17edo: Difference between revisions
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17edo's perfect fifth is around 4 cents sharp of just, and around 6 cents sharp of [[12edo]]'s, lending itself to an expressive [[diatonic scale]]. Meanwhile, it approximates [[harmonic]]s [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] to reasonable degrees, despite completely missing harmonic [[5/1|5]]. Thus it can plausibly be treated as a 2.3.25.7.11.13.23 [[subgroup temperament]], for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers). Because these harmonics are all tempered sharp, it adapts well to octave shrinking; [[27edt]] (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is [[44ed6]]. | 17edo's perfect fifth is around 4 cents sharp of just, and around 6 cents sharp of [[12edo]]'s, lending itself to an expressive [[diatonic scale]]. Meanwhile, it approximates [[harmonic]]s [[7/1|7]], [[11/1|11]], [[13/1|13]], and [[23/1|23]] to reasonable degrees, despite completely missing harmonic [[5/1|5]]. Thus it can plausibly be treated as a 2.3.25.7.11.13.23 [[subgroup temperament]], for which it is quite accurate (though the 7-limit ratios are generally not as well-represented as those of the other integers). Because these harmonics are all tempered sharp, it adapts well to octave shrinking; [[27edt]] (a variant of 17edo in which the octaves are flattened by ~2.5 cents) is a good alternative. Another one is [[44ed6]]. | ||
Because the 5th harmonic is not well-approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating. | Because the 5th harmonic is not well-approximated, using timbres with attenuated 5th harmonics (and its multiples) may reduce audible beating. | ||
The standard major triad is quite dissonant as the major third is closer to 9/7 than the traditional 5/4. | The standard major triad is quite dissonant as the major third is closer to [[9/7]] than the traditional [[5/4]]. Instead, the tonic chords of 17edo could be considered to be the tetrad 6:7:8:9 and its utonal inversion (representing 14:16:18:21 as [[64/63]] is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the [[wikipedia: Mu chord|mu chord]] of Steely Dan fame). These are realized in 17edo as 0-4-7-10 and 0-3-6-10, respectively. Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add the 0-3-7-10 (which is a sus4 with added second, or sus2 with added fourth). These three chords comprise the three ways to divide the 17edo perfect fifth into two whole tones and one subminor third. Chromatic alterations of them also exist, for example, the 0-3-7-10 chord may be altered to 0-2-7-10 (which approximates 12:13:16:18) or 0-3-8-10 (which approximates 8:9:11:12). The 0-3-8-10 chord is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 0-3-6-10. | ||
Instead, the tonic chords of 17edo could be considered to be the tetrad 6:7:8:9 and its utonal inversion (representing 14:16:18:21 as [[64/63]] is tempered out), the former of which is a subminor chord with added fourth, and the latter a supermajor chord with added second (resembling the [[wikipedia: Mu chord|mu chord]] of Steely Dan fame). These are realized in 17edo as 0-4-7-10 and 0-3-6-10, respectively. Both of these have distinct moods, and are stable and consonant, if somewhat more sophisticated than their classic 5-limit counterparts. To this group we could also add the 0-3-7-10 (which is a sus4 with added second, or sus2 with added fourth). These three chords comprise the three ways to divide the 17edo perfect fifth into two whole tones and one subminor third. Chromatic alterations of them also exist, for example, the 0-3-7-10 chord may be altered to 0-2-7-10 (which approximates 12:13:16:18) or 0-3-8-10 (which approximates 8:9:11:12). The 0-3-8-10 chord is impressive-sounding, resembling a sus4 but with even more tension; it resolves quite nicely to 0-3-6-10. | |||
