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Message: 11375 - Contents - Hide Contents Date: Tue, 13 Jul 2004 18:32:48 Subject: Re: 50 From: monz hi Gene, --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote: >>>> have you ever seen this? >> >> W. S. B. Woolhouse's 'Essay on musical interva... * [with cont.] (Wayb.) >> you quote Woolhouse as saying: > > [Woolhouse 1835, p 46:] > > This system is precisely the same as that which > Dr. Smith, in his _Treatise on harmonics_ [Smith 1759], > calls the scale of equal harmony. > It is decidedly the most perfect of any systems in > which the tones are all alike. > > Is Smith's tuning 50-equal? yes.but i haven't read his book.> > I'd also cut Woolhouse some slack on 5-limit vs 5-odd-limit. The place > where he discusses that you *might* interpret to say that 2 and 3/2 > are consonances, but 4, 3 or 6/5 are not, but I would presume his > assumption of octave equivalence would be clear from elsewhere. > > I also find this comment interesting: > > He then analyzes the resources of a 53-EDO 'enharmonic organ', built > by J. Robson and Son, St. Martin's-lane, but says that the number of > keys is too much to be practicable, and settles again on 19-EDO. > > Is this the first time someone built a 53-edo instrument?i don't think so, but can't cite anything concrete. Mercator studied 53edo and recommended it for instruments, but i don't know what was/wasn't built.> Incidentally, convergents to the Woolhouse fifth go 7/12, > 11/19, 18/31, 29/50, 76/131, 257/443 ... . It would be > interesting to dig up someone who advocated 131-equal for > meantone!hmmm ... there's a blank space in my equal-temeperaments table, just waiting for that! maybe Gene will be the first. -monz
Message: 11376 - Contents - Hide Contents Date: Tue, 13 Jul 2004 18:52:06 Subject: Re: 50 From: monz hi Paul, --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: >>> but you quote >> Woolhouse as saying: >> >> [Woolhouse 1835, p 46:] >> >> This system is precisely the same as that which >> Dr. Smith, in his Treatise on harmonics [Smith 1759], calls >> the scale of equal harmony. >> It is decidedly the most perfect of any systems in which >> the tones are all alike. >> >> Is Smith's tuning 50-equal? >> It's close, but it's much closer to 5/18-comma meantone > than to 50-equal. Search the tuning list for more info ;) > Also see Jorgenson.hmm ... Woolhouse *did* write exactly what Gene quoted from my webpage. it's too much of a pain to search the list archives. can you give us some details? exactly how did Smith measure his tuning? here's a very interesting webpage about Smith's "Equal Harmony" tuning (described as 50et), and its application to harpsichords. * [with cont.] (Wayb.). uk/russell/conference/robertsmithkirckman.html+%22robert+smith%22+% 2250%22+tuning%22&hl=en or & * [with cont.] (Wayb.) -monz
Message: 11377 - Contents - Hide Contents Date: Tue, 13 Jul 2004 19:42:31 Subject: Re: 50 From: Gene Ward Smith>> >ut it's much closer to 5/18-comma meantone >> than to 50-equal. Search the tuning list for more info ;) >> Also see Jorgenson. > > >> hmm ... Woolhouse *did* write exactly what Gene quoted > from my webpage.There's very little difference between 50-equal and 5/18-comma meantone, especially if you are doing your computing in the 18th century. The convergents to 5/18 meantone go 12, 19, 50, 1219 ..., whereas the convergents to 50-et in terms of fractions of a comma go 1/3, 1/4, 2/7, 3/11, 5/18, 13/47, 18/65... The fifth of 50-equal is a mere 0.019 cents sharper than 5/18-comma. It would be interesting to know why Smith chose the tuning he did, because it does seem he could have intended to close at 50, or to do something equivalent such as making the ratio between diatonic and chromatic semitones 5/3.
Message: 11378 - Contents - Hide Contents
Date: Tue, 13 Jul 2004 20:24:48
Subject: Hahn reduced scales
From: Gene Ward Smith
By a Hahn reduced scale I mean a 5 or 7 limit scale reduced according
to a comma set which defines a val for the number of notes in the
scale to the correct prime limit. The smallest Hahn distance is
picked, and if there is a tie, Tenney distance is used to break the tie.
Since Hahn distance and symmetrical Euclidean distance are so similar
this will be similar if not the same to the Euclidean reduction I've
mentioned already, but Hahn distance might be preferred; in any case,
I'm giving it; I give Scala files for the scales, but before that I
list some pertinant data. The numbers on the second line are major
tetrads and major tetrads tempered by marvel, and the same for the
other pairs, which are minor tetrads, supermajor tetrads, subminor
tetrads and 9-limit quintads in otonal and utonal flavors. Since
225/224 is a scale step for Hahn16, the count is a little confused there.
The third line gives the scale by giving its major/minor tetrads in
lattice format, union all the notes left over.
Hahn12 is noteable for being one of the
(15/14)^3 (16/15)^4 (21/20)^3 (25/24)^2 = 2 scales.
