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| -17
| -17
|-
|-
! colspan="2" | [[nearest edomapping]]
! colspan="2" | [[Nearest edomapping]]
| 41
| 41
| 24
| 24
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| 10
| 10
|-
|-
! colspan="2" | [[fifthspan]]
! colspan="2" | [[Fifthspan]]
| 0
| 0
| +1
| +1
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| -3
| -3
|}
|}
41-ET can be seen as a tuning of the [[Schismatic family #Garibaldi|garibaldi temperament]]<ref>[http://x31eq.com/schismic.htm Schismic Temperaments] at x31eq.com, the website of [[Graham Breed]]</ref><ref>[http://x31eq.com/decimal_lattice.htm Lattices with Decimal Notation] at x31eq.com</ref><ref>[[Wikipedia: Schismatic temperament]]</ref>, the [[Magic family #Magic|magic temperament]]<ref>[[Wikipedia: Magic temperament]]</ref> and the [[Superkleismic|superkleismic (26&41) temperament]]. It is the second smallest equal temperament (after [[29edo]]) whose perfect fifth is closer to just intonation than that of [[12edo|12-ET]], and is the seventh [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral edo]] after 31; it is not, however, a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta gap edo]]. This has to do with the fact that it can deal with the [[11-limit]] fairly well, and the [[13-limit]] perhaps close enough for government work, though its [[13/10]] is 14 cents sharp. Various 13-limit [[magic extensions]] are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in 22edo.
41edo can be seen as a tuning of the [[Schismatic family #Garibaldi|garibaldi temperament]]<ref>[http://x31eq.com/schismic.htm Schismic Temperaments] at x31eq.com, the website of [[Graham Breed]]</ref><ref>[http://x31eq.com/decimal_lattice.htm Lattices with Decimal Notation] at x31eq.com</ref><ref>[[Wikipedia: Schismatic temperament]]</ref>, the [[Magic family #Magic|magic temperament]]<ref>[[Wikipedia: Magic temperament]]</ref> and the [[Superkleismic|superkleismic (26&41) temperament]]. It is the second smallest equal division (after [[29edo]]) whose perfect fifth is closer to just intonation than that of [[12edo]], and is the seventh [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral edo]] after 31; it is not, however, a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta gap edo]]. This has to do with the fact that it can deal with the [[11-limit]] fairly well, and the [[13-limit]] perhaps close enough for government work, though its [[13/10]] is 14 cents sharp. Various 13-limit [[magic extensions]] are supported by 41: 13-limit magic, and less successfully necromancy and witchcraft, all merge into one in 41edo tuning. The 41f val provides a superb tuning for sorcery, giving a less-complex version of the 13-limit, and the 41ef val likewise works well for telepathy; telepathy and sorcery merging into one however not in 41edo but in 22edo.


