Tritave talk

Besides all that comes with tritave equivalence and square timbres,the tritave concept has one feature that makes it different than octave scales on a abstract plane.It is that while the basis for octave consonances are the rather complex prime relations of the whole number harmonic series,the odd numbered forms a rather linear series of primes:3 5 7 (9) 11 13 (15) 17 19...

This makes me think that the tritave system does not lend itself to a scale composed of combination intervals like 9/8,5/3 and 15/8,rather the basis of the scale because the primes up to at least 13.An interesting scale is created by taking the first harmonics up to 27,tritave reduced:

1/1
11/9
13/9
5/3
17/9
19/9
7/3
23/9
25/9
3/1 (repeat)

The 5/3 takes on the role as"most consonant interval",quite like the 3/2 in octave scales.However,i do not find it appropriate to create a scale(perhaps approximating the other intervals)through a circle of 5/3s,like the octave cycle of fifths.

Rather,the next most consonant interval proves to be the 7/3,thus forming a triad with the root and the 5/3.Note that the most consonant interval gets in the middle,as opposed to the 4-5-6 major triad where it is on top.These two ratios are excellently approximated by a 13 note equal division of the tritave.

Since three quite widely spaced notes is hardly enough for melodic variety,more notes need to be introduced to form a melodic scale.Various modes can be created through combinations of the two basic intervals,however,none of the new ratios introduced are particularly consonant and thus gives no possibilities for new harmonic counterpoint.

Since both 25/9 and 35/27 are both harmonically quite harsh,the harmonic language is still restricted to the 3:5:7 triad and it�s utonal/minor/subharmonic/upside-down version.No possibilities for"extended"chords like adding 9/8 or 15/8 to a major chord.Also,the base interval of 146 cents is rather large,smaller intervals probably being needed for melodies.

Thus the need for including 11 and 13-limit intervals.Again,these do not create any particularly interesting"combination tones",so adding the next pair of primes(17 and 19)we are still left with a gap between the 7/3 and the tritave.These hold the prime number 23/9,and the first non prime tone,25/9.One more odd-harmonic step back to the tritave,technically the same as the tonic.

This basic 9-note harmonic(and the respective subharmonic)series is a grand repository of both consonant and distinctive harmonies,however they all give a strong feel of the same one tonic.A scale is needed that encompasses different ways to combine the intervals,preferably an equal scale,which hold all possible combinations of the ratios within a set of microintervals.

So the question is:which equal scale is the best to approximate the 9-27 odd harmonic series?The following table is a basic attempt to display some equal divisions of the tritave,44 being the highest mentioned here.

To the left of each row is the just values for each one of the 7 basic prime pitches given in OCTAVES.The practicality of this can probably be discussed,and some have been reduced to their subharmonic version(when it is lower).In any case the degree of accuracy of each ET approximation to each interval can be clearly seen.

Just 13 17 18 20 21 22

.7369 .7315 .7458 .7044 .7132 .7547 .7204
.3625 .3658 .3729 .3522 .3962 .3774 .3602
.2895 .2797 .2641 .3019 .2881
.5305 .5593 .5283 .5548 .5283
.6674 .6526 .6792 .6484
.5070 .4877 .5043
.2313 .2438 .2377 .2264

Just 23 26 27 28 29 30

.7369 .7315 .7358 .7105 .7396
.3625 .3445 .3658 .3522 .3962 .3825 .3698
.2895 .2756 .3048 .2935 .2830 .2732 .2642
.5305 .5512 .5486 .5283 .5465 .5283
.6674 .6891 .6706 .6457 .6792 .6558 .6868
.5070 .4823 .4877 .5094 .4919
.2313 .2438 .2348 .2264 .2113

Just 32 33 34 35 36 38

.7369 .7429 .7204 .7458 .7245 .7484 .7508
.3625 .3467 .3842 .3623 .3522 .3753
.2895 .2972 .2882 .2796 .2717 .3082 .2920
.5305 .5448 .5283 .5594 .5434 .5283 .5422
.6674 .6439 .6724 .6526 .6792 .6604 .6673
.5070 .4953 .5128 .4981 .5005
.2313 .2477 .2401 .2330 .2264 .2201

Just 39 40 41 42 43 44

.7369 .7315 .7528 .7344 .7547 .7372 .7204
.3625 .3658 .3566 .3479 .3774 .3686 .3602
.2895 .2844 .2774 .3093 .3019 .2949 .2882
.5305 .5283 .5547 .5412 .5283
.6674 .6502 .6736 .6572 .6793 .6635 .6844
.5070 .4876 .5151 .5025 .4906 .5160 .5043
.2313 .2438 .2377 .2319 .2264 .2211

The"gaps"in some rows indicate that the particular ETs approximation of the interval is too far away to even be considered.A revised version of this would hold the values in _difference from the just value_,which i found too troublesome to calculate this time.The most interesting ones are 17,22,26,28,30,36,39 and 43.

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Mats �ljare
Eskilstuna,Sweden
http://www.angelfire.com/mo/oljare
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Mats wrote,

>Thus the need for including 11 and 13-limit intervals.

That leads inevitably to the triple-BP scale, with 39 steps per tritave. I
wouldn't put too much importance on 17-limit and 19-limit when you're
excluding all even numbers from your chords.