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Message: 11350 - Contents - Hide Contents Date: Fri, 09 Jul 2004 23:36:19 Subject: Re: 43 7-limit planar temperaments From: Paul Erlich That's not quite what I was asking for, but thanks! That's awesome. What I was asking for was the simplest possible criterion for determining that it's *this* set of generators and not any of the other equivalent sets. *That's* what I feel I owe the readers, even more than a method for the mathematical ones to be able to do it themselves. Of course, I could just give the readers the results without explaining the criterion, but that's a last resort.
Message: 11351 - Contents - Hide Contents Date: Fri, 09 Jul 2004 00:01:14 Subject: Re: Joining Post From: Carl Lumma>some of you know me from tuning list and Make Micro Music. >Just to say, I'll be listening on this list for while. Hiya Mark!For some resaon, I thought you were already onboard. Anyway, welcome (or welcome back)! -Carl
Message: 11352 - Contents - Hide Contents
Date: Fri, 09 Jul 2004 17:50:32
Subject: Some Sagittal intervals in arrow
From: Gene Ward Smith
Here are the 13-limit intervals of the "Selected Sagittal Sympols"
table, mapped via the generator pair (81/80, 64/63) to arrow.
32805/32768 [5, -4]
352/351 [-1, 1]
896/891 [-2, 2]
2048/2025 [-4, 4]
81/80 [1, 0]
64/63 [0, 1]
55/54 [-1, 2]
45927/45056 [4, -2]
45/44 [3, -1]
1701/1664 [3, -1]
36/35 [1, 1]
1053/1024 [1, 1]
250/243 [1, 1]
33/32 [0, 2]
729/704 [4, -1]
8505/8192 [3, 0]
27/26 [3, 0]
531441/512000 [3, 0]
________________________________________________________________________
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Message: 11353 - Contents - Hide Contents Date: Sat, 10 Jul 2004 00:44:20 Subject: Re: 43 7-limit planar temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> What I was asking for was the simplest possible criterion for > determining that it's *this* set of generators and not any of the > other equivalent sets. *That's* what I feel I owe the readers, even > more than a method for the mathematical ones to be able to do it > themselves.Don't the rules I give do that job? They force the result.
Message: 11354 - Contents - Hide Contents Date: Sat, 10 Jul 2004 00:50:30 Subject: Re: 43 7-limit planar temperaments From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote: >>> What I was asking for was the simplest possible criterion for >> determining that it's *this* set of generators and not any of the >> other equivalent sets. *That's* what I feel I owe the readers, even >> more than a method for the mathematical ones to be able to do it >> themselves. >> Don't the rules I give do that job? They force the result.Yes, but it requires a lot more math than I've introduced in the paper. If you were able to implement a criterion like the one I sketched out, then I could provide a complete explanation for the reader. But don't worry if it's not easy . . . I'll just use what you did, with your rules in a footnote.
Message: 11355 - Contents - Hide Contents
Date: Sat, 10 Jul 2004 10:29:33
Subject: The 36 {50/49, 64/63, 81/80} Fokker blocks
From: Gene Ward Smith
Below I give the scale, in the transposition with minimax Tenney
height, and the intervals of the scale.
