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Message: 11400 - Contents - Hide Contents Date: Fri, 16 Jul 2004 20:20:42 Subject: Re: Beep and bug From: Herman Miller Gene Ward Smith wrote:> --- 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.What I'm saying is that's the one that would logically get the name "bug" if we extend the 5-limit names to higher limits.
Message: 11401 - Contents - Hide Contents Date: Fri, 16 Jul 2004 01:14:23 Subject: Re: Naming temperaments From: Carl Lumma>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, ....How do you choose a period/generator representation? -Carl
Message: 11402 - Contents - Hide Contents Date: Fri, 16 Jul 2004 08:53:44 Subject: Re: Naming temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> 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.Right; what I call 5+7 as opposed to 5&7. The convergents to 7/5 are 1, 3/2, 7/5, from which we have that the generator is the mediant of 2/5 and 3/7, or (2+3)/(5+7) = 7/12. Nothing about JI or mappings to primes appears directly.
Message: 11403 - Contents - Hide Contents Date: Fri, 16 Jul 2004 19:46:47 Subject: Re: Extreme precison (Olympian) Sagittal From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- 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?Gene, I've also had trouble wrestling with this, but since I've had a few months to ponder it, let me give you my take on it. I believe that Dave is trying to arrive at a Sagittal JI notation with the highest precision possible *without* introducing any new symbol-elements beyond the left and right barb, arc, scroll, and boathook in combination with no/up/down (5-schisma) accent marks. This precision would hopefully be something on the order of 1-cent increments. Dave has determined that 1171-ET is the highest one that can be notated with these constraints, and the extreme-precision (olympian) JI that he's proposing is modeled after that. Personally, I feel that going very much beyond the precision offered by 612-ET is overkill in the extreme and that the economy sacrificed by using separate symbols for 13-limit consonances is a high price to pay for this. Still, if someone really thinks they might want it, then I have no objection to having it available. BTW, I should mention that we *actually use* one of these olympian- level symbols in the 12-relative (trojan) symbol set, for 5 degrees of 144-edo. Since there will probably be other instances in which one of these rarely used symbols will need to be pressed into service, I'll concede that olympian symbol development can have benefits. Gene, please see what you can make of this. I don't think that you'll be able to find a good alternative to 1171; 1224 was my choice as best division 1-cent resolution, but it can't be done completely with our existing symbol elements (although we come very close! -- we lack only 35, 55, and 57 steps up to 1/2-apotome). The best division I could find that makes a distinction between 351:351 and 5103:5120 (11:13s & 5:7s), between 45:46 and 1664:1701 (115S & 7:13S), and also between 6400:6561 and 39:40 (25S & 5:13S) is 742-ET (612+130). Instead of distinguishing between 1024:1053 and 35:36 (13M and 35M) and their complements (8192:8505 and 26:27, 13L and 35L), it distinguishes between 1892:8505 and 512000:531441 (35L and 125L, the latter being (81/80)^3). This retains the vanishing status of the "linchpin" schismina 4095:4096. But it doesn't even come close to the 1-cent resolution that Dave desired, so it is doubtful that this one really buys us anything. --George
