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Message: 25 - Contents - Hide Contents Date: Thu, 24 May 2001 18:13:47 Subject: Re: Hypothesis From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:>>>> Usually the chromatic vectors are an octave and a twelfth. >>>> Lost me there. >> The top of the equation will look like > > (1 0 0 ...) (1 0 0 ...) > (0 1 0 ...)H' = (0 1 0 ...)H' > > So the prime axes 2 and 3 aren't being tempered out.Still confused. Since when are octaves and twelfths "chromatic"?
Message: 26 - Contents - Hide Contents Date: Thu, 24 May 2001 19:29:20 Subject: Re: Fwd: optimizing octaves in MIRACLE scale.. From: Paul Erlich 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.
Message: 27 - Contents - Hide Contents Date: Fri, 25 May 2001 16:42:23 Subject: Re: Fwd: optimizing octaves in MIRACLE scale.. From: monz --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:> I wrote, > >> [monz]>> But for sure, the 12-integer-limit is in _Harmonielehre_. >> Really? So ratios such as 16:9 would have fallen outside it?Paul, I started a response to this but it is getting long and interesting. I'll post it tonight. -monz
Message: 28 - Contents - Hide Contents Date: Fri, 25 May 2001 18:25 +0 Subject: Re: Hypothesis From: graham@m... In-Reply-To: <9ejj0r+mrao@e...> Paul wrote:>>>> Usually the chromatic vectors are an octave and a twelfth. >>>>>> Lost me there. >>>> The top of the equation will look like >> >> (1 0 0 ...) (1 0 0 ...) >> (0 1 0 ...)H' = (0 1 0 ...)H' >> >> So the prime axes 2 and 3 aren't being tempered out. >> Still confused. Since when are octaves and twelfths "chromatic"?They're intervals used to specify the temperament that aren't being tempered out. I thought this was the definition of "chromatic unison vector". Graham
Message: 29 - Contents - Hide Contents Date: Fri, 25 May 2001 18:53:52 Subject: Re: Hypothesis From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> In-Reply-To: <9ejj0r+mrao@e...> > Paul wrote: >>>>>> Usually the chromatic vectors are an octave and a twelfth. >>>>>>>> Lost me there. >>>>>> The top of the equation will look like >>> >>> (1 0 0 ...) (1 0 0 ...) >>> (0 1 0 ...)H' = (0 1 0 ...)H' >>> >>> So the prime axes 2 and 3 aren't being tempered out. >>>> Still confused. Since when are octaves and twelfths "chromatic"? >> They're intervals used to specify the temperament that aren't being > tempered out. I thought this was the definition of "chromatic unison > vector".A unison vector defines an equivalence relation in the lattice. Hence the name "unison". A periodicity block results from N independent unison vectors in an N-dimensional lattice. A twelfth or fifth as a unison vector would be really crazy, as far as I can tell. In the diatonic scale, the commatic unison vector is 81:80, and the chromatic unison vector is 25:24 (or 135:128). In the decatonic scale, the commatic unison vectors are any two of {225:224, 64:63, 50:49}; and the chromatic unison vector is any one of {49:49, 28:27, 25:24}.
Message: 30 - Contents - Hide Contents Date: Fri, 25 May 2001 18:59:39 Subject: Re: Hypothesis From: Paul Erlich I wrote,> and the chromatic unison vector is any one > of {49:49Oops -- should be 49:48, of course!
Message: 31 - Contents - Hide Contents Date: Fri, 25 May 2001 19:59:12 Subject: Re: Fwd: optimizing octaves in MIRACLE scale.. From: Paul Erlich 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¢.