[1, 15/14, 9/8, 7/6, 5/4, 4/3, 10/7, 3/2, 63/40, 5/3, 7/4, 40/21]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 21/20, 8/7, 6/5, 9/7, 4/3, 7/5, 3/2, 8/5, 12/7, 9/5, 40/21]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 16/15, 8/7, 6/5, 9/7, 27/20, 7/5, 32/21, 8/5, 12/7, 9/5, 27/14]
{16/15, 28/27, 15/14, 160/147, 21/20}
[1, 28/27, 10/9, 7/6, 5/4, 21/16, 7/5, 40/27, 14/9, 5/3, 7/4, 15/8]
{16/15, 200/189, 28/27, 15/14, 21/20}
[1, 15/14, 9/8, 189/160, 80/63, 4/3, 10/7, 3/2, 45/28, 27/16, 16/9,
40/21]
{256/243, 15/14, 12800/11907, 21/20}
[1, 21/20, 10/9, 7/6, 5/4, 21/16, 10/7, 3/2, 14/9, 5/3, 7/4, 15/8]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 21/20, 10/9, 7/6, 5/4, 4/3, 10/7, 3/2, 14/9, 5/3, 7/4, 40/21]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 21/20, 8/7, 6/5, 9/7, 4/3, 10/7, 3/2, 8/5, 12/7, 9/5, 40/21]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 21/20, 8/7, 6/5, 9/7, 27/20, 10/7, 3/2, 8/5, 12/7, 9/5, 27/14]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 21/20, 10/9, 7/6, 5/4, 4/3, 7/5, 3/2, 14/9, 5/3, 7/4, 40/21]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 15/14, 9/8, 6/5, 80/63, 4/3, 10/7, 3/2, 63/40, 12/7, 16/9, 40/21]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 21/20, 10/9, 7/6, 5/4, 21/16, 7/5, 3/2, 14/9, 5/3, 7/4, 15/8]
{16/15, 200/189, 28/27, 15/14, 21/20}
[1, 28/27, 10/9, 7/6, 49/40, 4/3, 7/5, 40/27, 14/9, 5/3, 7/4, 28/15]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 21/20, 9/8, 189/160, 80/63, 4/3, 7/5, 3/2, 63/40, 27/16, 16/9,
28/15]
{256/243, 15/14, 12800/11907, 21/20}
[1, 21/20, 9/8, 189/160, 80/63, 4/3, 10/7, 3/2, 63/40, 27/16, 16/9,
40/21]
{256/243, 15/14, 12800/11907, 21/20}
[1, 21/20, 9/8, 189/160, 80/63, 4/3, 7/5, 3/2, 63/40, 27/16, 16/9,
40/21]
{256/243, 15/14, 12800/11907, 21/20}
[1, 16/15, 8/7, 6/5, 9/7, 27/20, 10/7, 32/21, 8/5, 12/7, 9/5, 27/14]
{16/15, 200/189, 28/27, 15/14, 21/20}
[1, 28/27, 49/45, 7/6, 49/40, 4/3, 7/5, 40/27, 14/9, 49/30, 7/4,
28/15]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 15/14, 8/7, 6/5, 9/7, 27/20, 10/7, 3/2, 8/5, 12/7, 9/5, 27/14]
{16/15, 200/189, 28/27, 15/14, 21/20}
[1, 21/20, 9/8, 6/5, 80/63, 4/3, 10/7, 3/2, 63/40, 12/7, 16/9, 40/21]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 15/14, 10/9, 7/6, 5/4, 21/16, 10/7, 3/2, 14/9, 5/3, 7/4, 15/8]
{16/15, 28/27, 15/14, 160/147, 21/20}
[1, 15/14, 9/8, 135/112, 80/63, 4/3, 10/7, 3/2, 45/28, 27/16, 16/9,
40/21]
{256/243, 15/14, 21/20}
[1, 21/20, 9/8, 7/6, 80/63, 4/3, 7/5, 3/2, 63/40, 5/3, 16/9, 28/15]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 15/14, 9/8, 189/160, 80/63, 4/3, 10/7, 3/2, 63/40, 27/16, 16/9,
40/21]
{256/243, 15/14, 12800/11907, 21/20}
[1, 21/20, 9/8, 7/6, 80/63, 4/3, 7/5, 3/2, 63/40, 5/3, 16/9, 40/21]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 21/20, 8/7, 6/5, 9/7, 4/3, 7/5, 3/2, 8/5, 12/7, 9/5, 28/15]
{16/15, 28/27, 15/14, 160/147, 21/20}
[1, 15/14, 9/8, 7/6, 80/63, 4/3, 10/7, 3/2, 63/40, 5/3, 16/9, 40/21]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 21/20, 9/8, 6/5, 80/63, 4/3, 7/5, 3/2, 63/40, 12/7, 16/9, 28/15]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 21/20, 9/8, 7/6, 80/63, 4/3, 10/7, 3/2, 63/40, 5/3, 16/9, 40/21]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 28/27, 10/9, 7/6, 49/40, 4/3, 7/5, 40/27, 14/9, 49/30, 7/4, 28/15]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 21/20, 9/8, 7/6, 5/4, 4/3, 7/5, 3/2, 63/40, 5/3, 7/4, 40/21]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 21/20, 9/8, 6/5, 80/63, 4/3, 7/5, 3/2, 63/40, 12/7, 16/9, 40/21]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 21/20, 10/9, 7/6, 5/4, 4/3, 7/5, 3/2, 14/9, 5/3, 7/4, 28/15]