Message: 11404 - Contents - Hide Contents Date: Fri, 16 Jul 2004 20:59:45 Subject: Re: Extreme precison (Olympian) Sagittal From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:> Gene, I've also had trouble wrestling with this, but since I've had a > few months to ponder it, let me give you my take on it.I was just hoping for a statement of the problem which did not require me to read things I have to squint at and still can't make out. What are the barbs, arcs, scrolls, boathooks and accent marks supposed to do, in numerical terms?> The best division I could find that makes a distinction between > 351:351 and 5103:5120 (11:13s & 5:7s), between 45:46 and 1664:1701 > (115S & 7:13S), and also between 6400:6561 and 39:40 (25S & 5:13S) is > 742-ET (612+130).(5120/5103)/(352/351) = 2080/2079 (1701/1644)/(46/45) = 76545/76544 (6561/6400)/(40/39) = 256000/255879 The first and last are 13-limit schisminas, with a TM reduction {2080/2079, 59319/59290}. The other is tiny, and drags us all of the way up to the 23-limit; it TM reduces to {2025/2024, 2080/2079, 35000/34983}. This defines a temperament with six generators, but we can cut that down to four if we ignore 17 and 19, which aren't in the picture. However, we are looking ets in which *none* of these commas vanish, which are quite common, though 742 with its best tuning isn't one of them, so I'm a little confused. Possibilies include 270, 311, 684, 1178, 1506, 2190, 2684 etc. For the "linchpin" comma to vanish while the rest of these do not is also not uncommon; 270, 311, 581, 764, 1012, 1106, 1236, 1506, 1600, 2742 etc. etc. It looks to me that 1506, a strong 13-17 limit system, would be a good choice; also 1600 and 2742 are examples of decent 23-limit systems which fit the bill and which clearly would give good accuracy. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 11405 - Contents - Hide Contents Date: Fri, 16 Jul 2004 04:03:06 Subject: Re: Beep and bug From: monz --- 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. -monz
Message: 11406 - Contents - Hide Contents Date: Fri, 16 Jul 2004 20:15:05 Subject: Re: Naming temperaments From: Carl Lumma At 06:37 PM 7/16/2004, you wrote:>--- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:>>> 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, .... >>>> How do you choose a period/generator representation? >>You don't have to. You just base it directly on the map -- any map >that's valid. i.e. the generator doesn't have to be in lowest >(period-reduced terms).So there'll be multiple names for each temperament?>For any of the ET/MOS/DE-based names you need to choose specific >values of period and generator. In most cases, different kinds of >optima do not change the period and generator enough to make much >difference, but I just found that while minimax and RMS versions of >5-limit Diminished can be described as 12&16-LT or 12&28-LT, the TOP >version cannot. It could be described as 12&20-LT or 12&32-LT but 20 >and 32 are not 5-limit consistent, so the best you can do >is 8&12-LT.I think that's why Gene is proposing to use ETs that represent the extreme ranges of the generator.>Back to the map-based method. > >The map tells you how many periods to the octave. That's all you >need to know about the period to know whether the temperament is >twin or triple etc.What about temperaments that map 2 through a combination of both the "period" and generator?>For the rest of it, lets look at the simplest case first -- an LT >with one period to the octave, and one generator to some prime. i.e >there's a "1" (or a "-1") there staring at you from one of the >generators-per-prime slots. > >If the 1 is in the prime-3 slot then it's "fourths". I'll say more >later about differentiating multiple temperaments having the same >name of generator.Hmm... I thought one could refactor these maps is several annoying ways. Thus, the reason for something called hermite normal form -- whatever that is. -Carl
Message: 11407 - Contents - Hide Contents Date: Fri, 16 Jul 2004 22:29:58 Subject: Re: Naming temperaments From: Herman Miller Dave Keenan wrote:> For any of the ET/MOS/DE-based names you need to choose specific > values of period and generator. In most cases, different kinds of > optima do not change the period and generator enough to make much > difference, but I just found that while minimax and RMS versions of > 5-limit Diminished can be described as 12&16-LT or 12&28-LT, the TOP > version cannot. It could be described as 12&20-LT or 12&32-LT but 20 > and 32 are not 5-limit consistent, so the best you can do is 8&12-LT.The notation "12&16" doesn't need to be restricted to tunings with exact octaves; it's just a