Message: 32 - Contents - Hide Contents Date: Fri, 25 May 2001 22:36 +0 Subject: Re: More temperaments (from Tuning List) From: graham@m... Paul wrote:>> Yes, the 11- and 15-limits are riddled with them. There's even one > that>> divides the octave into 29 equal parts! 4.7 cents for the 15- > limit, only>> needs 1 generator, needing 30 for a complete chord? Mighty > strange, but>> it does mean you can use it with two manuals tuned to 29-equal a > generator>> apart. As I've shown before that 29-equal works quite well mapped > to>> Halberstadt, this is far from arbitrary. >> My proposal of two 12-equal keyboards 15 cents apart must look pretty > good in the 5-limit, yes?Doesn't make the list, but only extremely accurate temperaments do in the 5-limit. I think the 29+ one should be more complex than that. I'm calculating the number of steps for a chord as the number of generators to the most complex interval times the number of equivalence intervals plus one. I think the addition should be done before the multiplication, but I think that makes the octave-splitting ones look worse than they really are. This particular one looks much better, though. I'll try and sort it out in the morning. I've changed the scoring which alters the 11 and 15-limit ordering. This was to balance the 9-limit for simplicity rather than accuracy. The program and output files should be at <Automatically generated temperaments * [with cont.] (Wayb.)> by now. Graham
Message: 33 - Contents - Hide Contents Date: Fri, 25 May 2001 21:46:10 Subject: Re: More temperaments (from Tuning List) From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> The program and output files should be at > <Automatically generated temperaments * [with cont.] (Wayb.)> by now. >Excuse me if I'm misunderstanding, but isn't your top 11-limit generator, "11/113", just the MIRACLE generator?
Message: 34 - Contents - Hide Contents Date: Fri, 25 May 2001 21:52:18 Subject: Re: More temperaments (from Tuning List) From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> The program and output files should be at > <Automatically generated temperaments * [with cont.] (Wayb.)> by now.On that page it says, "I've written a Python script to systematically find equal temperaments consistent within a given prime limit." You mean odd limit, right?
Message: 35 - Contents - Hide Contents Date: Sat, 26 May 2001 09:47 +0 Subject: Re: More temperaments (from Tuning List) From: graham@m... Paul Erlich wrote:> Excuse me if I'm misunderstanding, but isn't your top 11-limit > generator, "11/113", just the MIRACLE generator?Yes, with the new scoring system. It's also bottom of the 9-limit and around the middle of the 7-limit. The scoring could still do with some tweaking, to ensure the best temperaments make the list. The real challenge will be removing the octave invariance. Graham
Message: 36 - Contents - Hide Contents
Date: Sat, 26 May 2001 21:53 +0
Subject: Temperament program issues
From: graham@m...
I've half-updated the stuff at
<Automatically generated temperaments * [with cont.] (Wayb.)>
mostly to clean up the code. Also temperaments that too complex or
inaccurate get left out, which stops the 5-limit being swamped by
impractical temperaments.
On the question of the number of notes needed for a complete chord.
Properly, for an equal number of generators to each equivalence interval,
the formula should be (d+1)*n where d is the generator and n is the
number of divisions of an octave. For the strangest example (now top of
the 15-limit list!) which needs 29 equivalence intervals and 1 generator,
this gives 58 notes.
However, you get 29 different complete chords within those notes. So the
temperament is much more efficient than one with an equivalence interval
of an octave that requires 58 notes to get a complete chord.
For that simple case, you get a new complete chord with each new note. So
the number of notes to a complete chord tells you how many you'll get with
however many notes are in the MOS. The equivalent measure where the
equivalence interval is a division of the octave is d*n+1, which is what
my program uses.
In some cases, like that 29 step octave, it gives somewhat eccentric
results. But I think a human looking at the data can use their
intelligence to work this out, instead of expecting the dumb computer to
produce a magic number that explains it all.
If you're curious, you could try looking through the 7- and 9-limit lists.
There are some interesting scales in there I haven't seen before, along
with some old friends.
The magic connection between Miracle and diaschismic I had in mind is that
they both have a generator of a semitone, approximating 16:15. This is
the same but different to the connection between schismic and diaschismic
(single and double-positive).
For the following discussion I'll use "formal octave" for the interval we
choose to stand in for an octave, and "equivalence interval" for the MOS
parameter that will be a fraction of the formal octave.
To get the program to work for inharmonic timbres, you need to express all
intervals in terms of the formal octave. This shouldn't be difficult to
do. But how do you get the formal octave? That is our next challenge.
One approach would be to extend Dave Keenan's brute force to try all
generators and equivalence intervals. This is probably too complex for a
spreadsheet, but should be easy enough as a program. The problem is that
it'd need quadratic time to check all combinations. For checking an
octave to 0.1 cents precision, that's 144 million combinations. Okay,
less than that, because the generator doesn't have to be greater than the
equivalence interval. But we still have a lot of combinations. It
wouldn't be efficient, but it would work. I don't think I'll try this,
but somebody should. I'll have a go at the more complex method below.
