Tuning-Math Archive Section 6: 5750

Message: 5750 - Contents - Hide Contents Date: Sat, 14 Dec 2002 05:28:39 Subject: Re: Relative complexity and scale construction From: Carl Lumma

>> >raham complexity tells me the minimum number of notes of the >> temperament I need to play all the identities in question. >> Does relative complexity? >

>Graham complexity along both generators, and thier product, >might be what we need. We could attempt to minimize the product.

Sounds right. I wonder how this follows relative and/or geometric complexity... I should point out that I *don't know* the difference between this notes measure, and geometric complexity measures -- I'm asking. What is it exactly that we're trying to measure about temperaments, from a music-theoretical POV? -Carl

Message: 5751 - Contents - Hide Contents Date: Sat, 14 Dec 2002 05:29:12 Subject: Re: Relative complexity and scale construction From: Carl Lumma

>>>> >erhaps you don't have enough uvs to close a block, but you're >>>> certainly on you're way. No? >>>

>>> If you think 5-limit JI is on the way to being a block. >>

>> Maybe Paul can shed some light on this, when he's feeling >> better. >> >> -Carl > >i can't. >

>now answer the question, carl! What question? -Carl

Message: 5753 - Contents - Hide Contents Date: Sat, 14 Dec 2002 09:28:28 Subject: Re: Relative complexity and scale construction From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:

> Sounds right. I wonder how this follows relative and/or > geometric complexity...

It's not clear to me how to really get a Graham generalization yet. A simple function computing numbers of complete p-limit chords in a parallelogram would be good. I looked at 11-limit chords leaving off 9 for the "Wonder" temperament I just uploaded a graph for; any of the pairs of generators <5, 15/14>, <5, 7/6>, <5, 4/3> <15/14, 4/3>, <15/14, 12/7> will work, and give the minimal value of 9 for the product. So, for instance, the block based on generators of <33/32, 3087/3025> works, giving a Marvel-tempered block of 30 notes which should have (3-2)*(10-2) = 8 of these major chords. This doesn't seem very satisfactory; better would be something which takes a Fokker block and gives a quick answer, which I'll think about.

Message: 5756 - Contents - Hide Contents Date: Sun, 15 Dec 2002 14:38:05 Subject: Re: Relative complexity and scale construction From: Graham Breed Me:

>> So you can then add planar temperaments to that graph of size against >> accuracy. They'll appear for high accuracies with a large but not >> absurdly large number of notes. You can also say the number of notes >> beyond which a planar temperament is equivalent to a linear temperament >> that's consistent with it. Gene:

> Why a large number of notes? The Pauline tempered Duodene only has 12, and I was wrong about then all being covered by tetrads. You can cut the corners off, and get the following scale:

Oh, I was assuming you'd need a lot of notes to get an accurate temperament that's still simpler than JI.

> ! pship.scl > ! > Pauline (225/224) tempered 10 note scale

Why is this still being called the Pauline temperament? I showed that it isn't the temperament Pauline gave.

> This has 4 major and 4 minor triads, 2 major and 2 minor tetrads, > 2 supermajor and 2 subminor triads, one each of what I call supermajor and subminor tetrads (1-9/7-3/2-9/5 and 1-7/6-3/2-5/3), and > a 1-7/6-7/5-5/3 dimininished 7th chord. I wouldn't call this a lot of notes, but the tempering is very much in evidence.

It isn't that many notes, but it isn't that accurate either. It's simple as planar temperaments go. I make it equivalent to h12&h60 beyond 33 notes. Or 34 notes if you lop the corners off. 36 in practice as h12&h60 goes up in steps of 12 notes. Graham

Message: 5757 - Contents - Hide Contents Date: Sun, 15 Dec 2002 18:56:26 Subject: Re: Relative complexity and scale construction From: Graham Breed I wrote:

> It isn't that many notes, but it isn't that accurate either. It's > simple as planar temperaments go. I make it equivalent to h12&h60 > beyond 33 notes. Or 34 notes if you lop the corners off. 36 in > practice as h12&h60 goes up in steps of 12 notes.

Well, that's wrong because my simplification of planar temperaments is wrong. It looks like you get more chords per note if you have equal numbers of steps in each generator direction, instead of holding one at its smallest value and expanding the scale in only one direction. Here's a 10 note scale with 2 utonal and 2 otonal 7-limit tetrads when you temper out 225:224 D# A# E# E B F# C# C G D Expanding in one direction, you get a 13 note scale with 3 of each tetrad D# A# E# B# E B F# C# G# C G D A And the next in the pattern is a 16 note scale with 4 of each tetrad G# D# A# E# B# A E B F# C# G# F C G D A But there's also this 14 note scale that has 4 of each tetrad D# A# E# E B F# C# C G D A Ab Eb Bb So it can be done with 14 notes when the theory predicts 16 (and the even cruder theory 18) 18 notes can give 6 of each tetrad G# D# A# E# A E B F# C# F C G D A Db Ab Eb Bb This is the point where we get extra tetrads with meantone. Then there's a 23 note scale with 9 of each tetrad B# Fx Cx Gx C# G# D# A# E# A E B F# C# F C G D A Db Ab Eb Bb By my original formula, it would take 30 notes to get this many tetrads. So, the complexity of a planar temperament is (i, j) and the scale is (I, J). The number of notes is then I*J. The number of complete chords of each type is (I-i)*(J-j). This still slightly overestimates the number of notes, because some notes in the rectangle won't be used in any chords, but that's only a minor correction. I'm going to set I-i = J-j = n. Then the number of chords is n**2 (n squared) and the number of notes is ij + n(i+j) + n**2. I've worked out that the number of notes where the number of chords in an equal and planar temperament are the same is ((c-ij)/(i+j))**2 + c where c is the complexity of the linear temperament and (i, j) is the complexity of the planar temperament. Using that formula, the 225:224 planar temperament doesn't become equivalent to the h12&h60 linear temperament until they both have 49 notes. So the planar temperament holds it's own for larger scales than I said before. Ennealimmal (which is much more accurate) becomes simpler around 60 notes. It also takes more notes before the planar temperament becomes simpler than JI. This scale has 12 notes and 3 of each tetrad, a feat that requires 13 notes for a 225:224 tempered scale. You'll really need to view source and turn off word wrapping for this to come out right. C# / \ B#------Fx------Cx \ / / \ \ / \ A-/---\-E / \ / \ / / D#------A#\ / \ \ / / \ F-----\-C-/-----G \ / F# You can also get 4 of each tetrad from 14 notes by filling in the two 5-limit triads in the middle (is that a stellated hexany?) So the planar temperament rules where you have more than 14 up to around 49 or 60 notes -- a large, but not absurdly large number of notes to me. I may still be wrong on the details, but I think I'm getting closer Graham

