Topic: From Commas to Generators
1 scales
| File | Description | Notes | Period (¢) |
|---|---|---|---|
| ozan80_tuning-math_11729_11784 | 80-et version of Ozan Yarman scale | 12 | 1200.0 |
Thread (32 messages)
From: Kalle Aho (2005-03-08) Subject: From Commas to Generators Hi! Let's say I have a set of commas that define a 2D temperament. How can I get the period-generator-mappings of the primes from it? I believe they can be found by manipulating a set of linear equations but I guess you folks have a more straightforward method. Maybe a simple 7-limit example would be nice, thank you. Kalle
From: Rich Holmes (2005-03-08)
Subject: Re: [tuning-math] From Commas to Generators
"Kalle Aho" <[email protected]> writes:
> Hi!
>
> Let's say I have a set of commas that define a 2D temperament. How
> can I get the period-generator-mappings of the primes from it?
>
> I believe they can be found by manipulating a set of linear
> equations but I guess you folks have a more straightforward method.
>
> Maybe a simple 7-limit example would be nice, thank you.
What a coincidence, I just worked this out myself the other night.
And all my notes are at home... so let's see if I can reconstruct.
I'll give you your 7-limit, but I'll take away your 5. That is, I'll
use the primes 2, 3, and 7. Using four primes (2, 3, 5, and 7) puts
you in a higher dimension and complicates things, but I'm guessing the
principles are similar. If not, we'll hear about it, I'm sure.
In cents, these are 1200, 1901.96, and 3368.83.
Here's our comma: 64/63 = 2^6 * 3^-2 * 7^-1 = | 6 -2 -1 >. In cents,
it's 27.26.
To temper this comma out, we make the 2 smaller and the 3 and 7 larger
(since the signs of the exponents are +, -, and -). The amount we
temper by is
2 smaller by 27.26 * ln(2) / ln (64*63) = 2.28 cents
3 larger by 27.26 * ln(3) / ln (64*63) = 3.61 cents
7 larger by 27.26 * ln(7) / ln (64*63) = 6.39 cents
So the tempered values are
2 : 1200 - 2.28 = 1197.72
3 : 1901.96 + 3.61 = 1905.56
7 : 3368.83 + 6.39 = 3375.22
To verify: 6 * 1197.72 - 2 * 1905.56 - 1 * 3375.22 = 0 .
All the above is following Paul Erlich's new paper.
Now to get the period and generator. The period is easy. Note the
exponents of 3 and 7 in our comma: -2 and -1. The greatest common
divisor of (the absolute value of) these is 1, so the period is the
tempered octave, 1197.72, divided by 1. The tempered octave is the
period in this case.
For the generator, we have two equations of the form
(tempered interval) = (integer) * period + (integer) * generator
namely
1905.56 = n3 * 1197.72 + m3 * g
3375.22 = n7 * 1197.72 + m7 * g
For m3 and m7 use the exponents of 7 and 3, respectively, in the
comma, with the sign of one changed: -1 and 2. (I think it's
legitimate to change the sign of either.)
1905.56 = n3 * 1197.72 + -1 * g
3375.22 = n7 * 1197.72 + 2 * g
Eliminate g to get
(2*1905.56+3375.22)/1197.72 = 6 = 2 * n3 + n7
Different choices of integers for n3 and n7 give different generators,
all related by period equivalence (note you don't need n7 to compute
g, directly, but you do need the relationship between n3 and n7 to
discover which integers are valid values for n3, in general -- though
in this case all integers are):
n3 n7 = 6 - 2 * n3 g = n3 * 1197.72 - 1905.56
1 4 -707.84
2 2 489.88
3 0 1687.61
So we can use e.g. period = 1197.72, generator = 489.88.
I think I got that right.