The first step is to get hold of all the ranges of consistent ETs for the
intervals we're looking at. Paul Erlich has already done this for the
traditional limits. The best way is probably brute force. It'd take a
while but nowhere near as long as brute forcing the linear temperaments.
I think the linear temperaments can still be expressed in step sizes for
any pair of temperaments. The challenge is getting at the formal octave.
Optimising the tuning and assessing the complexity will be harder
otherwise. Because schismic and Miracle temperaments can work with
different equivalence intervals, there's obviously no unique way of
getting an equivalence interval when you know the temperament.
The first step would be to express each temperament using an arbitrary
formal octave, probably the first one in the list of prime intervals.
Then we can use that specification to weed out duplicates for when we come
to the hard work. This would mean taking each consonant interval for a
formal octave, and seeing which ones work best.
So, what makes a good formal octave? It's tempting to say it should be a
prime, rather than derived, interval. But schismic works with a fourth
and Miracle with a fifth, so that obviously won't work. Perhaps we only
need to check combinations of the first few prime intervals, assuming them
to be the most important. But can we be sure this is always sufficient?
A post on the practical microtonality list (I don't have the reference,
sorry) did suggest deliberately choosing poor consonances for formal
octaves. So we will probably have to have a lot of freedom in how the
choice is made.
Other factors are that the scale should have a "nice" number and pattern
of notes to its formal octave. I'm not sure how to express this for an
open ended temperament.
There we go, ideas are welcome.
Graham
Message: 37 - Contents - Hide Contents Date: Sat, 26 May 2001 21:44:40 Subject: Re: More temperaments (from Tuning List) From: Dave Keenan --- In tuning-math@y..., graham@m... wrote:> The scoring could still do with some tweaking, to ensure the best > temperaments make the list. The real challenge will be removing the > octave invariance.What do you mean by this? Do you mean you want to find MA errors at an N+1 integer limit instead of N odd limit when the octave is tempered too?
Message: 38 - Contents - Hide Contents Date: Sat, 26 May 2001 22:49:34 Subject: Re: Temperament program issues From: Dave Keenan --- In tuning-math@y..., graham@m... wrote:> On the question of the number of notes needed for a complete chord. ...Ah yes. I see your point. So how about simply using n*d, the number of generators in a complete chord. Its a more useful number. If you want to know how many complete otonalities you get for N notes, it's simply N - n*d.> If you're curious, you could try looking through the 7- and 9-limit lists. > There are some interesting scales in there I haven't seen before, along > with some old friends.It would be a lot easier to recognise any old friends if you had your program express the basis as "N chains of G cent generators".> For the following discussion I'll use "formal octave" for the interval we > choose to stand in for an octave, and "equivalence interval" for the MOS > parameter that will be a fraction of the formal octave.This is too confusing. How about we use the term "period" instead of "equivalence interval". If you thing that might be confused with a time period you could call it the "log period", but notice that we already use it in this way, without the "log" qualifier, in the term "periodicity block".> To get the program to work for inharmonic timbres, you need to express all > intervals in terms of the formal octave. This shouldn't be difficult to > do. But how do you get the formal octave? That is our next challenge.I'm personally not very interested in that completely general case just yet. Most inharmonic timbres are short lived and so coinciding partials isn't all that important. I'd be more interested in the cases where the period is a whole-number fraction of another highly consonant (harmonic) interval, such as the BP 1:3, or 1:4, 2:3, 1:5, 1:6, 2:5, 3:4 etc. And even more interested in the case where we stick to fractions of a real octave but allow the octave to be tempered and use N+1 integer limits in place of N odd-limits. -- Dave Keenan
Message: 39 - Contents - Hide Contents Date: Sun, 27 May 2001 06:40:57 Subject: Re: Temperament program issues From: Dave Keenan --- In tuning-math@y..., graham@m... wrote:> Okay, > less than that, because the generator doesn't have to be greater than the > equivalence interval.The generator doesn't have to be greater than _half_ the period (equivalence interval). But yes, it's still O(n^2).