Another construction of septimal chords involves 4:7:12 and its inversion 7:12:21. These triads span a twelfth, realized in 17edo as 0-14-27 and 0-13-27, respectively. To this we may add 0-12-14-27, representing 8:13:14:24, or 0-13-15-27, representing 7:12:13:21. | Another construction of septimal chords involves 4:7:12 and its inversion 7:12:21. These triads span a twelfth, realized in 17edo as 0-14-27 and 0-13-27, respectively. To this we may add 0-12-14-27, representing 8:13:14:24, or 0-13-15-27, representing 7:12:13:21. | ||
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! Quality | ! Quality | ||
! Color | ! Color | ||
! | ! Monzo Format | ||
! Examples | ! Examples | ||
|- | |- | ||
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|- | |- | ||
| 2.3 | | 2.3 | ||
| {{| 27 -17 }} | | {{monzo| 27 -17 }} | ||
| [{{val| 17 27 }}] | | [{{val| 17 27 }}] | ||
| -1.24 | | -1.24 | ||
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! [[Harmonic limit|Prime<br>Limit]] | ! [[Harmonic limit|Prime<br>Limit]] | ||
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref> | ! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref> | ||
! [[]] | ! [[Monzo]] | ||
! [[Cent]]s | ! [[Cent]]s | ||
! [[Color name]] | ! [[Color name]] | ||
| Line 507: | Line 506: | ||
| 3 | | 3 | ||
| [[134217728/129140163|(18 digits)]] | | [[134217728/129140163|(18 digits)]] | ||
| {{| 27 -17 }} | | {{Monzo| 27 -17 }} | ||
| 66.765 | | 66.765 | ||
| Sasawa | | Sasawa | ||
| Line 514: | Line 513: | ||
| 5 | | 5 | ||
| [[25/24]] | | [[25/24]] | ||
| {{| -3 -1 2 }} | | {{Monzo| -3 -1 2 }} | ||
| 70.762 | | 70.762 | ||
| Yoyo | | Yoyo | ||
| Line 521: | Line 520: | ||
| 5 | | 5 | ||
| [[32805/32768]] | | [[32805/32768]] | ||
| {{| -15 8 1 }} | | {{Monzo| -15 8 1 }} | ||
| 1.9537 | | 1.9537 | ||
| Layo | | Layo | ||
| Line 528: | Line 527: | ||
| 7 | | 7 | ||
| [[525/512]] | | [[525/512]] | ||
| {{| -9 1 2 1 }} | | {{Monzo| -9 1 2 1 }} | ||
| 43.408 | | 43.408 | ||
| Lazoyoyo | | Lazoyoyo | ||
| Line 535: | Line 534: | ||
| 7 | | 7 | ||
| [[64/63]] | | [[64/63]] | ||
| {{| 6 -2 0 -1 }} | | {{Monzo| 6 -2 0 -1 }} | ||
| 27.264 | | 27.264 | ||
| Ru | | Ru | ||
| Line 542: | Line 541: | ||
| 7 | | 7 | ||
| [[245/243]] | | [[245/243]] | ||
| {{| 0 -5 1 2 }} | | {{Monzo| 0 -5 1 2 }} | ||
| 14.191 | | 14.191 | ||
| Zozoyo | | Zozoyo | ||
| Line 549: | Line 548: | ||
| 7 | | 7 | ||
| [[1728/1715]] | | [[1728/1715]] | ||
| {{| 6 3 -1 -3 }} | | {{Monzo| 6 3 -1 -3 }} | ||
| 13.074 | | 13.074 | ||
| Triru-agu | | Triru-agu | ||
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| 7 | | 7 | ||
| <abbr title="420175/419904">(12 digits)</abbr> | | <abbr title="420175/419904">(12 digits)</abbr> | ||
| {{| -6 -8 2 5 }} | | {{Monzo| -6 -8 2 5 }} | ||
| 1.1170 | | 1.1170 | ||
| Quinzo-ayoyo | | Quinzo-ayoyo | ||
| Line 563: | Line 562: | ||
| 11 | | 11 | ||
| [[99/98]] | | [[99/98]] | ||
| {{| -1 2 0 -2 1 }} | | {{Monzo| -1 2 0 -2 1 }} | ||
| 17.576 | | 17.576 | ||
| Loruru | | Loruru | ||
| Line 570: | Line 569: | ||
| 11 | | 11 | ||
| [[896/891]] | | [[896/891]] | ||
| {{| 7 -4 0 1 -1 }} | | {{Monzo| 7 -4 0 1 -1 }} | ||
| 9.6880 | | 9.6880 | ||
| Saluzo | | Saluzo | ||
| Line 577: | Line 576: | ||
| 11 | | 11 | ||
| [[243/242]] | | [[243/242]] | ||
| {{| -1 5 0 0 -2 }} | | {{Monzo| -1 5 0 0 -2 }} | ||
| 7.1391 | | 7.1391 | ||
| Lulu | | Lulu | ||
| Line 584: | Line 583: | ||
| 11 | | 11 | ||
| [[385/384]] | | [[385/384]] | ||
| {{| -7 -1 1 1 1 }} | | {{Monzo| -7 -1 1 1 1 }} | ||
| 4.5026 | | 4.5026 | ||
| Lozoyo | | Lozoyo | ||
| Line 591: | Line 590: | ||
| 13 | | 13 | ||
| [[1352/1331]] | | [[1352/1331]] | ||
| {{| 3 0 0 0 -3 2 }} | | {{Monzo| 3 0 0 0 -3 2 }} | ||
| 27.101 | | 27.101 | ||
| Bithotrilu | | Bithotrilu | ||
| Line 598: | Line 597: | ||
| 13 | | 13 | ||
| [[364/363]] | | [[364/363]] | ||
| {{| 2 -1 0 1 -2 1 }} | | {{Monzo| 2 -1 0 1 -2 1 }} | ||
| 4.763 | | 4.763 | ||
| Tholuluzo | | Tholuluzo | ||