7-limit Hahn reduced scales
12: epimorphic, strictly proper, CS, superparticular
2/3, 2/2, 1/1, 0/1; 0/1, 0/1; smallest = 25/24
{[-1, 0, -1], [0, 0, 0], [0, 0, 1], [-1, -1, -1]} U {5/3}
15: epimorphic, improper, CS, superparticular
3/4, 2/3; 1/2, 2/3; 2/3, 1/2; smallest = 50/49
{[-1, 0, -1], [0, 0, 0], [0, -1, 0], [0, -1, -1], [0, 0, -1]} U
{16/15,10/9,9/5,15/8}
16: epimorphic, improper, not CS, not superparticular
2/4, 1/4; 1/2, 2/2; 1/2, 1/2; smallest = 225/224 (vanishes in marvel)
{[-1, 0, -1], [0, 1, 1], [0, 0, 0], [0, 0, 1], [-1, -1, -1]} U
{25/16,5/3,28/15}
19: epimorphic, improper, CS, superparticular
4/5, 4/6; 4/5, 3/4; 3/4, 3/4; smallest = 50/49
{[-1, 0, -1], [0, 0, 0], [-1, 1, 0], [-1, 1, 1], [0, -1, 0],
[0, -1, -1], [0, 0, -1], [0, 0, 1]} U {14/9,35/18}
22: epimorphic, improper, CS, superparticular
6/7, 5/6; 5/6, 5/6; 4/5, 3/4; smallest = 126/125
{[-1, 0, -1], [1, -2, -1], [1, -1, -1], [0, 0, 0],
[1, -1, 0], [0, -1, -1],
[0, 0, -1], [0, 0, 1], [-1, -1, -1], [-1, 0, 1],
[-1, 0, 0]} U {14/9,9/5}
! hahn12.scl
Hahn-reduced 12 note scale
12
!
15/14
8/7
6/5
5/4
4/3
7/5
3/2
8/5
5/3
7/4
15/8
2
! hahn15.scl
Hahn-reduced 15 note scale
15
!
16/15
10/9
7/6
6/5
5/4
4/3
7/5
10/7
3/2
8/5
5/3
7/4
9/5
15/8
2
! hahn16.scl
Hahn-reduced 16 note scale
16
!
15/14
9/8
8/7
6/5
5/4
21/16
4/3
7/5
3/2
25/16
8/5
5/3
7/4
28/15
15/8
2
! hahn19.scl
Hahn-reduced 19 note scale
19
!
21/20
15/14
9/8
7/6
6/5
5/4
9/7
4/3
7/5
10/7
3/2
14/9
8/5
5/3
7/4
9/5
15/8
35/18
2
! hahn22.scl
Hahn-reduced 22 note scale
22
!
25/24
15/14
10/9
8/7
7/6
6/5
5/4
9/7
4/3
25/18
7/5
35/24
3/2
14/9
8/5
5/3
12/7
7/4
9/5
15/8
35/18
2
Message: 11379 - Contents - Hide Contents Date: Tue, 13 Jul 2004 23:54:30 Subject: Extreme precison (Olympian) Sagittal From: Dave Keenan Hi Gene, Sorry for gaving you a hard time about jargon on "tuning". But I think you'd rather get it from me, now, than have it build up until you get attacked by a serious mob of maurauding math-phobes. :-) If you're not too pissed off with me to still be interested, here's one way of approaching the extreme-precision Sagittal question. The raw data you need is I think contained in Table 1 and Figure 3 of the XH paper. http://users.bigpond.net.au/d.keenan/sagittal/Sagittal.pdf - Type Ok * [with cont.] (Wayb.) In table 1, you can ignore everything but the first comma listed for each symbol (not necessarily the bold one, but the topmost one, the one on the same line as the symbol). All the other roles for a symbol are allowed to break, and be taken over by another (possibly accented) symbol in the extreme-precision set. And ignore the second row of symbols in Figure 3 since their definitions are purely as apotome complements of those on the first line no matter what set we are using. The idea then is to treat each of the 9 flags (left and right of barb, arc, scroll, boathook; plus accent) as a generator and find the optimum value in cents for each so as to minimise the maximum error over all the symbol/comma relationships in table 1. There may be some other important symbol-comma relationships not shown in table 1, that we don't want to break, but I'd like to see what happens when we just base it on those first. Due to the apotome complementarity we have one constraint on the flag values initially, so we are down to 8 degrees of freedom. That is that /| + |\ + (| + |) = apotome. Although you could ignore that and see what happens. Then once you have the value of each flag you can calculate the value of each symbol in the top row of figure 3 and the same with up accents and the same with down accents. And maybe figure out what 23- limit comma might be assigned as the primary role of each. Commas which notate more popular (simpler or lower-prime-limit) ratios are to be preferred, as are commas which notate ratios without having to go too far up or down the chain of fifths.