Message: 40 - Contents - Hide Contents Date: Sun, 27 May 2001 09:00:05 Subject: Re: Fwd: optimizing octaves in MIRACLE scale.. From: monz --- 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. ...? 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? Or will it be perceived as a nuisance to those of you who are ready to discuss nitty-gritty tuning math? I know that Paul is generous with his help, and so I can keep this stuff relegated to private email if others prefer. My hope is that the Tuning-math List can be a place for people of all mathematical levels to be able discuss aspects of this subject, but I would perfectly understand if most subscribers want to keep discussion on a high level. -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0"
Message: 41 - Contents - Hide Contents Date: Sun, 27 May 2001 13:27 +0 Subject: Re: Fwd: optimizing octaves in MIRACLE scale.. From: graham@m... monz wrote:> My hope is that the Tuning-math List can be a place for people > of all mathematical levels to be able discuss aspects of this > subject, but I would perfectly understand if most subscribers > want to keep discussion on a high level.I agree (with the first bit) Graham
Message: 42 - Contents - Hide Contents Date: Sun, 27 May 2001 13:28 +0 Subject: Re: More temperaments From: graham@m... Dave Keenan wrote:> --- In tuning-math@y..., graham@m... wrote:>> The scoring could still do with some tweaking, to ensure the best >> temperaments make the list. The real challenge will be removing the >> octave invariance. >> What do you mean by this? Do you mean you want to find MA errors > at an N+1 integer limit instead of N odd limit when the octave is > tempered too?I mean taking an arbitrary set of intervals and finding some generators and periods that fit them. Graham
Message: 43 - Contents - Hide Contents Date: Sun, 27 May 2001 13:28 +0 Subject: Re: Temperament program issues From: graham@m... Dave Keenan wrote:> --- In tuning-math@y..., graham@m... wrote:>> On the question of the number of notes needed for a complete chord. > ... >> Ah yes. I see your point. So how about simply using n*d, the number of > generators in a complete chord. Its a more useful number. If you want > to know how many complete otonalities you get for N notes, it's simply > N - n*d.Equally useful, they're only 1 note apart. Originally I was using n*d for the scoring anyway.>> If you're curious, you could try looking through the 7- and 9-limit > lists.>> There are some interesting scales in there I haven't seen before, > along>> with some old friends. >> It would be a lot easier to recognise any old friends if you had your > program express the basis as "N chains of G cent generators".Uh, yeah, sometime. You'll have to convert from octaves for now, or hack the code yourself (Python's really easy to learn, you can download the interpreter)>> For the following discussion I'll use "formal octave" for the > interval we>> choose to stand in for an octave, and "equivalence interval" for the > MOS>> parameter that will be a fraction of the formal octave. >> This is too confusing. How about we use the term "period" instead of > "equivalence interval". If you thing that might be confused with a > time period you could call it the "log period", but notice that we > already use it in this way, without the "log" qualifier, in the term > "periodicity block". Okay.>> To get the program to work for inharmonic timbres, you need to > express all>> intervals in terms of the formal octave. This shouldn't be > difficult to>> do. But how do you get the formal octave? That is our next > challenge. >> I'm personally not very interested in that completely general case > just yet. Most inharmonic timbres are short lived and so coinciding > partials isn't all that important.How about bells? My wok makes some good sound if I thwack it right as well. I don't think the harmonic series has many more surprises for us. It was worth a look to make sure we hadn't overlooked any Miracles, but I think the program already shows that. The 15-limit temperaments are too complex for me right now, and there's no point in finding them again. You could try it with different sets of intervals, but that's really easy, no more programming. One thing is it might be good for the program to recognize scales that approximate a subset of the intervals you gave it. Thinking about it, for the non-octave case, we don't even need a brute-force search for ETs. So long as we only take consistent ones, we can choose an arbitrary formal octave and do the same search as currently. It may be best to choose a large interval for that formal octave, as there may be more than one consistent ET that divides a small interval into the same number of steps.