{16/15, 200/189, 28/27, 15/14, 21/20}
[1, 21/20, 9/8, 7/6, 5/4, 4/3, 10/7, 3/2, 63/40, 5/3, 7/4, 40/21]
{16/15, 200/189, 28/27, 15/14, 160/147, 21/20}
[1, 28/27, 10/9, 7/6, 5/4, 21/16, 10/7, 40/27, 14/9, 5/3, 7/4, 15/8]
{16/15, 28/27, 15/14, 160/147, 21/20}
[1, 21/20, 8/7, 6/5, 9/7, 27/20, 7/5, 3/2, 8/5, 12/7, 9/5, 27/14]
{16/15, 28/27, 15/14, 160/147, 21/20}
________________________________________________________________________
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Message: 11356 - Contents - Hide Contents Date: Sun, 11 Jul 2004 02:15:20 Subject: Malcolm's monochord revisted From: Gene Ward Smith If we Tenney reduce 12 notes of 5-limit 12-et over the range -600 to 600 cents, we get 1, 16/15, 9/8, 6/5, 5/4, 4/3, 25/18, 3/2, 8/5, 5/3, 16/9, 15/8 I was a little surprised to discover that this scale wasn't already in my collection of Scala scales. Since 25/18 and 18/25 have the same Tenney height, the inverted version of this might also be cosidered. The biggest deviation from 12-equal occurs with 25/18, which is 31.3 cents flat. If we band-limit the scale, so that the biggest deviation cannot be greater than the 15.6 cents of 6/5, we get something more familiar: 1, 16/15, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 16/9, 15/8 This is Malcolm's monochord, which has already popped up in several connections. Since 45/32 and 32/45 have the same Tenney height, we can put 64/45 in place of 45/32, in which case we get Riley's New Albion scale. All of these scales are suitable candidates for tempering via marvel (225/224-planar) where a considerable amount of 7-limit harmony appears. If we play the same game in the 7-limit, we get 1, 15/14, 8/7, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 28/15 The biggest deviation here is now for 8/7; if we band-limit this we get instead of 8/7 and 7/4 9/8 and 16/9. Here's Malcolm's monochord in tempered in marvel, where its six major and six minor triads are joined by two major and two minor tetrads, three supermajor and two subminor triads, three 1-6/5-7/5 diminished triads, three 1-5/4-14/9 augmented triads, two 1-9/7-3/2-9/5 supermajor tetrads and two 1-7/6-3/2-5/3 subminor tetrads, all in marvel tuning which isn't bad. If you'd rather have subminor triads than supermajor triads inverting it will give marvel-tempered New Albion instead. ! malcolmm.scl Malcolm's monochord in 1/4-kleisma marvel 12 ! 115.587047 200.054240 315.641288 384.385833 499.972880 584.440075 700.027121 815.614167 884.358713 999.945760 1084.412954 1200.000000 Here's the marvel-tempered verison of the Tenney-reduced scale without band limitation, so that we have a 7/5. To five major and five minor triads (excuding the 12-et like ones) we can add two major and one minor tetrad, two supermajor and two subminor triads, two 1-6//5-7/5 diminished triads, two 1-7/6-7/5 diminished triads, three 1-5/4-14/9 augmented triads, two 1-9/7-3/2-9/5 supermajor tetrads and two 1-7/6-3/2-5/3 subminor tetrads. The greater irregularity of this scale is not netting any benefits; harmonically it is slightly less rich. ! tenred5_12m.scl Tenney reduced in 1/4-kleisma marvel 12 ! 115.587047 200.054240 315.641288 384.385833 499.972880 568.717426 700.027121 815.614167 884.358713 999.945760 1084.412954 1200.000000