- Rich Holmes
From: monz (2005-03-08) Subject: Re: From Commas to Generators --- In [email protected], Rich Holmes<rsholmes@m...> wrote: > I'll give you your 7-limit, but I'll take away your 5. > That is, I'll use the primes 2, 3, and 7. Using four > primes (2, 3, 5, and 7) puts you in a higher dimension > and complicates things, but I'm guessing the principles > are similar. If not, we'll hear about it, I'm sure. > > In cents, these are 1200, 1901.96, and 3368.83. > > Here's our comma: 64/63 = 2^6 * 3^-2 * 7^-1 = | 6 -2 -1 >. i think it would be really good to standardize the monzo notation. any objections to writing that as [6 -2, 0 -1> ? i suppose the straight pipe symbol | is preferred to the square bracket [ on the left, so as to make the bra-ket implication clear ... but i really like to see the comma-mark after the 3-exponent, as well as the zero holding the 5-exponent place. -monz
From: Paul Erlich (2005-03-08) Subject: Re: From Commas to Generators --- In [email protected], Rich Holmes<rsholmes@m...> wrote: > "Kalle Aho" <kalleaho@m...> writes: > > > Hi! > > > > Let's say I have a set of commas that define a 2D temperament. How > > can I get the period-generator-mappings of the primes from it? > > > > I believe they can be found by manipulating a set of linear > > equations but I guess you folks have a more straightforward method. > > > > Maybe a simple 7-limit example would be nice, thank you. > > What a coincidence, I just worked this out myself the other night. > And all my notes are at home... so let's see if I can reconstruct. > > I'll give you your 7-limit, but I'll take away your 5. That is, I'll > use the primes 2, 3, and 7. Using four primes (2, 3, 5, and 7) puts > you in a higher dimension and complicates things, but I'm guessing the > principles are similar. You're right. > If not, we'll hear about it, I'm sure. > > In cents, these are 1200, 1901.96, and 3368.83. > > Here's our comma: 64/63 = 2^6 * 3^-2 * 7^-1 = | 6 -2 -1 >. In cents, > it's 27.26. > > To temper this comma out, we make the 2 smaller and the 3 and 7 larger > (since the signs of the exponents are +, -, and -). The amount we > temper by is > > 2 smaller by 27.26 * ln(2) / ln (64*63) = 2.28 cents > 3 larger by 27.26 * ln(3) / ln (64*63) = 3.61 cents > 7 larger by 27.26 * ln(7) / ln (64*63) = 6.39 cents > > So the tempered values are > > 2 : 1200 - 2.28 = 1197.72 > 3 : 1901.96 + 3.61 = 1905.56 > 7 : 3368.83 + 6.39 = 3375.22 > > To verify: 6 * 1197.72 - 2 * 1905.56 - 1 * 3375.22 = 0 . > > All the above is following Paul Erlich's new paper. Glad someone's following it :) :) :) > Now to get the period and generator. The period is easy. Note the > exponents of 3 and 7 in our comma: -2 and -1. The greatest common > divisor of (the absolute value of) these is 1, so the period is the > tempered octave, 1197.72, divided by 1. The tempered octave is the > period in this case. Yup! > For the generator, we have two equations of the form > > (tempered interval) = (integer) * period + (integer) * generator > > namely > > 1905.56 = n3 * 1197.72 + m3 * g > 3375.22 = n7 * 1197.72 + m7 * g > > For m3 and m7 use the exponents of 7 and 3, respectively, in the > comma, with the sign of one changed: -1 and 2. Yes, this is what I mentioned to Kalle on the tuning list. > (I think it's > legitimate to change the sign of either.) > > 1905.56 = n3 * 1197.72 + -1 * g > 3375.22 = n7 * 1197.72 + 2 * g > > Eliminate g to get > > (2*1905.56+3375.22)/1197.72 = 6 = 2 * n3 + n7 > > Different choices of integers for n3 and n7 give different generators, > all related by period equivalence (note you don't need n7 to compute > g, directly, but you do need the relationship between n3 and n7 to > discover which integers are valid values for n3, in general -- though > in this case all integers are): > > n3 n7 = 6 - 2 * n3 g = n3 * 1197.72 - 1905.56 > 1 4 -707.84 > 2 2 489.88 > 3 0 1687.61 > > So we can use e.g. period = 1197.72, generator = 489.88. > > I think I got that right. Probably. What do Gene and Graham get? Hey, Rich, BTW, great to have you on board, we're a pretty rarefied group so every new member makes us all much richer.
From: Rich Holmes (2005-03-08) Subject: Notation, was Re: [tuning-math] Re: From Commas to Generators "monz" <[email protected]> writes: > --- In [email protected], Rich Holmes<rsholmes@m...> wrote: > > > > Here's our comma: 64/63 = 2^6 * 3^-2 * 7^-1 = | 6 -2 -1 >. > > > i think it would be really good to standardize the monzo > notation. any objections to writing that as [6 -2, 0 -1> ? > > i suppose the straight pipe symbol | is preferred to > the square bracket [ on the left, so as to make the > bra-ket implication clear ... but i really like to see > the comma-mark after the 3-exponent, as well as the zero > holding the 5-exponent place. Well, dropping the unused exponents is something I've seen done. Granted it has potential for confusion, if you don't make it clear what you're doing; but I think it can also simplify discussion and notation <-> computer code conversion in cases like this where e.g. we're considering a 5-less system. Certainly the inclusion of the 5-exponent in this case would obscure the fact that we're talking about a 3-dimensional lattice. Really, to me it's neither clearly good nor clearly bad to drop the unused exponent(s). I'd never noticed use of the comma after the 3-exponent before. I'm curious as to its function. I'm used to bra-ket notation that uses a vertical line, but I suppose actually one could argue in favor of bracket instead of pipe, since we're often reading these things off computer screens and the contents of the vector often are integers, and [ is arguably less likely than | to be confused with 1, at least in some fonts. - Rich Holmes
From: Gene Ward Smith (2005-03-08)
Subject: Re: From Commas to Generators
--- In [email protected], "Paul Erlich" <perlich@a...> wrote:
> Probably. What do Gene and Graham get?
I get confused; you are trying to find period and generator for a planar
(3D) temperament? 64/63 bridges the 5-limit to the 7-limit, so
generators can be taken as {2,3,5}; in other words, we are using a
retuned 64/9 for 7, so we want a sharp fifth to keep the 7 from being
too sharp. What 5 does is irrelevant, so we may as well leave it pure.