> I'd be more interested in the cases where the period is a > whole-number fraction of another highly consonant (harmonic) interval, > such as the BP 1:3, or 1:4, 2:3, 1:5, 1:6, 2:5, 3:4 etc.So long as you choose the interval first, that should be easy. You need to specify each prime direction in terms of that interval. Something like: newprimes = [] for interval in primes: newprimes.append(newprimes/pitchOfFormalOctave) where primes is a list of the intervals in terms of octaves (current standard) and pitchOfFormalOctave is the pitch of this interval you chose in octaves (log2(3) for 3:1).> And even more interested in the case where we stick to fractions of a > real octave but allow the octave to be tempered and use N+1 integer > limits in place of N odd-limits.I don't really care what happens after the linear temperament's found. The program can be hacked for that easily enough. If you have an algorithm for the optimization, it can be added as a method like optimizeMinimax. Graham
Message: 44 - Contents - Hide Contents Date: Sun, 27 May 2001 17:55:20 Subject: Re: Fwd: optimizing octaves in MIRACLE scale.. From: monz --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote: Yahoo groups: /tuning-math/message/24 * [with cont.]> I wrote, >>>> hence as 33/32 and 13/12 -- differed by virtually an entire >>> semitone (i.e., Schoenberg assumed a "unison vector" of 143:128). >> Oops! That should be 104:99, not 143:128! >>> But for sure, the 12-integer-limit is in _Harmonielehre_. >> Really? So ratios such as 16:9 would have fallen outside it? (early response:)Oops... Schoenberg doesn't actually claim that the 12th harmonic is any kind of limit... it's simply where his musical illustration and its accompanying explanation end. I suppose he implies that it continues beyond into inaudibility. The musical illustration uses the 1st thru 12th harmonics on F, C, and G. So using a 12-integer-limit here would relate it to Schoenberg's illustration, but not necessarily to his actual theory. In the later article, "Problems of Harmony", which BTW was written in 1927 then revised in 1934 for presentation in America, Schoenberg definitely explains harmony as being based on a 13-integer-limit as harmonics 1 thru 13 on F, C, and G. I would label this system as (1...13)/(3^(-1...1)). Is there a better notation for that? Still later, in _Structural Functions of Harmony_ [1949], his "Chart of the Regions" (2 versions, in major and minor) uses terms such as "mediant" which imply more extended 5-limit derivations for some notes than the ratios implied by the overtone model. ==== That was my first response to this. I was going to concede to Paul that I had been in error, and to some extent I *was*, but guess what?... The scale of approximated ratios implied by Schoenberg's diagram provides only one 16:9!, between d-27 and c-48. There are 3 other varieties of "minor 7th": 11:6 (really a "neutral 7th") between d-54 and c-99, 9:5 between b-45 and a-80, and 7:4 between g-36 and f-63. I was getting concerned that this thread was veering off-topic, but this gives me the opportunity to remedy that situation. :) (My quotes of Schoenberg are from the English translation of _Harmonielehre_ by Roy Carter, and the page numbers refer to that edition.) Schoenberg [p 23] posits the existences of two "forces", one pulling downward and one pulling upward around the tonic, which he illustrates as: F <- C -> G and likens to resistance against gravity. In mathematical terms, he is referring to the harmonic relationships of 3^-1 and 3^1, respectively.