Message: 11380 - Contents - Hide Contents Date: Wed, 14 Jul 2004 07:53:14 Subject: Re: Hahn reduced scales From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: Some of these scales can be adjusted, retaining epimorphicity, so that they have more complete tetrads and all of the notes can be given in terms of tetrads. If we look for triads of the 11 notes of the 12 note scale below, minus 5/3, we find that adjusting 5/3 to 12/7 allows it to be harmonized by the tonic minor, 1-6/5-3/2-12/7, which is given by [-1,0,0], so we could add that to the set of triads for 12 notes. Similarly, for 22 we can adjust 14/9 up a 50/40 to 100/63, and 9/5 up a 250/243 to 50/27, and add the [1,-3,-1] tetrad to the list. This sort of thing improves things for 7-limit tetrads but not always in general. I give the adjusted scales below. ! hen12.scl Adjusted Hahn12 12 ! 15/14 8/7 6/5 5/4 4/3 7/5 3/2 8/5 12/7 7/4 15/8 2 ! hen22.scl Adjusted Hahn22 22 ! 25/24 15/14 10/9 8/7 7/6 6/5 5/4 9/7 4/3 25/18 7/5 35/24 3/2 100/63 8/5 5/3 12/7 7/4 50/27 15/8 35/18 2
Message: 11381 - Contents - Hide Contents Date: Wed, 14 Jul 2004 10:25:40 Subject: Re: Extreme precison (Olympian) Sagittal From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:> The idea then is to treat each of the 9 flags (left and right of > barb, arc, scroll, boathook; plus accent) as a generator and find > the optimum value in cents for each so as to minimise the maximum > error over all the symbol/comma relationships in table 1.I can't decypher this well enough to tackle the problem, and looking at the Sagittal symbols and trying to figure out flags is just going to confuse me. Could you post, very explicitly, what you are seeking? What are the specific symbol/comma relationships you want approximated, in terms of the nine generators?
Message: 11382 - Contents - Hide Contents
Date: Wed, 14 Jul 2004 21:48:48
Subject: Beep and ennealimmal
From: Gene Ward Smith
There's a curious connection between beep/bug, the 7-limit 4&5
temperament (wedgie <<2 3 1 0 -4 -6||) and ennealimmal, the 7-limit
171&270 temperament (wedgie <<18 27 18 1 -22 -34||.) If we find the TM
basis for beep, we get {27/25, 21/20}, which happens to be a
period/generator pair for ennealimmal.
The 4-et and 5-et, which give the beep generator of 8/7 as 300 and 240
cents respectively, define its extreme range, but more importantly
4-et is the et of the tetrad, since 1-5/4-3/2-7/4 is an epimorphic
scale with <4 6 9 11| as its val, and 5-et is the et of the quintad,
1-9/8-5/4-3/2-7/4 being epimorphic with <5 8 12 14| as its val. This
means we can use beep to lift to 9 odd limit JI, in something like the
way I've used 4 and 5. In terms of the 8/7 generator, we have
-4: 9/8
-3: 5/4, 9/7
-2: 10/7, 3/2
-1: 5/3, 12/7, 7/4, 9/5
The major quintad is [0,-4,-3,-2,-1], the major and supermajor tetrads
are [0,-3,-2,-1], the minor and subminor tetrads [0,1,-2,-1]; these
are just contiguous sets of generators if we ignore the ordering.
Adding octaves, of course, gives us a note of beep.
The mapping for the generators 27/25 and 21/20 of ennealimma is
[<9 13 19 24|, <0 2 3 2|]. If we add mappings for the 4 and 5 vals,
we get a unimodular matrix which we can invert to get a matrix which
in terms of monzos is
[|0 3 -2 0>, |-2 1 -1 1>, |-6 -8 2 5>, |5 1 2 -4>]
In fractions, this is [27/25, 21/20, 420175/419994, 2400/2401]. The
perhaps unfamiliar comma is 420175/419994 = (2401/2400)(4375/4375).
This defines a "notation" for 7-limit JI; a note of ennealimmal plus a
note of beep giving the interval, where roughly speaking the note of
ennealimmal tells you the tuning, and the note of beep the chord position.
By the way, is it beep or bug, and why?
Message: 11383 - Contents - Hide Contents Date: Wed, 14 Jul 2004 19:58:48 Subject: Re: Beep and ennealimmal From: Herman Miller Gene Ward Smith wrote:> There's a curious connection between beep/bug, the 7-limit 4&5 > temperament (wedgie <<2 3 1 0 -4 -6||) and ennealimmal, the 7-limit > 171&270 temperament (wedgie <<18 27 18 1 -22 -34||.) If we find the TM > basis for beep, we get {27/25, 21/20}, which happens to be a > period/generator pair for ennealimmal. > By the way, is it beep or bug, and why?"Beep" was supposed to bring to mind "BP" (as in "Bohlen-Pierce"); 27/25 is a frequent step size of the untempered BP scale. But it's as if we were to call father "diatonic" because it tempers out the diatonic semitone. "Bug" appears to be the older name, and I don't know why it was dropped, unless the name was just forgotten. So I vote for "bug".