On the other hand, I suppose you could be asking for period and
generator for the corresponding no-fives system, which has a period of
an octave and a generator a fourth or fifth. You want two fourths to
be an approximate 7/4, so the fourths are flat and the fifths sharp,
but you know all this. Also a Pythagorean third of 81/64 is equated to
9/7. and if we take two of these, we get close to a 5/3, depending on
the tuning, and adding in (5/3)/(9/7)^2 = 245/243 to the commas makes
it into superpyth, which seems like a reasonable way to bring 5 into
the picture.
From: Paul Erlich (2005-03-08)
Subject: Re: From Commas to Generators
--- In [email protected], "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In [email protected], "Paul Erlich" <perlich@a...>
wrote:
>
> > Probably. What do Gene and Graham get?
>
> I get confused; you are trying to find period and generator for a
planar
> (3D) temperament?
No -- you must have missed all the discussion about prime 5 being
omitted and this being a temperament of {2,3,7}-JI, thus a rank-two
(2D) temperament is the desired result.
> On the other hand, I suppose you could be asking for period and
> generator for the corresponding no-fives system,
Yes, Rich couldn't have been more explicit that this is what he had
in mind, or at least so it seemed to me.
> which has a period of
> an octave and a generator a fourth or fifth. You want two fourths to
> be an approximate 7/4, so the fourths are flat and the fifths sharp,
> but you know all this.
I asked you because I thought you'd be a good candidate to both
verify the exact results Rich got, and to relate his method to any
more-abstract formulations you might have put forth at some point for
finding generators.
From: Paul Erlich (2005-03-08) Subject: Re: From Commas to Generators --- In [email protected], "monz" <monz@t...> wrote: > > --- In [email protected], Rich Holmes<rsholmes@m...> wrote: > > > > I'll give you your 7-limit, but I'll take away your 5. > > That is, I'll use the primes 2, 3, and 7. Using four > > primes (2, 3, 5, and 7) puts you in a higher dimension > > and complicates things, but I'm guessing the principles > > are similar. If not, we'll hear about it, I'm sure. > > > > In cents, these are 1200, 1901.96, and 3368.83. > > > > Here's our comma: 64/63 = 2^6 * 3^-2 * 7^-1 = | 6 -2 -1 >. > > > i think it would be really good to standardize the monzo > notation. any objections to writing that as [6 -2, 0 -1> ? The problem with this is that it makes remembering the how to do the complement operation harder, if you know you're dealing with a system without prime 5. Didn't you suggest the notation [6 -2, * -1> at some point for this purpose?
From: Gene Ward Smith (2005-03-08) Subject: Re: From Commas to Generators --- In [email protected], "monz" <monz@t...> wrote: > i think it would be really good to standardize the monzo > notation. any objections to writing that as [6 -2, 0 -1> ? I've never liked that. I think the <...| and |...>, aside from being more standard, better convey the idea that you can put them together as <...|...>. The comma seems to me just added stuff which gets in the way.
From: Gene Ward Smith (2005-03-08) Subject: Re: From Commas to Generators --- In [email protected], "Kalle Aho" <kalleaho@m...> wrote: > > Hi! > > Let's say I have a set of commas that define a 2D temperament. How > can I get the period-generator-mappings of the primes from it? First can you explain what you mean by 2D?
From: Kalle Aho (2005-03-09) Subject: Re: From Commas to Generators --- In [email protected], "Gene Ward Smith" <gwsmith@s...> wrote: > > --- In [email protected], "Kalle Aho" <kalleaho@m...> wrote: > > > > Hi! > > > > Let's say I have a set of commas that define a 2D temperament. How > > can I get the period-generator-mappings of the primes from it? > > First can you explain what you mean by 2D? Formerly known as "linear" temperament. :) Kalle
From: Gene Ward Smith (2005-03-09) Subject: Re: From Commas to Generators --- In [email protected], "Kalle Aho" <kalleaho@m...> wrote: > > First can you explain what you mean by 2D? > > Formerly known as "linear" temperament. :) There are different ways to do this, and probably you would like the ways other people do it better than my system. I first find the wedgie for the temperament, and then find what I call the "subgroup vals" belonging to it, which are vals defined in terms of the coordinates of the wedgie which send one of the primes to zero. For instance, for 7-limit wedgies, this would take <<l[1] l[2] l[3] l[4] l[5] l[6]|| and from it derive <[0, -l[1], -l[2], -l[3]|, <l[1], 0, -l[4], -l[5]|, <l[2], l[4], 0, -l[6]|, <l[3], l[5], l[6], 0]| We want two independent reduced vals, one of which sends 2 to the number of periods in an octave, and the other of which sends 2 to zero. We can get this by taking the Hermite normal form of the above. We then can find optimal generators using, for instance, rms optimization, and then can standarize our result by making the generator larger than one but less than sqrt(period). This describes what my Maple program does, but it isn't the only way to solve the problem, and for people not using a computer algebra package, not likely to prove a good system. However, since I have such a system it is very fast for me. Simply by inspecting the wedgie, you can find a lot of relevant information; if we have n primes in the prime limit, then the first n entries in the wedgie define the generator val, and their gcd defines the period. Thus from the wedgie <<2 -4 -4 -11 -12 2|| we can see that <0 1 -2 -2| is the generator val, up to sign, and the period is defined by the gcd of (2,-4,-4), or 2; in other words, a half-octave. Hence the mapping will look like [<2 * * *|, +-<0 1 -2 -2|] Filling in the blanks involves finding a reduced size for the generator, which can be done by hand, or coded.