Message: 11357 - Contents - Hide Contents Date: Sun, 11 Jul 2004 20:23:08 Subject: Modmos From: Gene Ward Smith Here's some more jargon for you jargon lovers, and I know you're out there. Suppose there is an N-note MOS for some linear temperament, which for simplicity we can assume has period an octave. Suppose we have an N note scale, described in terms of a set N generator steps in our temperament, in which the graph of the scale (connecting consonances in an odd limit of the temperament) is connected, and which reduces modulo N to the N-note MOS. We could call such a beast a modmos. Modmos are interesting because the most interesting N-note scales of our temperament will be modmos; another way of defining it is that the scale is in the given temperament and is N-epimorphic and connected. If the tuning of the temperament is N-et, then this simply reduces to N-equal; but if it is close to but not identical to N-et then the modmos will be reasonably regular in terms of evenness and step size. One can find modmos by tempering a block or working with a relatively prime pair of intervals (measured in terms of generator steps) among other methods. What the best method of finding them and then assessing them is a question worth exploring, I think.
Message: 11358 - Contents - Hide Contents Date: Sun, 11 Jul 2004 06:14:41 Subject: Re: monz back to math school From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> hey guys, > > > i've finally decided to enroll in school again > to study the math that i'm sorely lacking.Apparently it's never too late: Art intersects science: Dabby creates music ou... * [with cont.] (Wayb.) Musical variations from a chaotic mapping * [with cont.] (Wayb.)
Message: 11359 - Contents - Hide Contents Date: Sun, 11 Jul 2004 22:08:23 Subject: Vals (was: Please stop the jargon explosion From: Graham Breed Moved to tuning-math because it belongs here. Gene Ward Smith wrote:> A {2,3,7}-val would be fine, so long as you called it that; otherwise > you would have no way of knowing <118 187 331| means the mapping > applies to 2, 3 and 7 except by guessing. If "val", without > qualification, is what you say then you know what the prime limit is > by counting.What about inharmonic timbres? If the "prime intervals" were empirically determined, can we still talk about vals? All your definitions require rationals. I work with prime intervals whose magnitudes happen to be the logarithms of prime numbers by default. Graham
Message: 11360 - Contents - Hide Contents Date: Sun, 11 Jul 2004 07:16:31 Subject: Tuning the diminished temperament From: Gene Ward Smith The diminished temperament seems to have a pretty good degree of consistency in TOP tuning between the 5-limit and 7/9; but if we use pure octaves things are different. The top tunings are less than a cent apart, [299.16, 101.67] for 5-limit, and [298.53, 101.46] for 7/9 limit. Poptimal generators are a different story; for the 5-limit we get 2^(-15/8) sqrt(15), at 94.13 cents; for the 7-limit, 76/1068 is poptimal, and that is 85.39 cents; the poptimal range extends from here up to 85.7 cents. Quite a difference! In the 9-limit, things change once again; the poptimal generator is sqrt(9/8), at 101.96 cents. Speaking as someone with no experience using this temperament, these numbers suggest to me that tuning it in TOP might be the best practical solution. Has anyone (Herman?) taken the diminished plunge? What do you think, if so?
