. But you might be able to work it out anyway with the code and an interpreter. You don't need to know the language, just use a bit of pattern recognition. Optimising for inharmonic timbres with a perfect octave will be easy. The only assumption made is that you'll have "prime" and "derived" intervals. If all intervals are prime (no consonance is a combination of other consonances) consistency isn't defined, so it defaults to brute force for the ETs but might still work. Searching inharmonic timbres for a given formal octave is straightforward too. It's only feeding in an arbitrary set of intervals and saying "give me a period and generator" that isn't defined.
> By the way, your Blackjack chord progression totally blew away > and inspired Joseph Pehrson. You should take this as a sign that > you're on the right track!
That's good. One of the things I listened to was that piece of his played by Johnny Reinhard. I think he's somebody really worth inspiring. Graham
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Message: 59 - Contents - Hide Contents

Date: Tue, 29 May 2001 00:16:04

Subject: Re: Temperament program issues

From: Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>> Doesn't the fact that meantone, (the single most popular 5-limit >> temperament of all time), doesn't even make the top-ten, mean that >> there is something very wrong with our figure of demerit? >
> Very wrong. Very, very wrong.
When I wrote the above I hadn't got around to converting all Graham's decimal octaves into something I could understand. Here they are. Min Gens Num of MA per Figure chains Gen err compl of per oct (cents)(cents) otonal demerit --------------------------------------- 15-limit 29 15.93 4.7 29 3948 1 497.24 10.0 23 5269 1 351.71 7.8 28 6089 3 83.02 2.8 48 6426 1 497.89 4.8 37 6619 2 83.16 3.1 52 8431 1 83.42 6.4 38 9190 1 116.72 5.0 45 10111 1 234.41 5.8 42 10162 2 165.81 3.9 52 10575 11-limit 1 116.72 3.3 22 1608 2 183.21 2.4 30 2168 2 216.74 3.1 30 2747 1 271.14 9.3 17 2693 29 16.37 3.7 29 3124 2 83.25 2.9 34 3302 1 193.24 2.8 35 3417 1 310.20 5.4 25 3404 1 585.14 4.1 29 3454 2 165.21 5.1 26 3472 9-limit 1 380.39 5.9 12 853 1 497.81 4.3 16 1109 1 351.45 1.9 25 1188 1 503.42 10.8 10 1075 1 116.72 3.3 19 1200 miracle 2 108.77 17.5 8 1119 1 316.76 2.7 22 1322 2 92.21 19.5 8 1248 1 251.71 20.5 8 1314 1 271.62 5.3 17 1544 7-limit 1 193.87 1.4 16 366 1 116.59 2.4 13 410 miracle 1 271.14 4.3 11 516 1 503.42 5.4 10 538 1 310.20 5.4 10 545 1 77.76 4.6 12 662 2 111.04 17.5 6 630 paultone 1 316.99 17.8 6 643 1 380.39 5.9 12 853 1 232.19 5.4 13 909 5-limit 1 498.26 0.2 9 18 schismic? 1 316.99 1.4 6 49 kleismic 1 387.74 1.4 8 92 1 271.56 1.0 10 101 1 503.42 5.4 4 86 meantone 1 442.92 1.5 9 121 2 105.21 3.3 6 117 disachismic 1 380.39 5.9 5 148 3 84.36 13.7 3 123 1 176.26 3.1 9 249
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Message: 60 - Contents - Hide Contents

Date: Tue, 29 May 2001 03:18:37

Subject: Re: Temperament program issues

From: monz

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

Yahoo groups: /tuning-math/message/56 * [with cont.] 

> ... By the way, your Blackjack chord progression totally > blew away and inspired Joseph Pehrson.
Me too!!!! I could tell by looking at it that I simply *had* to hear it... that's exactly why *I* made the audio file! Just wish I currently had a polyphonic keyboard to map it too, so that I could play around with it too. (well, I'd use Canasta instead...) Pretty soon... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0"
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Message: 61 - Contents - Hide Contents

Date: Tue, 29 May 2001 07:01:36

Subject: Re: Temperament program issues

From: Dave Keenan

This follows from
Yahoo groups: /tuning-math/message/59 * [with cont.] 

