Daniel Wolf's observations on the arrival of horological accuracy, and the declining use of meantones tunings, with moves towards well temperaments and eventually in favor of 12tET, does have a certain irony. (see end of post *)
Assuming that he is correct (I have no reason to doubt him on this), and the fashion moved from meantone into well temperaments, and eventually 12tET, I wonder why meantone tunings were abandoned.
Could it have been due to the limitations imposed on composers and performers, when using twelve fixed pitches per octave?
The difficulty of quickly retuning meantones would have severely restricted the harmonic possibilities, yet well temperaments immediately offered keyboard players both tuning and intervallic novelties, in different keys.
So the musical innovators of the time gravitated towards well temperaments or to providing more notes per octave with instrument developments, resulting in 19tET, 31tET, 53tET etc.
Could 12tET's ubiquity be blamed on the need to simplify the well-tempered subtleties for subsequent less tonally sophisticated generations?
Meanwhile the JI advocates were still plugging away with an obsolete system, which had already been thrashed to death by earlier generations, and have regularly temporarily resuscitated it for later generations, as historical curiosities.
I appreciate that there is a strong bias against meantones tunings, by many tuning list subscribers, who still praise, rank, map, and compare their JI and ET tunings to "beatless perfection".
I seem to be seeing/hearing a revival of interest in meantones since their possibilities now made available by recent technological developments.
Despite their unfortunately ugly name, "meantone tunings" to my biased mind have many redeeming characteristics which can now be used practically.
1) Modulation and transposition is easy and limitless. 2) The tunings are clearly and easily defined by: a) size of Large interval (or difference between Fourth (IVth) and Fifth (Vth) interval) plus b) Octave Ratio. 3) All spiral and most circular tunings can also be described in these terms. Eg. Pythagorean, and almost all ET systems. 4) Harmonic and intervallic relationships are clear. 5) Scales can easily be unambiguously coded. 6) Degrees (or levels) of consonance and dissonance are obvious. 7) Current technology allows rapid retuning with conventional fingering. 8) All intervals can be described in terms of Large (L) and small (s) intervals to know precision.
9) The notation system is conventional and well-established.
BTW Has anyone else looked carefully at Bill and Anne Collins Collinsian Concepts notation system (May 1995)?
So maybe it's time to take a second and deeper look at the "neglected" meantone potentials.
Charles Lucy lucy@hour.com http://www.wonderlandinorbit.com/projects/lullaby
(*Ironic in connection with John "Longitude" Harrison's ideas on time measurement and tunings. One "world beating" the other neglected.)
For download of his writings go to: http://www.wonderlandinorbit.com/lucytuning/harrison/harrison.htm
Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sun, 6 Jul 1997 15:05 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA05704; Sun, 6 Jul 1997 15:06:10 +0200 Date: Sun, 6 Jul 1997 15:06:10 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA05691 Received: (qmail 8075 invoked from network); 6 Jul 1997 13:06:05 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 6 Jul 1997 13:06:05 -0000 Message-Id: <199707060902_MC2-1A52-E588@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu
>>If you allow for the octave equivalence of intervals, all triadic >>inversions contain the same intervals. >How so? The way I larnt it, a simple C major triad in root position consists >of a minor third stacked on top of a major third. In first inversion, it >consists of a perfect fourth stacked on top of the minor third.
Well, this premise is certainly clear you instead think of a root position major triad as a P4 atop a m3 atop a M3, and first inversion an M3 atop a P4 atop a m3, and so forth.
Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 8 Jul 1997 17:49 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA01173; Tue, 8 Jul 1997 17:49:49 +0200 Date: Tue, 8 Jul 1997 17:49:49 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA17512 Received: (qmail 2869 invoked from network); 8 Jul 1997 15:49:39 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 8 Jul 1997 15:49:39 -0000 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu
>''>>If you allow for the octave equivalence of intervals, all triadic >>>inversions contain the same intervals. >>How so? The way I larnt it, a simple C major triad in root position consists >>of a minor third stacked on top of a major third. In first inversion, it >>consists of a perfect fourth stacked on top of the minor third.