> [Schoenberg, p 24:] > > ...thus it is explained how the scale that finally emerged > is put together from the most important components of a > fundamental tone and its nearest relatives. These nearest > relatives are just what gives the fundamental tone stability; > for it represents the point of balance between their opposing > tendencies. This scale appears as the residue of the properties > of the three factors, as a vertical projection, as addition:Schoenberg then presents a diagram of the overtones and the resulting scale, which I have adaptated, adding the partial-numbers which relate all the overtones together as a single set: b-45 g-36 e-30 d-27 c-24 a-20 g-18 g-18 f-16 c-12 c-12 f-8 f c g a d e b 8 12 18 20 27 30 45> [Schoenberg:] > > Adding up the overtones (omitting repetitions) we get the seven > tones of our scale. Here they are not yet arranged consecutively. > But even the scalar order can be obtained if we assume that the > further overtones are also in effect. And that assumption is > in fact not optional; we must assume the presence of the other > overtones. The ear could also have defined the relative pitch > of the tones discovered by comparing them with taut strings, > which of course become longer or shorter as the tone is lowered > or raised. But the more distant overtones were also a > dependable guide. Adding these we get the following:Schoenberg then extends the diagram to include the following overtones: fundamental partials F 2...12, 16 C 2...11 G 2...12 (Note, therefore, that he is not systematic in his employment of the various partials.) Again, I adapt the diagram by adding partial-numbers: d-108 c-99 b-90 a-81 g-72 f-66 f-64 (f-63) e-60 d-54 d-54 c-48 c-48 b-45 b-44 (bb-42) a-40 g-36 g-36 g-36 f-32 e-30 (eb-28) d-27 c-24 c-24 a-20 g-18 g-18 f-16 c-12 c-12 f-8 (eb) (bb) c d e f g a b c d e f g a b c d [44] [64] (28) (42) [66] 24 27 30 32 36 40 45 48 54 60 63 72 81 90 99 108 (Note also that Schoenberg was unsystematic in his naming of the nearly-1/4-tone 11th partials, calling 11th/F by the higher of its nearest 12-EDO relatives, "b", while calling 11th/C and 11th/G by the lower, "f" and "c" respectively. This, ironically, is the reverse of the actual proximity of these overtones to 12-EDO: ~10.49362941, ~5.513179424, and ~0.532729432 Semitones, respectively). The partial-numbers are also given for the resulting scale at the bottom of the diagram, showing that 7th/F (= eb-28) is weaker than 5th/C (= e-30), and 7th/C (= bb-42) is weaker than 5th/G (= b-45). Also note that 11th/F (= b-44), 16th/F (= f-64) and 11th/C (= f-66) are all weaker still, thus I have included them in square brackets. These overtones are not even mentioned by Schoenberg. Schoenberg does take note of the ambiguity present in this collection of ratios, in his later article _Problems of Harmony_. I won't go into that here because this is focusing on his 1911 theory. Here is an interval matrix of Schoenberg's scale (broken in half to fit the screen), with implied proportions given along the left and the bottom, and Semitone values of the intervals in the body. Because Schoenberg's implied proportions form an "octave"-specific pitch-set in his presentation (not necessarily in his theory), this matrix has no "bottom" half. Interval Matrix of Schoenberg's implied JI scale: 108 26.04 24.00 23.37 22.18 21.06 19.02 17.20 16.35 15.55 15.16 14.04 99 24.53 22.49 21.86 20.67 19.55 17.51 15.69 14.84 14.04 13.65 12.53 90 22.88 20.84 20.21 19.02 17.90 15.86 14.04 13.19 12.39 12.00 10.88 81 21.06 19.02 18.39 17.20 16.08 14.04 12.22 11.37 10.57 10.18 9.06 72 19.02 16.98 16.35 15.16 14.04 12.00 10.18 9.33 8.53 8.14 7.02 66 17.51 15.47 14.84 13.65 12.53 10.49 8.67 7.82 7.02 6.63 5.51 64 16.98 14.94 14.31 13.12 12.00 9.96 8.14 7.29 6.49 6.10 4.98 63 16.71 14.67 14.04 12.84 11.73 9.69 7.86 7.02 6.21 5.83 4.71 60 15.86 13.82 13.19 12.00 10.88 8.84 7.02 6.17 5.37 4.98 3.86 54 14.04 12.00 11.37 10.18 9.06 7.02 5.20 4.35 3.55 3.16 2.04 48 12.00 9.96 9.33 8.14 7.02 4.98 3.16 2.31 1.51 1.12 0.00 45 10.88 8.84 8.21 7.02 5.90 3.86 2.04 1.19 0.39 0.00 44 10.49 8.45 7.82 6.63 5.51 3.47 1.65 0.81 0.00 42 9.69 7.65 7.02 5.83 4.71 2.67 0.84 0.00 40 8.84 6.80 6.17 4.98 3.86 1.82 0.00 36 7.02 4.98 4.35 3.16 2.04 0.00 32 4.98 2.94 2.31 1.12 0.00 30 3.86 1.82 1.19 0.00 28 2.67 0.63 0.00 27 2.04 0.00 24 0.00 24 27 28 30 32 36 40 42 44 45 48 --- 108 12.00 10.18 9.33 9.06 8.53 7.02 4.98 3.16 1.51 0.00 99 10.49 8.67 7.82 7.55 7.02 5.51 3.47 1.65 0.00 90 8.84 7.02 6.17 5.90 5.37 3.86 1.82 0.00 81 7.02 5.20 4.35 4.08 3.55 2.04 0.00 72 4.98 3.16 2.31 2.04 1.51 0.00 66 3.47 1.65 0.81 0.53 0.00 64 2.94 1.12 0.27 0.00 63 2.67 0.84 0.00 60 1.82 0.00 54 0.00 54 60 63 64 66 72 81 90 99 108 -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0"