I listed the temperaments in the order that Graham ranked them but I 
gave a different figure-of-demerit. I used the MA error times the 
square of the max number of notes giving zero complete otonal chords 
(instead MA error times square of min number of notes giving one 
complete otonal chord). 

You will notice that, for 5-limit, meantone is 5th on Graham's ranking 
and 3rd on mine.

At the 7-limit, Miracle is 2nd on both rankings, but Paultone is 6th 
on mine, 7th on Graham's. The single chain 125 c generator that I 
posted many months ago doesn't make it into the top ten by Grahams 
ranking, but probably would on mine.

I think that, before we start writing a paper re Miracle, we should 
spend some time fine-tuning our figure of demerit (FoD) so that 
meantone ends up on top of the 5-limit list. Then I will have more 
confidence that we haven't missed something that is even better than 
Miracle at the 7 and 11 limits.

First I'd like to confirm that squaring numbers of notes is entirely 
justified by the fact that this is proportional to a number of 
_intervals_. By using a number one less than Graham was using, I'm 
favouring scales with fewer notes, a little more. 

In thinking about why other stuff, that is obvious junk, is beating 
meantone, I came up with two ideas.
(a) We need some melodic factor(s) in our FoD
(b) There is a critical range of errors. Errors greater than that 
range might as well be infinite. Errors less than that range, might as 
well be zero (or might as well be equal to the bottom of the range).

(a) One crude way of taking melodic efficiency (not Rothenberg's) into 
account would be to use the (square of the) number of notes in the 
smallest proper MOS that contains a complete otonality, instead of 
merely the (square of the) number of notes (of any propriety) that 
contain a complete otonality.

(b) Although one doesn't like to have too many free parameters in 
these models, I think the parameter we need here is the same one that 
informs harmonic entropy; i.e. the standard deviation of the 
probability-of-recognition curve (or whatever it is called). 

Instead of multiplying by the raw error in cents, whether it be MA or 
RMS, I think we should divide by the probability of recognition of 
such an error. i.e. exp(-(error/std_error)^2). We should do two runs; 
one for the average listener and one for the critical listener. Paul I 
think the values you use correspond to std_errors of about 17c and 10c 
respectively. Is that right?

Graham, whenever you have time, I'd be very interested in the top 10's 
using the above FoD (MA and RMS making 4 runs in total). If you need 
more explanation of how to calculate these numbers, don't hesitate to 
ask.

But maybe folks can suggest other psychoacoustically justifiable FoD's 
that might give the right answer for meantone.

Regards,
-- Dave Keenan


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Message: 62 - Contents - Hide Contents

Date: Tue, 29 May 2001 07:12:30

Subject: Re: Temperament program issues

From: Dave Keenan

Notice how 16.6 cents (approx one step of 72-EDO) is by far the best 
"generator generator". i.e. the vast majority of top-10 generators are 
very close to a multiple of it. I haven't figured out what the others 
are a multiple of.


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Message: 63 - Contents - Hide Contents

Date: Tue, 29 May 2001 10:04 +0

Subject: Re: Temperament program issues

From: graham@m...

In-Reply-To: <9ev4ed+ntv4@e...>
monz wrote:

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > > Yahoo groups: /tuning-math/message/56 * [with cont.] >
>> ... By the way, your Blackjack chord progression totally >> blew away and inspired Joseph Pehrson. > >
> Me too!!!! I could tell by looking at it that I simply > *had* to hear it... that's exactly why *I* made the audio file! > > Just wish I currently had a polyphonic keyboard to map it > too, so that I could play around with it too. (well, I'd use > Canasta instead...) Pretty soon...
Which progression are we talking about here? The one I worked out by improvising and posted notation and an MP3 for, or the one I worked out on the lattice to show it could handle a particular acoustic equivalence? Graham
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Message: 64 - Contents - Hide Contents

Date: Tue, 29 May 2001 10:04 +0

Subject: Re: Temperament program issues

From: graham@m...

