Marvel woo: Difference between revisions

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__TOC__

== Math ==

Marvel woo is the marvel tuning with 10/3, 7/2 and 11 as [[eigenmonzo]]s (unchanged-intervals). This gives three monzos with eigenvalue 1, and two with eigenvalue 0, allowing us to construct a projection matrix whose columns (or rows if you prefer) are fractional monzos, which defines the tuning. This matrix is [{{monzo| 0 -4 4 4 4 }}, {{monzo| -21 6 -6 15 8 }}, {{monzo| 7 -18 18 11 4 }}, {{monzo| -28 -4 4 32 4 }}, {{monzo| 0 0 0 0 28 }}]/28. It leads to a tuning where the octave is sharp by {{sfrac|{{monzo| -7 -1 1 1 1 }}|7}}~~ ~~= (385/384)1/7, about 0.643 [[cent]]s. In this tuning, 9/5 and 12/7 are sharp by only {{sfrac|{{monzo| -49 -26 -2 19 12 }}|28}}~~ ~~= {{sfrac|(385/384)3/7|(225/224)1/4}}, about 0.0018 cents. Putting 10/3, 7/2, 11, and 9/5 together with 2 leads to the full 11-limit. This means every interval in the 11-limit tonality diamond is either pure, ±0.0018 cents from pure, or a certain number of octaves away from an interval which is within 0.0018 cents of pure. Because of this, the [[beat ratio]]s of everything in the [[11-limit diamond]] are closely approximated by small integer ratios. For instance, for every eight beats of the [[octave]] in the chord 4:5:6:7:8, the approximate [[5/4]] beats approximately 20 times, [[3/2]] 12 times, and [[7/4]] 7 times; the actual numbers being 8, 19.968, 11.977 and 6.997 respectively.

Marvel woo is the marvel tuning with 10/3, 7/2 and 11 as [[eigenmonzo]]s (unchanged-intervals). This gives three monzos with eigenvalue 1, and two with eigenvalue 0, allowing us to construct a projection matrix whose columns (or rows if you prefer) are fractional monzos, which defines the tuning. This matrix is [{{monzo| 0 -4 4 4 4 }}, {{monzo| -21 6 -6 15 8 }}, {{monzo| 7 -18 18 11 4 }}, {{monzo| -28 -4 4 32 4 }}, {{monzo| 0 0 0 0 28 }}]/28. It leads to a tuning where the octave is sharp by {{sfrac|{{monzo| -7 -1 1 1 1 }}|7}} = (385/384)1/7, about 0.643 [[cent]]s. In this tuning, 9/5 and 12/7 are sharp by only {{sfrac|{{monzo| -49 -26 -2 19 12 }}|28}} = {{sfrac|(385/384)3/7|(225/224)1/4}}, about 0.0018 cents. Putting 10/3, 7/2, 11, and 9/5 together with 2 leads to the full 11-limit. This means every interval in the 11-limit tonality diamond is either pure, ±0.0018 cents from pure, or a certain number of octaves away from an interval which is within 0.0018 cents of pure. Because of this, the [[beat ratio]]s of everything in the [[11-limit diamond]] are closely approximated by small integer ratios. For instance, for every eight beats of the [[octave]] in the chord 4:5:6:7:8, the approximate [[5/4]] beats approximately 20 times, [[3/2]] 12 times, and [[7/4]] 7 times; the actual numbers being 8, 19.968, 11.977 and 6.997 respectively.

== Scales ==

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 __TOC__
 == Math ==
-Marvel woo is the marvel tuning with 10/3, 7/2 and 11 as [[eigenmonzo]]s (unchanged-intervals). This gives three monzos with eigenvalue 1, and two with eigenvalue 0, allowing us to construct a projection matrix whose columns (or rows if you prefer) are fractional monzos, which defines the tuning. This matrix is [{{monzo| 0 -4 4 4 4 }}, {{monzo| -21 6 -6 15 8 }}, {{monzo| 7 -18 18 11 4 }}, {{monzo| -28 -4 4 32 4 }}, {{monzo| 0 0 0 0 28 }}]/28. It leads to a tuning where the octave is sharp by {{sfrac|{{monzo| -7 -1 1 1 1 }}|7}}&nbsp;=&nbsp;(385/384)<sup>1/7</sup>, about 0.643 [[cent]]s. In this tuning, 9/5 and 12/7 are sharp by only {{sfrac|{{monzo| -49 -26 -2 19 12 }}|28}}&nbsp;=&nbsp;{{sfrac|(385/384)<sup>3/7</sup>|(225/224)<sup>1/4</sup>}}, about 0.0018 cents. Putting 10/3, 7/2, 11, and 9/5 together with 2 leads to the full 11-limit. This means every interval in the 11-limit tonality diamond is either pure, ±0.0018 cents from pure, or a certain number of octaves away from an interval which is within 0.0018 cents of pure. Because of this, the [[beat ratio]]s of everything in the [[11-limit diamond]] are closely approximated by small integer ratios. For instance, for every eight beats of the [[octave]] in the chord 4:5:6:7:8, the approximate [[5/4]] beats approximately 20 times, [[3/2]] 12 times, and [[7/4]] 7 times; the actual numbers being 8, 19.968, 11.977 and 6.997 respectively.
+Marvel woo is the marvel tuning with 10/3, 7/2 and 11 as [[eigenmonzo]]s (unchanged-intervals). This gives three monzos with eigenvalue 1, and two with eigenvalue 0, allowing us to construct a projection matrix whose columns (or rows if you prefer) are fractional monzos, which defines the tuning. This matrix is [{{monzo| 0 -4 4 4 4 }}, {{monzo| -21 6 -6 15 8 }}, {{monzo| 7 -18 18 11 4 }}, {{monzo| -28 -4 4 32 4 }}, {{monzo| 0 0 0 0 28 }}]/28. It leads to a tuning where the octave is sharp by {{sfrac|{{monzo| -7 -1 1 1 1 }}|7}} =&nbsp;(385/384)<sup>1/7</sup>, about 0.643 [[cent]]s. In this tuning, 9/5 and 12/7 are sharp by only {{sfrac|{{monzo| -49 -26 -2 19 12 }}|28}} =&nbsp;{{sfrac|(385/384)<sup>3/7</sup>|(225/224)<sup>1/4</sup>}}, about 0.0018 cents. Putting 10/3, 7/2, 11, and 9/5 together with 2 leads to the full 11-limit. This means every interval in the 11-limit tonality diamond is either pure, ±0.0018 cents from pure, or a certain number of octaves away from an interval which is within 0.0018 cents of pure. Because of this, the [[beat ratio]]s of everything in the [[11-limit diamond]] are closely approximated by small integer ratios. For instance, for every eight beats of the [[octave]] in the chord 4:5:6:7:8, the approximate [[5/4]] beats approximately 20 times, [[3/2]] 12 times, and [[7/4]] 7 times; the actual numbers being 8, 19.968, 11.977 and 6.997 respectively.
 == Scales ==