From: Ozan Yarman (2005-03-09) Subject: Re: [tuning-math] Re: From Commas to Generators Finally, I now understand how the exponent notation is written. But in order to distinguish primes, can I suggest something like this: Septimal Comma = [ A6 B-2 D-1 > Where the primes are represented by the letters of the latin alphabet. Cordially, Ozan ----- Original Message ----- From: Paul Erlich To: [email protected] Sent: 08 Mart 2005 Salı 19:47 Subject: [tuning-math] Re: From Commas to Generators --- In [email protected], "monz" <monz@t...> wrote: > > --- In [email protected], Rich Holmes<rsholmes@m...> wrote: > > > > I'll give you your 7-limit, but I'll take away your 5. > > That is, I'll use the primes 2, 3, and 7. Using four > > primes (2, 3, 5, and 7) puts you in a higher dimension > > and complicates things, but I'm guessing the principles > > are similar. If not, we'll hear about it, I'm sure. > > > > In cents, these are 1200, 1901.96, and 3368.83. > > > > Here's our comma: 64/63 = 2^6 * 3^-2 * 7^-1 = | 6 -2 -1 >. > > > i think it would be really good to standardize the monzo > notation. any objections to writing that as [6 -2, 0 -1> ? The problem with this is that it makes remembering the how to do the complement operation harder, if you know you're dealing with a system without prime 5. Didn't you suggest the notation [6 -2, * -1> at some point for this purpose?
From: monz (2005-03-09) Subject: Notation, was Re: [tuning-math] Re: From Commas to Generators hi Rich, --- In [email protected], Rich Holmes<rsholmes@m...> wrote: > Well, dropping the unused exponents is something I've seen > done. Granted it has potential for confusion, if you don't > make it clear what you're doing; but I think it can also > simplify discussion and notation <-> computer code conversion > in cases like this where e.g. we're considering a 5-less system. > Certainly the inclusion of the 5-exponent in this case would > obscure the fact that we're talking about a 3-dimensional > lattice. Really, to me it's neither clearly good nor clearly > bad to drop the unused exponent(s). yes, i can easily agree to everything you say here. i've often used the form which drops unused exponents myself. in talking about Boethius's theory, for instance, you certainly don't want to include the unused exponents, because his theory used prime-factors 2, 3, 19, and 499. when i do this i make it a standard practice to label the monzo clearly with the prime-factors involved, i.e., a monzo used to describe a ratio in Boethius would be labeled a 2,3,19,499-monzo, and written [a b c d> where those variables are the exponents of those four primes in that sequence. it would be ridiculous to inlcude all the unused prime-factors in this case. etc. > I'd never noticed use of the comma after the 3-exponent > before. I'm curious as to its function. this was a proposal which i enthusiastically embraced and put into the Encyclopedia definition. http://tonalsoft.com/enc/monzo.htm putting the comma-mark after 3 is a useful idea because the exponent of 2 is so often ignored in many theoretical concerns, and the comma-mark helps the reader to see whether this is the case or not. (notice that i'm always careful to write "comma-mark" instead of just plain "comma" ... since "comma" is a word loaded with musical tuning meaning, and monzos are often used to describe "commas", it could become really confusing in these discussions about monzos.) putting subsequent comma-marks after every third exponent simply helps the eye to pinpoint which prime-factor an exponent belongs to, and it just so happens by coincidence (or is it a coincidence?) that for a while in the prime series, every third prime-factor after 3 is somewhat important in the history / theory / practice of rational tunings. viz, 11, 19, 31, 43. > I'm used to bra-ket notation that uses a vertical line, > but I suppose actually one could argue in favor of bracket > instead of pipe, since we're often reading these things off > computer screens and the contents of the vector often are > integers, and [ is arguably less likely than | to be confused > with 1, at least in some fonts. i prefer to use the bracket for exactly that reason. when a complet bra-ket is notated, then fine, the pipe symbol is OK. by for a bra or ket by itself, i think the bracket is better. -monz
From: monz (2005-03-09) Subject: Re: From Commas to Generators hi Paul, --- In [email protected], "Paul Erlich" <perlich@a...> wrote: > --- In [email protected], "monz" <monz@t...> wrote: > > > > > > i think it would be really good to standardize the monzo > > notation. any objections to writing that as [6 -2, 0 -1> ? > > The problem with this is that it makes remembering the how > to do the complement operation harder, if you know you're > dealing with a system without prime 5. Didn't you suggest > the notation [6 -2, * -1> at some point for this purpose? hmm, this is funny ... yes, actually, that was exactly how i wrote that post in the first place ... but the asterisk is supposed to be a "wildcard" which may represent any integer and it wouldn't affect the results, and i reasoned that since 5 was absent then that exponent should just be zero. but i certainly preferred the notation with both the comma-mark and the asterisk. then i guess this is valid for the usage in this thread? -monz
From: Gene Ward Smith (2005-03-09) Subject: Re: From Commas to Generators --- In [email protected], "monz" <monz@t...> wrote: > hmm, this is funny ... yes, actually, that was exactly > how i wrote that post in the first place ... but the asterisk > is supposed to be a "wildcard" which may represent any integer > and it wouldn't affect the results, and i reasoned that since > 5 was absent then that exponent should just be zero. I'd use "*" as a wildcard and "0" if 5 is absent.
