Einstein, linear temps

I had forgotten Bose-Einstein statistics, which is strange
because I had just read recently of the observation of interference
between two Bose-Einstein condensates and an "atom" laser.

I do believe that it is accepted that Einstein wasted the last
years of his career on an attempt to unify electromagnetism and
gravity while ignoring the strong and weak nuclear forces.

Following up Paul Erlich's suggestion, I have computed
the LST SQR positive tunings with 7/6 and 5/3 in addition
to 3/2, 5/4, and 7/4. The error functions are (F-X), (T+8X-5oct),
(S+14X-9oct), (MS+9X-6oct) and (ts+15X-9oct). F2,
T4, S4, MS3 and ts6. TFmpered fifth.


LST Pos 3, 5, 7, 5/3, 7/6 702.0909887095729
TF(2*F-16*T-28*S-30*ts-18*MS+710*OCT)/1134
E3.1359878 E321.849269E-02
E53.041624 E529.251473
E71.900252 E723.610956
E353.177611 E3529.269966
E372.036239 E3723.629449
E574.941875 E57212.86243
E3575.077863 E357212.88092
EFETEMS6.355223 EFETEMS219.36718
EFESEts3.800503 EFESEts26.742075
EFETESEMSEts10.01973812073231 EFETESEMSEts226.09076


LST Pos 3, 7, 7/6 702.216418782563
TF(2*F-28*S-30*ts+522*OCT)/844
E3.2614179 E326.833933E-02
E54.045064 E5216.36254
E7.1442306 E722.080246E-02
E354.306482 E35216.43088
E37.4056485 E3728.914179E-02
E574.189295 E57216.38335
E3574.450713 E357216.45168
EFETEMS8.612964 EFETEMS234.97667
EFESEts.5228359 EFESEts2.1028747
EFETESEMSEts8.874382002202765 EFETESEMSEts235.01121


LST Pos 3, 5, 5/3 701.7275128284361
TF(2*F-16*T-18*MS+188*OCT)/292
E3.227488 E325.175081E-02
E5.1338165 E521.790685E-02
E76.988914 E7248.84492
E35.3613045 E3526.965766E-02
E377.216402 E37248.89667
E577.12273 E57248.86282
E3577.350218 E357248.91458
EFETEMS.4549761 EFETEMS27.843202E-02
EFESEts14.4328 EFESEts2100.9731
EFETESEMSEts14.66029197639307 EFETESEMSEts2100.9998


These tunings, like the rest of the singly-positive analogs, resemble
53- or 41-tet. They might be practical as a 24-tone set on two keyboards.
With more difficulty, tunings analogous to 22-, 34-, 46-tet may be
constructed.


--John


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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

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