Message: 11384 - Contents - Hide Contents Date: Wed, 14 Jul 2004 21:19:54 Subject: Naming temperaments From: Herman Miller I've been thinking about names for linear temperaments and ways of classifying them; there's probably nothing much new here, but a summary might have some use. We know that wedgies are useful for identifying linear temperaments (and any temperament in which two independent commas vanish, like 11-limit planar temperaments). But they have two serious drawbacks as names: they're not easily recognized or understood by looking at them. Even the 7-limit wedgies are pretty long, and higher limit wedgies are too long to use as names. Take this one for instance: <<1, 9, -2, 12, -6, -30|| You've probably seen it, but do you recognize it? You can guess that it has a mapping of (0, 1, 9, -2) or (0, -1, -9, 2), which might be of some use, but then you might as well give the whole map, which isn't much longer: [<1, 2, 6, 2|, <0, -1, -9, 2|] Or you can give the mapping by steps, which tells you more directly something about the structure of the scale, but isn't as easily recognizable, and is even less suitable as a name. [(22, 5), (35, 8), (51, 12), (62, 14)] For most of the better temperaments, if you just give the mapping by steps of the octave (which is equivalent to a combination of two octave-based ET's), you can fill in the others by taking the best approximation of the primes in each of the ET's. So we could call this the 5&22 temperament. But there's another alternative that looks somewhat interesting. Notice that the first thing in Graham Breed's temperament finder results is a fraction of an octave: in this case: 11/27, 489.3 cent generator If you start with a generator like this, you end up with a closed temperament; it seems like there should be one best way to map this to a linear temperament, if the ET is consistent. You can figure out how many iterations to get to any interval by just adding 11 mod 27 until you end up with the best approximation, which will take a maximum of 26 iterations (subtract the result from 27 if it ends up more than 27/2 steps away). One drawback is that you also need to specify the period if it isn't an approximate octave; e.g., 1/6 (1/2) or 2/11 (1/2) for pajara. The nice thing about calling this the "11/27-LT" is that none of the other methods give you the size of the generator, but this method gives you all that *plus* if you consult a Stern-Brocot tree you can find out which ET's it supports. It's basically giving it the name of a branch on the scale tree. You could even go up a branch and call it the "9/22-LT", since 22-ET is consistent; you don't actually need *two* consistent ETs to name it. If you want to be more precise and narrow it down, you could also call it 20/49-LT; all of these share the same mapping. Things get more complicated if the ET is inconsistent, but it should be manageable. And of course the other way of naming these things is by giving them actual names. The big advantage is recognizability: if you've heard of "superpyth", you take one look at the name and say "Ah yes, *that* temperament". You might not immediately recognize "7/31" or "9&22", but probably anyone who's familiar with it knows the name "orwell". The disadvantage, of course, is that if you haven't heard of it, it doesn't mean anything. So you might as well use the name in combination with one or more of the other methods, especially if it's a less familiar one like 3/8 (1/2) 10&16 lemba <<6, -2, -2, -17, -20, 1||.
Message: 11385 - Contents - Hide Contents Date: Wed, 14 Jul 2004 20:11:22 Subject: Re: Beep and ennealimmal From: Carl Lumma>"Beep" was supposed to bring to mind "BP" (as in >"Bohlen-Pierce"); 27/25 is a frequent step size of >the untempered BP scale. But it's as if we were to >call father "diatonic" because it tempers out the >diatonic semitone. "Bug" appears to be the older >name, and I don't know why it was dropped, unless >the name was just forgotten. So I vote for "bug".I too prefer bug, but Joseph Pehrson has done a piece called Beepy, so I think we should stick with that name. -Carl
Message: 11386 - Contents - Hide Contents Date: Wed, 14 Jul 2004 22:38:51 Subject: Re: Beep and ennealimmal From: Herman Miller Carl Lumma wrote:>> "Beep" was supposed to bring to mind "BP" (as in >> "Bohlen-Pierce"); 27/25 is a frequent step size of >> the untempered BP scale. But it's as if we were to >> call father "diatonic" because it tempers out the >> diatonic semitone. "Bug" appears to be the older >> name, and I don't know why it was dropped, unless >> the name was just forgotten. So I vote for "bug". > >> I too prefer bug, but Joseph Pehrson has done a piece > called Beepy, so I think we should stick with that name.hmm... but wasn't Beepy in the actual BP scale? I don't know if he used the JI or ET version, but the JI version features 27/25 as a step, not a vanishing comma. As a linear temperament, it could be mapped [<x, 1, 1, 2|, <x, 0, 2, -1|] or [<x, 1, 3, 3|, <x, 0, -5, -4|] (no octaves), which isn't anything like [<1, 2, 3, 3|, <0, -2, -3, -1|]. The first one (5&19) results from tempering out the minor BP diesis 245;243, and the second one (4&12) from tempering out the major BP diesis 3125;3087. You could also temper out any combination of these intervals. Equal tempered BP is 41&49, [<1, 0, 0, 0|, <0, 13, 19, 23|]. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 11387 - Contents - Hide Contents
Date: Wed, 14 Jul 2004 06:08:02
Subject: 30 8-note 7-limit JI scales with three tetrads
From: Gene Ward Smith
I took all subsets of three tetrads out of the 27 in the [0,0,0]
centered 27 chord chord cube in 7-limit JI. Counting the notes showed
that these can amount to 8, 9, 10, 11 or 12 notes; I selected the ones
with eight notes, obtaining 171. I then eliminated transpositions,
getting 30 scales, which I reduced to minimal Tenney height. A lot of
them are simply subsets of the 7-limit consonances; the first two
listed have the largest smallest interval, 16/15, and no interval
smaller than 7/6, giving them a fair degree of regularity.
