Edϕ: Difference between revisions
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7 steps of 10ed2 | The octave-reduced 13th harmonic, 13/8 (≈840.5276618¢), is closely approximated by 7 steps of 10ed2 (840¢). | ||
13/8 (1.625) is also close to acoustic phi (≈1.618033989). | |||
So, if we divide acoustic phi into 7 steps, then 10 of those steps will bring us quite close to an octave. | |||
This tuning therefore could also be thought of as 10edo with a stretched octave. Since the ratio between acoustic phi and the 10ed2 approximation of the 13th harmonic is ≈0.9917741623, the resulting stretched octave is 2^0.9917741623 = 1.988629015, or ≈1190.128995¢. | |||
{| class="wikitable" | {| class="wikitable" | ||
| Line 5: | Line 11: | ||
| | | | ||
| colspan="4" |'''10ed2''' | | colspan="4" |'''10ed2''' | ||
| colspan="4" |'''7edφ''' | | colspan="4" |'''7edφ or 10ed(<math>2^{\frac{10log_2{φ}}{7}}</math>)''' | ||
|- | |- | ||
|'''scale step''' | |'''scale step''' | ||
| Line 22: | Line 28: | ||
|120 | |120 | ||
|120 | |120 | ||
|<math>φ^{\frac{1}{7}}</math> | |<math>φ^{\frac{1}{7}}</math> or <math>2^{\frac{1log_2{φ}}{7}}</math> | ||
|1.071162542 | |1.071162542 | ||
|119.0128995 | |119.0128995 | ||
| Line 32: | Line 38: | ||
|240 | |240 | ||
|120 | |120 | ||
|<math>φ^{\frac{2}{7}}</math> | |<math>φ^{\frac{2}{7}}</math> or <math>2^{\frac{2log_2{φ}}{7}}</math> | ||
|1.147389191 | |1.147389191 | ||
|238.025799 | |238.025799 | ||
| Line 42: | Line 48: | ||
|360 | |360 | ||
|120 | |120 | ||
|<math>φ^{\frac{3}{7}}</math> | |<math>φ^{\frac{3}{7}}</math> or <math>2^{\frac{3log_2{φ}}{7}}</math> | ||
|1.229040323 | |1.229040323 | ||
|357.0386984 | |357.0386984 | ||
| Line 52: | Line 58: | ||
|480 | |480 | ||
|120 | |120 | ||
|<math>φ^{\frac{4}{7}}</math> | |<math>φ^{\frac{4}{7}}</math> or <math>2^{\frac{4log_2{φ}}{7}}</math> | ||
|1.316501956 | |1.316501956 | ||
|476.0515979 | |476.0515979 | ||
| Line 62: | Line 68: | ||
|600 | |600 | ||
|120 | |120 | ||
|<math>φ^{\frac{5}{7}}</math> | |<math>φ^{\frac{5}{7}}</math> or <math>2^{\frac{5log_2{φ}}{7}}</math> | ||
|1.410187582 | |1.410187582 | ||
|595.0644974 | |595.0644974 | ||
| Line 72: | Line 78: | ||
|720 | |720 | ||
|120 | |120 | ||
|<math>φ^{\frac{6}{7}}</math> | |<math>φ^{\frac{6}{7}}</math> or <math>2^{\frac{6log_2{φ}}{7}}</math> | ||
|1.510540115 | |1.510540115 | ||
|714.0773969 | |714.0773969 | ||
| Line 82: | Line 88: | ||
|840 | |840 | ||
|120 | |120 | ||
|<math>φ^{\frac{7}{7}}</math> | |<math>φ^{\frac{7}{7}}</math> or <math>2^{\frac{7log_2{φ}}{7}}</math> | ||
|1.618033989 | |1.618033989 | ||
|833.0902964 | |833.0902964 | ||
| Line 92: | Line 98: | ||
|960 | |960 | ||
|120 | |120 | ||
|<math>φ^{\frac{8}{7}}</math> | |<math>φ^{\frac{8}{7}}</math> or <math>2^{\frac{8log_2{φ}}{7}}</math> | ||
|1.7331774 | |1.7331774 | ||
|952.1031958 | |952.1031958 | ||
| Line 102: | Line 108: | ||
|1080 | |1080 | ||
|120 | |120 | ||
|<math>φ^{\frac{9}{7}}</math> | |<math>φ^{\frac{9}{7}}</math> or <math>2^{\frac{9log_2{φ}}{7}}</math> | ||
|1.85651471 | |1.85651471 | ||
|1071.116095 | |1071.116095 | ||
| Line 112: | Line 118: | ||
|1200 | |1200 | ||
|120 | |120 | ||
|<math>φ^{\frac{10}{7}}</math> | |<math>φ^{\frac{10}{7}}</math> or <math>2^{\frac{10log_2{φ}}{7}}</math> | ||
|1.988629015 | |1.988629015 | ||
|1190.128995 | |1190.128995 | ||
|119.0128995 | |119.0128995 | ||
|} | |} | ||