Message: 5758 - Contents - Hide Contents Date: Sun, 15 Dec 2002 02:14:44 Subject: Re: Relative complexity and scale construction From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

> So you can then add planar temperaments to that graph of size against > accuracy. They'll appear for high accuracies with a large but not > absurdly large number of notes. You can also say the number of notes > beyond which a planar temperament is equivalent to a linear temperament > that's consistent with it.

Why a large number of notes? The Pauline tempered Duodene only has 12, and I was wrong about then all being covered by tetrads. You can cut the corners off, and get the following scale: ! pship.scl ! Pauline (225/224) tempered 10 note scale 10 ! 268.145800 384.152140 500.158479 583.835182 699.841521 768.304279 884.310618 967.987322 1083.993661 2/1 This has 4 major and 4 minor triads, 2 major and 2 minor tetrads, 2 supermajor and 2 subminor triads, one each of what I call supermajor and subminor tetrads (1-9/7-3/2-9/5 and 1-7/6-3/2-5/3), and a 1-7/6-7/5-5/3 dimininished 7th chord. I wouldn't call this a lot of notes, but the tempering is very much in evidence.

Message: 5759 - Contents - Hide Contents Date: Sun, 15 Dec 2002 03:45:34 Subject: Re: Relative complexity and scale construction From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:

>>>>> Perhaps you don't have enough uvs to close a block, but you're >>>>> certainly on you're way. No? >>>>

>>>> If you think 5-limit JI is on the way to being a block. >>>

>>> Maybe Paul can shed some light on this, when he's feeling >>> better. >>> >>> -Carl >> >> i can't. >>

>> now answer the question, carl! > > What question?

do you think 5-limit JI is on the way to being a block?

Message: 5761 - Contents - Hide Contents Date: Sun, 15 Dec 2002 06:54:26 Subject: Re: Relative complexity and scale construction From: Carl Lumma

>>> > >>> now answer the question, carl! >> >> What question? >

> do you think 5-limit JI is on the way to being a block? No.

The lattices Gene are posting contain pitches outside of 5-limit JI. -C.

Message: 5763 - Contents - Hide Contents Date: Mon, 16 Dec 2002 12:26:43 Subject: Re: Relative complexity and scale construction From: Graham Breed Gene Ward Smith wrote:

> The 225/224 planar is pretty accurate; you can use 72-et or for more accuracy even 228-et. Why do you say it isn't accurate?

I was comparing it with other 7-limit planar temperaments, not equal temperaments. It's only a marginal improvement on Miracle, whereas some planar temperaments can do a lot better (such as those Schismic-with-explicit-schisma notations). But this isn't important. You can call it "pretty accurate" if you like and say that planar temperaments occupy the "pretty accurate" zone.

> I don't have a name listed for this temperament yet, but Duodecimal ought to do.

Why does it need a name? I didn't think anybody was writing in it. I've a feeling it came up on Usenet earlier in the year.

> It is equivalent in the sense that if you use 72-et or > 228-et as above, they are Duodecimal compatible; you can describe it as 225/224 plus the Pythagorean comma, though its actual TM reduced basis is <225/224, 250047/250000>. If you use the mapping > [[12,19,28,34],[0,0,-1,-2]] for it, you get generators of the Pythagorean minor second (256/243) and the 81/80~126/25 comma; this not neccessarily the best setup for 225/224 alone, it seems to me.

It's equivalent in the sense that it tempers out 225:224, and gets within 98% of the accuracy of 225:224.

> You can also add commas to get, for instance, Catakleismic or Miracle, so this isn't uniquely attached to 225/224, though the connection is close.

225:224 planar's worst error is 79.4% of Miracle's. You can call that equivalent if you like. In which case the equivalence occurs beyond 18 notes. Catakleismic is more complex and less accurate, so what's the point? h31&h63 gets to within nearly 85%, and has a complexity of 23. But as h12&h60's so much more accurate, and only slightly more complex (24) I went with that. Graham

Message: 5764 - Contents - Hide Contents Date: Mon, 16 Dec 2002 18:08:06 Subject: Re: A common notation for JI and ETs From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...> wrote:

> At 12:44 PM 12/12/2002 -0800, George Secor wrote: >> [DK:]

>>> But I will say: Now that you've centered those right triangles, the filled >>> ones look too much like concave flags. >> [GS:]

>> I also concluded that they're all too hard to read -- the triangles are >> too small. I made them a little larger and discarded the filled ones. >> I put these in the same file with the previous ones, so we can make a >> comparison with what I had: >> >> Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/Schisma.gif >> ... >> I don't know how quickly these could eventually be read, but I think >> the meanings are clear enough. > [DK:]

> I think they suffer from the problem that the size of the modification is > visually _way_ out of proportion with the size (and direction) of the > alteration in pitch.

