I always preferred the simple explanation that if you play only white keys on the piano in ascending order, the different modes are based upon which key you start from.
~Kyle
On 8/16/06, Kyle Klipowicz kyleklip@gmail.com wrote:
Only if you study abstract algebra (and I think that musicians are a subset of mathematicians less than the converse). :-)
~Kyle
On 8/15/06, Mathieu Bouchard matju@artengine.ca wrote:
On Tue, 15 Aug 2006, Alexandre Quessy wrote:
All seven modes follow the exact same spacing between degrees. There is a convention using bows and circumflex accents for respectively tones and semi-tones.
I prefer the digits 2 and 1, which represent the number of semitones between degrees. This is because they are more readily parseable by pd and by other computer systems.
All other modes are just the same intervals starting at an other point.
in other words, let Z denote semitone-multiple intervals, and 12Z denote octave-multiple intervals, then 12Z is an Ideal Subring of Z, so the Quotient Z/12Z is a Ring, which is the chromatic scale. Then take a subset of that chromatic scale that includes its zero (zero represents the fundamental). For example, {0,2,4,5,7,9,11}, which is made of the partial sums of the sequence (2,2,1,2,2,2,1), is the Ionian mode. A mode is a scale that contains its zero. Then define the addition of an interval and a scale: the resulting scale contains each sum of the interval with each element of the scale. The seven ecclesiastic modes appear as each of the sums of the Ionian mode and an interval that results in a mode (that is, that contains the zero).
Isn't that clearer? ;)
http://en.wikipedia.org/wiki/Ring_%28mathematics%29
_ _ __ ___ _____ ________ _____________ _____________________ ... | Mathieu Bouchard - tél:+1.514.383.3801 - http://artengine.ca/matju | Freelance Digital Arts Engineer, Montréal QC Canada
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