Marvel woo: Difference between revisions

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'''Marvel woo''' is a particular tuning of Marvel which is optimized for synchronized ~~beating~~, and which also happens to be very close to the [[Tenney-Euclidean_Tuning|TE tuning]] for Marvel.

'''Marvel woo''' is a particular tuning of Marvel which is optimized for synchronized [[beat]]ing, and which also happens to be very close to the [[Tenney-Euclidean_Tuning|TE tuning]] for Marvel.

[[Marvel family|Marvel]] is the rank-3 [[Tour of Regular Temperaments|temperament]] tempering out 225/224, the [https://en.wikipedia.org/wiki/Septimal_kleisma septimal kleisma]. For Marvel woo, we extend Marvel into the 11-limit by tempering out [[385/384]].

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== Math ==

Marvel woo is the marvel tuning with 10/3, 7/2 and 11 as ~~eigenmonzos~~. 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 [|0 -4 4 4 4>, |-21 6 -6 15 8>, |7 -18 18 11 4>, |-28 -4 4 32 4>, |0 0 0 0 28>]/28. It leads to a tuning where the octave is sharp by |-7 -1 1 1 1>/7 = (385/384)^(1/7), about 0.643 ~~cents~~. In this tuning, 9/5 and 12/7 are sharp by only |-49 -26 -2 19 12>/28 = (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 ~~ratios~~ 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 1-5/4-3/2-7/4-2, the approximate 5/4 beats approximately 20 times, 3/2 12 times, and 7/4 7 times; the actual numbers being 19.968, 11.977, 6.997 ~~and 8~~ respectively..

Marvel woo is the marvel tuning with 10/3, 7/2 and 11 as [[eigenmonzo]]s. 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 [|0 -4 4 4 4>, |-21 6 -6 15 8>, |7 -18 18 11 4>, |-28 -4 4 32 4>, |0 0 0 0 28>]/28. It leads to a tuning where the octave is sharp by |-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 |-49 -26 -2 19 12>/28 = (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 1-5/4-3/2-7/4-2, 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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-'''Marvel woo''' is a particular tuning of Marvel which is optimized for synchronized beating, and which also happens to be very close to the [[Tenney-Euclidean_Tuning|TE tuning]] for Marvel.
+'''Marvel woo''' is a particular tuning of Marvel which is optimized for synchronized [[beat]]ing, and which also happens to be very close to the [[Tenney-Euclidean_Tuning|TE tuning]] for Marvel.
 [[Marvel family|Marvel]] is the rank-3 [[Tour of Regular Temperaments|temperament]] tempering out 225/224, the [https://en.wikipedia.org/wiki/Septimal_kleisma septimal kleisma]. For Marvel woo, we extend Marvel into the 11-limit by tempering out [[385/384]].
@@ Line 7: / Line 7: @@
 == Math ==
-Marvel woo is the marvel tuning with 10/3, 7/2 and 11 as eigenmonzos. 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 [|0 -4 4 4 4&gt;, |-21 6 -6 15 8&gt;, |7 -18 18 11 4&gt;, |-28 -4 4 32 4&gt;, |0 0 0 0 28&gt;]/28. It leads to a tuning where the octave is sharp by |-7 -1 1 1 1&gt;/7 = (385/384)^(1/7), about 0.643 cents. In this tuning, 9/5 and 12/7 are sharp by only |-49 -26 -2 19 12&gt;/28 = (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 ratios 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 1-5/4-3/2-7/4-2, the approximate 5/4 beats approximately 20 times, 3/2 12 times, and 7/4 7 times; the actual numbers being 19.968, 11.977, 6.997 and 8 respectively..
+Marvel woo is the marvel tuning with 10/3, 7/2 and 11 as [[eigenmonzo]]s. 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 [|0 -4 4 4 4&gt;, |-21 6 -6 15 8&gt;, |7 -18 18 11 4&gt;, |-28 -4 4 32 4&gt;, |0 0 0 0 28&gt;]/28. It leads to a tuning where the octave is sharp by |-7 -1 1 1 1&gt;/7 = (385/384)^(1/7), about 0.643 [[cent]]s. In this tuning, 9/5 and 12/7 are sharp by only |-49 -26 -2 19 12&gt;/28 = (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 1-5/4-3/2-7/4-2, 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 ==