Well, you're right. I had been experimenting with small triangles on the assumption that small straight flags wouldn't be readable. However, I tried some small straight flags and think I have something that works -- see second staff that I have added to this file: Yahoo groups: /tuning- * [with cont.] math/files/secor/notation/Schisma.gif I have these along with a bunch of other symbols in order of size for comparison, including the new 3-flag 7:17 (or 7+17') comma. The 5' symbols are the same width as the 19-comma symbol, but they are thinner. There are some flag combinations that I didn't label, but you should be able to figure out what they mean. Here is a conclusion that I've reached. Beginning with the 13' symbol I've been calling the notational commas dieses, since they're the sum of two commas. Now that we're notating the 125:128 diesis, I think that anything larger than the 5:11 comma (|( also ought to be called a diesis rather than a comma. This would mean that any symbol having two "large" comma flags would be classified as a diesis, with a large comma flag being anything larger than the 17 comma (there being a rather large size difference between the 17 and 23 commas). Another basis for establishing a boundary between large and small commas (which agrees with this) goes back to the original definition of comma: the difference in size between the two largest steps in a diatonic tetrachord. About the smallest that these steps can get is in Ptolemy's diatonic hemiolon, where they are 9:10 and 10:11, with a comma of 99:100 (~17.399 cents). The next smallest superparticular pair are 11:12 and 10:11, making a lesser comma of 120:121 (~14.367 cents, which is not only significantly smaller than 1deg72 (~16.667 cents), but also closer in size to 1deg94 (~12.766 cents), in which system both the 5 and 7 commas are 2deg (and 120:121 is only slightly more than one-half the size of a 7 comma.) So I think this is getting a bit small to be considered a comma in the original sense. What we really need is a separate name for commas smaller than ~1deg72, and I don't think "kleisma" fills the bill. Does anyone out there have any ideas for a categorical name for commas less than ~16 cents? --George

Message: 5765 - Contents - Hide Contents Date: Mon, 16 Dec 2002 23:15:01 Subject: Re: Relative complexity and scale construction From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>" <clumma@y...> wrote:

>>>> >>>> now answer the question, carl! >>> >>> What question? >>

>> do you think 5-limit JI is on the way to being a block? > > No. >

> The lattices Gene are posting contain pitches > outside of 5-limit JI. > > -C.

but they're planar arrangements of pitches, with like vectors corresponding to like intervals. so there's no essential difference . . .

Message: 5766 - Contents - Hide Contents Date: Mon, 16 Dec 2002 02:56:18 Subject: Re: Relative complexity and scale construction From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:

>> ! pship.scl >> ! >> Pauline (225/224) tempered 10 note scale >

> Why is this still being called the Pauline temperament? I showed that > it isn't the temperament Pauline gave.

Sorry, I missed that. This is a very important temperament, and needs a name.

>> This has 4 major and 4 minor triads, 2 major and 2 minor tetrads, >> 2 supermajor and 2 subminor triads, one each of what I call supermajor and subminor tetrads (1-9/7-3/2-9/5 and 1-7/6-3/2-5/3), and >> a 1-7/6-7/5-5/3 dimininished 7th chord. I wouldn't call this a lot of notes, but the tempering is very much in evidence.

> It isn't that many notes, but it isn't that accurate either.

The 225/224 planar is pretty accurate; you can use 72-et or for more accuracy even 228-et. Why do you say it isn't accurate? 72-et: [-1.955001, -2.980380, -2.159239] 228-et: [-1.955001, -2.103187, -.404853] It's

> simple as planar temperaments go. I make it equivalent to h12&h60 > beyond 33 notes.

I don't have a name listed for this temperament yet, but Duodecimal ought to do. It is equivalent in the sense that if you use 72-et or 228-et as above, they are Duodecimal compatible; you can describe it as 225/224 plus the Pythagorean comma, though its actual TM reduced basis is <225/224, 250047/250000>. If you use the mapping [[12,19,28,34],[0,0,-1,-2]] for it, you get generators of the Pythagorean minor second (256/243) and the 81/80~126/25 comma; this not neccessarily the best setup for 225/224 alone, it seems to me. You can also add commas to get, for instance, Catakleismic or Miracle, so this isn't uniquely attached to 225/224, though the connection is close. If we used [331, 524, 768, 929] [-2.257116, -2.023683, -.850075] or [631, 999, 1464, 1771] [-2.113480, -2.161574, -.838584] as mappings, on the grounds they are near the rms optimum, of course the Pythagorean comma would not be a factor.

