... I see by the dates on my Soundcloud page that it must be about nine months (!) since I mentioned on this list tempering 36/35 and 25/24 to one step forMessage 1 of 124 , Feb 22, 2012View Source--- In email@example.com, "gbreed@..." <gbreed@...> wrote:
>I see by the dates on my Soundcloud page that it must be about nine months (!) since I mentioned on this list tempering 36/35 and 25/24 to one step for septimal music. So I can hardly expect anyone to remember that.
> If you're using runs of 36:35 or 25:24 then yes you'll tend towards >Magic temperament. But you didn't say that before so I didn't >comment on it.
>You'll also tend towards the MOS scales.If that were the only consideration, yes. Introducing the syntonic comma via Pythagorean structures messes up this tidiness, however.
> The message was "Seventh chords, intonation, and function." The >chords as I worked them out use twenty thirds piled up above the >tonic. All but one outlying pitch fit in the sixteen note MOS. >Seventh degrees above ii would be more remote but I don't know if >they're required or how they're tuned.But the chords are not all "above the tonic". I explained this carefully, but okay, more slowly this time.
(When we say above or below, we are of course talking about first or root position).
Let us call our tonic C. A tertian chord is built on, above, this C. Now, this C remains as a pedal tone, and a tertian chord forms BELOW the C. In the example I presented, C became the 11th partial of the following chord.
In traditional terms, a common-tone modulation which is also an enharmonic modulation.
Even in tempered systems using 41 or 53 tones, you already have both an "F#" and a "Gb", and a lot of different pitches. In the example I more recently presented, in which the "C" is a pedal and the chords move such that C is the eleventh partial, then the ninth, then seventh, and so on down, more than 30 discrete pitches are required even without full 11th chords at each new identity for C.
The scheme itself is obviously very simple. But it generates masses of pitches very quickly, and adding root movement by 5ths/4ths will generate even more, offset by syntonic commas.
53-edo- all of it- is perfect for implementing this scheme up to the 11th partial and probably the 13th as well.
The case of distinguishing between harmonic and "dominant" seventh chords also generates many tones and easily steps out of MOS structures. You need numerous instances of full harmonic structures, and Pythagorean root movement will entail needing these structures offset by the syntonic comma as well.
Once again, not once have I criticized the temperaments themselves. Quite the contrary- the examples I give could be titled "reasons to use temperament" (specifically 41 and 53 equal temperaments).
No, the problem is that of practical application, and the unavoidable fact that temperaments cannot both reach every prime at lightening speed and be accurate at the same time.
... The link shows that Wilson was still looking at the Cassandra/Schismatic (29&41) mapping. There s something to be said for that because it maps the 43Message 124 of 124 , Mar 14, 2012View Sourcekraiggrady <kraiggrady@...> wrote:
> I have uploadedThe link shows that Wilson was still looking at the
> this document which is not yet cataloged in the archives
> [i will probably add to the treasure section at some
> point] http://anaphoria.com/PartchMappedTo41.pdf if you
> look especially on page 6 even though the document was
> done in 2001 he references Secor. So in this case i think
> George might be giving more credit elsewhere than maybe
> he should.
Cassandra/Schismatic (29&41) mapping. There's something to
be said for that because it maps the 43 notes of the Genesis
scale to 41 distinctly tempered notes with the only
ambiguity being the unavoidable one from 11/10 and 10/9
mapping to the same pitch. This means we have to ignore
100:99. (Two notes disappear because each pitch occurs
with its octave complement.)
You may recall that each of the 43 pitches is distinct in
Miracle (31&41) temperament. For the earlier Exposition on
Monophony scale, these 43 distinct pitches are included in
the first 45 Miracle generated pitches.
I've been looking at the ambiguous 41 note temperings
again. The coincidence of 43 just pitches mapping to a 41
note MOS doesn't hold with Cassandra and the Exposition
scale. 49/48 maps to 29 fifths.
To get exactly 41 tempered pitches, a rank 2 mapping has to
be consistent with 41-equal and temper out 100:99. It must
have an 11-limit complexity of no more than 20. The
highest complexity for an 11-limit pitch will then be 20
generators from the 1/1. Each pitch occurs with its octave
complement in the diamond, and the 11-limit diamond is a
subset of either Partch scale, so there will be 40
generators between these two worst-case pitches. 40
generators means 41 pitches.
The only mappings that comply with these conditions are
those for Cassandra, Octacot (31&27e), and Magic (19&22).
It turns out that Octacot needs 57 generated notes to
approximate either 43 note scale. But Magic needs exactly
41 for either scale. So both scales correspond to
periodicity blocks that detemper the 41 note Magic MOS with
two ambiguous pairs of pitches. Only Magic works like this
for both scales. Only Magic and Cassandra work like this
for either scale.
The discrepancy between the Genesis and Exposition scales
amounts to an interval of 245:243. Tempering out 245:243
is the same as ignoring the difference between the two
scales. Combining this with the 100:99 that allows us to
fit the 11-limit diamond into 41 pitches is consistent with
The most accurate are Bohpier (41&8d), Magic, Octacot, and
Varan (41&5e). This is another way of producing a shortlist
that includes Magic.