From: monz (2005-03-09) Subject: Re: From Commas to Generators hi Ozan, --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote: > Finally, I now understand how the exponent notation is > written. But in order to distinguish primes, can I suggest > something like this: > > Septimal Comma = [ A6 B-2 D-1 > > > Where the primes are represented by the letters of the > latin alphabet. hmm, that's an interesting idea. here's the mapping to the whole alphabet: A = 2 B = 3 C = 5 D = 7 E = 11 F = 13 G = 17 H = 19 I = 23 J = 29 K = 31 L = 37 M = 41 N = 43 O = 47 P = 53 Q = 59 R = 61 S = 67 T = 71 U = 73 V = 79 W = 83 X = 89 Y = 97 Z = 101 of course, we run into a slight problem with composers and theorists who have used prime-factors higher than 101, and as i just posted, they do exist. a simple solution is to use 2-digit letters after Z: AA = 103 AB = 107 AC = 109 AD = 113 AE = 127 AF = 131 AG = 137 AH = 139 . . . AZ = 239 BA = 241 . . . BZ = 397 CA = 401 but it's not likely that anyone's going to remember the mapping of more than just the first few ... so in the end it's better to just use the prime-factors themselves, i.e.: 2^a 3^b 5^c 7^d ... p^n instead of [a b c d ...n>. for me, the two best methods are the ones i always use: the comma-marks and asterisk-wildcard if the prime-series covers a spread that is not to far (i.e., most cases), and the explicit prime-factor label for those cases in which the spread is far (as with Boethius's 3,19,499 system). -monz
From: Ozan Yarman (2005-03-09) Subject: Re: [tuning-math] Re: From Commas to Generators Hi Monz, If the intention is to skip the primes in between primes, then I suggest the following usage: Septimal Comma = [ 6 -2, -1 > Where the comma-mark (,) indicates one skip in the prime sequence and a semi colon (;) indicates indefinite amount of skips up to the next prime. So, Boethius's prime factors can be represented like this: 2^a; 19^b; 499^c ----- Original Message ----- From: monz To: [email protected] Sent: 09 Mart 2005 Çarşamba 11:37 Subject: [tuning-math] Re: From Commas to Generators hi Ozan, --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote: > Finally, I now understand how the exponent notation is > written. But in order to distinguish primes, can I suggest > something like this: > > Septimal Comma = [ A6 B-2 D-1 > > > Where the primes are represented by the letters of the > latin alphabet. hmm, that's an interesting idea. here's the mapping to the whole alphabet: A = 2 B = 3 C = 5 D = 7 E = 11 F = 13 G = 17 H = 19 I = 23 J = 29 K = 31 L = 37 M = 41 N = 43 O = 47 P = 53 Q = 59 R = 61 S = 67 T = 71 U = 73 V = 79 W = 83 X = 89 Y = 97 Z = 101 of course, we run into a slight problem with composers and theorists who have used prime-factors higher than 101, and as i just posted, they do exist. a simple solution is to use 2-digit letters after Z: AA = 103 AB = 107 AC = 109 AD = 113 AE = 127 AF = 131 AG = 137 AH = 139 . . . AZ = 239 BA = 241 . . . BZ = 397 CA = 401 but it's not likely that anyone's going to remember the mapping of more than just the first few ... so in the end it's better to just use the prime-factors themselves, i.e.: 2^a 3^b 5^c 7^d ... p^n instead of [a b c d ...n>. for me, the two best methods are the ones i always use: the comma-marks and asterisk-wildcard if the prime-series covers a spread that is not to far (i.e., most cases), and the explicit prime-factor label for those cases in which the spread is far (as with Boethius's 3,19,499 system). -monz
From: Ozan Yarman (2005-03-09) Subject: From Commas to Generators I do not wish to approximate temperaments by any EDOs at this stage. That is the easy part once you have a temperament. I rather liked the methodology used by Rich (at least things don’t get messed up once in order). Now that I have understood how to prime factorize 4000:3993, let me review step by step how to temper it out. 1. 4000:3993 = (2*2*2*2*2*5*5*5) / (3*11*11*11) = (2^5*5^3) * (3^-1*11^-3) 2. In other words, 4000:3993 = [5 -1 3 0 -3> where zero represents the absent prime number 7. 3. If the primes 2,3,5,11 are represented by generators g1, g2, g3 and g5 respectfully, then these would have the just ratios 2:1, 3:2, 5:4 and 11:10. The cent values for g1 is 1200.000, g2 is 701.9550, g3 is 386.3137 and g5 is 165.0042. I admit that cent calculations appeal to me more. 4. What I fail to comprehend as yet is by what amount the tempering ought to be done. Can someone please give me a few examples from this point forward on how to do it? And Paul, you ask: Hi Ozan, I'm impressed that you're sensitive to a 3 cent change in the intonation of one note. What means are you using to render and listen to these intervals? I’m using a SBLive audio card with Scala. The piano sample sound is ok for theoretical verification of intervals. (1) How do you know it's 9801/8000 and not 49/40 that your ear is guiding you towards? I find 49/40 to be too low when the hammer hits the string in the first micro-second and 27/22 to be too high likewise. It must be a matter of timbral artifacts as Monz pointed out earlier. And I