[1, 8/7, 6/5, 5/4, 10/7, 3/2, 12/7, 7/4]
[1, 15/14, 7/6, 5/4, 7/5, 3/2, 7/4, 15/8]
[1, 7/6, 5/4, 4/3, 7/5, 3/2, 5/3, 7/4]
[1, 8/7, 6/5, 4/3, 7/5, 10/7, 8/5, 12/7]
[1, 7/6, 5/4, 4/3, 7/5, 10/7, 5/3, 7/4]
[1, 8/7, 6/5, 7/5, 10/7, 3/2, 8/5, 12/7]
[1, 7/6, 6/5, 5/4, 7/5, 3/2, 8/5, 7/4]
[1, 7/6, 6/5, 5/4, 7/5, 3/2, 12/7, 7/4]
[1, 8/7, 6/5, 4/3, 7/5, 3/2, 8/5, 12/7]
[1, 6/5, 5/4, 10/7, 3/2, 5/3, 12/7, 7/4]
[1, 8/7, 5/4, 4/3, 10/7, 8/5, 5/3, 12/7]
[1, 7/6, 5/4, 7/5, 10/7, 3/2, 5/3, 7/4]
[1, 8/7, 5/4, 10/7, 3/2, 5/3, 12/7, 7/4]
[1, 8/7, 6/5, 5/4, 10/7, 3/2, 5/3, 12/7]
[1, 21/20, 8/7, 6/5, 10/7, 3/2, 12/7, 9/5]
[1, 7/6, 5/4, 4/3, 10/7, 3/2, 5/3, 7/4]
[1, 7/6, 6/5, 7/5, 3/2, 8/5, 12/7, 7/4]
[1, 8/7, 7/6, 6/5, 4/3, 7/5, 8/5, 5/3]
[1, 8/7, 7/6, 5/4, 4/3, 10/7, 8/5, 5/3]
[1, 7/6, 6/5, 4/3, 7/5, 8/5, 5/3, 7/4]
[1, 8/7, 7/6, 5/4, 4/3, 10/7, 5/3, 12/7]
[1, 8/7, 7/6, 4/3, 7/5, 8/5, 5/3, 7/4]
[1, 8/7, 7/6, 4/3, 35/24, 8/5, 5/3, 35/18]
[1, 8/7, 7/6, 4/3, 10/7, 8/5, 5/3, 12/7]
[1, 21/20, 5/4, 21/16, 10/7, 3/2, 5/3, 7/4]
[1, 15/14, 6/5, 9/7, 7/5, 3/2, 8/5, 12/7]
[1, 25/24, 8/7, 5/4, 10/7, 35/24, 5/3, 12/7]
[1, 6/5, 5/4, 7/5, 3/2, 8/5, 12/7, 7/4]
[1, 8/7, 6/5, 4/3, 10/7, 3/2, 8/5, 12/7]
[1, 8/7, 7/6, 6/5, 4/3, 7/5, 8/5, 7/4]
________________________________________________________________________
________________________________________________________________________
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Message: 11388 - Contents - Hide Contents
Date: Thu, 15 Jul 2004 19:48:05
Subject: Beep and bug
From: Gene Ward Smith
The 13-limit temperament 4&9 is a 13-limit extension of 7-limit beep
with TM basis {27/25, 21/20, 33/32, 65/64}. It has mapping given by
[<1 2 3 3 3 3|, <0 -2 -3 -1 2 3|]
We can fill in the gap, getting a prime to go with +1 generators, by
adding the comma 39/34, which is of dubious vintage, being a bit
larger than 8/7; however it gives a mapping with very nice properties:
[<1 2 3 3 3 3 4|, <0 -2 -3 -1 2 3 1|]
This assigns a unique number of contiguous generator steps to each odd
number from 1 to 17:
-5: 15
-4: 9
-3: 5
-2: 3
-1: 7
0: 1
1: 17
2: 11
3: 13
Any subset of a chain of nine generators, together with a selection of
one of the chain (which need not be an element of of the subset) as
root, determines a 17-limit otonal chord up to octave equivalence.
Utonal chords can then be notated as the inverses of otonal chords.
Since we seem to have two names, "beep" and "bug", I propose to call
this bug, unless there is an objection. It isn't important so much as
a temperament, but because it can be lifted so readily to 17-limit JI;
its usefulness in that department seems to make it deserving of a name.
Message: 11389 - Contents - Hide Contents Date: Thu, 15 Jul 2004 21:28:41 Subject: Re: Naming temperaments From: Herman Miller Dave Keenan wrote:>> fraction of an octave: in this case: >> >> 11/27, 489.3 cent generator > >> Right. But why _is_ this 11/27 and not 20/49?It could be either one: a 27-note scale with an approximate 11/27 generator or a 49-note scale with an approximate 20/49 generator. But a 27-note scale is simpler, and it's good to have an idea of the simplest useful scale associated with a temperament. Note that in the 11-limit these represent different temperaments: 11/27 [<1, 2, 6, 2, 1|, <0, -1, -9, 2, 6|] 20/49 [<1, 2, 6, 2, 10|, <0, -1, -9, 2, -16|] To be more precise, since 27-ET isn't 11-limit consistent, you could call it 11/(27,43,63,76,93). But unless there turns out to be a lot of useful-looking temperaments that can't be notated any other way (which could very well be the case once we start dealing with the 11-limit), it would be simpler just to assume that 11/27 denotes a temperament based on the best mapping of each of the primes to degrees of 27-EDO, which is (27,43,63,76,93,100,110,115,122,131,134,...).>> One drawback is that you also need to specify the period if >> it isn't an approximate octave; e.g., 1/6 (1/2) or 2/11 (1/2) for > > pajara. > >> That's a pretty serious drawback. At least with the two-ETs method > you can see if they are both even and know immediately that the > period is a half octave. > > But you could just call Pajara "twin 4/46" or is it "twin 19/46", > or "twin 27/46", or should the denominator be some other ET?4/46 gives you [<2, 3, 5, 7|, <0, 1, -2, -8|], not pajara (see A third kind of linear temperament * [with cont.] (Wayb.)). Pajara is historically associated with 22-ET, of course. But you can think of the denominator as representing the size of a typical MOS scale associated with the temperament, rather than an ET. In that case, the minimum is 10 steps, which matches Paul's decatonic scale.>> if you've heard of >> "superpyth", you take one look at the name and say "Ah yes, *that* >> temperament". You might not immediately recognize "7/31" >> or "9&22", but > > Why wouldn't you immediately recognise one of these, if that's what > you've been used to seeing it called.Combinations of numbers aren't especially easy to remember. It would be like using ZIP codes to refer to cities in the US, instead of names; they all look alike.