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| Another use for the Riemann zeta function is to determine the optimal tuning for an EDO, meaning the optimal octave stretch. This is because the zeta peaks are typically not integers. The fractional part can give us the degree to which the generator diverges from what you would need to have the octave be a perfect 1200 cents. Here is a list of successively higher zeta peaks, taken to five decimal places: | | Another use for the Riemann zeta function is to determine the optimal tuning for an EDO, meaning the optimal octave stretch. This is because the zeta peaks are typically not integers. The fractional part can give us the degree to which the generator diverges from what you would need to have the octave be a perfect 1200 cents. Here is a list of successively higher zeta peaks, taken to five decimal places: |
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|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">0.00000</span>
| | 0.00000 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">1.12657</span>
| | 1.12657 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">1.97277</span>
| | 1.97277 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">3.05976</span>
| | 3.05976 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">3.90445</span>
| | 3.90445 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">5.03448</span>
| | 5.03448 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">6.95669</span>
| | 6.95669 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">10.00846</span>
| | 10.00846 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">12.02318</span>
| | 12.02318 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">18.94809</span>
| | 18.94809 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">22.02515</span>
| | 22.02515 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">27.08661</span>
| | 27.08661 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">30.97838</span>
| | 30.97838 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">40.98808</span>
| | 40.98808 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">52.99683</span>
| | 52.99683 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">71.95061</span>
| | 71.95061 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">99.04733</span>
| | 99.04733 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">117.96951</span>
| | 117.96951 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">130.00391</span>
| | 130.00391 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">152.05285</span>
| | 152.05285 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">170.99589</span>
| | 170.99589 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">217.02470</span>
| | 217.02470 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">224.00255</span>
| | 224.00255 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">270.01779</span>
| | 270.01779 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">341.97485</span>
| | 341.97485 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">422.05570</span>
| | 422.05570 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">441.01827</span>
| | 441.01827 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">494.01377</span>
| | 494.01377 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">742.01093</span>
| | 742.01093 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">764.01938</span>
| | 764.01938 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">935.03297</span>
| | 935.03297 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">953.94128</span>
| | 953.94128 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">1012.02423</span>
| | 1012.02423 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">1105.99972</span>
| | 1105.99972 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">1177.96567</span>
| | 1177.96567 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">1236.02355</span>
| | 1236.02355 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">1394.98350</span>
| | 1394.98350 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">1447.97300</span>
| | 1447.97300 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">1577.98315</span>
| | 1577.98315 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">2459.98488</span>
| | 2459.98488 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">2683.99168</span>
| | 2683.99168 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">3395.02659</span>
| | 3395.02659 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">5585.00172</span>
| | 5585.00172 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">6079.01642</span>
| | 6079.01642 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">7032.96529</span>
| | 7032.96529 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">8268.98378</span>
| | 8268.98378 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">8539.00834</span>
| | 8539.00834 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">11664.01488</span>
| | 11664.01488 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">14347.99444</span>
| | 14347.99444 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">16807.99325</span>
| | 16807.99325 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">28742.01019</span>
| | 28742.01019 |
|
| |
|
| <span style="background-color: #ffffff; color: #1d2129; font-family: Helvetica,Arial,sans-serif; font-size: 14px;">34691.00191</span>
| | 34691.00191 |
|
| |
|
| ==Removing primes== | | ==Removing primes== |