convenient shorthand for a temperament with map [<4, 6, 9|, <0, 1, 1|]. Certainly, to get from the name "12&16" to the map [<4, 6, 9|, <0, 1, 1|], you need to adopt the convention of exact octave ET's, but the values of the period and generator can be any values that are consistent with the map. But it seems that what you're saying is that 12+16 isn't an MOS scale in TOP diminished; it skips from 12+8 to 12+20. So I was clearly wrong about the MOS interpretation in this case. Well, once you get up to 12 notes TOP diminished has a better 2/1 at (3, 3), so you're dealing with an inconsistent temperament in any case; you might as well close it at 12 notes and call it a well- temperament.> Similarly the minimax and RMS versions can be called 4/16-oct, 8/28- > oct, 12/40-oct, ... but the TOP version cannot. It has to be 4/12- > oct, 8/20-oct, 12/32-oct, .... Notice that I have not reduced these > fractions to lowest terms. This lets you extract the number of > periods per octave as the GCD of numerator and denominator."4/12" is a nice convention; I think I'll adopt it. Apparently the generator/period ratio of 5-limit TOP diminished is somewhere around 0.33985, which puts it on an entirely different branch of the scale tree from the ET-based version of diminished with the largest g/p ratio, which is 12&16 (g/p = 0.285714). So essentially there are two different kinds of scales that fit the same temperament map; the one with a better JI approximation has a different scale structure from the pure octave-based one. I'm wondering if these different scale structures are similar enough to be given the same name.... At least I can see that it would be misleading to include TOP diminished in the category of "4&12" or "4/16"....
Message: 11408 - Contents - Hide Contents
Date: Sat, 17 Jul 2004 00:36:33
Subject: TM bases for 1506
From: Gene Ward Smith
My posting seems not to have made it.
7-limit
{962072674304/961083984375, 250047/250000, 645700815/645657712}
11-limit
{2097152/2096325, 9801/9800, 151263/151250, 1771561/1771470}
13-limit
{4096/4095, 105644/105625, 371293/371250, 9801/9800, 6656/6655}
17-limit
{4096/4095, 4914/4913, 5832/5831, 6656/6655, 9801/9800, 28561/28560}
Message: 11409 - Contents - Hide Contents Date: Sat, 17 Jul 2004 07:27:18 Subject: Re: names and definitions: meantone From: monz --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: re: Frame Index for Tuning Dictionary * [with cont.] (Wayb.) meantone> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:>> thanks, Gene ... this is great! >> But please use the revised version. done.i put it at the very bottom of the page. does anyone think it should be at the top, or closer to the top? my feeling is that i should leave the verbal description at the top and let the math come at the end. -monz
Message: 11410 - Contents - Hide Contents Date: Sat, 17 Jul 2004 17:12:46 Subject: Re: names and definitions: schismic From: Herman Miller Gene Ward Smith wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote: > >>>> This should be "schismic". It's consistent with all those ETs > > except 17 >>>> (which isn't 7-limit consistent in itself). >>>> I agree; "schismic" is essentially a 5-limit name (based on tempering >> out the [-15 8 1> "schisma"), so the 7-limit extension (if any) >> shouldn't be more complex than it needs to be. > >> This reasoning is backwards, and leads to bizarre conclusions. If > schismic is essentially a 5-limit temperament, then you'd better keep > close to the 5-limit tuning seems like the way to reason from your > premise. Moreover, it leads to the conclusion that dominant should get > the name "meantone" in the 7-limit, and that is a conclusion no one > seems to buy. We'd also end up renaming pajara to diaschismic, I suppose.12&19 meantone isn't excessively complex, and it's not a good idea to change established names in any case. If you have to go all the way to 118&171 for schismic, it's way too complex to be of much use to anyone; very few people will bother with it at all, so why give it the familiar name when there are better options? Even the 13-limit Cassandra 1 is only 41&94.