>> probably anyone who's familiar with it knows the name "orwell". > >> That's a tautology.The pronoun "it" refers to the temperament represented by "7/31" and "9&22", which happens to be named "orwell". I think it's a fairly safe assumption that most people who've heard of this temperament will recognize that name. (Certainly "19/84" is more familiar, but it implies a greater degree of complexity, and could easily be overlooked by people who don't care for highly complex scales.)>> So you might as well use the name in combination with one >> or more of the other methods, especially if it's a less familiar > > one >>> like 3/8 (1/2) 10&16 lemba <<6, -2, -2, -17, -20, 1||. > >> So who needs the "lemba". It adds absolutely nothing, for me. For > some reason it suggests "unleavened bread" to me. Huh?Anyone who's vaguely heard of it, but doesn't know much about it. You might not know how to recognize a beech tree if you see one, or how it differs from other trees, but you probably know that the word "beech" represents a kind of tree. I don't know the mapping of "nonkleismic" off the top of my head, and probably wouldn't recognize it if I saw it (it's [<1, -1, 0, 1|, <0, 10, 9, 7|]), but I do recall it being (in theory) a good temperament. "8/31" might give some idea of its usefulness, but doesn't distinguish it from the many other "n/31" temperaments. So even a questionable name like "nonkleismic" has some use. But "myna" is better because it links it with the starling family of temperaments. Along those lines, Gene's "Japanese monster" names also provide useful hints to similarities between temperaments. Why does anything have a name? Why do we talk about "major thirds" when we could call them 5/4's? Language works by naming things; the problem is that the study of linear and higher-dimensional temperaments is so new that we haven't settled on the best names for things. So in the meantime, names which may end up being changed will have to be supplemented by numerical keys of one kind or another. I'm leaning toward the fractional generator + period notation for unfamiliar temperaments, with wedgies for those few that can't easily be symbolized in this way. But I still find names easier to remember, and I don't want to discourage the naming of temperaments that look like they might be useful.
Message: 11390 - Contents - Hide Contents Date: Thu, 15 Jul 2004 21:54:09 Subject: Re: Beep and bug From: Herman Miller Gene Ward Smith wrote:> The 13-limit temperament 4&9 is a 13-limit extension of 7-limit beep > with TM basis {27/25, 21/20, 33/32, 65/64}. It has mapping given by > > [<1 2 3 3 3 3|, <0 -2 -3 -1 2 3|] > > We can fill in the gap, getting a prime to go with +1 generators, by > adding the comma 39/34, which is of dubious vintage, being a bit > larger than 8/7; however it gives a mapping with very nice properties: > > [<1 2 3 3 3 3 4|, <0 -2 -3 -1 2 3 1|] > > This assigns a unique number of contiguous generator steps to each odd > number from 1 to 17: > > -5: 15 > -4: 9 > -3: 5 > -2: 3 > -1: 7 > 0: 1 > 1: 17 > 2: 11 > 3: 13 > > Any subset of a chain of nine generators, together with a selection of > one of the chain (which need not be an element of of the subset) as > root, determines a 17-limit otonal chord up to octave equivalence. > Utonal chords can then be notated as the inverses of otonal chords. > > Since we seem to have two names, "beep" and "bug", I propose to call > this bug, unless there is an objection. It isn't important so much as > a temperament, but because it can be lifted so readily to 17-limit JI; > its usefulness in that department seems to make it deserving of a name.The simplest bug-compatible temperament by the combined-ET method or the fractional-generator method is: 1/5 4&5 [<1, 2, 3, 3, 4, 4, 4|, <0, -2, -3, -1, -3, -1, 0|]. So there could be better alternatives for approximating JI. Try this one: 1/7 [<1, 2, 2, 3, 4, 4, 4|, <0, -3, 2, -1, -4, -2, 1|]
Message: 11391 - Contents - Hide Contents Date: Thu, 15 Jul 2004 03:31:45 Subject: Re: Beep and ennealimmal From: Carl Lumma>X-eGroups-Return: perlich@xxx.xxxx.xxx >Date: Thu, 15 Jul 2004 03:20:28 -0000 >From: "Paul Erlich" <perlich@xxx.xxxx.xxx> >To: Carl Lumma <ekin@xxxxx.xxx> >Subject: Re: Beep and ennealimmal >User-Agent: eGroups-EW/0.82 >X-Mailer: Yahoo Groups Message Poster >X-Originating-IP: 199.103.208.200 > >--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> "Beep" was supposed to bring to mind "BP" (as in >>> "Bohlen-Pierce"); 27/25 is a frequent step size of >>> the untempered BP scale. But it's as if we were to >>> call father "diatonic" because it tempers out the >>> diatonic semitone. "Bug" appears to be the older >>> name, and I don't know why it was dropped, unless >>> the name was just forgotten. So I vote for "bug". >>>> I too prefer bug, but Joseph Pehrson has done a piece >> called Beepy, so I think we should stick with that name. >> >> -Carl >>Joseph's piece wasn't in beep/bug; it was in BP! Eek- sorry! -C.