In-Reply-To: <9evi4u+e5pu@e...>
In article <9evi4u+e5pu@e...>, D.KEENAN@U... (Dave Keenan) 
wrote:

> Notice how 16.6 cents (approx one step of 72-EDO) is by far the best > "generator generator". i.e. the vast majority of top-10 generators are > very close to a multiple of it. I haven't figured out what the others > are a multiple of.
If you look down my tables, you should see which were produced using 72-equal. Graham
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Message: 66 - Contents - Hide Contents

Date: Mon, 29 May 2000 03:52:35

Subject: Graham's pump progression

From: Joe Monzo

----- Original Message -----
From: <graham@m...>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Tuesday, May 29, 2001 2:04 AM
Subject: [tuning-math] Re: Temperament program issues


> In-Reply-To: <9ev4ed+ntv4@e...> > monz wrote: >
>> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >> >> Yahoo groups: /tuning-math/message/56 * [with cont.] >>
>>> ... By the way, your Blackjack chord progression totally >>> blew away and inspired Joseph Pehrson. >> >>
>> Me too!!!! I could tell by looking at it that I simply >> *had* to hear it... that's exactly why *I* made the audio file! >> >> Just wish I currently had a polyphonic keyboard to map it >> too, so that I could play around with it too. (well, I'd use >> Canasta instead...) Pretty soon... >
> Which progression are we talking about here? The one I worked out by > improvising and posted notation and an MP3 for, or the one I worked out on > the lattice to show it could handle a particular acoustic equivalence?
The latter, which I've added near the bottom of: Blackjack scale: 21-out-of-72-EDO (c) 2001 by... * [with cont.] (Wayb.) _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
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Message: 67 - Contents - Hide Contents

Date: Tue, 29 May 2001 14:15 +0

Subject: Re: Temperament program issues

From: graham@m...

In-Reply-To: <9evhgg+5rf0@e...>
Dave Keenan wrote:

> Instead of multiplying by the raw error in cents, whether it be MA or > RMS, I think we should divide by the probability of recognition of > such an error. i.e. exp(-(error/std_error)^2). We should do two runs; > one for the average listener and one for the critical listener. Paul I > think the values you use correspond to std_errors of about 17c and 10c > respectively. Is that right? > > Graham, whenever you have time, I'd be very interested in the top 10's > using the above FoD (MA and RMS making 4 runs in total). If you need > more explanation of how to calculate these numbers, don't hesitate to > ask.
I've updated the files at <Automatically generated temperaments * [with cont.] (Wayb.)> with a few changes: Simpler ETs are listed. The +1 is taken out for measuring complexity. The size of generator in cents is shown. The minimax function should ignore octave-divisions as suggested, although I'm not sure it's working. getFigureOfDemerit has been added, to make this easier to change. You can get it to work using the suggestion above by commenting out the last line in the function and uncommenting the one before. If that's too difficult for you, the results are at <3 4 5 6 7 8 9 10 12 15 16 18 19 22 23 24 25 26... * [with cont.] (Wayb.)> <4 5 6 9 10 12 15 16 18 19 22 26 27 29 31 35 36... * [with cont.] (Wayb.)> <5 12 19 22 26 27 29 31 41 46 50 53 58 60 68 70... * [with cont.] (Wayb.)> <22 26 29 31 41 46 58 72 80 87 89 94 * [with cont.] (Wayb.)> <29 41 58 72 80 87 94 * [with cont.] (Wayb.)> It does make meantone the best 5-limit temperament. It leaves Miracle as the best 11-limit, but doesn't treat it well otherwise. Graham
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Message: 68 - Contents - Hide Contents

Date: Tue, 29 May 2001 20:08:45

Subject: Re: Temperament program issues

From: Paul Erlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: > > Yahoo groups: /tuning-math/message/56 * [with cont.] >
>> ... By the way, your Blackjack chord progression totally >> blew away and inspired Joseph Pehrson. > >
> Me too!!!! I could tell by looking at it that I simply > *had* to hear it... that's exactly why *I* made the audio file!
You deserve a lot of credit also for the voice-leading (which Joseph and I meticulously reproduced on the keyboard) and for such a good choice of timbre.
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Message: 70 - Contents - Hide Contents