Message: 11411 - Contents - Hide Contents Date: Sat, 17 Jul 2004 17:20:41 Subject: Re: names and definitions: schismic From: monz hi Gene (and Graham and everyone else too), --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> family name: schismic > period: octave > generator: fourth or fifth > > 5-limit > > name: schismic > comma: 32805/32768 > mapping: [<1 2 1|, <0 -1 8|] > poptimal generator: 120/289 > MOS: 12, 17, 29, 41, 53, 65, 118, 171 > > 7-limit > > <etc. -- snip>thanks much for that. i was going to copy it into my webpage right away, as i did with your meantone defitions, but i see that there's already been a lot of discussion about what you posted here. what do you guys think? should i put Gene's data into my "schismic" Encyclpaedia webpage, and correct as we go along? or should i wait until there's been more of a consensus here? while we're at it, i'd like to do something about the schismic/schismatic business. i have webpages under both names, with different content. i'm appealing to all of you to read both of them and give me advice on how to combine them or separate them more clearly, whichever is advisable. -monz
Message: 11412 - Contents - Hide Contents
Date: Sat, 17 Jul 2004 00:59:08
Subject: Re: TM bases for 1506
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
19-limit
{3971/3969, 2432/2431, 2926/2925, 4096/4095, 4914/4913,
5832/5831, 6656/6655}
Message: 11413 - Contents - Hide Contents Date: Sat, 17 Jul 2004 07:38:04 Subject: Re: Naming temperaments From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote: >>> I only proposed using cents when the temp is so complex that the >> diamond ratio with the fewest generators has say 5 or more >> generators in it. >> Speaking of jargon, what is a diamond ratio?Thanks for asking. I shouldn't have been using the term at all. That was stupid of me. All I mean is "a ratio in the odd-limit of the temperament", as opposed to the prime limit. If we're talking about optimising according to Tenney Harmonic Distance (TOP) then only a prime limit may be given, in which case, for the p-prime-limit it would make sense to consider the n-odd- limit where n is the largest odd number that is less than the next prime after p. e.g. for the 7-prime limit, we should consider ratios in the 9-odd-limit as possible candidates for the generator-as- fraction-of-ratio method of naming temperaments. Similarly for a temperament described as 13-prime-limit, we should probably consider 15-odd-limit ratios. "Tonality diamond" comes from Partch. You can look up what it really means, here: Frame Index for Tuning Dictionary * [with cont.] (Wayb.)
Message: 11414 - Contents - Hide Contents Date: Sat, 17 Jul 2004 12:37:59 Subject: Re: names and definitions: meantone From: Herman Miller Gene Ward Smith wrote:> name: meantone, 12&19 > comma: 81/80 > mapping: [<1 2 4|, <0 -1 -4|] > poptimal generator: 34/81 > MOS: 5, 7, 12, 19, 31, 50, 81This MOS list is valid for quarter-comma meantone and Kornerup's golden meantone (among others), but as we've seen, various tunings of the same temperament can have different MOS structures. In particular, meantone with a 23/55 generator/period ratio (Mozart's tuning) has a 43-note MOS (12L+31s), but not one with 50 notes. 1/5-comma meantone technically has a 43-note MOS, but it's so close to 43-ET that the step sizes are roughly equal.
