source file: m1584.txt Date: Mon, 16 Nov 1998 18:18:39 -0500 Subject: A strange 9-limit temperament From: "Paul H. Erlich" Dave Keenan described his 12-note scale with 8 7-limit tetrads, etc. >As offsets from 12-tET it is >C +12.19 >C# +17.41 >D -17.41 >D# -12.19 >E -6.98 >F +6.98 >F# +12.19 >G +17.41 >G# -17.41 >A -12.19 >A# -6.98 >B +6.98 This tuning certainly seems to be a genuinely new and interesting development. Whoever's keeping track of those out there, pay heed! I'm assuming that no one else has paid any attention to my contest, but in case anyone gave it a try, at this point I'm willing to allow for some rule-bending. For example, a couple of 14 out of 26-tET scales look good if you drop the requirement that three complete chords suffice to cover the scale. If also you drop the requirement that otonal and utonal chords be constructed from the same pattern of scale steps, a whole array of many-noted scale open up. A couple of very well-tuned ones have occured to us: 22 out of 41-tET to me, and 19 out of 31-tET to Dave. Let me describe these scales a bit . . . A maximally even 22 out of 41-tET has a very interesting melodic structure. First, note that any octave span (C-c, D-d, etc.) of the normal diatonic scale (maximally even 7 out of 12-tET, 19-tET, or 26-tET) has two tetrachords, each spanning a 4:3 frequency ratio and consisting of several equally-sizes large steps plus one small step, and that an additional large step (either at the bottom, at the top, or between the two tetrachords) completes the octave. Well, any 2:1 span of the 22-out-of-41 scale has three "octachords", each spanning a 5:4 frequency ratio and consisting of several equally-sized large steps (2/41 oct) plus one small step (1/41 oct), and an additional large step (either at the bottom, at the top, between the bottom two octachords, or between the top two octachords). This is a remarkable similarity and may have something to with the fact that 22 out of 41 is an MOS scale generated by 41's 5:4 approximation, 13/41 oct. Now the harmonic properties of this scale are at least as remarkable. First, note that the normal diatonic scale in any meantone tuning (including 12-tET, 19-tET, 26-tET, 31-tET, and many non-closed temperaments) has close approximations to three Otonalities complete through the 5-limit (major triads) and three Utonalities complete through the 5-limit (minor triads). These have a maximum error of 15.6 cents in 12-tET and a maximum error of 5.4 cents in quarter-comma meantone. Well, the 22-out-of-41 scale has ten Otonalities complete through the 9-limit and ten Utonalities complete through the 9-limit (two of each can be completed through the 11-limit). The maximum error is 6.8 cents, which is the error of the 9:5. The largest error within the 7-limit is that of the 5:3, namely, 6.3 cents. A maximally even 19 out of 31-tET can be thought of as being constructed from identical tetrachord-like blocks. But there are several instances of both large (2/31 oct.) and small (1/31 oct.) step sizes in each block. Therefore it may be difficult to know where one is in such a scale just from the melodic succession. Richmond Browne has pointed out the importance of rare intervals for position finding. The closest thing to such a signpost is the three-step sequence small-large-small, which only happens twice in the scale. But the scale is generated by the 4:3 approximation, something it has in common with the traditional diatonic scale. Harmonically, the scale is wonderful, having nine Otonalities complete through the 9-limit and nine Utonalities complete through the 9-limit, all with a maximum error of 11.1 cents (in the 9:5) and a maximum 7-limit error of 6 cents (in the 5:3). A while back Andrzej Gawel described a maximally even 19 out of 36-tET scale which was summarily ignored by the List. This scale is a 19-note MOS generated by the 19/36 oct interval, much like the standard diatonic scale is a 7-note MOS generated by the 7/12 oct interval, and including the latter scale within it. Thus the tetrachord-like structure spans the approximate 7:5, and only one small interval occurs within each of the tetrachord-like blocks, which bodes well for position finding. The scale has seven Otonalities complete through the 7-limit and seven Utonalities complete through the 7-limit, with a maximum error of 15.8 cents (in the 7:5). This almost half the size of the smallest step. Attemting to construct 9-limit chords forces one to have an error of more than half the smallest step, since 36-tET is not consistent through the 9-limit. Can anyone think of a way of improving Gawel's proposal?