41edo is consistent in the [[15-odd-limit]]. In fact, ''all'' of its intervals between 100 and 1100 cents in size are 15-odd-limit consonances, although 16\41 as 13/10 is debatable. (In comparison, [[31edo]] is only consistent up to the 11-limit, and the intervals 12\31 and 19\31 have no 11-odd-limit approximations). Treated as a no-seventeens tuning, it is consistent all the way up to 21-odd-limit.  
41edo is consistent in the [[15-odd-limit]]. In fact, ''all'' of its intervals between 100 and 1100 cents in size are 15-odd-limit consonances, although 16\41 as 13/10 is debatable. (In comparison, [[31edo]] is only consistent up to the 11-limit, and the intervals 12\31 and 19\31 have no 11-odd-limit approximations). Treated as a no-seventeens tuning, it is consistent all the way up to 21-odd-limit.  
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! Approximate Ratios*
! Approximate Ratios*
! colspan="3" | [[Ups and Downs Notation]]
! colspan="3" | [[Ups and Downs Notation]]
! Andrew's <br> Solfege Syllables
! Andrew's<br>Solfege
![[KiteGiedraitis|Kite]]'s
! [[Kite Giedraitis|Kite]]'s<br>Solfege
Solfege
|-
|-
| 0
| 0
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| D
| D
| do
| do
|do
| do
|-
|-
| 1
| 1
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| ^D
| ^D
| di
| di
|da
| da
|-
|-
| 2
| 2
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| ^^D, vEb
| ^^D, vEb
| ro
| ro
|ru
| ru
|-
|-
| 3
| 3
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| vD#, Eb
| vD#, Eb
| rih
| rih
|ro
| ro
|-
|-
| 4
| 4
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| D#, ^Eb
| D#, ^Eb
| ra
| ra
|ra
| ra
|-
|-
| 5
| 5
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| ^D#, vvE
| ^D#, vvE
| ru
| ru
|ruh
| ruh
|-
|-
| 6
| 6
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| vE
| vE
| reh
| reh
|reh
| reh
|-
|-
| 7
| 7
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| E
| E
| re
| re
|rih
| rih
|-
|-
| 8
| 8
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| ^E
| ^E
| ri
| ri
|ri
| ri
|-
|-
| 9
| 9
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| vF
| vF
| ma
| ma
|mu
| mu
|-
|-
| 10
| 10
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| F
| F
| meh
| meh
|mo
| mo
|-
|-
| 11
| 11
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| ^F
| ^F
| me
| me
|ma
| ma
|-
|-
| 12
| 12
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| ^^F, vGb
| ^^F, vGb
| mu
| mu
|muh
| muh
|-
|-
| 13
| 13
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| vF#, Gb
| vF#, Gb
| mi
| mi
|meh
| meh
|-
|-
| 14
| 14
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| F#, ^Gb
| F#, ^Gb
| maa
| maa
|mih
| mih
|-
|-
| 15
| 15
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| ^F#, vvG
| ^F#, vvG
| mo
| mo
|mi
| mi
|-
|-
| 16
| 16
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| vG
| vG
| fe
| fe
|fu
| fu
|-
|-
| 17
| 17
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| G
| G
| fa
| fa
|fo
| fo
|-
|-
| 18
| 18
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| ^G
| ^G
| fih
| fih
|fa
| fa
|-
|-
| 19
| 19
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| ^^G, vAb
| ^^G, vAb
| fu
| fu
|fuh
| fuh
|-
|-
| 20
| 20
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| vG#, Ab
| vG#, Ab
| fi
| fi
|feh / so
| feh / so
|-
|-
| 21
| 21
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| G#, ^Ab
| G#, ^Ab
| se
| se
|fih / sa
| fih / sa
|-
|-
| 22
| 22
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| vvA
| vvA
| su
| su
|suh
| suh
|-
|-
| 23
| 23
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| vA
| vA
| sih
| sih
|seh
| seh
|-
|-
| 24
| 24
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| A
| A
| sol
| sol
|sih
| sih
|-
|-
| 25
| 25
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| ^A
| ^A
| si
| si
|si
| si
|-
|-
| 26
| 26
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| ^^A, vBb
| ^^A, vBb
| lo
| lo
|lu
| lu
|-
|-
| 27
| 27
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| vA#, Bb
| vA#, Bb
| leh
| leh
|lo
| lo
|-
|-
| 28
| 28
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| A#, ^Bb
| A#, ^Bb
| le
| le
|la
| la
|-
|-
| 29
| 29
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| ^A#, vvB
| ^A#, vvB
| lu
| lu
|luh
| luh
|-
|-
| 30
| 30
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| vB
| vB
| la
| la
|leh
| leh
|-
|-
| 31
| 31
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| B
| B
| laa
| laa
|lih
| lih
|-
|-
| 32
| 32
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| ^B
| ^B
| li
| li
|li
| li
|-
|-
| 33
| 33
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| vC
| vC
| ta
| ta
|tu
| tu
|-
|-
| 34
| 34
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| C
| C
| teh
| teh
|to
| to
|-
|-
| 35
| 35
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| ^C
| ^C
| te
| te
|ta
| ta
|-
|-
| 36
| 36
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| ^^C, vDb
| ^^C, vDb
| tu
| tu
|tuh
| tuh
|-
|-
| 37
| 37
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| vC#, Db
| vC#, Db
| ti
| ti
|teh
| teh
|-
|-
| 38
| 38
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| C#, ^Db
| C#, ^Db
| taa
| taa
|tih
| tih
|-
|-
| 39
| 39
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| C#^, vvD
| C#^, vvD
| to
| to
|ti
| ti
|-
|-
| 40
| 40
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| vD
| vD
| da
| da
|du
| du
|-
|-
| 41
| 41
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| D
| D
| do
| do
|do
| do
|}
|}
<nowiki>*</nowiki> Based on treating 41-edo as a 2.3.5.7.11.13.19 subgroup temperament; other approaches are possible.  
<nowiki>*</nowiki> Based on treating 41-edo as a 2.3.5.7.11.13.19 subgroup temperament; other approaches are possible.  


Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:


{| class="wikitable" style="text-align:center"
{| class="wikitable" style="text-align:center"
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|}
|}


=== Chord Names  ===
=== Chord names ===


All 41edo chords can be named using ups and downs. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are. Here are the zo, gu, ilo, yo and ru triads:
All 41edo chords can be named using ups and downs. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are. Here are the zo, gu, ilo, yo and ru triads:
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For a more complete list, see [[41edo Chord Names]] and [[Ups and Downs Notation #Chords and Chord Progressions]].  
For a more complete list, see [[41edo Chord Names]] and [[Ups and Downs Notation #Chords and Chord Progressions]].  


== Notations ==
== Notations ==
 
=== Red-Blue Notation ===
=== Red-Blue Notation ===
 
A red-note/blue-note system, similar to the one proposed for [[36edo]], is one option for notating 41edo. (This is separate from and not compatible with Kite's [[color notation]].) We have the "white key" albitonic notes A-G (7 in total), the "black key" sharps and flats (10 in total), a "red" and "blue" version of each albitonic note (14 in total), a "red" (dark red?) version of each sharp and a "blue" (dark blue?) version of each flat (10 in total), adding up to 41. This would result in quite a colorful keyboard! Note that there are no red flats or blue sharps. Using this nomenclature the notes are:
A red-note/blue-note system, similar to the one proposed for [[36edo|36edo]], is one option for notating 41edo. (This is separate from and not compatible with Kite's [[Kite's_color_notation|color notation]].) We have the "white key" albitonic notes A-G (7 in total), the "black key" sharps and flats (10 in total), a "red" and "blue" version of each albitonic note (14 in total), a "red" (dark red?) version of each sharp and a "blue" (dark blue?) version of each flat (10 in total), adding up to 41. This would result in quite a colorful keyboard! Note that there are no red flats or blue sharps. Using this nomenclature the notes are:


A, red A, blue Bb, Bb, A#, red A#, blue B, B, red B, blue C, C, red C, blue Db, Db, C#, red C#, blue D, D, red D, blue Eb, Eb, D#, red D#, blue E, E, red E, blue F, F, red F, blue Gb, Gb, F#, red F#, blue G, G, red G, blue Ab, Ab, G#, red G#, blue A, A.
A, red A, blue Bb, Bb, A#, red A#, blue B, B, red B, blue C, C, red C, blue Db, Db, C#, red C#, blue D, D, red D, blue Eb, Eb, D#, red D#, blue E, E, red E, blue F, F, red F, blue Gb, Gb, F#, red F#, blue G, G, red G, blue Ab, Ab, G#, red G#, blue A, A.
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=== Sagittal ===
=== Sagittal ===


From the appendix to [[The Sagittal Songbook]] by [[JacobBarton|Jacob A. Barton]], a diagram of how to notate 41-EDO in the Revo flavor of Sagittal:
From the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], a diagram of how to notate 41-EDO in the Revo flavor of Sagittal:


[[File:41edo Sagittal.png|800px]]
[[File:41edo Sagittal.png|800px]]


== Just approximation ==
== JI approximation ==
 
=== 15-odd-limit interval mappings ===
=== Selected just intervals ===
 
The following table shows how [[15-odd-limit intervals]] are represented in 41edo. Prime harmonics are in '''bold'''.  
The following table shows how [[15-odd-limit intervals]] are represented in 41edo. Prime harmonics are in '''bold'''.  


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|}
|}


=== Temperament measures ===
== Relationship to 12-edo ==
The following table shows [[TE temperament measures]] (RMS normalized by the rank) of 41et.  
 
{| class="wikitable center-all"
Whereas 12-edo has a circle of twelve 5ths, 41-edo has a spiral of twelve 5ths (since 24\41 is on the 7\12 kite in the scale tree). This spiral of 5th shows 41-edo in a 12-edo-friendly format. Excellent for introducing 41-edo to musicians unfamiliar with microtonal music. There are 12 "-ish" categories, where "-ish" means ±1 edostep. The 6 mid intervals are uncategorized, since they are all so far from 12edo. The two innermost and two outermost intervals on the spiral are duplicates.
! colspan="2" |
 