Message: 11415 - Contents - Hide Contents
Date: Sat, 17 Jul 2004 22:33:53
Subject: Miracle
From: Gene Ward Smith
family name: miracle
period: octave
generator: secor, the neutral semitone
5-limit
name: miracle, ampersand, 31&41
comma: |-25 7 6>
mapping: [<1 1 3|, <0 6 -7|]
poptimal generator: 41/422
TOP period: 1200.631014 mapping: 116.720642
MOS: 10, 11, 21, 31, 41, 72, 103, 175
7-limit
name: miracle, 31&41
wedgie: <<6 -7 -2 -25 -20 15||
mapping: [<1 1 3 3|, <0 6 -7 -2|]
7-limit poptimal generator: 17/175
9-limit poptimal generator: 32/329
TOP period: 1200.631014 mapping: 116.720642
TM basis: {225/224, 1029/1024}
MOS: 10, 11, 21, 31, 41, 72
11-limit
name: miracle, 31&41
wedgie: <<6 -7 -2 15 -25 -20 3 15 59 49||
mapping: [<1 1 3 3 2|, <0 6 -7 -2 15|]
poptimal generator: 39/401
TOP period: 1200.631014 mapping: 116.720642
TM basis: {225/224, 243/242, 385/384}
MOS: 10, 11, 21, 31, 41, 72, 113, 185
Message: 11416 - Contents - Hide Contents Date: Sat, 17 Jul 2004 08:05:00 Subject: Re: Atomic temperament as a basis for a valhalla-level notation From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Atomic temperament, in 2460-et, has a period of exactly 100 cents and > a generator of size one step of 615-equal, a type of schisma. The > lowest Graham complexity for the 23-limit is given by > > [<12 19 28 34 42 43 47 50 52|, <0 1 -7 -16 -25 72 105 50 117|] > > so this I propose as the official 23-limit atomic mapping. 2460 is a > strong temperament up to the 27-odd-limit, and so makes a good basis > for defining atomic. Since 205 schismas make up 400 cents in this > tuning, we can obtain alternate mappings by raising or lowering the > generator steps by 205. We want -25 schismas for 11 at one end, and > +50 schismas for 19 at the other, which sticks us with this mapping if > we want minimal Graham complexity. 50+88 is 138, less than 117+25=142, > so we might bring 117 down to -88; however if we do that then 105+88 > is 193, whereas bringing it down to -100 leaves us with 100+88=188, > and clearly leaving 117 as it is means we have to leave 72 and 105 > alone also. > > An atomic notation would, I presume, have twelve nominals dividing the > octave into twelve parts exactly. Other symbols would be defined in > terms of schismas and semitones. One curious feature would be that in > the 5-limit, with the atom as a comma, it would be a very, very, very, > very precise notation for the 5-limit. One question I have is how, and > how easily, 2460 and atomic could be notated in Sagittal.Well it certainly can't notate every step of 2460-ET. But it should be able to notate the distributionally even (DE or "MOS") scales having the following numbers of steps to the octave: 612 and 624, and it would go close to doing 1236. If you really had to do 2460-ET you'd have to invent a new kind of accent mark that is defined as some comma (specifically a schismina) somewhere in the 0.3 to 1 cent range, preferably around 0.6 or 0.7 of a cent, that doesn't vanish in the 12-fold schismas (atomic) temperament.
Message: 11417 - Contents - Hide Contents Date: Sat, 17 Jul 2004 17:40:33 Subject: 136edo From: monz on this page (which can also be reached via a link on the "meantone" defintion) Definitions of tuning terms: meantone-from-JI ... * [with cont.] (Wayb.) i found that 136edo is some sort of optimal meantone for the 11-limit, regarding the amount of error from JI. (at least, using the prime-mappings i used) i found this visually using my graphic applet. but i don't recall anything ever being said about 136edo. comments? -monz
Message: 11418 - Contents - Hide Contents Date: Sat, 17 Jul 2004 22:48:14 Subject: Re: names and definitions: schismic From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:> 12&19 meantone isn't excessively complex..."Excessively complex" is a value judgment which only makes sense in the context of some projected use; hence it is really meaningless for us, since the possible uses are varied. So long as the *commas* are not complex it certainly might find a use in electronic music (and this *is* the 21st century; this aspect is likely to assume ever-increasing importance) if nowhere else. And from a comma point of view, this so-called "excessively complex temperament" is not that complex! It has commas of 4375/4374 (Hahn size 7) and 32805/32768 (Hahn size 9.) You are simply failing to think in terms of the 7-limit lattice when you call it "excessively complex". The prejudice against complex temperaments is obvious on this group, but it all comes from people who never actually *use* them. Until you've used them, it really resembles the people who kvetch to you that some of your favorite temperaments are excessively poorly tuned. and it's not a good idea to> change established names in any case.We've *been* changing established names; I don't think 7-limit schismic was ever as established as some Paul wants to deep-six. Moreover, sticking to a consistent scheme means we have a better idea what the name means; in this case, that the name in the higher limit has a tuning in accord with the lower limit. If you have to go all the way to> 118&171 for schismic, it's way too complex to be of much use to anyone;Wrong! I should probably write something in it; would that convince anyone of anything or is everyone hypotized by the idea that the only way to make music is to tune up a guitar?> very few people will bother with it at all, so why give it the familiar > name when there are better options?This is the ***21st century***. I would *not* assume people in the future are going to ignore complex tunings.
