I've 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 Minor Seconds (16:15) are added together AS STEPS they equal a Major Second (9:8). Very neat.
But if we add 16:15 + 16:15 we get 32:15 which is a Minor Ninth!!!
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 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.
It is presumed similar differences exist throughout the interval chart. 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 23 and will not create discords.
PS: As far as I know my idea of resolved and unresolved beats is new and unique. I've not read of it in any music theory I've come across.
11.02 - Attributes of the Scale of Infinite Ninths 11.03 - Development of the Scale of Infinite Ninths 11.11 - Explanations of the Scale of Infinite Ninths Harmonic Interval Music Part 11 - SVP Music Model for greater detail and development of this concept and scale Syntropy