think I understand why log(a^b) = b*log(a). The logarithm of, say, 100 (10^2) is 2, which is equal to this exponent times log 10 (1): (log 10^2 = 2*log 10) You say: OK . . . let's take a step back. Do you normally consider interval sizes in cents? Or in some other system of units? Would you agree than an octave is a 2:1 ratio, and that a triple-octave is an 8:1 ratio? Or does this mystify you? Not at all. You say: I wasn't thinking of a piano, because the scale in question will have too many notes per octave to be expressible on a piano. A piano has stretched overtones, which is why stretched octaves sound great on it. However, many other instruments, such as bowed strings, brass/wind instruments, and the human voice have harmonic overtones. For such instruments, a stretched octave won't sound much less discordant than a similarly compressed octave, based on my experience and knowledge. But perhaps you have found otherwise? Perhaps you're only speaking of *melodic* octaves, not harmonic (simultaneously- sounding) ones? To me, they are more or less the same. >Tempering out the syntonic comma will result in the upper-most >dimension (prime) to collapse, What do you mean uppermost? The largest? How do you know this? What is it about the syntonic comma that allows you to reach this conclusion? Oops, I did something wrong again, didn’t I? Oh, forget it… >creating a temperament of 2 dimensions. So, the first dimension is >the octave with 2:1 which will remain untouched for my purposes, and >the second dimension is the tempered perfect fifth with the ratio >3:2 and the third dimension is the tempered pure third with the >ratio 5:4. I thought you said 2 dimensions? No longer! I will say `rank` from now on… Cordially, Ozan
From: Kalle Aho (2005-03-09) Subject: Re: From Commas to Generators --- In [email protected], Rich Holmes<rsholmes@m...> wrote: > "Kalle Aho" <kalleaho@m...> writes: > > > Hi! > > > > Let's say I have a set of commas that define a 2D temperament. How > > can I get the period-generator-mappings of the primes from it? > > > > I believe they can be found by manipulating a set of linear > > equations but I guess you folks have a more straightforward method. > > > > Maybe a simple 7-limit example would be nice, thank you. > > What a coincidence, I just worked this out myself the other night. > And all my notes are at home... so let's see if I can reconstruct. <snip> > So we can use e.g. period = 1197.72, generator = 489.88. > > I think I got that right. > > - Rich Holmes That's very nice, Rich. Thank you for this demonstration! Now, I would very much like to understand the general case. Gene already demonstrated his method (thank you, Gene!) but that is just pretty hard to understand. I hope others will demonstrate their methods too. Kalle
From: Gene Ward Smith (2005-03-09)
Subject: Re: From Commas to Generators
--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> 4. What I fail to comprehend as yet is by what amount the tempering
ought to be done. Can someone please give me a few examples from this
point forward on how to do it?
When there is just one comma, as in the case of the 4000/3993
temperament, TOP tuning is particularly easy. However, there are other
ways to optimize, and a method which is useful and easy to caculate is
least-squares, so I'll explain that. We have five primes and one
comma, and therefore four generators. You gave these as 2,3/2,5/4, and
11/10, but this isn't quite right, as 3/2 is equivalent to
(3/2)(4000/3993) = 2000/1331, and we also need a generator for the 7
stuff. If we set g1=2, g3=5/4, g5=11/10 as you do, we can add g4=7/4,
and now we have a complete set of generators.
In terms of these, we represent the primes by
2 ~ g1
3 ~ g1^2*g5^(-3)
5 ~ g1^2*g3
7 ~ g1^2*g4
11 ~ g1^3*g3*g5
The exponents of this allow us to write down a matrix mapping from
generators to primes:
[[1 2 2 2 3], [0 0 1 0 1], [0 0 0 1 0], [0 -3 0 0 1]]
The vectors in the above being columns of the matrix, and representing
the exponents of g1, g3, g4 and g5 respectively. Now we take a finite
list of the consonances we want to approximate, and if we like, a
weighting on each consonance. If the consonances are c_i and the
weights are w_i, we define a function of g1, g3, g4, and g5 by
S(g1,g3,g4,g5) = sum w_i (rep(c_i)-cents(c_i))^2
Here rep(c_i) is the representation of an interval in terms of these
generators which we can define from the above matrix map by
multiplying it by its monzo; for instance to find rep(7/6), we
multiply |-1 -1 0 1 0>M = [1 0 1 3], where M is the mapping matrix.
This tells us rep(7/6) = g1+g4+3g5.
If now we take the partial derivatives of S with respect to g1, g3, g4
and g5, we get *linear* equations, which we may solve to obtain a
unique optimal solution by this definition of "optimal".
For instance, suppose we constrain g1 to be exactly 1200 cents, and
then take {3, 5, 7, 9, 11, 11/5, 5/3, 7/5, 11/3, 9/7, 9/5, 7/3, 11/9,
11/7} to be our set of consonances, weighted equally. Then
g3 = 385.912, g4 = 968.915, g5 = 165.985
are the optimal tunings of the other generators.