Message: 11419 - Contents - Hide Contents Date: Sat, 17 Jul 2004 01:37:49 Subject: Re: Naming temperaments From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma" <ekin@l...> wrote:>> 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, .... >> How do you choose a period/generator representation?You don't have to. You just base it directly on the map -- any map that's valid. i.e. the generator doesn't have to be in lowest (period-reduced terms). And you don't have to decide on specific optimum values of period and generator. That's the beauty of it. For any of the ET/MOS/DE-based names you need to choose specific values of period and generator. In most cases, different kinds of optima do not change the period and generator enough to make much difference, but I just found that while minimax and RMS versions of 5-limit Diminished can be described as 12&16-LT or 12&28-LT, the TOP version cannot. It could be described as 12&20-LT or 12&32-LT but 20 and 32 are not 5-limit consistent, so the best you can do is 8&12-LT. Similarly the minimax and RMS versions can be called 4/16-oct, 8/28- oct, 12/40-oct, ... but the TOP version cannot. It has to be 4/12- oct, 8/20-oct, 12/32-oct, .... Notice that I have not reduced these fractions to lowest terms. This lets you extract the number of periods per octave as the GCD of numerator and denominator. Back to the map-based method. The map tells you how many periods to the octave. That's all you need to know about the period to know whether the temperament is twin or triple etc. For the rest of it, lets look at the simplest case first -- an LT with one period to the octave, and one generator to some prime. i.e there's a "1" (or a "-1") there staring at you from one of the generators-per-prime slots. If the 1 is in the prime-3 slot then it's "fourths". I'll say more later about differentiating multiple temperaments having the same name of generator. If the 1 is in the prime-5 slot then it's "major thirds" or just "thirds" (assuming the convention that if it's not explicitly called minor or neutral or anything else, then it's major). If the 1 is in the prime-7 slot then it's "supermajor seconds". If the 1 is in the prime-11 slot then it's "super fourths". If there is 1 or -1 generators to more than one prime then you give them both, as in "fourth thirds". If there is no 1 or -1 _directly_ as entries in the generator mapping, then you look for two entries which differ by 1. e.g. Kleismic has <0 6, 5]. If the difference of 1 generator is between 3's and 5's then it's "minor thirds"; between 5's and 7's it's "augmented fourths"; between 3's and 7's it's "subminor thirds"; between 3's and 11's it's "neutral seconds";", between 5's and 11's it's "narrow neutral seconds"; between 7's and 11's it's "narrow supermajor thirds". It starts to get a bit hairy with those "narrow"s and "supermajor"s and we might prefer to just give the approximated ratio. Or you might prefer to give that every time. If we're looking at 9 or 11-limit temps then we also have to double the number in the 3's slot and see if anything differs from that by 1. If the difference of 1 is between 9's and prime 5's it's "narrow major seconds", although we could probably drop the "narrow" since it unlikely we'd ever have a generator that approximates an 8:9; between 9's and prime 7's it's "supermajor thirds"; between 9's and prime 11's it's "neutral thirds". Again if there's more than one, list them all, e.g. "minor major thirds". If you can't find any 1's or differences of 1 then go looking for 2's or differences of 2, in exactly the same way, and put "semi" in front of whatever you find. If no 2's then 3's and put "tri" or "tripartite" or "3-part" (I used to say "tertia") in front. If the n-limit diamond ratio with the fewest generators has 4 generators then put "quarter" in front. After that "5-part", "6- part" etc. But you also have to check that you're describing the correct octave inversion or octave extension of the diamond ratio, so that when you divide it into however many equal parts you really do get something that is a valid generator. For example if you have 1 generator to the prime-3 then you could call it fourths or fifths, but with 2 generators to the prime 3 then you have to check whether your generator is a semifourth or semififth. Only one of those will be correct. And with 4 generators to the prime-3, the generator might even be a quarter eleventh or quarter twelfth. When it comes to LTs with more than one period to the octave, you have to be a little more careful. You the have to look at the periods per prime as well as the generators per prime. You have to ensure that, as well as having the minimum number of generators, the diamond ratio being approximated has a number of periods which corresponds to an integral number of octaves, i.e. that comes to zero when taken modulo the number of periods in the octave. I note that Erv Wilson uses this kind of terminology for LTs, at least "semifourths" and "semififths".