Message: 5767 - Contents - Hide Contents Date: Mon, 16 Dec 2002 23:17:27 Subject: Re: A common notation for JI and ET's From: M. Schulter Hello, everyone, and for my first post on tuning-math, I'd like to propose a new sagittal symbol that might illustrate an approach to the notation of certain tempered systems not based on an equal division of an interval such as the octave and involving more than one chain of fifths. Before presenting my proposal, I should caution that I am still in my early beginner's stage of learning sagittal notation, and only within the last day or so realized that I hadn't been quite clear on the effect of comma signs in the region between an 11-diesis and a sharp or flat (apotome-less-comma rather than diesis-plus-comma). Happily, reading again the relevant portions of the draft by George and Dave for _Xenharmonikon_ 18 brought this nuance to my attention, so that I can at least construct my suggested new symbol correctly. Also, I would caution that especially given my inexperience, this article is not unlikely to include its share of obvious beginner's errors, and that I warmly welcome any corrections. As I discuss in some longer articles which I might want to proofread and possibly revise accordingly, one approach to the kind of tempered system I here discuss is a "quasi-JI" transcription which shows precise or approximate deviations from Pythagorean intonation, including significant commas which are dispersed in the system. Consider, for example, this diatonic scale and a possible "quasi-JI" representation, with octave numbers (C4=middle C) appearing before the note names and sagittal signs: 4C 4D|( 4E)|( 4F!( 4G 4A|( 4B)|( 5C 1/1 ~44/39 ~14/11 ~4/3 ~3/2 ~22/13 ~21/11 2/1 0 208.19 416.38 495.90 704.10 912.29 1120.48 1200 Mapping a regular temperament to a JI notation, like mapping a three-dimensional globe on a 2-D surface, inevitably involves some distortion. Thus the notation for this scale based on a regular temperament with fifths around 704.096 cents (the Wilson/Pepper "Noble Fifth") accurately represents the sizes of many intervals, including the whole-tones near 39:44 -- 8:9 + 351:352 or |( -- and the diatonic semitones near 21:22 -- 243:256 less 891:896 or )|(. From a vertical perspective, the notation here also accurately suggests that the major thirds are very close to 11:14, or 64:81 plus 891:896 -- e.g. 4C-4E)|( or 4F!(-4A|(. However, certain small vertical and melodic anomalies occur because the notation does not actually show the fractional commas by which each fifth or fourth is tempered -- not quite half of a 351:352, or 1/4 of an 891:896. Thus 4C-4G and 4E)|(-4B)|( are arbitrarily shown as pure, while 4C-4F!( and 4E)|(-4A are shown as tempered by about a full 351:352 (~4.925 cents), although in reality all of these intervals are tempered by an identical of amount of ~2.141 cents. Apart from the issue of such unavoidable distortions, the main drawback of this "quasi-JI" style of notation is that it is in tension with the usual sagittal principle of notating routine intervals along a single chain of fifths with routine symbols. Here 4C-4E)|( indeed suggests that the size of this regular major third is very close to a pure 11:14 (~417.51 cents), but the extra symbols might distract from the basic fact that it is the _usual_ major third to be found simply by pressing the keys 4C-4E. Of course, for many linear temperaments based on a single chain of fifths, one might simply pick a nearby equal temperament whose standard symbol set fits. However, I here consider the system called Peppermint 24, with two 12-note chains of fifths (~704.096 cents) at a distance of ~58.680 cents, the "quasi-diesis" which when added to the regular major second yields a pure 6:7 minor third (~266.871 cents). My solution for communicating some intonational information about the system while following the usual conventions of sagittal notation within a chain of fifths is to use a symbol rather accurately defining the size of the quasi-diesis as around 88:91 (11-diesis at 32:33 plus 11:13 comma at 351:352, ~58.036 cents) or 117:121 (32:33 plus 363:364 at about 4.763 cents, ~58.198 cents). The ratios of 88:91 and 117:121, like 351:352 and 363:364, differ by the harmonisma at 10647:10648 (~0.163 cents). In JI, 88:91 defines the difference for example between 39:44 and 6:7, 7:11 and 8:13, or 13:22 and 4:7; the very slightly larger 117:121 defines the difference between 11:13 and 9:11, or 22:39 and 6:11. In a tempered system such as Peppermint 24, these 88:91 or 117:121 relationships are closely approximated, with the first interval of each of the above pairs as a regular interval along the chain of fifths, and the second as an interval realized by the addition of a 58.68-cent quasi-diesis. To show the approximate size of the quasi-diesis, and thus to imply this type of intonational structure, I propose the following sign showing a modification of 32:33 or /|\ plus 351:352 or |(. /|\( I am tempted to call this an "intonational signature," since the sagittal symbol looks a bit like the usual 11-diesis arrow plus the sign "(" or "C" associated metrically with "common time" (4/4). Here the sign could be said to stand for "common temperament," a term applied to 24-note systems with two chains of fifths at around 704 cents or so spaced so as to optimize certain ratios, and especially those of 2-3-7-9 (e.g. 6:7, 7:9, 4:7). In such a scheme, the approximate 88:91 diesis or /|\( serves at once as the 11-diesis (e.g. 11:12 vs. 8:9) or /|\ in JI, and the septimal or Archytas semitone at 27:28 (~62.96 cents), or )/|\( as it might be written in JI (32:33 plus 891:896). The latter interval marks the difference, for example, between 8:9 and 6:7, or 7:9 and 3:4. Following this approach, we might notate some characteristic progressions as follows, with JI approximations given below the examples: 5E\!!/ 5F 5E\!!/ 5F 4B\!!/ 5C 4B\!!/ 5C 4G 4F 4E 4F 4G 4F/|\( 4F or 4G!!!) 4F 4C 4D\!/( 4E 4D\!