! 3-limit
[[File:41-edo spiral.png|673x673px]]
! 5-limit
 
! 7-limit
The same spiral, but with notes not intervals:
! 11-limit
 
! 13-limit
[[File:41-edo spiral with notes.png|730x730px]]
! 2.3.5.7.11.19
 
! 2.3.5.7.11.13.19
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
! colspan="2" |Octave stretch (¢)
| 2.3
| {{monzo| 65 -41 }}
| [{{val| 41 65 }}]
| -0.153
| -0.153
| +0.734
| 0.15
| +0.815
| 0.52
| +0.375
|-
| -0.060
| 2.3.5
| +0.502
| 3125/3072, 20000/19683
| [{{val| 41 65 95 }}]
| +0.734
| 1.26
| 4.31
|-
| 2.3.5.7
| 225/224, 245/243, 1029/1024
| [{{val| 41 65 95 115 }}]
| +0.815
| 1.10
| 3.76
|-
| 2.3.5.7.11
| 100/99, 225/224, 243/242, 245/242
| [{{val| 41 65 95 115 142 }}]
| +0.375
| 1.32
| 4.51
|-
| 2.3.5.7.11.13
| 100/99, 105/104, 144/143, 196/195, 243/242
| [{{val| 41 65 95 115 142 152 }}]
| -0.060
| 1.55
| 5.29
|-
| 2.3.5.7.11.13.19
| 100/99, 105/104, 133/132, 144/143, 171/169, 196/195
| [{{val| 41 65 95 115 142 152 174 }}]
| +0.111
| +0.111
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 0.15
| 1.26
| 1.10
| 1.32
| 1.55
| 1.24
| 1.49
| 1.49
|-
! [[TE simple badness|relative]] (%)
| 0.52
| 4.31
| 3.76
| 4.51
| 5.29
| 4.23
| 5.10
| 5.10
|}
|}


* 41et has a lower relative error than any previous ETs in the 3-, 13- and 19-limit. The next ET that does better in these subgroups is 53, 53, and 46, respectively.  
41et is lower in relative error than any previous equal temperaments in the 3-, 13- and 19-limit. The next ETs better in these subgroups are 53, 53, and 46, respectively. It is even more prominent in the 2.3.5.7.11.19 and 2.3.5.7.11.13.19 subgroup. The next ETs better in these subgroups are 72 and 53, respectively.  
* 41et is prominent in the 2.3.5.7.11.19 and 2.3.5.7.11.13.19 subgroup. The next ET that does better in these subgroups is 72 and 53, respectively.  
 
== Relationship to 12-edo ==
 
Whereas 12-edo has a circle of twelve 5ths, 41-edo has a spiral of twelve 5ths (since 24\41 is on the 7\12 kite in the scale tree). This spiral of 5th shows 41-edo in a 12-edo-friendly format. Excellent for introducing 41-edo to musicians unfamiliar with microtonal music. There are 12 "-ish" categories, where "-ish" means ±1 edostep. The 6 mid intervals are uncategorized, since they are all so far from 12edo. The two innermost and two outermost intervals on the spiral are duplicates.
 
[[File:41-edo spiral.png|673x673px]]
 
The same spiral, but with notes not intervals:
 
[[File:41-edo spiral with notes.png|730x730px]]


== Commas ==
=== Commas ===


41 EDO [[tempers out]] the following [[comma]]s using its patent [[val]], {{val|41 65 95 115 142 152 168 174 185 199 203}}.
41 EDO [[tempers out]] the following [[comma]]s using its patent [[val]], {{val| 41 65 95 115 142 152 168 174 185 199 203 }}.


{| class="commatable wikitable center-1 center-2 right-3 center-6"
{| class="commatable wikitable center-1 center-2 right-3 center-6"
Line 1,236: Line 1,243:
<references/>
<references/>


== Temperaments ==
=== Rank-2 temperaments ===
* [[List of edo-distinct 41et rank two temperaments]]
* [[List of edo-distinct 41et rank two temperaments]]
* [[Schismic-counterpyth equivalence continuum]]
* [[Schismic-counterpyth equivalence continuum]]
Line 1,374: Line 1,381:
A list of [[41edo modes]] (MOS and others). See also [[The Kite Guitar Scales|Kite Guitar Scales]] and [[Kite Giedraitis's Categorizations of 41edo Scales]].
A list of [[41edo modes]] (MOS and others). See also [[The Kite Guitar Scales|Kite Guitar Scales]] and [[Kite Giedraitis's Categorizations of 41edo Scales]].


