source file: mills3.txt Date: Sun, 11 Jan 1998 16:11:15 +0100 Subject: Clearer From: Carl Lumma >>There is no such thing as the odd limit of a pitch -- all intervals can be >>constructed above and below any pitch. What do you really mean? Odd limit of the interval, sorry. >>}Idea "A" seems to suggest that we temper the octave the most, the fifth the >>}next most, and so on up. But we can't do this without wrecking tonality, >>}since the low identities provide that, as explained in idea "B". >> >>}Both A and B are really one idea, in that they're both caused by the way >>}superparticular fractions get closer to eachother, so there's no >>}"contradiction", true. But when tempering, it presents a trade-off >>}situation, one which can't be well-addressed by just making the de-tuning >>}inversely proportional to limit. >> >>None of the above makes any sense to me. Perhaps you have something in mind >>that you'd like to clarify, but insofar as you're relying on (B), I honestly >>can't figure it out. Moreover, I don't see anything special about >>superparticulars. The superparticulars are just all the intervals between adjacent members of the series. You're supposed to be able to detune a 3/2 more than an 13/12, since there's less low-numbered ratios near the 3/2 to get confussed with, right? That's idea A. It suggests that if you've got a certain amount of tempering to do, you should do it more on the low identities. Yet the high identities rely on the low ones for context in music. High identities work in a tonality, and its the lower identities that play the bigger role in establishing tonality. That's idea B. It's the cons to tempering the lower identities more. While they alone can stand more tempering than the higher identities, how well the higher identities to work depends on the strength of the lower ones. >>How did you think I arrived at this list, if not by insisting that the >>candidates must be better in the 7-limit than 12 at the 5-limit? I see I made a mistake as to what your criteria were for the two candidate lists. I don't have the charts, but I didn't think that 22, 26, and 27 were as good at the 5-limit as 12... >>Can you suggest a way to make the paper less confusing on this point? Yes. Instead of... "if we wish to move to the 7-limit without decreasing the overall level of accuracy from that of 12-equal at the 5-limit" Say... "if we want our new tuning to approximate the 7-limit at least as well as 12-equal approximates the 5-limit" Instead of... "but now only two are still more accurate than 5-limit 12-equal" Say... "but now only two are still better at the 7-limit than 12 equal is at the 5-limit." >>}1) Conventional theory is full of holes. The tritone is spelled as an >>}augmented forth in certain contexts, and as a diminished fifth in others, >>}but it's the same pitch. >> >>First of all, it's an interval, not a pitch. This time my use of pitch is just fine. The intervals are the aug 4th and the dim 5th. I meant that, starting the two intervals on the same bottom pitch, the "two" upper pitches are the same. >>Secondly, the augmented fourth and diminished fifth _sound_ different in >>context, and one must resolve outwards while the other must resolve inwards. I agree. Playing Jazz got me to understand enharmonics in 12, especially the difference between the aug 4 and the dim 5. I can see the reason why someone might prefer to use enharmonics and proper spellings of intervals for certain work in 12. I just don't prefer using them for the type of work I do in 12. >>}Why you consider it a type of fifth in your >>}example was, in any case, not clear to me. >> >>Because you have to count 5 scale steps to get from the lower note to the >>upper note, inclusive. You mean scale degrees, not steps? If my scale is, in 12... C C# E F# G# B C ..then the tritone is the 4th degree. >>I did provide such a definition -- "step" means a step in the scale, not a >>step in the tuning. Without a basic, "generalized diatonic" scale, the >>concept is meaningless. I'm with you on the need for a generalized diatonic scale, but I missed the definition. I don't see why a fifth is ever anything but the fifth degree of a scale. The tritone isn't part of the "unaltered diatonic" scale, so I can't see how it can be a type of 5th or 4th degree in that scale. >>Wow, I thought my definition was fine. It certainly seems