/( 4C 16/9 2/1 3/2 4/3 14/11 4/3 3/2 4/3 3/2 1/1 12/11 14/11 12/11 1/1 28/27 1/1 4A/|\( 4G/|\( 4B!!!) 4A!!!) 4A/|\( 4A 4B!!!) 4A 4G 4G/|\( 4G 4A!!!) 4G 4A 4G 4A 4D/|\( 4C/|\( 4E!!!) 4D!!!) 4E/|\( 4D 4F!) 4D 4C 4C/|\( or 4C 4D!!!) 4C 4D or 4C 4D 7/4 14/9 7/4 27/16 3/2 14/9 3/2 27/16 7/6 28/27 21/16 9/8 1/1 28/27 1/1 9/8 From the alternative notations for some of these examples, it can be seen that in this kind of tempered system, the "limma complement" of the 88:91 or /|\( is the representation of the 63:64 Archytas comma or 7-comma, !) -- thus D/|\( is equivalent to E!!!) at a 6:7 above C. One "refinement" -- if that is the right word -- to go along with this innovation is an "apotome complement" of sorts for /|\(. On this point, I will propose a poetic liberty with the sagittal system. Just as /|\( closely approximates the _absolute_ size of the quasi-diesis in a tempered system like Peppermint 24, so its proposed apotome complement also represents an absolute size close to that obtaining in such a system, where the apotome /||\ or \!!/ has a size of around 13:14 or ~128.30 cents (in Peppermint 24, actually ~128.67 cents). Thus this "system-specific" apotome complement is equal to about 169:176 (~70.262 cents), the difference between 13:14 and 88:91, and also the sum of 26:27 (the 13' diesis, ~65.337 cents) and 351:352. I suggest this symbol for the 169:176, or also 121:126 (~70.100 cents) as the difference of 13:14 and 117:121, or sum of 26:27 and 363:364; (|\( This is the usual 13' diesis sign plus a 351:352. While its use as an apotome complement, at least in relation to 88:91 or 117:121, is highly "system specific," the 169:176-like size could also represent the Pythagorean "tricomma" of ~70.380 cents or (531441:524288)^3, a ratio of 150094635296999121:144115188075855872. The schisma between this tricomma and 169:176 is 25365993365192851449:25364273101350633472 or ~0.117 cents, well within usual sagittal tolerances. A nice touch here, at least for those of us with a taste for this kind of thing, is that both the 88:91 sign /|\( and its system-specific apotome complement at around 169:176 of (|\( share a 351:352 comma as a distinguishing mark, also the intonational signature for a "common temperament" of this sort. In Peppermint 24, the quasi-diesis and its apotome complement have sizes of around 58.680 cents and 69.990 cents, quite close to the advertised rational representations. Using (|\(, we can conveniently write a progression like the following using single-symbol notation, with a double-symbol version also shown: 5E\!!/ 5E(!\( 5Eb 5Eb/|\( 5C/|\( 4B(!\( 5C/|\( 5Bb/|\( 4B\!!/ 4B(!\( 4Bb 4Bb/|\( 4F/|\( 4E(!\( or 4F/|\( 4Eb/|\( 27/14 2/1 27/16 3/2 81/56 3/2 9/8 1/1 I must admit that the second style of notation looks much more intuitive to me: it quickly reveals the location of regular whole-tone and near-27:28 steps -- e.g. 4F/|\(-4Eb/|\( and 4Bb-4Bb/|\(. This consideration might motivate the use of some double-symbol notation in pieces where the single-symbol style is generally preferred. If we wish to stick with a single-symbol style, another solution is also possible: 5E\!!/ 5F!!!) 5D!!!) 5C!!!) 4B\!!/ 5C!!!) 4G!!!) 4F!!!) Here 4F!!!) -- or 4Fb!) in double-symbol style -- is the system's Archytas comma lower than the diminished fourth 4C-4F\!!/ or 4C-4Fb at ~367.235 cents, yielding an interval of about 346.393 cents, also written conveniently in double-symbol notation as 4C-4Eb/|\(. Thus while a regular diminished fourth approximates 14:17 (~365.825 cents), the diminished fourth less Archytas comma (or minor third plus quasi-diesis) approximates 9:11 (~347.408 cents). This notational solution involves more "remote" accidentals like F!!!), but shares with the double-symbol example the advantage of readily communicating usual whole-tone steps and approximate 27:28 steps. Getting back to my main proposal for the symbol /|\( to show the "quasi-diesis" of around 88:91 in a tempered system with two chains of fifths at about 704 cents, this approach seeks to analyze some important intonational and harmonic relationships in the system and then give them a "shorthand" signature. For example, /|\( signals that the near-88:91 is serving as a representation of both 32:33 and 27:28, and that regular major and minor thirds are rather close to 11:14 and 11:13 or 28:33, regular whole-tones to 39:44, and diatonic semitones to 21:22. Also implied are "39:44 + 88:91 = 6:7" and "7:11 + 88:91 = 8:13" -- and so forth. As compared to a "quasi-JI" style of transcription showing approximate deviations from Pythagorean intonation, this solution follows the usual sagittal approach of regarding the reference intervals as those following the chain of fifths -- thus 4C-4E for a regular major third, rather than 4C-4E)|( to show that this ~11:14 interval is ~891:896 wider than a Pythagorean 64:81 (~407.820 cents). As compared to the approach of attempting to map a non-equal system to some very large equal temperament, the "intonational signature" approach reflects the structure of two regular chains of fifths placed an arbitrary distance to produce certain ratios of 2-3-7-9-11-13 complementing the ratios represented within each chain (e.g. 22:26:33 and 14:17:21). The resulting sagittal notation looks rather like a style of keyboard notation showing simply which note is played on which keyboard, with a symbol like the asterisk (*) used to show a note on the upper keyboad raised by a quasi-diesis, e.g. F*4-Bb4 for a near-7:9 third, written as 4F/|\(-4B\!!/ or 4F/|\(-4Bb in this sagittal approach. One could also write 4G!!!)-4B\!!/ to show the represented 7:9 relationship. In view of the accustomed subject of this thread, "A Common Notation for JI and ET's," I might seem in a curious position devoting my first post to a "not-so-common" sagittal approach to a nonequal tempered system based on two chains of fifths at an arbitrary distance. Such is what can happen when a good system falls into certain not so experienced hands. Most appreciatively, Margo Schulter mschulter@xxxxx.xxx

