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.

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. 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.

Keep in mind this scale is not a "music scale". It is a scale of vibration or oscillation as found in nature - and not in man's sensorial audio pleasures known as music.

Harmonic / Enharmonic Scale

All difference tones are base 2³ 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

2002, 09/01 Below is a "new" scale (subject to modification) that accounts for Harmony in such a way that all notes are equivalent and proportional. Any note and its immediate harmonics will form Harmony with all other notes and their immediate harmonics. No unresolved beats will occur - unresolved beats are a source of dissonance. A beat will resolve when it forms Harmony with the notes around it. All beats generated with this scale are multiples of two as are all the notes as also immediate harmonics. So any beats will resolve (form harmony) between themselves and the surrounding notes. If we use conventional music intervals beats occur which are unresolveable which means they may have a base of 1, 2, 3, 5 or 7. These odd numbers do not form Harmony or unison with anything other than themselves. Theoretically resolved beats will "disappear" to the ear as they merge with neighboring sounds. I don't know this for a practical thing but arithmetically and vibratorily this appears to be the case. The proof will come in the actually playing and hearing of it.

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.

The Dale Scale

Interval	Note	cps/120
Octave	C"""	2048
Minor 8th	Cb	1920
Major 7th	B	1856
Minor 7th	Bb	1792
Dim. 7th	A#	1728
Major 6th	A	1664
Minor 6th	G#	1600
Major 5th	G	1536
Minor 5th	Gb	1472
Major 4th	F#	1408
Minor 4th	F	1344
Major 3rd	E	1280
Minor 3rd	D#	1216
Major 2nd	D	1152
Minor 2nd	C#	1088
Octave	C""'	1024
Minor 8th	Cb	960
Major 7th	B	928
Minor 7th	Bb	896
Dim. 7th	A#	864
Major 6th	A	832
Minor 6th	G#	800
Major 5th	G	768
Minor 5th	Gb	736
Major 4th	F#	704
Minor 4th	F	672
Major 3rd	E	640
Minor 3rd	D#	608
Major 2nd	D	576
Minor 2nd	C#	544
Octave	C""	512
Minor 8th	Cb	480
Major 7th	B	464
Minor 7th	Bb	448
Dim. 7th	A#	432
Major 6th	A	416
Minor 6th	G#	400
Major 5th	G	384
Minor 5th	Gb	368
Major 4th	F#	352
Minor 4th	F	336
Major 3rd	E	320
Minor 3rd	D#	304
Major 2nd	D	288
Minor 2nd	C#	272
Octave	C'	256
Minor 8th	Cb	240
Major 7th	B	232
Minor 7th	Bb	224
Dim. 7th	A#	216
Major 6th	A	208
Minor 6th	G#	200
Major 5th	G	192
Minor 5th	Gb	184
Major 4th	F#	176
Minor 4th	F	168
Major 3rd	E	160
Minor 3rd	D#	152
Major 2nd	D	144
Minor 2nd	C#	136
Octave	C	128
Minor 8th	Cb	120
Major 7th	B	116
Minor 7th	Bb	112
Dim. 7th	A#	108
Major 6th	A	104
Minor 6th	G#	100
Major 5th	G	96
Minor 5th	Gb	92
Major 4th	F#	88
Minor 4th	F	84
Major 3rd	E	80
Minor 3rd	D#	76
Major 2nd	D	72
Minor 2nd	C#	68
Octave	C'	64
Minor 8th	Cb	60
Major 7th	B	58
Minor 7th	Bb	56
Dim. 7th	A#	54
Major 6th	A	52
Minor 6th	G#	50
Major 5th	G	48
Minor 5th	Gb	46
Major 4th	F#	44
Minor 4th	F	42
Major 3rd	E	40
Minor 3rd	D#	38
Major 2nd	D	36
Minor 2nd	C#	34
Unison	C	32

2005, 01/04: This scale was initially called "The Dale Scale" at that time for lack of a better name. That name was changed to "Scale of Infinite Ninths" some time in 2003.

See Also

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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