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Topic: Dale Pond
Collected Articles Section: Nonequivalence of Music Intervals 1 of 2 Table of Contents to this Topic |
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Been playing around with that interval chart this morning. I "discovered" it only addresses counting by steps. This works pretty well but does not address frequency, etc. When frequency is applied it does not work as expected but totally unexpected. So there is something of importance to be discovered here, I presume. My thoughts are running to things concerning the very most basic premises of what intervals are or supposed to represent. The issues are so convoluted they are appearing as a paradox and the solving of paradoxes always leads to new revelations. The problem is in the definition of "step". Each step throughout the octave has a different size - and we know we can't add apples and oranges. Steps are a generalization and not arithmetical certainties. For instance if two Mn2 (16:15) are added together AS STEPS they equal a Mj2 (9:8). Very neat. But if we add 16:15 + 16:15 we get 32:15 which is a Mn9th!!! The error is apparently the result of a misinterpretation of what 16:15 is. The reality is a Mn2 is NOT 16:15 (in this case). It is 1/15 plus the whole of 1:1 or 15:15 (Unison) which equals 16:15. So 1/15 + 15:15 = 16:15. Since a Mj2 is 9:8 we must find the common denominator which is 120. 1/15 * 120 = 8. A Mn2 is 8 larger than Unison (120/120)+8/1 = 128/120. So each Mn2 step = 8. Which suggests a Mj2 is 8 larger than a Mn2 which works out to 136/120 where a Mn2 is 128/120. However error begins to creep in due to nonequivalence of fractional parts. In this case the error is 1/120. Because 16:15*120=128/120 and 9:8*120=135/120. The bottom line is a Mn2 = 1/15 (8/120) over and above Unison. and a Mj2 = 1/8 (15/120) over and above Unison. A Mn2 step (from Unison) = 8/120 A Mj2 step (from Unison) = 15/120 The Mj2 which we consider as 2 X Mn2 is actually smaller than this by 1/120 of the octave. The two intervals are not octave harmonically equivalent or proportional by 1/120th of the octave. It is presumed similar differences exist throughout the interval chart. Hope the arithmetic doesn't turn you two off. This may not seem important and may even seem a waste of time. But I submit neither premise is correct. Musicians count by steps because that is the way we've been taught - and it works for practical music purposes. The fact is, steps are unequal and disproporational arithmetically and therefore when we think they create harmony they are actually creating discord. In such suppositions we are operating from a place of illusion and nonreality and therefore we find ourselves fumbling around in the dark and not getting what we want or expect. Harmonic / Enharmonic Scale all difference tones are base 2^3 and will not create discords. Octave C" 240/120 Minor 8th Cb 232/120 Major 7th B 224/120 Minor 7th Bb 216/120 Dim. 7th A# 208/120 Major 6th A 200/120 Minor 6th G# 192/120 Major 5th G 184/120 Minor 5th Gb 176/120 Major 4th F# 168/120 Minor 4th F 160/120 Major 3rd E 152/120 Minor 3rd D# 144/120 Major 2nd D 136/120 Minor 2nd C# 128/120 Unison C 120/120 |
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