Message: 5768 - Contents - Hide Contents Date: Mon, 16 Dec 2002 03:37:05 Subject: Re: Relative complexity and scale construction From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <genewardsmith@j...>" <genewardsmith@j...> wrote:

> I don't have a name listed for this temperament yet, but Duodecimal ought to do. It is equivalent in the sense that if you use 72-et or > 228-et as above, they are Duodecimal compatible; you can describe it as 225/224 plus the Pythagorean comma, though its actual TM reduced basis is <225/224, 250047/250000>. If you use the mapping > [[12,19,28,34],[0,0,-1,-2]] for it, you get generators of the Pythagorean minor second (256/243) and the 81/80~126/25 comma; this not neccessarily the best setup for 225/224 alone, it seems to me.

A closely related situation happens with the 126/125 temperament, which is covered by 108 and 120, and done to perfection by [336, 532, 781, 943]. We can tune it using the linear temperament of 126/125 and the Pythagorean comma, for which a mapping is [[12,19,28,34], [0,0,-1,-3]], with generators a Pythagorean minor second and a 81/80~225/224 comma.

Message: 5770 - Contents - Hide Contents Date: Tue, 17 Dec 2002 21:51:55 Subject: Re: A common notation for JI and ET's From: gdsecor --- In tuning-math@xxxxxxxxxxx.xxxx "M. Schulter" <mschulter@m...> wrote:

> Hello, everyone, and for my first post on tuning-math, I'd like to > propose a new sagittal symbol that might illustrate an approach to the > notation of certain tempered systems not based on an equal division of > an interval such as the octave and involving more than one chain of > fifths. > > Before presenting my proposal, I should caution that I am still in my > early beginner's stage of learning sagittal notation, and only within > the last day or so realized that I hadn't been quite clear on the > effect of comma signs in the region between an 11-diesis and a sharp > or flat (apotome-less-comma rather than diesis-plus-comma). Happily, > reading again the relevant portions of the draft by George and Dave > for _Xenharmonikon_ 18 brought this nuance to my attention, so that I > can at least construct my suggested new symbol correctly.

I'll have to express my apologies to just about anyone else who might be reading this (and who hasn't seen the paper), that this probably isn't going to mean a whole lot to you, and hopefully I'll have a presentation on this soon. Margo asked me a question in a private communication about whether the new notation would provide a way to do such-and-such, and my explanation wouldn't have been very meaningful without sending along the latest draft of the paper to her. An apology is also extended to you, Margo, since I haven't yet responded to a message from last Thursday -- after looking at what you had to say, I started getting bogged down in the mathematics and couldn't find a quick answer.

> Also, I would caution that especially given my inexperience, this > article is not unlikely to include its share of obvious beginner's > errors, and that I warmly welcome any corrections.

We're all beginners with this notation, and we're all going to be learning from one another how best to use it for various tunings.

> As I discuss in some longer articles which I might want to proofread > and possibly revise accordingly, one approach to the kind of tempered > system I here discuss is a "quasi-JI" transcription which shows > precise or approximate deviations from Pythagorean intonation, > including significant commas which are dispersed in the system. > Consider, for example, this diatonic scale and a possible "quasi-JI" > representation, with octave numbers (C4=middle C) appearing before the > note names and sagittal signs: > > 4C 4D|( 4E)|( 4F!( 4G 4A|( 4B)|( 5C > 1/1 ~44/39 ~14/11 ~4/3 ~3/2 ~22/13 ~21/11 2/1 > 0 208.19 416.38 495.90 704.10 912.29 1120.48 1200

This is very close to 46-ET, which is notated very simply using the naturals (as you also noted): deg46: 8 16 19 27 35 43 46 C D E F G A B C 0 208.70 417.39 495.65 704.35 913.04 1121.74 1200 Just as meantone can be notated with conventional notation, any temperament with a regular chain of fifths can be notated similarly. If you need fractional alteration symbols, the 46-ET notation should do the job. For example, if you needed to notate the tone 11 fifths along the chain ~545.10c or ~11/8), instead of E# or E/||\ you could use the symbol for 21deg46, or F/|\. (But after figuring out something else below and reading this again, I might now do this differently -- just a precaution -- as I said, I'm still learning.) Since the symbols for a 2-digit ET are inevitably going to be much simpler than the ones for rational intervals (or JI), it would be much easier to read the ET notation, *provided* that you have specified how the symbols are being used and that anyone who is to perform your music reads and understands your explanation.

> Mapping a regular temperament to a JI notation, like mapping a > three-dimensional globe on a 2-D surface, inevitably involves some > distortion.

That's why I don't recommend doing that -- it can get very complicated. But I do recommend mapping JI or a regular temperament to an ET notation, as long as you define your symbols consistently. The rational symbols that we have are division-independent, i.e., they can be used without mapping to any ET, but if one modulates too far there won't be enough rational symbols to handle the requirements, and even if there were, they might not be very meaningful. So an ET mapping is the only practical way. One of the purposes of this notation is to keep the symbols as simple as possible, with much of the complexity of a tuning being put in an explanation of symbols (or note-symbol combinations) that should accompany a composition. If a regular temperament uses a generator other than an approximation of a fifth, then an ET mapping is almost certainly a foregone conclusion. For example, the Miracle temperament would be done by mapping it to 72-ET and using that division's standard symbol set.

> ... Apart from the issue of such unavoidable distortions, the main > drawback of this "quasi-JI" style of notation is that it is in tension > with the usual sagittal principle of notating routine intervals along > a single chain of fifths with routine symbols. Here 4C-4E)|( indeed > suggests that the size of this regular major third is very close to a > pure 11:14 (~417.51 cents), but the extra symbols might distract from > the basic fact that it is the _usual_ major third to be found simply > by pressing the keys 4C-4E.

Keyboard, guitar, or other polyphonically scored music is one place where it's essential to keep the symbols simple; otherwise it would be very difficult to read. For the one-note-at-a-time flexible-pitch instruments precise pitch information is necessary. As to whether it is more useful to have it appearing with each note (per Johnny Reinhard) or placed in an accompanying explanation (as is required for sagittal symbols) is dependent on the variables of a particular situation -- the tuning, the players, their level and type of experience, the instruments (off-the-shelf vs. specially built or retuned), so we have decided to supplement the sagittal symbols with Reinhard 1200-ET (cents) notation to accommodate either methodology (or a combination of the two).

