2500/2499: Difference between revisions

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== Temperaments ==
== Temperaments ==
Tempering out this comma in the 17-limit results in the rank-6 '''sperasmic''' temperament, or in the the 2.3.5.7.17 subgroup, the rank-4 '''speric''' temperament. You may find a list of good equal temperaments that support these temperaments below.  
Tempering out this comma in the 17-limit results in the rank-6 '''sperasmic''' temperament, or in the the 2.3.5.7.17 subgroup, the rank-4 '''speric''' temperament. You may find a list of good equal temperaments that support these temperaments below.  
=== Speric ===
[[Subgroup]]: 2.3.5.7.17
{{Mapping|legend=2| 1 0 0 0 2 | 0 1 0 0 -1 | 0 0 1 0 4 | 0 0 0 1 -2 }}
: mapping generators: ~2, ~3, ~5, ~7
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9902{{c}}, ~3/2 = 701.9772{{c}}, ~5/4 = 386.2274{{c}}, ~7/4 = 968.9230{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9754{{c}}, ~5/4 = 386.2178{{c}}, ~7/4 = 968.9184{{c}}
{{Optimal ET sequence|legend=1| 19, 27g, 31, 41g, 56, 60, 68, 72, 99, 171, 494, 525, 581, 593, 665, 764, 836, 935, 1771, 1942, 2706, 2877, 4477, 4648 }}
[[Badness]] (Sintel): 0.122


=== Sperasmic ===
=== Sperasmic ===
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: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13
: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9677{{c}}, ~5/4 = 386.2047{{c}}, ~7/4 - 968.9056{{c}}, ~11/8, ~13/8
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9902{{c}}, ~3/2 = 701.9772{{c}}, ~5/4 = 386.2274{{c}}, ~7/4 = 968.9230{{c}}, ~11/8 = 551.3474{{c}}, ~13/8 = 840.5571{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9754{{c}}, ~5/4 = 386.2178{{c}}, ~7/4 = 968.9184{{c}}, ~11/8 = 551.3352{{c}}, ~13/8 = 840.5461{{c}}


{{Optimal ET sequence|legend=1| 19, 24d, 27eg, 29g, 31, 41g, 43, 53g, 58g, 60e, 68, 72, 103, 111, 140, 171, 183, 239f, 243e, 270, 311, 354, 414, 422, 494, 581, 764, 935, 1075, 1106, 1178, 1672, 1942, 2113, 2253, 2535, 2805g, 3299d, 3713, 3983g, 4477 }}
{{Optimal ET sequence|legend=1| 19, 24d, 27eg, 29g, 31, 41g, 43, 53g, 58g, 60e, 68, 72, 103, 111, 140, 171, 183, 239f, 243e, 270, 311, 354, 414, 422, 494, 581, 764, 935, 1075, 1106, 1178, 1672, 1942, 2113, 2253, 2535, 2805g, 3299d, 3713, 3983g, 4477 }}


=== Speric ===
[[Badness]] (Sintel): 1.13
[[Subgroup]]: 2.3.5.7.17
 
{{Mapping|legend=2| 1 0 0 0 2 | 0 1 0 0 -1 | 0 0 1 0 4 | 0 0 0 1 -2 }}
: mapping generators: ~2, ~3, ~5, ~7
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 701.9677{{c}}, ~5/4 = 386.2047{{c}}, ~7/4 = 968.9056{{c}}
 
{{Optimal ET sequence|legend=1| 19, 27g, 31, 41g, 56, 60, 68, 72, 99, 171, 494, 525, 581, 593, 665, 764, 836, 935, 1771, 1942, 2706, 2877, 4477, 4648 }}


== Etymology ==
== Etymology ==

Latest revision as of 07:49, 29 March 2026

Interval information
Ratio 2500/2499
Factorization 22 × 3-1 × 54 × 7-2 × 17-1
Monzo [2 -1 4 -2 0 0 -1⟩
Size in cents 0.6926322¢
Name sperasma
Color name 17urry4-3, subiruyoyo negative 3rd
FJS name [math]\displaystyle{ \text{ddd}{-3}^{5,5,5,5}_{7,7,17} }[/math]
Special properties square superparticular,
reduced
Tenney norm (log2 nd) 22.5748
Weil norm (log2 max(n, d)) 22.5754
Wilson norm (sopfr(nd)) 58
Comma size unnoticeable
S-expression S50
Open this interval in xen-calc

2500/2499, the sperasma, is an unnoticeable 17-limit superparticular comma with a size of about 0.693 cents. Among other things, it is the difference between the small septimal sixth-tone of 50/49 and the large septendecimal sixth-tone of 51/50. It also represents the little gap between 17/12 and a stack of two 25/21's.

Commatic relations

This comma is the difference between the following superparticular pairs:

Not to mention some nonsuperparticular but useful relations:

It factors into the following superparticular pairs:

Temperaments

Tempering out this comma in the 17-limit results in the rank-6 sperasmic temperament, or in the the 2.3.5.7.17 subgroup, the rank-4 speric temperament. You may find a list of good equal temperaments that support these temperaments below.

Speric

Subgroup: 2.3.5.7.17

Subgroup-val mapping: [⟨1 0 0 0 2], ⟨0 1 0 0 -1], ⟨0 0 1 0 4], ⟨0 0 0 1 -2]]

mapping generators: ~2, ~3, ~5, ~7

Optimal tunings:

  • WE: ~2 = 1199.9902 ¢, ~3/2 = 701.9772 ¢, ~5/4 = 386.2274 ¢, ~7/4 = 968.9230 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9754 ¢, ~5/4 = 386.2178 ¢, ~7/4 = 968.9184 ¢

Optimal ET sequence: 19, 27g, 31, 41g, 56, 60, 68, 72, 99, 171, 494, 525, 581, 593, 665, 764, 836, 935, 1771, 1942, 2706, 2877, 4477, 4648

Badness (Sintel): 0.122

Sperasmic

Subgroup: 2.3.5.7.11.13.17

Mapping:

[⟨ 1 0 0 0 0 0 2 ],
⟨ 0 1 0 0 0 0 -1 ],
⟨ 0 0 1 0 0 0 4 ],
⟨ 0 0 0 1 0 0 -2 ],
⟨ 0 0 0 0 1 0 0 ],
⟨ 0 0 0 0 0 1 0 ]]
mapping generators: ~2, ~3, ~5, ~7, ~11, ~13

Optimal tunings:

  • WE: ~2 = 1199.9902 ¢, ~3/2 = 701.9772 ¢, ~5/4 = 386.2274 ¢, ~7/4 = 968.9230 ¢, ~11/8 = 551.3474 ¢, ~13/8 = 840.5571 ¢
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9754 ¢, ~5/4 = 386.2178 ¢, ~7/4 = 968.9184 ¢, ~11/8 = 551.3352 ¢, ~13/8 = 840.5461 ¢

Optimal ET sequence: 19, 24d, 27eg, 29g, 31, 41g, 43, 53g, 58g, 60e, 68, 72, 103, 111, 140, 171, 183, 239f, 243e, 270, 311, 354, 414, 422, 494, 581, 764, 935, 1075, 1106, 1178, 1672, 1942, 2113, 2253, 2535, 2805g, 3299d, 3713, 3983g, 4477

Badness (Sintel): 1.13

Etymology

The sperasma was named by Aura in 2023. It comes from the Latin verb "spērāre" meaning "to hope" or "to fear"[1].

See also

References

  1. ↑ Wiktionary: spero #Latin
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