Tempering out: Difference between revisions
add related term “tempering together”, which was already widely used but not specifically defined throughout the wiki |
m Fix inverted superscripts in denominator of prime factors fraction of 81/80. |
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== Overview == | == Overview == | ||
For a tone measured as a ratio to "disappear", it must become equal to 1/1, so that multiplying by the ratio does not change anything. For a tone measured in cents to "disappear", it must become 0 cents, so that adding it does not change anything. | For a tone measured as a ratio to "disappear", it must become equal to 1/1, so that multiplying by the ratio does not change anything. For a tone measured in cents to "disappear", it must become 0 cents, so that adding it does not change anything. | ||
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== Tempering together == | == Tempering together == | ||
Two or more [[chord]]s or intervals are said to be ''tempered together'' if the commas that relate all of their corresponding steps are ''tempered out''. If two chords are tempered together, then the representation of each of those chords in the temperament is identical. | |||
For example, in [[meantone]], which tempers out [[81/80]], the chords [[54:64:81]] (with steps 32/27, 81/64) and [[10:12:15]] (with steps 6/5, 5/4) are tempered together. Since {{nowrap|{{Frac|32|27}}{{dot}}{{Frac|81|80}} {{=}} {{Frac|6|5}}}} and {{nowrap|{{Frac|81|64}} {{=}} {{Frac|5|4}}{{dot}}{{Frac|81|80}}}}, equating 81/80 with 1/1 also equates 32/27 with 6/5, and 81/64 with 5/4. | |||
For example, in [[meantone]], which tempers out [[81/80]], the chords [[54:64:81]] (with steps 32/27, 81/64) and [[10:12:15]] (with steps 6/5, 5/4) are tempered together. Since {{Frac|32|27}}{{dot}}{{Frac|81|80}} = {{Frac|6|5}} and {{Frac|81|64}} = {{Frac|5|4}}{{dot}}{{Frac|81|80}}, equating 81/80 with 1/1 also equates 32/27 with 6/5, and 81/64 with 5/4. | |||
== Example == | == Example == | ||
The syntonic comma is 81/80. That is {{sfrac|3<sup>4</sup>|2<sup>4</sup> × 5}} or, in [[monzo]] form, {{monzo| -4 4 -1}}. | |||
The syntonic comma is 81/80. That is | |||
19edo tempers out 81/80. (Technically, we should say that 19edo tempers out 81/80 when you use the [[patent val]].) You can see this in several ways: | 19edo tempers out 81/80. (Technically, we should say that 19edo tempers out 81/80 when you use the [[patent val]].) You can see this in several ways: | ||
=== 1. Counting steps of the val === | === 1. Counting steps of the val === | ||
Because there are no primes larger than 5 in 81/80, we say it is a 5-limit comma. The 5-limit patent val for 19edo is {{val| 19 30 44 }}. That means that you add 19 steps of 19edo to get to 2/1, 30 steps to get closest to 3/1, and 44 steps to get closest to 5/1. | Because there are no primes larger than 5 in 81/80, we say it is a 5-limit comma. The 5-limit patent val for 19edo is {{val| 19 30 44 }}. That means that you add 19 steps of 19edo to get to 2/1, 30 steps to get closest to 3/1, and 44 steps to get closest to 5/1. | ||
Note that, because this is an edo, 19 steps gets you precisely to 2/1. We say that 30 steps of 19edo gets you to 3/1, but that is only an approximation. Same with 5/1, etc. This is where the error in the primes gets introduced. Don't worry, though, it is very useful error. | Note that, because this is an edo, 19 steps gets you precisely to 2/1. We say that 30 steps of 19edo gets you to 3/1, but that is only an approximation. Same with 5/1, etc. This is where the error in the primes gets introduced. Don't worry, though, it is very useful error. | ||
Getting to 81 is 3×3×3×3, or, with 19edo steps, 30 + 30 + 30 + 30 = 120 steps of 19edo. | Getting to 81 is 3×3×3×3, or, with 19edo steps, {{nowrap|30 + 30 + 30 + 30 {{=}} 120}} steps of 19edo. | ||
Getting to 80 is 5×2×2×2×2, or, with 19edo steps, 44 + 19 + 19 + 19 + 19 = 120 steps of 19edo. | Getting to 80 is 5×2×2×2×2, or, with 19edo steps, {{nowrap|44 + 19 + 19 + 19 + 19 {{=}} 120}} steps of 19edo. | ||
Getting to 81/80 means adding the steps needed to get to 81, and subtracting the steps needed to get to 80. 120 steps | Getting to 81/80 means adding the steps needed to get to 81, and subtracting the steps needed to get to 80. {{nowrap|120 steps − 120 steps {{=}} 0 steps}}. | ||
Applying the monzo to the val (also called getting the ''homomorphism'') is easier. Multiply the first number in the monzo (which represents the number of 2/1's in the comma) and by the first number in the val (which represents the number of steps it takes to get to 2/1), then multiply the second number in the monzo by the second number in the val, then the third by the third, and add them all together: ( | Applying the monzo to the val (also called getting the ''homomorphism'') is easier. Multiply the first number in the monzo (which represents the number of 2/1's in the comma) and by the first number in the val (which represents the number of steps it takes to get to 2/1), then multiply the second number in the monzo by the second number in the val, then the third by the third, and add them all together: {{nowrap|(−4 × 19) + (4 × 30) + (−1 × 44) {{=}} 0 steps}}. | ||
Therefore, adding 81/80 to any interval in 19edo means adding 0 steps of 19edo to it. In other words, 81/80 is effectively zero: 81/80 is ''tempered out''. | Therefore, adding 81/80 to any interval in 19edo means adding 0 steps of 19edo to it. In other words, 81/80 is effectively zero: 81/80 is ''tempered out''. | ||
=== 2. Painstakingly doing the math === | === 2. Painstakingly doing the math === | ||
We say that 30 steps of 19edo gets you to 3/1, but, as we say above, that is an error. One step of 19edo is the 19th root of 2, or 2<sup>1/19</sup>, or approximately 1.037155. (That is 63.157895 cents.) If you multiply that by itself 19 times, you get exactly 2. But if you multiply that by itself 30 times, you do not get 3: You get 2.987518. Similarly, multiplying it by 44 steps gets you 4.97877 instead of 5. | |||
We say that 30 steps of 19edo gets you to 3/1, but, as we say above, that is an error. One step of 19edo is the 19th root of 2, or 2<sup>1/19</sup>, or approximately 1. | |||
If we plug in these values into 81/80, we see that 81/80 is tempered out: | If we plug in these values into 81/80, we see that 81/80 is tempered out: | ||
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81/80 = 3*3*3*3 / 5*2*2*2*2 = (3^4) / (5)*(2^4). // Substitute our values and you get | 81/80 = 3*3*3*3 / 5*2*2*2*2 = (3^4) / (5)*(2^4). // Substitute our values and you get | ||
(2. | (2.987518 ^ 4) / (4.97877)*(2^4) | ||
= 79. | = 79.660326 / (4.97877 * 16) | ||
= 79. | = 79.660326 / 79.660326 | ||
= 1/1 | = 1/1 | ||
</pre> | </pre> | ||
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== Notes == | == Notes == | ||
<references/> | <references /> | ||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||