Message: 11361 - Contents - Hide Contents Date: Sun, 11 Jul 2004 09:56:16 Subject: The five 128/125 250/243 Fokker blocks From: Gene Ward Smith [1, 25/24, 10/9, 9/8, 6/5, 5/4, 4/3, 25/18, 36/25, 3/2, 25/16, 5/3, 125/72, 9/5, 15/8] [25/24, 16/15, 81/80, 16/15, 25/24, 16/15, 25/24, 648/625, 25/24, 25/24, 16/15, 25/24, 648/625, 25/24, 16/15] [1, 25/24, 27/25, 9/8, 6/5, 5/4, 4/3, 27/20, 36/25, 3/2, 8/5, 5/3, 27/16, 9/5, 15/8] [25/24, 648/625, 25/24, 16/15, 25/24, 16/15, 81/80, 16/15, 25/24, 16/15, 25/24, 81/80, 16/15, 25/24, 16/15] [1, 25/24, 10/9, 9/8, 6/5, 5/4, 4/3, 25/18, 36/25, 3/2, 8/5, 5/3, 16/9, 9/5, 15/8] [25/24, 16/15, 81/80, 16/15, 25/24, 16/15, 25/24, 648/625, 25/24, 16/15, 25/24, 16/15, 81/80, 25/24, 16/15] [1, 25/24, 27/25, 9/8, 6/5, 5/4, 4/3, 27/20, 36/25, 3/2, 25/16, 5/3, 27/16, 9/5, 15/8] [25/24, 648/625, 25/24, 16/15, 25/24, 16/15, 81/80, 16/15, 25/24, 25/24, 16/15, 81/80, 16/15, 25/24, 16/15] [1, 25/24, 27/25, 9/8, 6/5, 5/4, 125/96, 27/20, 45/32, 3/2, 25/16, 5/3, 27/16, 9/5, 15/8] [25/24, 648/625, 25/24, 16/15, 25/24, 25/24, 648/625, 25/24, 16/15, 25/24, 16/15, 81/80, 16/15, 25/24, 16/15]
Message: 11362 - Contents - Hide Contents Date: Sun, 11 Jul 2004 10:21:04 Subject: Re: The five 128/125 250/243 Fokker blocks From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote: Of these five scales, all benefit from tempering by 225/224, and the first and last by tempering by 15625/15552 as well, from whence they get an extra minor third. These together produce hanson/catakleismic, and I list them in terms of (minor third) generators of below.> [1, 25/24, 10/9, 9/8, 6/5, 5/4, 4/3, 25/18, 36/25, 3/2, 25/16, 5/3, > 125/72, 9/5, 15/8][-7, -6, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12]> [1, 25/24, 27/25, 9/8, 6/5, 5/4, 4/3, 27/20, 36/25, 3/2, 8/5, 5/3, > 27/16, 9/5, 15/8][-6, -5, -1, 0, 1, 2, 4, 5, 6, 7, 8, 11, 12, 13, 18]> [1, 25/24, 10/9, 9/8, 6/5, 5/4, 4/3, 25/18, 36/25, 3/2, 8/5, 5/3, > 16/9, 9/5, 15/8][-12, -7, -6, -5, -2, -1, 0, 1, 2, 4, 5, 6, 7, 11, 12]> [1, 25/24, 27/25, 9/8, 6/5, 5/4, 4/3, 27/20, 36/25, 3/2, 25/16, 5/3, > 27/16, 9/5, 15/8][-6, -1, 0, 1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 18]> [1, 25/24, 27/25, 9/8, 6/5, 5/4, 125/96, 27/20, 45/32, 3/2, 25/16, > 5/3, 27/16, 9/5, 15/8][-1, 0, 1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 18] ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ 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: 11363 - Contents - Hide Contents
Date: Mon, 12 Jul 2004 09:13:12
Subject: A 12 note meantone modmos
From: Gene Ward Smith
I was trying to figure out, by fiddling with an example, what methods
might be suitable for constructing modmos. It doesn't seem to me that
limiting the number of islands, which would be the simplest case after
MOS, is necessarily the most interesting. Below I give a
paper-and-pencil constructed modmos, which has three major tetrads and
one minor tetrad, rather than the two and two of Meantone[12]. It is
in 50-et, a tuning range with excellent 11 and 13 harmonies, and some
of that comes with this scale. While one may still prefer
Meantone[12], clearly a scale like this has its merits. It also has
five separate islands; the chain of fifths being
0,1,2;4,5,6;8,9,10,11;15;19.
