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{{Infobox ET}}
{{Infobox ET}}
The '''301 equal divisions of the octave''' ('''301edo'''), or the '''301(-tone) equal temperament''' ('''301tet''', '''301et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 301 parts of about 3.99 [[cent]]s each.
{{ED intro}}


== Theory ==
== Theory ==
301edo is a strong 7-limit system, and distinctly consistent through the [[17-odd-limit]]. It tempers out [[32805/32768]] in the 5-limit, [[2401/2400]] in the 7-limit, [[3025/3024]], 5632/5625, [[8019/8000]] in the 11-limit, [[729/728]], [[847/845]], [[1001/1000]], [[1716/1715]], [[2200/2197]] in the 13-limit, and 561/560, [[833/832]], [[1089/1088]], [[1156/1155]], 1275/1274 and [[1701/1700]] in the 17-limit. Because it tempers out both 32805/32768 and 2401/2400, it [[support]]s the [[sesquiquartififths]] temperament.
301edo is a strong 7-limit system, and distinctly [[consistent]] through the [[17-odd-limit]]. The equal temperament [[tempering out|tempers out]] [[32805/32768]] in the 5-limit, [[2401/2400]] in the 7-limit, [[3025/3024]], [[5632/5625]], [[8019/8000]] in the 11-limit, [[729/728]], [[847/845]], [[1001/1000]], [[1716/1715]], [[2200/2197]] in the 13-limit, and 561/560, [[833/832]], [[1089/1088]], [[1156/1155]], 1275/1274 and [[1701/1700]] in the 17-limit. Since it tempers out both 32805/32768 and 2401/2400, it [[support]]s the [[sesquiquartififths]] temperament.


301 is a composite number, since 301 = 7 × 43. This is related to the proposal of the deaf French mathematician and acoustician [[Wikipedia: Joseph Sauveur|Joseph Sauveur]] to divide the octave in 43 parts called ''merides'', and those into seven more parts called ''heptamerides''. Back in the days of slide rules and log tables, this made sense since by multiplying the log base ten of the interval in question by 1000, one came close to how many heptamerides it constituted.
=== Prime harmonics ===
{{Harmonics in equal|301}}
 
=== Subsets and supersets ===
Since 301 factors into {{factorisation|301}}, 301edo has [[7edo]] and [[43edo]] as its subsets. This is related to the proposal of the deaf French mathematician and acoustician {{w|Joseph Sauveur}} to divide the octave in 43 parts called ''merides'', and those into seven more parts called ''heptamerides''. Back in the days of slide rules and log tables, this made sense since by multiplying the log base ten of the interval in question by 1000, one came close to how many heptamerides it constituted.


301edo also tempers out {{monzo| 168 -43 -43 }} and 5250987/5242880, so it supports the [[Mitonismic temperaments #Meridic|meridic temperament]].  
301edo also tempers out {{monzo| 168 -43 -43 }} and 5250987/5242880, so it supports the [[Mitonismic temperaments #Meridic|meridic temperament]].  
=== Prime harmonics ===
{{Primes in edo|edo=301|columns=11|prec=3}}


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 25: Line 27:
| 2.3
| 2.3
| {{monzo| -477 301 }}
| {{monzo| -477 301 }}
| [{{val| 301 477 }}]
| {{mapping| 301 477 }}
| +0.0927
| +0.0927
| 0.0927
| 0.0927
Line 32: Line 34:
| 2.3.5
| 2.3.5
| 32805/32768, {{monzo| 3 45 -32 }}
| 32805/32768, {{monzo| 3 45 -32 }}
| [{{val| 301 477 699 }}]
| {{mapping| 301 477 699 }}
| +0.0048
| +0.0048
| 0.1456
| 0.1456
Line 39: Line 41:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 32805/32768, 1959552/1953125
| 2401/2400, 32805/32768, 1959552/1953125
| [{{val| 301 477 699 845 }}]
| {{mapping| 301 477 699 845 }}
| +0.0085
| +0.0085
| 0.1262
| 0.1262
Line 46: Line 48:
| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 3025/3024, 5632/5625, 8019/8000
| 2401/2400, 3025/3024, 5632/5625, 8019/8000
| [{{val| 301 477 699 845 1041 }}]
| {{mapping| 301 477 699 845 1041 }}
| +0.0734
| +0.0734
| 0.1720
| 0.1720
Line 53: Line 55:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 729/728, 847/845, 1001/1000, 1716/1715, 3025/3024
| 729/728, 847/845, 1001/1000, 1716/1715, 3025/3024
| [{{val| 301 477 699 845 1041 1114 }}]
| {{mapping| 301 477 699 845 1041 1114 }}
| +0.0310
| +0.0310
| 0.1834
| 0.1834
Line 60: Line 62:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 561/560, 729/728, 833/832, 847/845, 1001/1000, 1089/1088
| 561/560, 729/728, 833/832, 847/845, 1001/1000, 1089/1088
| [{{val| 301 477 699 845 1041 1114 1230 }}]
| {{mapping| 301 477 699 845 1041 1114 1230 }}
| +0.0721
| +0.0721
| 0.1973
| 0.1973
Line 68: Line 70:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per octave
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
Line 78: Line 81:
| 25\301
| 25\301
| 99.67
| 99.67
| 200/189
| 18/17
| [[Quintaschis]]
| [[Quintaschis]]
|-
|-
Line 103: Line 106:
| 498.34
| 498.34
| 4/3
| 4/3
| [[Helmholtz]]
| [[Helmholtz (temperament)|Helmholtz]]
|-
|-
| 7
| 7
| 125\301<br>(4\301)
| 125\301<br />(4\301)
| 498.34<br>(15.95)
| 498.34<br />(15.95)
| 4/3<br>(245/243)
| 4/3<br />(245/243)
| [[Septant]]
| [[Septant]]
|-
|-
| 43
| 43
| 125\301<br>(1\301)
| 125\301<br />(1\301)
| 498.34<br>(3.99)
| 498.34<br />(3.99)
| 4/3<br>(540/539)
| 4/3<br />(540/539)
| [[Meridic]]
| [[Meridic]]
|}
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Meridic]]
[[Category:Meridic]]

