Two equal temperaments scale

The equal tempered scale should be modified to agree with the natural and perfect consonances F = (4/3) = 1,3333... and G = (3/2) = 1,5 that were known a long time before its proposal. Since hundred years ago, perfect fifths and fourths are the references used to classify the scales; their consonance degrees are unquestionable.

If we have to accept the equal tempered scale imperfection at least we could introduce in this scale the two perfect consonances. It means two temperaments: The first one controls the tone frequencies that work between C=1 and the new F = 1,3333.. The second section would comprise F, F# and G; this section has a range of (3/2)/(4/3) = (9/8) = 1,125. The third section is comprised between C = 2 and G = 1,5 with a range of (4/3) = 1,33333., the same range of the first section. So the first and third sections include five equal intervals each. Let us see the detailed proposal :

MODIFIED EQUAL TEMPERED SCALE

First section semitone factor = (4/3)0,2= 1,05922384105:

C ... 1

C# ... 1,05922384105

D ... 1,12195514545

Eb .. 1,18840163865

E ... 1,2587833484***

F ... 1,33333333333

Second section semitone factor = (9/8)0,5= 1,06066017178:

F# .. 20,5= 1,41421356237

G ... (3/2) = 1,5

Third section semitone factor = (4/3)0,2= 1,05922384105. The same as for the first section:

Ab ... 1,58883576157

A ... 1,68293271817

Bb ... 1,78260245796

B .. 1,88817502259

2C ... 2

Also the major thirds problem might be reduced.

It is much better to make an approach to the final scale than to maintain a rather imperfect one.

I suggest you to make a comparison with the equal tempered scale.

I would appreciate it if you could give me your opinion on this matter.

Thanks

Mario Pizarro

Lima, October 02

piagui@...