Here's a Scala scl for it:
! modmos12a.scl
A 12-note modmos in 50-et meantone
12
!
24.000000
192.000000
264.000000
384.000000
456.000000
576.000000
696.000000
768.000000
840.000000
960.000000
1080.000000
1200.000000
________________________________________________________________________
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Message: 11364 - Contents - Hide Contents Date: Tue, 13 Jul 2004 00:24:24 Subject: 50 From: Gene Ward Smith Here are the TM bases for 50-et up to the 19-limit. It might be noted that in terms of the 50-equal version of meantone, 20/13 maps to -11, 11/8 to -13, and 16/13 to -15, leading to the 12, 14, and 16 note wolves of the same size. 5-limit [81/80, 1207959552/1220703125] 7-limit [16807/16384, 81/80, 126/125] 11-limit [81/80, 245/242, 126/125, 385/384] 13-limit [81/80, 245/242, 105/104, 126/125, 144/143] 17-limit [81/80, 105/104, 126/125, 144/143, 170/169, 221/220] 19-limit [81/80, 105/104, 126/125, 133/132, 144/143, 153/152, 170/169]
Message: 11365 - Contents - Hide Contents Date: Tue, 13 Jul 2004 02:57:10 Subject: Re: 50 From: monz hi Gene, have you ever seen this? W. S. B. Woolhouse's 'Essay on musical interva... * [with cont.] (Wayb.) Woolhouse and Paul Erlich both (independently, about 160 years apart) discovered that 7/26-comma meantone is an optimal meantone by one type of measure. Woolhouse then advocates 50-et as a very good approximation to that tuning, and then 19-et as a more practical (but not as close) alternative. good luck trying to find Woolhouse's book itself if you want it. the only copy i've ever discovered is at the U. of Pennsylvania library in Philadelphia. -monz --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Here are the TM bases for 50-et up to the 19-limit. It > might be noted that in terms of the 50-equal version of > meantone, 20/13 maps to -11, 11/8 to -13, and 16/13 to -15, > leading to the 12, 14, and 16 note wolves of the same size. > > > 5-limit > [81/80, 1207959552/1220703125] > > 7-limit > [16807/16384, 81/80, 126/125] > > 11-limit > [81/80, 245/242, 126/125, 385/384] > > 13-limit > [81/80, 245/242, 105/104, 126/125, 144/143] > > 17-limit > [81/80, 105/104, 126/125, 144/143, 170/169, 221/220] > > 19-limit > [81/80, 105/104, 126/125, 133/132, 144/143, 153/152, 170/169]
Message: 11366 - Contents - Hide Contents Date: Tue, 13 Jul 2004 04:17:08 Subject: Re: 50 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> hi Gene, > > > have you ever seen this? > > W. S. B. Woolhouse's 'Essay on musical interva... * [with cont.] (Wayb.) > > Woolhouse and Paul Erlich both (independently, about > 160 years apart) discovered that 7/26-comma meantone > is an optimal meantone by one type of measure.It's worth noting that Woolhouse found it. It's not too surprising Paul did, or I did a little later for that matter. It's what anyone would find who applied unweighted least squares. Gauss invented this method as a teenager, proved some important facts about it when it is used to estimate a noisy signal, and it was still of pretty recent vintage when Woolhouse used it. I don't know if anyone had used it before Woolhouse to approximate something which was *not* a noisy signal, but known and exact quantities; this could be an idea original to him. Certainly, applying it to music must have been, but you quote Woolhouse as saying: [Woolhouse 1835, p 46:] This system is precisely the same as that which Dr. Smith, in his Treatise on harmonics [Smith 1759], calls the scale of equal harmony. It is decidedly the most perfect of any systems in which the tones are all alike. Is Smith's tuning 50-equal? I'd also cut Woolhouse some slack on 5-limit vs 5-odd-limit. The place where he discusses that you *might* interpret to say that 2 and 3/2 are consonances, but 4, 3 or 6/5 are not, but I would presume his assumption of octave equivalence would be clear from elsewhere. I also find this comment interesting: He then analyzes the resources of a 53-EDO 'enharmonic organ', built by J. Robson and Son, St. Martin's-lane, but says that the number of keys is too much to be practicable, and settles again on 19-EDO. Is this the first time someone built a 53-edo instrument? Incidentally, convergents to the Woolhouse fifth go 7/12, 11/19, 18/31, 29/50, 76/131, 257/443 ... . It would be interesting to dig up someone who advocated 131-equal for meantone! 29/50 is slightly (8/27 of a cent) to the south of the smallest poptimal meantone at 47/81, but 0.19 cents *sharper* than Zarlino's 2/7 comma. It occurs to me that Zarlino's advocacy of 2/7-comma could be taken as evidence for Paul's contention that the poptimal range ought to be brought all the way down to exponent 1; if we did that 50 becomes the smallest poptimal meantone et. I think I'll at least code it. In any case 50 also has the distinction of being the last to appear in the convergents to both Zarlino's and Woolhouse's optimal fifths.
