Hemiseven: Difference between revisions
From Xenharmonic Reference
Created page with "'''Hemiseven''' is a temperament generated by a very sharp diatonic fifth, with 6 of them mapped to prime 3. This generator is very close to sqrt(16/7), equivalent to sqrt(7), hence the name. It is traditionally considered to be at least 7-limit 72 & 77, but may more broadly be interpreted as a 2.3.7.{5.13.17} temperament (meaning it has a good 13/5, 17/5, and 17/13) also called '''Island Slendric'''. It it notable for being present in most edos containing generators for..." |
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'''Hemiseven''' is a temperament generated by a very sharp diatonic fifth, with 6 of them mapped to prime 3. This generator is very close to sqrt(16/7), equivalent to sqrt(7), hence the name. It is traditionally considered to be | '''Hemiseven''' is a temperament generated by a very sharp diatonic fifth, with 6 of them mapped to prime 3. This generator is very close to sqrt(16/7), equivalent to sqrt(7), hence the name. It is traditionally considered to be 7-limit or higher 72 & 77, but the more common interpretation covers a broader range including the 2.3.7.{5.13.17} temperament (2.3.7.13/5.17/5.17/13) also called '''Island Slendric'''. It is notable for being present in most edos containing generators for both Island (~15/13) and Slendric (~8/7). | ||
=== Hexamech superset === | === Hexamech superset === | ||
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{{UserTag|g_|Ground|7766ff|Some of my favorite rank-2 temperaments are alpha-dicot (semiquartal, including Island and its [[Intergan]] extension) and tricot (specifically Slendric). This makes anything in the Hemiseven range useful. There are two others, beta-tricot/omega-tricot (specifically Porcupine) and delta-triseph (like Vengeance and other antidiatonic tunings of [[Pentagoth]]). | {{UserTag|g_|Ground|7766ff|Some of my favorite rank-2 temperaments are alpha-dicot (semiquartal, including Island and its [[Intergan]] extension) and tricot (specifically Slendric). This makes anything in the Hemiseven range useful. There are two others, beta-tricot/omega-tricot (specifically Porcupine) and delta-triseph (like Vengeance and other antidiatonic tunings of [[Pentagoth]]). | ||
The best way to add Porcupine to Hemiseven is to have the sharp | The best way to add Porcupine to Hemiseven is to have the typical sharp 3/2 be used as the Hemiseven generator, making it a [[Straddle primes|straddle-3 temperament]]. This splits 7/4 into six parts, making it '''hexamech''' in [[Ploidacot|ploidacot]] using the proposed -mech suffix for 7/4. I propose that the most straightforward name is '''Sextanseventh''' because while it does not split 7/1 into six parts, 7/4 is the harmonic seventh. I cannot escape 6-7 jokes. It's even more obnoxious that 67edo is a great tuning of it. | ||
Vengeance is easy to add by making this temperament half-octave. | The GTO string is 1200 | 2 <3 >3 5 7 11 <13 >13 <17 >17 || 0 -18 -3 -5 6 -4 20 -17 8 -29 using 52edo's straddled 13 and 17 with the Slendric mapping of 17/13. This is not the case in tunings closer to 15edo such as 67edo, where the flat 3 becomes more accurate. A version without the straddled 3 is 11-limit 15 & 67. The range in which this 3 is diatonic is 17\126 to 2\15. Some smaller edo tunings in order of generator size are 89, 52, 119, 67, 82, 97, and 112. The only other viable single-octave generator for the subset temperaments Island, Slendric, and Porcupine in these edos is the difference between the Island and Slendric generators with various mappings. | ||
Vengeance is easy to add by making this temperament half-octave. Perhaps it could be called Bisectanseventh as a pun. It can be shown that the generator is the same by approximating the subset temperament generators in a subgroup: 2.7.15/13.11/5.25/17 52 & 82. 52 & 66 provides a similarly useful neutral third generator but is unrelated. | |||
}} | }} | ||
{{Navbox regtemp}} | {{Navbox regtemp}} | ||
{{Cat|temperaments}} | {{Cat|temperaments}} | ||
Revision as of 06:09, 13 August 2026
Hemiseven is a temperament generated by a very sharp diatonic fifth, with 6 of them mapped to prime 3. This generator is very close to sqrt(16/7), equivalent to sqrt(7), hence the name. It is traditionally considered to be 7-limit or higher 72 & 77, but the more common interpretation covers a broader range including the 2.3.7.{5.13.17} temperament (2.3.7.13/5.17/5.17/13) also called Island Slendric. It is notable for being present in most edos containing generators for both Island (~15/13) and Slendric (~8/7).
Hexamech superset
g_
Some of my favorite rank-2 temperaments are alpha-dicot (semiquartal, including Island and its Intergan extension) and tricot (specifically Slendric). This makes anything in the Hemiseven range useful. There are two others, beta-tricot/omega-tricot (specifically Porcupine) and delta-triseph (like Vengeance and other antidiatonic tunings of Pentagoth).
The best way to add Porcupine to Hemiseven is to have the typical sharp 3/2 be used as the Hemiseven generator, making it a straddle-3 temperament. This splits 7/4 into six parts, making it hexamech in ploidacot using the proposed -mech suffix for 7/4. I propose that the most straightforward name is Sextanseventh because while it does not split 7/1 into six parts, 7/4 is the harmonic seventh. I cannot escape 6-7 jokes. It's even more obnoxious that 67edo is a great tuning of it.
The GTO string is 1200