=== Harmonic Scale ===
=== Harmonic scale ===


41edo is the first edo to do some justice to Mode 8 of the [[Overtone series|harmonic series]], which Dante Rosati calls the "[[overtone scale|Diatonic Harmonic Series Scale]]," consisting of overtones 8 through 16 (sometimes made to repeat at the octave).
41edo is the first edo to do some justice to Mode 8 of the [[harmonic series]], which Dante Rosati calls the "[[overtone scale|Diatonic Harmonic Series Scale]]," consisting of overtones 8 through 16 (sometimes made to repeat at the octave).


{| class="wikitable" style="text-align:center"
{| class="wikitable" style="text-align:center"
Line 1,390: Line 1,397:
| 16
| 16
|-
|-
| ...as JI Ratio from 1/1:
| … as JI Ratio from 1/1:
| 1/1
| 1/1
| 9/8
| 9/8
Line 1,401: Line 1,408:
| 2/1
| 2/1
|-
|-
| ...in cents:
| … in cents:
| 0
| 0
| 203.9
| 203.9
Line 1,423: Line 1,430:
| 41
| 41
|-
|-
| ...in cents:
| … in cents:
| 0
| 0
| 204.9
| 204.9
Line 1,437: Line 1,444:
While each overtone of Mode 8 is approximated within a reasonable degree of accuracy, the steps between the intervals are not uniquely represented. (41edo is, after all, a temperament.)
While each overtone of Mode 8 is approximated within a reasonable degree of accuracy, the steps between the intervals are not uniquely represented. (41edo is, after all, a temperament.)


* 7\41 (7 degrees of 41edo) (204.9 cents) stands in for just ratio 9/8 (203.9 cents) -- a close match.
* 7\41 (7 degrees of 41edo) (204.9 cents) stands in for just ratio 9/8 (203.9 cents) – a close match.
* 6\41 (175.6 cents) stands in for both 10/9 (182.4 cents) and 11/10 (165.0 cents).
* 6\41 (175.6 cents) stands in for both 10/9 (182.4 cents) and 11/10 (165.0 cents).
* 5\41 (146.3 cents) stands in for both 12/11 (150.6 cents) and 13/12 (138.6 cents).
* 5\41 (146.3 cents) stands in for both 12/11 (150.6 cents) and 13/12 (138.6 cents).
Line 1,444: Line 1,451:
The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.
The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.


=== Nonoctave Temperaments ===
=== Nonoctave temperaments ===


Taking every third degree of 41edo produces a scale extremely close to [[88cET|88cET]] or 88-cent equal temperament (or the 8th root of 3:2). Likewise, taking every fifth degree produces a scale very close to the equal-tempered <span style="">[[BP|Bohlen-Pierce]]</span>[[BP| Scale]] (or the 13th root of 3). See [[Relationship between Bohlen-Pierce and octave-ful temperaments]], and see this chart:
Taking every third degree of 41edo produces a scale extremely close to [[88cET]] or 88-cent equal temperament (or the 8th root of 3:2). Likewise, taking every fifth degree produces a scale very close to the equal-tempered <span style="">[[BP|Bohlen-Pierce]]</span>[[BP| Scale]] (or the 13th root of 3). See [[Relationship between Bohlen-Pierce and octave-ful temperaments]], and see this chart:


{| class="wikitable center-all right-3 right-4 right-5"
{| class="wikitable center-all right-3 right-4 right-5"
Line 1,935: Line 1,942:


[[File:Caleb's Kite guitar.jpg|480x640px]]
[[File:Caleb's Kite guitar.jpg|480x640px]]


A possible 41-edo keyboard design:
A possible 41-edo keyboard design:
[[File:41edo keyboard layout.png|none|thumb|484x484px]]
[[File:41edo keyboard layout.png|none|thumb|484x484px]]


Another possible system to tune keyboards in 41EDO is discussed in [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_73151.html#74155 http://launch.groups.yahoo.com/group/tuning/message/74155].
Another possible system to tune keyboards in 41EDO is discussed in [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_73151.html#74155 http://launch.groups.yahoo.com/group/tuning/message/74155].


== Music ==
== Music ==
 
* [http://soundcloud.com/cameron-bobro/eveninghorizon-cbobro EveningHorizon] [http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/EveningHorizon_CBobro.mp3 play] by Cameron Bobro
[http://soundcloud.com/cameron-bobro/eveninghorizon-cbobro EveningHorizon] [http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/EveningHorizon_CBobro.mp3 play] by Cameron Bobro


== Links ==
== Links ==
Retrieved from "https://en.xen.wiki/w/41edo"