Message: 11367 - Contents - Hide Contents Date: Tue, 13 Jul 2004 05:49:55 Subject: A 14-note modmos of meantone From: Gene Ward Smith This is interesting as one way to construct these, though it's cheating in a way because from another point of view it is mos, or at least ce. If you take my comma list for 50 and dispense with one of the TM basis commas, one of the temperaments you get (above the 7-limit) is 12&50, which is actually pretty good if you want higher limit consonances. The 50-et generators are 1/2 and 2/25, and if you take the 14-note (MOS? DE?) you get, when the result is translated into meantone, -24, -23, -22, -3, -2, -1, 0, 1, 2, 3, 22, 23, 24, 25 This is a 14-note modmos; it has two 50-et diatonic scales hence the name. ! bidiatonic.scl 14 note modmos of meantone, mos of 12&50 14 ! 96.000000 192.000000 288.000000 312.000000 408.000000 504.000000 600.000000 696.000000 792.000000 888.000000 912.000000 1008.000000 1104.000000 1200.000000
Message: 11369 - Contents - Hide Contents Date: Tue, 13 Jul 2004 06:05:33 Subject: Re: 50 From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> but you quote > Woolhouse as saying: > > [Woolhouse 1835, p 46:] > > This system is precisely the same as that which Dr. Smith, in his > Treatise on harmonics [Smith 1759], calls the scale of equal harmony. > It is decidedly the most perfect of any systems in which the tones are > all alike. > > Is Smith's tuning 50-equal?It's close, but it's much closer to 5/18-comma meantone than to 50- equal. Search the tuning list for more info ;) Also see Jorgenson.
Message: 11370 - Contents - Hide Contents Date: Tue, 13 Jul 2004 06:10:29 Subject: Re: A 14-note modmos of meantone From: Paul Erlich Is this distinct from Injera in some way? The two scales I originally proposed for Injera both had 14 notes: the DE one and the omnitetrachordal variant. They're both "double diatonic". --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> This is interesting as one way to construct these, though it's > cheating in a way because from another point of view it is mos, or at > least ce. > > If you take my comma list for 50 and dispense with one of the TM basis > commas, one of the temperaments you get (above the 7-limit) is 12&50, > which is actually pretty good if you want higher limit consonances. > The 50-et generators are 1/2 and 2/25, and if you take the 14-note > (MOS? DE?) you get, when the result is translated into meantone, > > -24, -23, -22, -3, -2, -1, 0, 1, 2, 3, 22, 23, 24, 25 > > This is a 14-note modmos; it has two 50-et diatonic scales hence the name. > > ! bidiatonic.scl > 14 note modmos of meantone, mos of 12&50 > 14 > ! > 96.000000 > 192.000000 > 288.000000 > 312.000000 > 408.000000 > 504.000000 > 600.000000 > 696.000000 > 792.000000 > 888.000000 > 912.000000 > 1008.000000 > 1104.000000 > 1200.000000
Message: 11371 - Contents - Hide Contents Date: Tue, 13 Jul 2004 08:20:12 Subject: Re: A 14-note modmos of meantone From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:> Is this distinct from Injera in some way?Conceptually, at any rate. The tuning map I was proposing was 12&50, which in the 19-limit is [<2 0 -8 -26 -31 39 5 -1|, <0 1 4 10 12 -10 1 3|] For a 14-note MOS this suffers from the defect that you don't actually get to use the 7, 11, and 13 much; for a 26-note MOS it's a lot better, and the tuning is considerably more accurate than 26-equal, which is what injera would more or less amount to. The two scales I originally> proposed for Injera both had 14 notes: the DE one and the > omnitetrachordal variant. They're both "double diatonic".The 7/9 copop generator is (35/24)^(1/7), which translates to 2/5-comma meantone. That's a far more hefty dose of tempering than 5-equal. I'd say 12%26 (injera) was a sibling to 12&50, but no more than that.