> Of course, for many linear temperaments based on a single chain of > fifths, one might simply pick a nearby equal temperament whose > standard symbol set fits. However, I here consider the system called > Peppermint 24, with two 12-note chains of fifths (~704.096 cents) at a > distance of ~58.680 cents, the "quasi-diesis" which when added to the > regular major second yields a pure 6:7 minor third (~266.871 cents). > > My solution for communicating some intonational information about the > system while following the usual conventions of sagittal notation > within a chain of fifths is to use a symbol rather accurately defining > the size of the quasi-diesis as around 88:91 (11-diesis at 32:33 plus > 11:13 comma at 351:352, ~58.036 cents) or 117:121 (32:33 plus 363:364 > at about 4.763 cents, ~58.198 cents). The ratios of 88:91 and 117:121, > like 351:352 and 363:364, differ by the harmonisma at 10647:10648 > (~0.163 cents). > > In JI, 88:91 defines the difference for example between 39:44 and 6:7, > 7:11 and 8:13, or 13:22 and 4:7; the very slightly larger 117:121 > defines the difference between 11:13 and 9:11, or 22:39 and 6:11. > > In a tempered system such as Peppermint 24, these 88:91 or 117:121 > relationships are closely approximated, with the first interval of > each of the above pairs as a regular interval along the chain of > fifths, and the second as an interval realized by the addition of a > 58.68-cent quasi-diesis. > > To show the approximate size of the quasi-diesis, and thus to imply > this type of intonational structure, I propose the following sign > showing a modification of 32:33 or /|\ plus 351:352 or |(. > > /|\( > ... > One "refinement" -- if that is the right word -- to go along with this > innovation is an "apotome complement" of sorts for /|\(. On this > point, I will propose a poetic liberty with the sagittal system. > > Just as /|\( closely approximates the _absolute_ size of the > quasi-diesis in a tempered system like Peppermint 24, so its proposed > apotome complement also represents an absolute size close to that > obtaining in such a system, where the apotome /||\ or \!!/ has a size > of around 13:14 or ~128.30 cents (in Peppermint 24, actually ~128.67 > cents). > > Thus this "system-specific" apotome complement is equal to about > 169:176 (~70.262 cents), the difference between 13:14 and 88:91, and > also the sum of 26:27 (the 13' diesis, ~65.337 cents) and 351:352. > I suggest this symbol for the 169:176, or also 121:126 (~70.100 cents) > as the difference of 13:14 and 117:121, or sum of 26:27 and 363:364; > > (|\(

If I understand you correctly, you have: /|\( at around 58 cents, and (|\( at around 70 cents which are supposed to be apotome complements. But in order for this to be so, they must add up to an apotome, ~113.7 cents. So your usage of symbols does not seem to be internally consistent. You mentioned something like a "system-specific apotome complement". We tried a concept called "alternate complements" for some ET notations, but that was eventually dropped, because it was much simpler to have a one-to-one correspondence between a symbol and its complement without exception. We are very reluctant to define symbols with three flags unless there is a very good reason for doing so (as we just did for the 7:17 comma, but that was the first one). It's not necessary to define new symbols if you consider that it would still be necessary for you to describe your temperament apart from the symbols themselves. The 46-ET notation would actually do quite nicely in this instance, the standard symbol sequence being: 46: /| /|\ (|) ||\ /||\ Since you're not using ratios of 5 and since ratios of 7 are so prominent in Peppermint, you would probably want to replace the 5- comma symbols with the 7-comma ones: 46: |) /|\ (|) ||) /||\ These are simple symbols that everyone who used the sagittal notation would be familiar with. One problem that you might encounter with this is if you extended one chain of fifths to D# or D/||\ and wished to respell that as E(!), whereas you already have an E(!) in the other chain of fifths. Avoiding the respelling would sidestep the problem, but I don't know if that would be suitable for your purposes. But I see another way to avoid this problem, one that involves a quasi-217 mapping, and I believe that the end result would be preferable to the above. In the 46-mapped notation, equivalent pitches in your upper chain of fifths are D/|\ and E!!!). In 217 these equivalents are 47 and 48 degrees, respectively. The equivalent tones for ~14/11 in the 46 mapping are E and F(!), but in 217 these are 74 and 79 degrees, respectively, far enough off to indicate that the F(!) is not related to E in the same way as with the other equivalents. If we were to use the 217 symbol for 75deg217 as the equivalent of E (~14/11), then we would have F\!! or Fb|\ for this purpose; better yet, using the rational symbol for 14/11 -- F)!! ~ or Fb(| -- which would involve the same number of degrees in 217 (which confirms that its use here would be appropriate). So then the symbols for your two chains of fifths (single-symbol version, with "enharmonic" equivalents -- maybe not quite the right term) would be: E(!) B(!) F/|\ C/|\ G/|\ D/|\ A/|\ E/|\ B/|\ F/|||\ C/|||\ G/|||\ F!!!) C!!!) G!!!) D!!!) A!!!) E!!!) B!!!) F!) C!) G!) D!) A!) D)||~ A)||~ E)||~ B)||~ F)X~ C)X~ G)X~ D)X~ A)X~ E)X~ B)X~ E\!!/ B\!!/ F C G D A E B F/||\ C/||\ G/||\ F)x~ C)x~ G)x~ D)x~ A)x~ E)x~ B)x~ F)!!~ C)!!~ G)!!~ D)!!~ A)!!~ This uses 11-limit rational symbols that are harmonically meaningful and, taken together with the information that your major thirds are ~11:14 and minor thirds ~11:13, these would be easily understood by anyone already familiar with the notation. Otherwise, we already have a couple of symbols between /|\ and (|), which you might want to look at: (/| - the 31' comma: 31:32 (~54.964c), and |\) - the (7+11-5) comma: 1701:1760 (~59.031c) These are for all practical purposes apotome complements of one another, and the larger of these is close to /|\(. These two symbols are in the 494-ET standard set. But I think the foregoing solution is the one to go with. Let me know how this works out. --George

Message: 5771 - Contents - Hide Contents Date: Tue, 17 Dec 2002 04:21:27 Subject: Re: Relative complexity and scale construction From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> but they're planar arrangements of pitches, with like vectors > corresponding to like intervals. so there's no essential > difference . . .