From: Ozan Yarman (2005-03-11)
Subject: Re: [tuning-math] Re: From Commas to Generators
Gene, widening the fifth by 4000/3993 is enough to temper all instances of this kleisma?
I cannot make the proper correlations between temperament and matrix yet. Since you are math-savvy, I'm sure you can explain. I'm still in the dark as to the cent values that should be added to or subtracted from consonant ratios. Please remember that I am not knowledged in equations and algebra as you are! Gentler introductions are in order.
Cordially,
Ozan
----- Original Message -----
From: Gene Ward Smith
To: [email protected]
Sent: 10 Mart 2005 Perşembe 1:01
Subject: [tuning-math] Re: From Commas to Generators
--- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote:
> 4. What I fail to comprehend as yet is by what amount the tempering
ought to be done. Can someone please give me a few examples from this
point forward on how to do it?
When there is just one comma, as in the case of the 4000/3993
temperament, TOP tuning is particularly easy. However, there are other
ways to optimize, and a method which is useful and easy to caculate is
least-squares, so I'll explain that. We have five primes and one
comma, and therefore four generators. You gave these as 2,3/2,5/4, and
11/10, but this isn't quite right, as 3/2 is equivalent to
(3/2)(4000/3993) = 2000/1331, and we also need a generator for the 7
stuff. If we set g1=2, g3=5/4, g5=11/10 as you do, we can add g4=7/4,
and now we have a complete set of generators.
In terms of these, we represent the primes by
2 ~ g1
3 ~ g1^2*g5^(-3)
5 ~ g1^2*g3
7 ~ g1^2*g4
11 ~ g1^3*g3*g5
The exponents of this allow us to write down a matrix mapping from
generators to primes:
[[1 2 2 2 3], [0 0 1 0 1], [0 0 0 1 0], [0 -3 0 0 1]]
The vectors in the above being columns of the matrix, and representing
the exponents of g1, g3, g4 and g5 respectively. Now we take a finite
list of the consonances we want to approximate, and if we like, a
weighting on each consonance. If the consonances are c_i and the
weights are w_i, we define a function of g1, g3, g4, and g5 by
S(g1,g3,g4,g5) = sum w_i (rep(c_i)-cents(c_i))^2
Here rep(c_i) is the representation of an interval in terms of these
generators which we can define from the above matrix map by
multiplying it by its monzo; for instance to find rep(7/6), we
multiply |-1 -1 0 1 0>M = [1 0 1 3], where M is the mapping matrix.
This tells us rep(7/6) = g1+g4+3g5.
If now we take the partial derivatives of S with respect to g1, g3, g4
and g5, we get *linear* equations, which we may solve to obtain a
unique optimal solution by this definition of "optimal".
For instance, suppose we constrain g1 to be exactly 1200 cents, and
then take {3, 5, 7, 9, 11, 11/5, 5/3, 7/5, 11/3, 9/7, 9/5, 7/3, 11/9,
11/7} to be our set of consonances, weighted equally. Then
g3 = 385.912, g4 = 968.915, g5 = 165.985
are the optimal tunings of the other generators.
From: Gene Ward Smith (2005-03-11) Subject: Re: From Commas to Generators --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote: > Gene, widening the fifth by 4000/3993 is enough to temper all instances of this kleisma? Right. 3 appears only to exponent -1 in it, so we can use it to express 3 in terms of 2,5,7, and 11.
From: Ozan Yarman (2005-03-11) Subject: Re: [tuning-math] Re: From Commas to Generators Is this a correct scale where 4000:3993 is tempered? 0: 1/1 0.000 unison, perfect prime 1: 134.911 cents 134.911 2: 2000000/1771561 209.975 3: 344.886 cents 344.886 4: 419.949 cents 419.949 5: 1331/1000 495.013 6: 629.924 cents 629.924 7: 2000/1331 704.987 8: 839.899 cents 839.899 9: 914.962 cents 914.962 10: 1049.873 cents 1049.873 11: 1124.937 cents 1124.937 12: 2/1 1200.000 octave | Temperings of 4000/3993 0: 0.000: -3.0323 1: 134.911: -3.0323 2: 209.975: -3.0323 3: 344.886: -3.0323 4: 419.949: -3.0323 5: 495.013: -3.0323 6: 629.924: -3.0323 7: 704.987: -3.0323 8: 839.899: -3.0323 9: 914.962: -3.0323 10: 1049.873: -3.0323 11: 1124.937: -3.0323 12: 1200.000: -3.0323 Total abs. diff. : 36.3878 Average abs. diff.: 3.0323 Highest abs. diff.: 3.0323
From: Gene Ward Smith (2005-03-11) Subject: Re: From Commas to Generators --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote: > Is this a correct scale where 4000:3993 is tempered? It's hard to say, since you don't tell us what rational intervals you are tempering.