Message: 11420 - Contents - Hide Contents
Date: Sat, 17 Jul 2004 08:13:45
Subject: Re: names and definitions: schismic
From: Gene Ward Smith
family name: schismic
period: octave
generator: fourth or fifth
5-limit
name: schismic
comma: 32805/32768
mapping: [<1 2 1|, <0 -1 8|]
poptimal generator: 120/289
MOS: 12, 17, 29, 41, 53, 65, 118, 171
7-limit
name: schismic, 118&171
wedgie: <<1 -8 39 -15 59 113||
mapping: [<1 2 -1 19|, <0 -1 8 -39|]
7&9 limit copoptimal generator: 732/1763
TM basis: {4375/4374, 32805/32768}
MOS: 12, 17, 29, 41, 53, 65, 118, 171
name: garibaldi, 41&53
wedgie: <<1 -8 -14 -15 -25 -10||
mapping: [<1 2 -1 -3|, <0 -1 8 14|]
7&9 limit copoptimal generator: 39/94
TM basis: {3125/3087}
MOS: 12, 17, 29, 41, 53
name: schism, 12&17
wedgie: <<1 -8 -2 -15 -6 18||
mapping: [<1 2 -1 2|, <0 -1 8 2|]
7 limit poptimal generator: 27/65
9 limit poptimal generator: 22/53
TM basis: {64/63, 360/343}
MOS: 12, 17, 29, 41, 53
name: grackle, 65&77
wedgie: <<1 -8 -26 -15 -44 -38||
mapping: [<1 2 -1 -8|, <0 -1 8 26|]
7 limit poptimal generator: 170/409
9 limit poptimal generator: 133/320
TM basis: {126/125, 32805/32768}
MOS: 12, 17, 29, 41, 53, 65, 77, 89
11 limit
name: garibaldi, 41&53
wedgie: <<1 -8 -14 23 -15 -25 33 -10 81 113||
mapping: [<1 2 -1 -3 13|, <0 -1 8 14 -23|]
poptimal generator: 95/229
TM basis: {225/224, 385/384, 2200/2187}
MOS: 12, 17, 29, 41, 53, 94
name: garybald, 29&41
wedgie: <<1 -8 -14 -18 -15 -25 -32 -10 -14 -2||
mapping: [<1 2 -1 -3 -4|, <0 -1 8 14 18|]
poptimal generator: 63/152
TM basis: {100/99, 225/224, 245/242}
MOS: 12, 17, 29, 41, 70
Message: 11421 - Contents - Hide Contents Date: Sat, 17 Jul 2004 13:02:12 Subject: Re: names and definitions: schismic From: Herman Miller Graham Breed wrote:>> name: garibaldi, 41&53 >> wedgie: <<1 -8 -14 -15 -25 -10|| >> mapping: [<1 2 -1 -3|, <0 -1 8 14|] >> 7&9 limit copoptimal generator: 39/94 >> TM basis: {225/224, 3125/3087} >> MOS: 12, 17, 29, 41, 53 > >> This should be "schismic". It's consistent with all those ETs except 17 > (which isn't 7-limit consistent in itself).I agree; "schismic" is essentially a 5-limit name (based on tempering out the [-15 8 1> "schisma"), so the 7-limit extension (if any) shouldn't be more complex than it needs to be.>> name: schism, 12&17 >> wedgie: <<1 -8 -2 -15 -6 18|| >> mapping: [<1 2 -1 2|, <0 -1 8 2|] >> 7 limit poptimal generator: 27/65 >> 9 limit poptimal generator: 22/53 >> TM basis: {64/63, 360/343} >> MOS: 12, 17, 29, 41, 53 > >> That name's confusingly similar to "schismic". It also looks like a > white elephant. And certainly don't call it "12&17" because this is > ambiguous with the 7/9-limit optimal mapping of 17-equal, which happens > to give an all-round better temperament in this case.This looks like a strange hybrid of schismic and dominant; something you might use if you don't have enough notes for schismic, but you're willing to substitute a 16/9 for a 7/4. It would be nice to see some error values for comparison purposes.