You could even make zoomers and dualzoomers for 'em.

Message: 5772 - Contents - Hide Contents Date: Wed, 18 Dec 2002 20:57:42 Subject: Re: Ultimate 5-limit comma list From: wallyesterpaulrus still hoping for an answer to this . . . --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> what if we pushed up the badness limit until ampersand made it in? > the 5-limit comma does have a name, so it must be of some use . . . > > --- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

>> Not that any list is really ultimate, but with rms error < 40,

> geometric complexity < 500, and badness < 3500, it covers a lot of > ground. >>

>> 27/25 3.739252 35.60924 1861.731473 >> >> 135/128 4.132031 18.077734 1275.36536 >> >> 256/243 5.493061 12.759741 2114.877638 >> >> 25/24 3.025593 28.851897 799.108711 >> >> 648/625 6.437752 11.06006 2950.938432 >> >> 16875/16384 8.17255 5.942563 3243.743713 >> >> 250/243 5.948286 7.975801 1678.609846 >> >> 128/125 4.828314 9.677666 1089.323984 >> >> 3125/3072 7.741412 4.569472 2119.95499 >> >> 20000/19683 9.785568 2.504205 2346.540676 >> >> 531441/524288 13.183347 1.382394 3167.444999 >> >> 81/80 4.132031 4.217731 297.556531 >> >> 2048/2025 6.271199 2.612822 644.408867 >> >> 67108864/66430125 15.510107 .905187 3377.402314 >> >> 78732/78125 12.192182 1.157498 2097.802867 >> >> 393216/390625 12.543123 1.07195 2115.395301 >> >> 2109375/2097152 12.772341 .80041 1667.723301 >> >> 4294967296/4271484375 18.573955 .483108 3095.692488 >> >> 15625/15552 9.338935 1.029625 838.631548 >> >> 1600000/1594323 13.7942 .383104 1005.555381 >> >> (2)^8*(3)^14/(5)^13 21.322672 .276603 2681.521263 >> >> (2)^24*(5)^4/(3)^21 21.733049 .153767 1578.433204 >> >> (2)^23*(3)^6/(5)^14 21.207625 .194018 1850.624306 >> >> (5)^19/(2)^14/(3)^19 30.57932 .104784 2996.244873 >> >> (3)^18*(5)^17/(2)^68 38.845486 .058853 3449.774562 >> >> (2)^39*(5)^3/(3)^29 30.550812 .057500 1639.59615 >> >> (3)^8*(5)/(2)^15 9.459948 .161693 136.885775 >> >> (3)^45/(2)^69/(5) 48.911647 .026391 3088.065497 >> >> (2)^38/(3)^2/(5)^15 24.977022 .060822 947.732642 >> >> (3)^35/(2)^16/(5)^17 38.845486 .025466 1492.763207 >> >> (2)*(5)^18/(3)^27 33.653272 .025593 975.428947 >> >> (2)^91/(3)^12/(5)^31 55.785793 .014993 2602.883149 >> >> (3)^10*(5)^16/(2)^53 31.255737 .017725 541.228379 >> >> (2)^37*(3)^25/(5)^33 50.788153 .012388 1622.898233 >> >> (5)^51/(2)^36/(3)^52 82.462759 .004660 2613.109284 >> >> (2)^54*(5)^2/(3)^37 39.665603 .005738 358.1255 >> >> (3)^47*(5)^14/(2)^107 62.992219 .003542 885.454661 >> >> (2)^144/(3)^22/(5)^47 86.914326 .002842 1866.076786 >> >> (3)^62/(2)^17/(5)^35 72.066208 .003022 1131.212237 >> >> (5)^86/(2)^19/(3)^114 151.69169 .000751 2621.929721 >> >> (3)^54*(5)^110/(2)^341 205.015253 .000385 3314.979642 >> >> (2)^232*(5)^25/(3)^183 191.093312 .000319 2223.857514 >> >> (2)^71*(5)^37/(3)^99 104.66308 .000511 586.422003 >> >> (5)^49/(2)^90/(3)^15 74.858154 .000761 319.341867 >> >> (3)^69*(5)^61/(2)^251 143.055244 .000194 566.898668 >> >> (3)^153*(5)^73/(2)^412 235.664038 5.224825e-05 683.835625 >> >> (2)^161/(3)^84/(5)^12 100.527798 .000120 121.841527 >> >> (2)^734/(3)^321/(5)^97 431.645735 3.225337e-05 2593.925421 >> >> (2)^21*(3)^290/(5)^207 374.22268 2.495356e-05 1307.744113 >> >> (2)^140*(5)^195/(3)^374 423.433817 2.263360e-05 1718.344823 >> >> (3)^237*(5)^85/(2)^573 332.899311 5.681549e-06 209.60684

Message: 5773 - Contents - Hide Contents Date: Fri, 20 Dec 2002 12:05:10 Subject: Re: Tempering commas From: manuel.op.de.coul@xxxxxxxxxxx.xxx

>This worked! I then tried to get it to work via the command line, >and it didn't, until I entered "2" by itself. >It then asked me for the rest of the data.

It's not quite clear to me what exactly you typed, or how you misinterpreted the help file. It's always important to look at the first line of a help subject, to see what the command parameters are. Any other subsequent data will be prompted for. Manuel

Message: 5774 - Contents - Hide Contents Date: Fri, 20 Dec 2002 11:06:56 Subject: Proposed help file example From: Gene Ward Smith Instead of "enter 2, 2048/2025, 34171875/33554432, 2/1" I would propose you have pipedum <enter> 2 <enter> 25/24 <enter> 81/80 <enter> 2 <enter> as an example.