Date: Tue, 29 May 2001 20:13:57

Subject: Re: Temperament program issues

From: Paul Erlich

--- In tuning-math@y..., graham@m... wrote:

> Which progression are we talking about here? The one I worked out by > improvising and posted notation and an MP3 for, or the one I worked out on > the lattice to show it could handle a particular acoustic equivalence? > > > Graham
The latter. I must have missed the former -- where it it?
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Message: 71 - Contents - Hide Contents

Date: Mon, 29 May 2000 13:20:38

Subject: Re: Temperament program issues

From: Joe Monzo

----- Original Message ----- 
From: Paul Erlich <paul@s...>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Tuesday, May 29, 2001 1:08 PM
Subject: [tuning-math] Re: Temperament program issues


> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: >> >> Yahoo groups: /tuning-math/message/56 * [with cont.] >>
>>> ... By the way, your Blackjack chord progression totally >>> blew away and inspired Joseph Pehrson. >> >>
>> Me too!!!! I could tell by looking at it that I simply >> *had* to hear it... that's exactly why *I* made the audio file! >
> You deserve a lot of credit also for the voice-leading (which Joseph > and I meticulously reproduced on the keyboard) and for such a good > choice of timbre.
Wow, Paul, thanks! So the stuff I learned in school had a point behind it after all! ;-) -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.)
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Message: 72 - Contents - Hide Contents

Date: Tue, 29 May 2001 21:37:49

Subject: Re: Temperament program issues

From: Paul Erlich

I don't think it's essential to make meantone look like the best 5-
limit linear temperament. Meantone is important for other reasons, 
outside the scope of this program:

One of the first MOSs with complete chords is the diatonic scale. The 
diatonic scale has identical tetrachords in every octave species -- a 
very important melodic property. The diatonic scale allows both the 
complete utonalities and the complete otonalities to come from a 
single pattern in the scale -- something only my 7-limit decatonic 
scales share. The diatonic scale has been the basis of musical 
composition and notation back in the 3-limit era, when Pythagorean 
tuning was the norm, and 5-limit practice _evolved_ from this, rather 
than being a new creation. Etc, etc . . .


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Message: 73 - Contents - Hide Contents

Date: Tue, 29 May 2001 22:00:46

Subject: Re: Temperament program issues

From: Paul Erlich

> If that's too difficult for you, the results are at > > <3 4 5 6 7 8 9 10 12 15 16 18 19 22 23 24 25 26... * [with cont.] (Wayb.)> > <4 5 6 9 10 12 15 16 18 19 22 26 27 29 31 35 36... * [with cont.] (Wayb.)> > <5 12 19 22 26 27 29 31 41 46 50 53 58 60 68 70... * [with cont.] (Wayb.)> > <22 26 29 31 41 46 58 72 80 87 89 94 * [with cont.] (Wayb.)> > <29 41 58 72 80 87 94 * [with cont.] (Wayb.)> >
I looked at the 7-limit results and they're fascinating. While I don't see paultone, I see a 5/11 oct. generator with a half-octave interval of repetition. So 22-tET should work very well for it. Can you talk about the structure of this scale at all?
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Message: 74 - Contents - Hide Contents

Date: Wed, 30 May 2001 03:45:50

Subject: Re: Temperament program issues

From: David C Keenan

Graham wrote:
>If that's too difficult for you, the results are at ...
Thanks for doing that Graham. Much appreciated. Here's how to calculate the notes per octave of the smallest MOS containing a complete otonality. In pseudo Pascal. Given period p and generator g (both in octaves) and w as the width of the complete otonality (in generators). r := g/p m_prev := 0 m := 1 WHILE m <= w DO i := INT(r) r := 1/(r-i) temp := m m := m*i + m_prev m_prev := temp return m/p Regards, -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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