Message: 11392 - Contents - Hide Contents Date: Thu, 15 Jul 2004 03:32:32 Subject: Re: Beep and ennealimmal From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>> "Beep" was supposed to bring to mind "BP" (as in >> "Bohlen-Pierce"); 27/25 is a frequent step size of >> the untempered BP scale. But it's as if we were to >> call father "diatonic" because it tempers out the >> diatonic semitone. "Bug" appears to be the older >> name, and I don't know why it was dropped, unless >> the name was just forgotten. So I vote for "bug". >> I too prefer bug, but Joseph Pehrson has done a piece > called Beepy, so I think we should stick with that name.I think Paul suggested the change, if so he should know why. I kind of like bug too, though. Of course for my suggested use (to me this idea is a gold mine) 4&5 really says it all. We had this argument about whether beepbug was a real temperament, but surely we can regard it as a rough draft temperament for something we propose to refine. I'm going to use it, certainly. It should be possible, in bug, to get a draft that actually sounds almost like music, and it is not so complex that I can't manage to deal with it. Obviously, by changing the 14-et tune I'll be forced to change my tune about 14-et--I now think it's wonderful!
Message: 11393 - Contents - Hide Contents Date: Thu, 15 Jul 2004 23:09:33 Subject: Re: Naming temperaments From: Herman Miller Dave Keenan wrote:> Good summary Herman. But you left out my favourite method which > unlike some of the others does not treat ETs as if they were more > fundamental than LTs or JI, but instead relates to what this is all > about -- approximating JI.That's the nice thing about tuning maps. You can interpret some of the other methods as detempering ET's, but there are other interpretations. 5/12 could be a 12-ET based scale, or it could be a 12-note MOS with a 5-step generator. Even 5&7 could be interpreted in MOS terms, as a scale with 5 steps of one size and 7 of another size.> The name starts with a word for the number of periods per octave, if > more than one: twin, triple, quadruple, quintuple, 6-fold, 7- > fold, .... > > And then the generator is described in terms of the simplest n-odd- > limit consonance (from the diamond) (or its octave inversion or > extension, as required). That is the one that takes the fewest > generators to approximate according to the LTs mapping. > > I use the following words if there is more than one generator to the > consonance: semi, tri, quarter, 5-part, 6-part, .... > > Followed by the ratio or words for the consonance as given here: > The page cannot be found * [with cont.] (Wayb.) > ing.txt > > e.g. Miracle is "semi 7:8's" or "semi supermajor seconds".This could be potentially useful up to a point; certainly there's a mnemonic value in names like "semisixths". But I don't see how this can be generalized to the 7-limit and higher without being arbitrary. Which LT gets to be called "fourths" -- dominant (5&12), meantone (12&19), superpyth[agorean] (5&22), flattone (19&26), or schismic (12&29)? You could make good arguments at least for dominant, meantone, and schismic; then you need to figure out how to name the others. "Major thirds" could be either muggles (16&19) or magic (19&22), and so on.> This is used up to some point where the LT is so complex you just > describe the generator in cents. e.g. What used to be called > Aritoxenean is the 12-fold 15 cent LT. > > This at least works up to 11-limit.But giving the generator in cents doesn't determine a unique mapping; you can derive one from a rational generator/period ratio if you make some assumptions, but an arbitrary value in cents could represent more than one temperament. An LT with a 316.5 cent generator can be mapped as [<1, 0, 1, 2|, <0, 6, 5, 3|] or [<1, 0, 1, -3|, <0, 6, 5, 22|]. With rational generators and the naming conventions I've described, you can unambiguously describe the first mapping as 5/19 and the second as 19/72.