Message: 11372 - Contents - Hide Contents Date: Tue, 13 Jul 2004 09:31:02 Subject: Re: monz back to math school From: Graham Breed jjensen142000 wrote:> Trigonometry and calculus are really irrelevant to linear algebra, > and Grassmann algebra is (i think) abstract algebra (groups, rings, > homomorphisms, etc) applied to linear algebra. In other words, > it is *much* harder, as in you get a bachelors degree in math and > then you take it in graduate school. Of course, you wouldn't need > the full force of it to follow most of the tuning-math discussions.Trig and calculus are both handy to know, and do come up in tuning-math theory. So if they're pre-requisites, why not give them a try? It depends on how advanced the courses get. Easy trig and calculus are all you need. Grassman algebra is based on wedge products. It's not that difficult. But it isn't that well known, so they'll only expect hard core mathematicians to learn it. Get basic algebra, and you should be able to learn it from the online materials, and questions here. Also, you can avoid a lot of linear algebra if you know Grassman algebra. So choose your poison ... Graham
Message: 11373 - Contents - Hide Contents Date: Tue, 13 Jul 2004 08:56:34 Subject: That 13 note modmos from tuning From: Gene Ward Smith --- In tuning@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:> Hemiwuerschmidt (99&130) has a 7-limit Graham complexity of 16, so we > need to get up at least to the 19-not MOS before having complete tetrads > in a MOS. Here I give a 13-note modmos, found via more pen & paper > fiddling, which has three major tetrads and two minor tetrads, which > is not bad for 13 notes tempered in something more accurate than > miracle. It JI lattice terms, it has [-2,0,0], [0,0,0] and [1,1,0] for > major tetrads and [-1,0,0] and [0,1,0] for minor tetrads, so the > harmony links together nicely.The trouble with tuning this in hemiwuerschmidt is that you may as well detemper it to 7-limit JI, in which case valentine (15&31) or myna (27&31) actually get you somewhere. :( Here's the detempered version, which doesn't even manage to be epimorphic, though I am sure it must have tried. 36/35-21/20-35/32-6/5-5/4-21/16-48/35-3/2-49/32-12/7-7/4-9/5-2 It's a bit irregular, but it does have a nice collection of chords.
Message: 11374 - Contents - Hide Contents Date: Tue, 13 Jul 2004 09:01:28 Subject: Re: monz back to math school From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Graham Breed <graham@m...> wrote:> Grassman algebra is based on wedge products. It's not that difficult. > But it isn't that well known, so they'll only expect hard core > mathematicians to learn it. Get basic algebra, and you should be able > to learn it from the online materials, and questions here. > > Also, you can avoid a lot of linear algebra if you know Grassman > algebra. So choose your poison ...It's a subject normally taught to grad students which should be taught to sophmores. In a linear algebra class, the sophmores will need to learn the determinant. Textbook authors usually approach this from the point of view of computing them by hand using Gaussian reduction, which is pointless and does not advance your understanding much. They should teach it from wedge products.
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