Hemiseven: Difference between revisions
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). | ||
=== Half-octave === | |||
One of the greatest weaknesses of traditional Hemiseven is its very complex mapping of 5/1. Tunings across the range may use a different mapping, such as from Meantone or Rodan, which are also complex. One solution is to use a half-octave as the period instead, allowing for 600*-5+483*7 = 381 and 600*-9+483*12 = 396, with the sharp or flat 5/4 preferred depending on the generator. | |||
=== 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. | ||
The GTO string is 1200 <nowiki>| 2 <3 >3 5 7 11 <13 >13 <17 >17 || 0 -18 -3 -5 6 -4 20 -17 8 -29</nowiki> 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. | |||
Vengeance | In short: | ||
* 52+5n is Hemiseven (Island Slendric, the mapping of 5 varies) | |||
* 52+15n adds Porcupine by using the very sharp 3 as the Hemiseven generator | |||
* 52+30n adds Vengeance generated by Porcupine's extremely inflated 81/80 plus 1\2 | |||
}} | }} | ||
{{Navbox regtemp}} | {{Navbox regtemp}} | ||
{{Cat|temperaments}} | {{Cat|temperaments}} | ||
Latest revision as of 17:38, 23 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).
Half-octave
One of the greatest weaknesses of traditional Hemiseven is its very complex mapping of 5/1. Tunings across the range may use a different mapping, such as from Meantone or Rodan, which are also complex. One solution is to use a half-octave as the period instead, allowing for 600*-5+483*7 = 381 and 600*-9+483*12 = 396, with the sharp or flat 5/4 preferred depending on the generator.
Hexamech superset
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 | 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.
In short:
- 52+5n is Hemiseven (Island Slendric, the mapping of 5 varies)
- 52+15n adds Porcupine by using the very sharp 3 as the Hemiseven generator
- 52+30n adds Vengeance generated by Porcupine's extremely inflated 81/80 plus 1\2