> Well, this premise is certainly clear you instead think of a root >position major triad as a P4 atop a m3 atop a M3, and first inversion anM3 >atop a P4 atop a m3, and so forth.''
The three inversions of a triad (that is, three tones without octave duplications) are intervallically distinct: (in 12tet interval classes:) 4,3,7; 3,5,8 ; 5,4,9. Throw in an additional tone, doubling the lowest pitch at the octave, and you get: 4,3,5,7,8,12; 3,5,4,8,9,12; 5,4,3,9,7,12, which are also distinct, shuffling the two-out-of-three combinations of (7,8,9). Triadic inversions are equivalent (a) in terms of pitch class content, or(b) under some octave modulus operation where inversions are equivalent. Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 8 Jul 1997 19:25 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA20008; Tue, 8 Jul 1997 19:26:11 +0200 Date: Tue, 8 Jul 1997 19:26:11 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA20067 Received: (qmail 10644 invoked from network); 8 Jul 1997 17:24:35 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 8 Jul 1997 17:24:35 -0000 Message-Id: <199707081320_MC2-1A72-8173@compuserve.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu
>How so? The way I larnt it, a simple C major triad in root position consists >of a minor third stacked on top of a major third. In first inversion, it >consists of a perfect fourth stacked on top of the minor third. In second >inversion it's that perfect fourth with the major third stacked above it. > From what perspective are those the same set of intervals?
Um, yes. Technically there are six intervals in a major root position triad: plus a major third, plus a minor third, plus a perfect fifth, minus a major third, minus a minor third and minus a perfect fifth. In octave equivalent terms, minus a fifth is the same as plus a fourth. In these terms, then, all inversions of a major triad contain the same intervals. I had overlooked the obvious simplification of only taking positive intervals and this does, indeed, distinguish simple inversions. However, in the specific method David mentions the chords C-E-G-C, E-G-C-E and G-C-E-G have identical dissonance. I consider this to be a problem. Taking all positive intervals, the first chord can be distinguished from the last two (it's all in the sixths) which is less of a problem.
My mention of utonal/utonal degeneracy betrays some muddy thinking on my part. What I meant is that a major triad has the same intervals as a minor triad. This is a completely different issue to inversions and octave equivalence, and I shouldn't have mentioned it in that context.
What I mean by a "privileged root" is taking one note from the chord, calling it the root, and treating it in a privileged way when you analyse the chord. In traditional harmony, the root of an x major or minor chord is x. It is considered that a chord is more consonant if the root is the lowest note and the root is the most common note in the chord. The problem with privileged roots in general is that you have to specify why one note in particular should be the root, and why it should have that privilege conferred upon it.
The best way, IMHO, of having octave invariance without inversional invariance is to take the lowest note in the chord to be the root, and measuring only positive intervals that include it. Hence, C-E-G-C contains a major third, perfect fifth and perfect unison (octave invariant octave). This is a sensible simplification, distinguishes all simple inversions of major and minor triads and tetrads, and is in good agreement with traditional harmony. A fundamental theory, though, has to explain why this approximation works. Although I don't know of such a theory, it would have to be octave specific.
Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 9 Jul 1997 01:11 +0200 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA22258; Wed, 9 Jul 1997 01:11:38 +0200 Date: Wed, 9 Jul 1997 01:11:38 +0200 Received: from ella.mills.edu by ns (smtpxd); id XA22294 Received: (qmail 8339 invoked from network); 8 Jul 1997 23:11:32 -0000 Received: from localhost (HELO ella.mills.edu) (127.0.0.1) by localhost with SMTP; 8 Jul 1997 23:11:32 -0000 Message-Id: <970708191023_-892188239@emout06.mail.aol.com> Errors-To: madole@mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu