Don't hesitate to correct me where I am wrong, or at least confusing.
"Would the bps of all four at some location in the keyboard equally tempered coincide? The greatest specificity for my question is eliminated when I am forced to use certain notes to explain my question. I use a minor 6th above the fundamental to check both the 8:4 octave and the 8:6 fourth. Could we perfect the temperament by determining at some stage of tuning not only where the partials in the octave should coincide, or the two in the forth, but where all four partials against the check should possess coincident bps?"
Should I say of all three because both intervals utilize the same fundamental? Which more clearly demonstrates what I am trying to communicate, 3 or 4? The idea is that I am trying to match the partials of two intervals with the same fundamental note, not one note to the partial of an octave.
Perhaps removing numbers for intervals would better explain my contention about 2nds, 3rds, 6ths, and 7ths partial insignificance, but fundamental importance in relation to tonic.
After tonic, the names of the remaining scale degrees (of a diatonic scale) in order are as follows:
supertonic<http://en.wikipedia.org/wiki/Supertonic> - second scale degree (the scale degree immediately "above" the tonic);
mediant<http://en.wikipedia.org/wiki/Mediant> - third scale degree (the "middle" note<http://en.wikipedia.org/wiki/Note_(music)> of the tonic triad);
subdominant<http://en.wikipedia.org/wiki/Subdominant> - fourth scale degree (a fifth "below" the tonic);
dominant<http://en.wikipedia.org/wiki/Dominant_(music)> - fifth scale degree (the most "pronounced" harmonic<http://en.wikipedia.org/wiki/Harmonic> note after the tonic);
submediant<http://en.wikipedia.org/wiki/Submediant> - sixth scale degree (the "middle" note of the subdominant triad);
leading tone<http://en.wikipedia.org/wiki/Leading_tone> (or leading note) - seventh scale degree (the scale degree that "leads" to the tonic);
subtonic<http://en.wikipedia.org/wiki/Subtonic> - also seventh scale degree, but applying to the lowered 7th found in the natural minor scale<http://en.wikipedia.org/wiki/Minor_scale>.
http://en.wikipedia.org/wiki/Tonic_%28music%29
The terms mædiant, submædiant, supertonic, subtonic, and leading tone, while these lack a terse method of writing M/m, help indicate through language that we need not consider the coincident partials of these intervals from any single fundamental that makes them 2nds, 3rds, 6ths, or 7ths. Obviously these possess their own. Supertonic and subtonic indicate a relationship to the fundamental that is for the most part identified in tonic itself, not its own set of coincident partials. Mædiant is defined this way as well:
me·di·ant (md-nt)
n.
The third tone in a diatonic musical scale, determining the major or minor quality of the tonic chord.
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[Italian mediante, from Late Latin medins, mediant-, present participle of medire, to be in the middle, from Latin medius, middle; see medium.]
http://www.thefreedictionary.com/mediant
Thirds are just that. They are in the middle of the subdominant and supertonic, but within themselves in relation to the fundamental, are not something the tonal spectrum says anything about. On the contrary, it is the third and the sixth and the tenth and so on that tell us about the tonal spectrum, not the other way around.
From: caut-bounces at ptg.org [mailto:caut-bounces at ptg.org] On Behalf Of Sloane, Benjamin (sloaneba)
Sent: Saturday, March 21, 2009 9:33 PM
To: caut at ptg.org
Subject: Re: [CAUT] Temperment check inquiry
That was the quick answer. I thought it was incomplete enough to make an effort to explain more. The diagrams are microsoft office 2007, for windows vista, so I hope the compatibility issues of the attachment are tolerable.
Good luck with the 12:9 fourths!
________________________________
From: caut-bounces at ptg.org [caut-bounces at ptg.org] On Behalf Of Sloane, Benjamin (sloaneba) [sloaneba at ucmail.uc.edu]
Sent: Sunday, March 15, 2009 5:14 PM
To: caut at ptg.org
Subject: Re: [CAUT] Temperment check inquiry
The theory behind checks for intervals as I now understand it, though my understanding of aural tuning is constantly changing, is that the note we use to check the interval also has a coincident partial that agrees with the coincident partials of the interval we use the check for.
So with the first example you gave, more specifically, checking the fourth of E3 to A3 with C3, we find that the partials coincide for all these at E5, being the 5th partial of C3, the 4th partial of E3, and the 3rd partial of A3.
Technically, then, partials do coincide, more remotely, with the check you suggested for D# to G#, i.e. the C above both. E.g., we find that the 12th partial of D#3, and the 9th partial of G# 3, coincide with the 7th partial of C4 at A#6, I believe, if I calculated correctly. It is more remote than the more traditional check, but theoretically, it should work in some manner, because the partial coincide.
________________________________
From: caut-bounces at ptg.org [caut-bounces at ptg.org] On Behalf Of Dan Reed [pianoarts at tx.rr.com]
Sent: Sunday, March 15, 2009 2:36 PM
To: caut at ptg.org
Subject: [CAUT] Temperment check inquiry
I am taking a fresh look at refining my temperment...(hey..sounds like an ad for Xanax!)
Here is the question...Does anyone use a 'reverse', M3 / M6 check for checking the P4th width?
'Reverse', meaning, instead of, for example, C up to E(3rd) and C up to A(6th) to check and set the width of 'A' in the E -A P4th.
Will this work going down? Checking the P4th width.... C down to G# (3rd) and C down to D#(6th) to check/set the D# width in the G# down to D# P4...
The point is, this 3rd/6th test is useful, when 2 notes in a M3rd are known...as in any contiguous M3rd...
Why not use the test backwards, going down?
My question is...can this beat rate comparison be useful going in reverse?
Going down, the M3rd is faster than the M6th...So, in setting the M6th, it would beat 'slightly' slower than the M3rd...
Dan Reed
Dallas, Tx
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