Latest revision as of 13:31, 13 March 2026

← 300edo 301edo 302edo →
Prime factorization 7 × 43
Step size 3.98671 ¢ 
Fifth 176\301 (701.661 ¢)
Semitones (A1:m2) 28:23 (111.6 ¢ : 91.69 ¢)
Consistency limit 17
Distinct consistency limit 17

301 equal divisions of the octave (abbreviated 301edo or 301ed2), also called 301-tone equal temperament (301tet) or 301 equal temperament (301et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 301 equal parts of about 3.99 ¢ each. Each step represents a frequency ratio of 21/301, or the 301st root of 2.

Theory

301edo is a strong 7-limit system, and distinctly consistent through the 17-odd-limit. The equal temperament tempers out 32805/32768 in the 5-limit, 2401/2400 in the 7-limit, 3025/3024, 5632/5625, 8019/8000 in the 11-limit, 729/728, 847/845, 1001/1000, 1716/1715, 2200/2197 in the 13-limit, and 561/560, 833/832, 1089/1088, 1156/1155, 1275/1274 and 1701/1700 in the 17-limit. Since it tempers out both 32805/32768 and 2401/2400, it supports the sesquiquartififths temperament.

Prime harmonics

Approximation of prime harmonics in 301edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.29 +0.40 -0.06 -1.15 +0.67 -1.30 +1.49 +1.63 -1.01 -0.85
Relative (%) +0.0 -7.4 +10.0 -1.4 -28.9 +16.8 -32.6 +37.4 +40.8 -25.2 -21.3
Steps
(reduced) 		301
(0) 		477
(176) 		699
(97) 		845
(243) 		1041
(138) 		1114
(211) 		1230
(26) 		1279
(75) 		1362
(158) 		1462
(258) 		1491
(287)

Subsets and supersets

Since 301 factors into 7 × 43, 301edo has 7edo and 43edo as its subsets. This is related to the proposal of the deaf French mathematician and acoustician Joseph Sauveur to divide the octave in 43 parts called merides, and those into seven more parts called heptamerides. Back in the days of slide rules and log tables, this made sense since by multiplying the log base ten of the interval in question by 1000, one came close to how many heptamerides it constituted.

301edo also tempers out [168 -43 -43⟩ and 5250987/5242880, so it supports the meridic temperament.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-477 301⟩ [⟨301 477]] +0.0927 0.0927 2.33
2.3.5 32805/32768, [3 45 -32⟩ [⟨301 477 699]] +0.0048 0.1456 3.65
2.3.5.7 2401/2400, 32805/32768, 1959552/1953125 [⟨301 477 699 845]] +0.0085 0.1262 3.17
2.3.5.7.11 2401/2400, 3025/3024, 5632/5625, 8019/8000 [⟨301 477 699 845 1041]] +0.0734 0.1720 4.31
2.3.5.7.11.13 729/728, 847/845, 1001/1000, 1716/1715, 3025/3024 [⟨301 477 699 845 1041 1114]] +0.0310 0.1834 4.60
2.3.5.7.11.13.17 561/560, 729/728, 833/832, 847/845, 1001/1000, 1089/1088 [⟨301 477 699 845 1041 1114 1230]] +0.0721 0.1973 4.95

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 25\301 99.67 18/17 Quintaschis
1 44\301 175.42 448/405 Sesquiquartififths / sesquart (301e)
1 68\301 271.10 90/77 Quasiorwell
1 76\301 302.99 25/21 Quinmite
1 125\301 498.34 4/3 Helmholtz
7 125\301
(4\301) 		498.34
(15.95) 		4/3
(245/243) 		Septant
43 125\301
(1\301) 		498.34
(3.99) 		4/3
(540/539) 		Meridic

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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