Message: 45 - Contents - Hide Contents Date: Mon, 28 May 2001 03:56:34 Subject: Re: Temperament program issues From: Dave Keenan --- In tuning-math@y..., graham@m... wrote:> Dave Keenan wrote:>> It would be a lot easier to recognise any old friends if you had your >> program express the basis as "N chains of G cent generators". >> Uh, yeah, sometime. You'll have to convert from octaves for now, or hack > the code yourself (Python's really easy to learn, you can download the > interpreter)I'm getting too old for that. I've had to learn so many new programming languages (and GUI libraries) in my life that I'm heartily sick of it. I had a hard time recognising Paultone among your top-10 7-limit generators. I think there's a bug in your MA optimiser. 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.> I don't really care what happens after the linear temperament's found. > The program can be hacked for that easily enough. If you have an > algorithm for the optimization, it can be added as a method like > optimizeMinimax.Not by me. How about giving us the top-tens ranked according to RMS error times (n*d)^2, generators in cents. -- Dave Keenan
Message: 46 - Contents - Hide Contents Date: Mon, 28 May 2001 04:20:07 Subject: Re: Fwd: optimizing octaves in MIRACLE scale.. From: Dave Keenan --- In tuning-math@y..., "monz" <joemonz@y...> wrote:> There are 3 other varieties of "minor 7th": > > 11:6 (really a "neutral 7th") between d-54 and c-99, > > 9:5 between b-45 and a-80, and > > 7:4 between g-36 and f-63.7:4 (really a subminor 7th) between g-36 and f-63 :-)
Message: 47 - Contents - Hide Contents Date: Mon, 28 May 2001 05:01:00 Subject: Re: Temperament program issues From: Dave Keenan 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?
Message: 48 - Contents - Hide Contents Date: Mon, 28 May 2001 08:56 +0 Subject: Re: Temperament program issues From: graham@m... 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. It's this one: 66/157 basis: (1.0, 0.419517976278) mapping: ([1, 0], ([2, 4], [-1, -4])) ^^^^^^ primeApprox: ([88, 69], [(139, 109), (204, 160)]) highest interval width: 4 notes required: 5 highest error: 0.004480 (5.377 cents) The list I point to is the easiest way of showing that it's meantone. A third is four fifths. The figure of demerit isn't perfect but, now the really useless temperaments are excluded, it's good enough to bring the interesting ones into the 10. You can change that to 20 if you don't trust it. Graham
Message: 49 - Contents - Hide Contents Date: Mon, 28 May 2001 09:08:59 Subject: Re: Temperament program issues From: Paul Erlich --- In tuning-math@y..., graham@m... wrote:> The magic connection between Miracle and diaschismic I had in mind is that > they both have a generator of a semitone, approximating 16:15.Well, I usually think of diaschismic as having a ~3:2 generator and a half-octave "by-generator". But yes, replacing the ~3:2 with an approximate 16:15 is equivalent.> This is > the same but different to the connection between schismic and diaschismic > (single and double-positive).The same but different . . . pretty cryptic!> > For the following discussion I'll use "formal octave" for the interval we > choose to stand in for an octave, and "equivalence interval" for the MOS > parameter that will be a fraction of the formal octave. > > To get the program to work for inharmonic timbres, you need to express all > intervals in terms of the formal octave. This shouldn't be difficult to > do. But how do you get the formal octave? That is our next challenge.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.> > I think the linear temperaments can still be expressed in step sizes for > any pair of temperaments. The challenge is getting at the formal octave. Still? > Optimising the tuning and assessing the complexity will be harder > otherwise. Because schismic and Miracle temperaments can work with > different equivalence intervals, Really? > there's obviously no unique way of > getting an equivalence interval when you know the temperament. > > The first step would be to express each temperament using an arbitrary > formal octave, probably the first one in the list of prime intervals. > Then we can use that specification to weed out duplicates for when we come > to the hard work. This would mean taking each consonant interval for a > formal octave, and seeing which ones work best.You've lost me.
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