source file: mills2.txt Date: Sun, 16 Mar 1997 08:28:15 -0800 Subject: Re: Dominant 7th 4-5-6-7-8 ? From: Andrew Milne Paul Erlich wrote: > Andrew M. wrote, > > >the augmented sixth chord (e.g. g - b - d - e#) . . . is > >another chord that I think would be incorrectly represented by a 4-5-6-7 > >tuning > > But Andrew, the augmented sixth chord IS represented almost exactly by > 4:5:6:7 in meantone tuning. Do you therefore think meantone is not a good > tuning for music with augmented sixth chords? Remember that it was the > tuning in place when they arose. Well, first of all, thanks for such an interesting and penetrating question. Yes it is true that the augmented sixth as expressed in meantone (and even in 5-limit Just Intonation) is very close to the 4-5-6-7 chord. In 1/4 comma meantone the aug6 is only 3 cents flat of its 7-limit version (in J.I. it is just 7.7 cents sharp). The former certainly gives a very good (and the latter a pretty reasonable) aural analogue of 4-5-6-7, and as a 7-limit major tetrad it sounds very well tuned. So it does appear that there is a danger that an augmented sixth chord could lose its restlessness (i.e its function), but it must not be forgotten that the aug6th is a "highly chromatic" chord (as Paul Erlich correctly comments). In classical music of the meantone period, the augmented sixth is usually found built on the flat 2 or flat 6 of the scale. The spelling of bVIaug6 in C major would be Ab-C-D-F# or Ab-C-Eb-F#. The notes Ab, F# and Eb will seek to resolve by moving a minor second (diatonic semitone) in the direction of their alteration (this being the usual resolution of a chromatic note). Furthermore in the first version of the augmented sixth, the augmented fourth between Ab and D is highly dissonant (the meantone aug4, which is very close to 7/5, only seems to be consonant when it is part of a 4-5-6-7, 5-6-7, or 4-5-7 chord). The spelling of bIIaug6 in C major would be Db-F-G-B or Db-F-Ab-B. Once again a highly chromatic chord in C major - the first has an augmented fourth with the bass, the second has two chromatic notes. For these reasons it is unlikely that an augmented sixth can function as a tonic, despite its consonance. So, if a consonant (virtually 4-5-6-7) chord can be used as an aug6th chord why not use a 4-5-6-7 chord as a dominant (or indeed secondary dominant)? The augmented sixth chord sounds like a 4-5-6-7 chord by *chance*, there is no design here. The origin of this chord was not the result of a quest to maximise the consonance of a dominant seventh type chord, but to serve as a chromatic substitution for (alteration of) the second inversion of the dominant 7th whose *bass* note is a chromatic semitone above the aug6th's root (that, at least, is the conventional explanation for its origins). To actively choose 4-5-6-7 for a (secondary) dominant 7th, is to quite consciously introduce another note foreign to the major scale. In C major the 7/4 of the dominant chord is a note more than 27 cents below F, and as we have seen is well represented (in meantone) by the augmented 6th E#. If composers really wanted to minimise the dissonance of the dominant seventh, it would not have been beyond the notational possibilities of the time to spell G7 as G-B-D-E#. But no composer does this (unless of course the chord resolves *as an aug6th* (i.e. to F# major). To use this aug6th with the dominant chord consistently, one would have to reconstruct the C major scale as an 8 note scale - C, D, E, E#, F, G, A, B. But this E# only functions as a useful consonance in the G7 chord - it is a note which can only serve in one chord position (7/4 type seventh) in one chord (the 4-5-6-7 chord whose root is G). All the other notes can serve as root, third or fifth (and seventh etc.) except for F - which can't be a fifth, and B - which can't be a root. Furthermore it is only related to the other scale notes as a 7-limit ratio when the very foundation of the major (and minor scales) are the 3 and 5-limit ratios of the major and minor triads. It is therefore a somewhat superfluous and *unintegrated* note with respect to the C major scale. To use this extra-scalic E# (#3) is, therefore, a conscious and explicit device which is useful for only one purpose, and that purpose is to minimise the natural dissonance of the dominant seventh chord. BUT, it is important to remember that it should not be the purpose of a tuning system to remove all dissonance. Dissonance is the driving energy of music. In counterpoint the alternation of consonance and dissonance is fundamental to good form, and indeed without dissonance, creativity is, perhaps, impossible. In relation to a quite different discussion on the MTO-List, Nicholas Meeus wrote on 6th March 97: > I used to claim, when teaching the history of medieval > music, (and I was neither the first nor the only one to do so) that the > status of composer originated when dissonances came to be accepted as > possible intervals in composition. To write a consonant counterpoint > against a borrowed cantus firmus is but an automatic application of a > limited set of rules. To admit dissonances in the counterpoint is to give > oneself the freedom that transforms an act of ordinary craftmanship into > one of original creation. In this way, dissonances are the most essential > aspect of a composition. Yet, throughout the history of our music, > dissonances have indeed been presented as "supplementary" and > "unessential". Alternative tunings enable us to create strange and wonderful consonances not available to 5-limit tuning, but at the same time it is quite wrong to use tuning to eradicate all dissonances where-ever we find them. Dissonance must be embraced not discarded. > when I play around > in meantone, I often go to the augmented sixth and don't resolve -- it's a > nice bridge from 5-limit to 7-limit harmony. This is certainly an intriguing idea, and it makes one wonder if any classical composers or theorists of the time were aware of, or discussed, the unexpected consonance of the aug6th chord. Andrew Milne Islington London Received: from ns.ezh.nl [137.174.112.59] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sun, 16 Mar 1997 17:30 +0100 Received: by ns.ezh.nl; (5.65v3.2/1.3/10May95) id AA30148; Sun, 16 Mar 1997 17:30:19 +0100 Received: from ella.mills.edu by ns (smtpxd); id XA30147 Received: from by ella.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) id IAA23131; Sun, 16 Mar 1997 08:28:39 -0800 Date: Sun, 16 Mar 1997 08:28:39 -0800 Message-Id: Errors-To: madole@mills.edu Reply-To: tuning@ella.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@ella.mills.edu