Message: 11394 - Contents - Hide Contents Date: Thu, 15 Jul 2004 23:13:52 Subject: Re: Beep and bug From: Herman Miller monz wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: > >>> The 13-limit temperament 4&9 is a 13-limit extension >> of 7-limit beep with TM basis {27/25, 21/20, 33/32, 65/64}. >> It has mapping given by >> >> [<1 2 3 3 3 3|, <0 -2 -3 -1 2 3|] > > > >> this is great -- a model of how to describe a temperament. > > i only wish there was a way to distinguish between > the periods and the generators without labeling them.If the octave is mapped to an exact multiple of periods without any generators (which is the typical case unless you're doing something like a BP scale), the first number in the generator mapping will always be zero. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)
Message: 11395 - Contents - Hide Contents Date: Fri, 16 Jul 2004 07:42:39 Subject: Re: Naming temperaments From: Graham Breed Herman Miller wrote:> To be more precise, since 27-ET isn't 11-limit consistent, you could > call it 11/(27,43,63,76,93). But unless there turns out to be a lot of > useful-looking temperaments that can't be notated any other way (which > could very well be the case once we start dealing with the 11-limit), it > would be simpler just to assume that 11/27 denotes a temperament based > on the best mapping of each of the primes to degrees of 27-EDO, which is > (27,43,63,76,93,100,110,115,122,131,134,...).It depends on what you think is useful. I don't find the inconsistent search finds anything I'm interested in in the 11-limit. But you may prefer less accurate temperaments. When you get to the 17-limit, inconsistently composed temperaments score high. At least by my criteria that try to keep the number of notes manageable. You can test all this with the online script. Graham
Message: 11396 - Contents - Hide Contents Date: Fri, 16 Jul 2004 00:04:17 Subject: Re: Naming temperaments From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> I've been thinking about names for linear temperaments and ways of > classifying them; there's probably nothing much new here, but a summary > might have some use.Yes. This is good. Thanks Herman.> For most of the better temperaments, if you just give the mapping by > steps of the octave (which is equivalent to a combination of two > octave-based ET's), you can fill in the others by taking the best > approximation of the primes in each of the ET's. So we could call this > the 5&22 temperament.Well I'd prefer to call it the 22&27-LT. You can get the 5 by subtraction and you can get an even better ET for it (49) by addition. i.e. the two ETs should be two extremes of the generator value such that you would actually consider using those ETs for the LT. You wouldn't really use 5. The 5 just tells you that pentatonic scales make sense in this LT. And as I say, you can get that by subtraction.> But there's another alternative that looks somewhat interesting. Notice > that the first thing in Graham Breed's temperament finder results is a > fraction of an octave: in this case: > > 11/27, 489.3 cent generatorRight. But why _is_ this 11/27 and not 20/49?> One drawback is that you also need to specify the period if > it isn't an approximate octave; e.g., 1/6 (1/2) or 2/11 (1/2) for pajara. >That's a pretty serious drawback. At least with the two-ETs method you can see if they are both even and know immediately that the period is a half octave. But you could just call Pajara "twin 4/46" or is it "twin 19/46", or "twin 27/46", or should the denominator be some other ET?> And of course the other way of naming these things is by giving them > actual names. The big advantage is recognizability: if you've heard of > "superpyth", you take one look at the name and say "Ah yes, *that* > temperament". You might not immediately recognize "7/31"or "9&22", but Why wouldn't you immediately recognise one of these, if that's what you've been used to seeing it called. Superpyth isn't a good example of the problem, since it at least suggests something related to pythagorean. The real problem is with names like "keenan" or "porcupine" or "orwell".> probably anyone who's familiar with it knows the name "orwell".That's a tautology.> The > disadvantage, of course, is that if you haven't heard of it, it doesn't > mean anything. Exactly. > So you might as well use the name in combination with one > or more of the other methods, especially if it's a less familiar one > like 3/8 (1/2) 10&16 lemba <<6, -2, -2, -17, -20, 1||.So who needs the "lemba". It adds absolutely nothing, for me. For some reason it suggests "unleavened bread" to me. Huh?
Message: 11397 - Contents - Hide Contents Date: Fri, 16 Jul 2004 08:44:52 Subject: Re: Beep and bug From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> The simplest bug-compatible temperament by the combined-ET method or the > fractional-generator method is: > > 1/5 4&5 [<1, 2, 3, 3, 4, 4, 4|, <0, -2, -3, -1, -3, -1, 0|].This is full of equivalences we don't want.> So there could be better alternatives for approximating JI. Try this one: > > 1/7 [<1, 2, 2, 3, 4, 4, 4|, <0, -3, 2, -1, -4, -2, 1|]15/8 ~ 7/4 isn't ideal, and 9 is out in left field somewhere. I hadn't looked at that one, but had considered [<1, 2, 3, 3, 3, 3, 4|, <0, -2, -3, -1, 3, 2, 1|] and [<1, 1, 2, 2, 2, 2, 3|, <0, 2, 1, 3, 5, 6, 4|]
Message: 11398 - Contents - Hide Contents Date: Fri, 16 Jul 2004 00:53:26 Subject: Re: Naming temperaments From: Dave Keenan Good summary Herman. But you left out my favourite method which unlike some of the others does not treat ETs as if they were more fundamental than LTs or JI, but instead relates to what this is all about -- approximating JI. The name starts with a word for the number of periods per octave, if more than one: twin, triple, quadruple, quintuple, 6-fold, 7- fold, .... And then the generator is described in terms of the simplest n-odd- limit consonance (from the diamond) (or its octave inversion or extension, as required). That is the one that takes the fewest generators to approximate according to the LTs mapping. I use the following words if there is more than one generator to the consonance: semi, tri, quarter, 5-part, 6-part, .... Followed by the ratio or words for the consonance as given here: The page cannot be found * [with cont.] (Wayb.) ing.txt e.g. Miracle is "semi 7:8's" or "semi supermajor seconds". This is used up to some point where the LT is so complex you just describe the generator in cents. e.g. What used to be called Aritoxenean is the 12-fold 15 cent LT. This at least works up to 11-limit.
Message: 11399 - Contents - Hide Contents Date: Fri, 16 Jul 2004 08:46:28 Subject: Re: Beep and bug From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: >>> The 13-limit temperament 4&9 is a 13-limit extension >> of 7-limit beep with TM basis {27/25, 21/20, 33/32, 65/64}. >> It has mapping given by >> >> [<1 2 3 3 3 3|, <0 -2 -3 -1 2 3|]> this is great -- a model of how to describe a temperament. > > i only wish there was a way to distinguish between > the periods and the generators without labeling them.The period maps 2 to a positive integer, and the generator maps it to 0.
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