Message: 11422 - Contents - Hide Contents Date: Sat, 17 Jul 2004 23:09:37 Subject: Re: names and definitions: schismic From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> We've *been* changing established names; I don't think 7-limit > schismic was ever as established as some Paul wants to deep-six. > Moreover, sticking to a consistent scheme means we have a better idea > what the name means; in this case, that the name in the higher limit > has a tuning in accord with the lower limit.A compromise solution would be not to call anything in the 7-limit "schismic". "Groven" or "Helmholtz" are possible names; they did not have the 7-limit in mind, but the 1/7 to 1/10 schisma range suggested by 188&171 seems to accord pretty well with their 1/8 and 1/9 schisma tunings. However I suppose one could claim that 118&171 is the wrong name, since 118&224 is more like the extreme ranges.
Message: 11423 - Contents - Hide Contents Date: Sat, 17 Jul 2004 02:26:03 Subject: Atomic temperament as a basis for a valhalla-level notation From: Gene Ward Smith Atomic temperament, in 2460-et, has a period of exactly 100 cents and a generator of size one step of 615-equal, a type of schisma. The lowest Graham complexity for the 23-limit is given by [<12 19 28 34 42 43 47 50 52|, <0 1 -7 -16 -25 72 105 50 117|] so this I propose as the official 23-limit atomic mapping. 2460 is a strong temperament up to the 27-odd-limit, and so makes a good basis for defining atomic. Since 205 schismas make up 400 cents in this tuning, we can obtain alternate mappings by raising or lowering the generator steps by 205. We want -25 schismas for 11 at one end, and +50 schismas for 19 at the other, which sticks us with this mapping if we want minimal Graham complexity. 50+88 is 138, less than 117+25=142, so we might bring 117 down to -88; however if we do that then 105+88 is 193, whereas bringing it down to -100 leaves us with 100+88=188, and clearly leaving 117 as it is means we have to leave 72 and 105 alone also. An atomic notation would, I presume, have twelve nominals dividing the octave into twelve parts exactly. Other symbols would be defined in terms of schismas and semitones. One curious feature would be that in the 5-limit, with the atom as a comma, it would be a very, very, very, very precise notation for the 5-limit. One question I have is how, and how easily, 2460 and atomic could be notated in Sagittal.
Message: 11424 - Contents - Hide Contents Date: Sat, 17 Jul 2004 08:17:02 Subject: Re: names and definitions: meantone From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> > wrote: > > re: > Frame Index for Tuning Dictionary * [with cont.] (Wayb.) > meantoneIt doesn't seem to be up.
11000 11050 11100 11150 11200 11250 11300 11350 11400 11450 11500
11400 - 11425 -