From: Ozan Yarman (2005-03-11) Subject: Re: [tuning-math] Re: From Commas to Generators You lost me again. I thought you said widening the fifth by 4000:3993 was enough to temper all instances of this kleisma. ----- Original Message ----- From: Gene Ward Smith To: [email protected] Sent: 11 Mart 2005 Cuma 21:35 Subject: [tuning-math] Re: From Commas to Generators --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote: > Is this a correct scale where 4000:3993 is tempered? It's hard to say, since you don't tell us what rational intervals you are tempering.
From: Gene Ward Smith (2005-03-11) Subject: Re: From Commas to Generators --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote: > You lost me again. I thought you said widening the fifth by 4000:3993 was enough to temper all instances of this kleisma. Yes, but tempering is generally conceived of as an adjustment of just intonation. You can take anything in just intonation, widen the fifth by 4000/3993, leave everything else the same, and 4000/3993 will be tempered out. Hwever, you didn't start with something in just intonation.
From: Ozan Yarman (2005-03-12) Subject: Re: [tuning-math] Re: From Commas to Generators I surmise what you just said stands valid for `temperament`, not `specifically tempering an interval`. In this case, the question was: Is the 12-tone scale I produced a correct example where 4000:3993 is tempered regardless of a `just intonation reference`? Or does one need a simple integer ratio to compare a comma against? ----- Original Message ----- From: Gene Ward Smith To: [email protected] Sent: 11 Mart 2005 Cuma 22:41 Subject: [tuning-math] Re: From Commas to Generators --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote: > You lost me again. I thought you said widening the fifth by 4000:3993 was enough to temper all instances of this kleisma. Yes, but tempering is generally conceived of as an adjustment of just intonation. You can take anything in just intonation, widen the fifth by 4000/3993, leave everything else the same, and 4000/3993 will be tempered out. Hwever, you didn't start with something in just intonation.
From: Gene Ward Smith (2005-03-13) Subject: Re: From Commas to Generators --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote: > I surmise what you just said stands valid for `temperament`, not `specifically tempering an interval`. In this case, the question was: Is the 12-tone scale I produced a correct example where 4000:3993 is tempered regardless of a `just intonation reference`? Or does one need a simple integer ratio to compare a comma against? Your scale is quite close to a scale on 80-equal, namely ! ozan80.scl 80-et version of Ozan Yarman scale 12 ! 135.000000 210.000000 345.000000 420.000000 495.000000 630.000000 705.000000 840.000000 915.000000 1050.000000 1125.000000 1200.000000 Since 80-et is a 4000/3993 system if regarded as the regular temperament tempering out 176/175, 540/539, 896/891 and 1331/1323 your scale can reasonably be considered as a 4000/3993 scale. It is a MOS for the 63&80 temperament, which has a generator of a sharp fifth. I don't think this has been discussed before, but it does have some no-fives potential, as 5 is complex (of course, 3 is just the opposite.)
From: Gene Ward Smith (2005-03-13)
Subject: Re: From Commas to Generators
--- In [email protected], "Gene Ward Smith" <gwsmith@s...>
wrote:
> It is a MOS for the 63&80 temperament, which has a generator of a
> sharp fifth. I don't think this has been discussed before, but it
> does have some no-fives potential, as 5 is complex (of course, 3 is
> just the opposite.)
Incidentally, this "yarman" temperament, to suggest a possible name
for it, really makes more sense if you go up-limit, at least as far
as 13, if not higher. As a 13-limit temperament it has the mapping
[<1 1 -20 -6 -3 -1|, <0 1 38 15 11 8|]
if we take the generator to be a fifth. It has a TM comma basis of
{169/168, 352/351, 364/363, 540/539}. If I take a chain of fifths and
reduce by these commas, I get
13/12, 9/8, 13/11, 14/11, 4/3, 13/9, 3/2, 11/7, 22/13, 16/9, 21/11, 2
From: Gene Ward Smith (2005-03-14) Subject: Re: From Commas to Generators --- In [email protected], "Ozan Yarman" <ozanyarman@s...> wrote: > I surmise what you just said stands valid for `temperament`, not `specifically tempering an interval`. In this case, the question was: Is the 12-tone scale I produced a correct example where 4000:3993 is tempered regardless of a `just intonation reference`? Or does one need a simple integer ratio to compare a comma against? You need to interpret what you did to decide; you gave a chain of sharp fifths, and the most reasonable interpretation of that, it seems to me, is by way of a temperament in which, among other things, 4000/3993 vanishes. However, this is a rank 2 temperament, and I was thinking of a temperament which tempered out 4000/3993 and nothing else.
From: Ozan Yarman (2005-03-14) Subject: Re: [tuning-math] Re: From Commas to Generators Ah, but you still have not given me an example where 4000/3993 alone is tempered out. Please, I implore you, explain to me once more, in rudimentary math and in layman terms.... How can I discern which simple integer just ratios I tempered by sharpening the fifths and forming a chain? ----- Original Message ----- From: Gene Ward Smith To: [email protected] Sent: 14 Mart 2005 Pazartesi 3:57 Subject: [tuning-math] Re: From Commas to Generators You need to interpret what you did to decide; you gave a chain of sharp fifths, and the most reasonable interpretation of that, it seems to me, is by way of a temperament in which, among other things, 4000/3993 vanishes. However, this is a rank 2 temperament, and I was thinking of a temperament which tempered out 4000/3993 and nothing else.