robust to me. How >>would you suggest I improve it? Perhaps by spelling out every single interval >>occuring in each of the "generalized diatonic" scales? Maybe. How would you define diatonic? How do you define dissonace, as in "characteristic dissonance", at a given odd limit? An interval whose ratio involves odd numbers (after factors of two are taken out) bigger than the limit? >>}I noticed that. But why is it Microsoft's fault? Did you used some >>}automatic numbering scheme? I hate that trend in software nowadays! It's >>}not already so easy that you can't do it yourself? >> >>I couldn't figure out how to turn off the auto-numbering in Word 97. You can't just type the numbers in by hand? I'm sure you can turn off the auto-numbering with a short spelunk thru the preferences thingy. I hate Word to death, and I just removed it from my computer, or I'd tell you how exactly. >>Henry Cow was a British group whose music was very composed, a bit Zappa-like >>but much deeper emotionally (I feel). I'll keep my eyes peeled. >>Have you seen or heard Phish live? >>There's no comparison. Unfortunately, no. >>I'm sure you can transcribe all Partch music in 72TET and not lose anything >>in the modulatory effects. Oh, maybe so. At the cost of a pitch set twice the size of Partch's (roughly, but the pitch set used varies from work to work). >>}Please Note: How the intervals are heard depends on context!!! I'm just >>}throwing in my analysis here because sometimes a particular context seems >>}to be assumed when giving rational approximations for the 22TET intervals... >> >>what context are you assuming??? None in particular. That was the point. Bill Alves writes... >Looking at your 22TET JI approximation, it occurs to me that it will all >fit pretty neatly into the 11 limit. You have 31/16, and, by implication >its inversion 32/31, but 33/32 is only 1 cent off, and is already present >in the distance between the 4/3 and the 11/8 (or 3/2 and 16/11). Likewise >40/33, while a couple of cents further off than 23/19 that you use, also >uses this interval down from the 5/4. A 25/22 (11/10 down from 5/4) is just >2 cents worse than your 17/15. While the 600 cent interval is always a >problem in low limit JIs, 99/70 (7/6 down from 33/20), though not exactly a >pretty ratio, nails it. Here are my suggestions in your format, with >inversions filled in: > >> 1- 55 33/32 (1) >> 2- 109 16/15 (3) >> 3- 164 11/10 (1) >> 4- 218 25/22 (3) >> 5- 273 7/6 (6) >> 6- 327 40/33 (6) >> 7- 382 5/4 (4) >> 8- 436 9/7 (1) >> 9- 491 4/3 (7) >>10- 545 11/8 (6) >>11- 600 99/70 (0) >>12- 655 16/11 (6) >>13- 709 3/2 (7) >>14- 764 14/9 (1) >>15- 818 8/5 (4) >>16- 873 33/20 (6) >>17- 927 12/7 (6) >>18- 982 44/25 (3) >>19- 1036 20/11 (1) >>20- 1091 15/8 (3) >>21- 1145 64/33 (0) >>22- 1200 2/1 (0) > >I understand that you may be using different criteria for your >approximations, so I just offer these as another possible interpretation >and look forward to your paper. Thanks for work. This seems quite a bit more elegant to me. Truth be told, elegant was what I was trying to avoid. I said that I left out the inversions to reveal structure in the scale. Maybe this was misleading. I didn't pick the approximations as a scale. I was just trying to show what you might wind up hearing if you're not careful. Carl SMTPOriginator: tuning@eartha.mills.edu From: jpff@maths.bath.ac.uk Subject: Re: Our instruments PostedDate: 11-01-98 20:14:36 SendTo: CN=coul1358/OU=AT/O=EZH ReplyTo: tuning@eartha.mills.edu $MessageStorage: 0 $UpdatedBy: CN=notesrv2/OU=Server/O=EZH,CN=coul1358/OU=AT/O=EZH,CN=Manuel op de Coul/OU=AT/O=EZH RouteServers: CN=notesrv2/OU=Server/O=EZH,CN=notesrv1/OU=Server/O=EZH RouteTimes: 11-01-98 20:14:11-11-01-98 20:14:12,11-01-98 20:14:02-11-01-98 20:14:02 DeliveredDate: 11-01-98 20:14:02 Categories: $Revisions: Received: from ns.ezh.nl ([137.174.112.59]) by notesrv2.ezh.nl (Lotus SMTP MTA SMTP v4.6 (462.2 9-3-1997)) with SMTP id C1256589.0069A6AE; Sun, 11 Jan 1998 20:14:28 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA26563; Sun, 11 Jan 1998 20:14:36 +0100 Date: Sun, 11 Jan 1998 20:14:36 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA26596 Received: (qmail 17668 invoked from network); 11 Jan 1998 11:14:34 -0800 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 11 Jan 1998 11:14:34 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu