41edo: Difference between revisions
m The ratio of supermajor chord was wrong! |
Add the ratios of 19 and 21. Complete the ratios of 13. I speculated it consistent in 19- and 21-odd-limit if treated as a no-17s tuning. |
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The 41-tET, 41-EDO, 41-ET, or '''41-Tone Equal Temperament''' is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.268 [[cent|cent]]s, an [[interval|interval]] close in size to [[64/63|64/63]], the [[Septimal_comma|septimal comma]]. 41-ET can be seen as a tuning of the ''[[Schismatic_family#Garibaldi|Garibaldi temperament]]'' [[#cite_note-1|[1]]] , [[#cite_note-2|[2]]] , [[#cite_note-3|[3]]] the ''[[Magic_family|Magic temperament]]'' [[#cite_note-4|[4]]] and the superkleismic (41&26) temperament. It is the second smallest equal temperament (after [[29edo|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|11-limit]] fairly well, and the [[13-limit|13-limit]] perhaps close enough for government work, though its [[13/10|13/10]] is 14 cents sharp. Various 13-limit [[Magic_extensions|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. | The 41-tET, 41-EDO, 41-ET, or '''41-Tone Equal Temperament''' is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.268 [[cent|cent]]s, an [[interval|interval]] close in size to [[64/63|64/63]], the [[Septimal_comma|septimal comma]]. 41-ET can be seen as a tuning of the ''[[Schismatic_family#Garibaldi|Garibaldi temperament]]'' [[#cite_note-1|[1]]] , [[#cite_note-2|[2]]] , [[#cite_note-3|[3]]] the ''[[Magic_family|Magic temperament]]'' [[#cite_note-4|[4]]] and the superkleismic (41&26) temperament. It is the second smallest equal temperament (after [[29edo|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|11-limit]] fairly well, and the [[13-limit|13-limit]] perhaps close enough for government work, though its [[13/10|13/10]] is 14 cents sharp. Various 13-limit [[Magic_extensions|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|31edo]] is only consistent up to the 11-limit, and the intervals 12/31 and 19/31 have no 11-limit approximations). | 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|31edo]] is only consistent up to the 11-limit, and the intervals 12/31 and 19/31 have no 11-limit approximations). Treated as a no-seventeens tuning, it is consistent all the way up to 21-odd-limit. | ||
41-ET forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41-ET as the basic [[13-limit|13-limit]] intervals requiring fine tuning +/- 1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41-ET circle in [[205edo|205edo]]. 41-ET is also used by the [[The Kite Guitar|Kite Guitar]], see below in "Instruments". | 41-ET forms the foundation of the [http://www.h-pi.com/theory/huntsystem1.html H-System], which uses the scale degrees of 41-ET as the basic [[13-limit|13-limit]] intervals requiring fine tuning +/- 1 [http://www.h-pi.com/theory/huntsystem2.html average JND] from the 41-ET circle in [[205edo|205edo]]. 41-ET is also used by the [[The Kite Guitar|Kite Guitar]], see below in "Instruments". | ||
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41edo is the 13th [[prime_numbers|prime]] edo, following [[37edo|37edo]] and coming before [[43edo|43edo]]. | 41edo is the 13th [[prime_numbers|prime]] edo, following [[37edo|37edo]] and coming before [[43edo|43edo]]. | ||
= | =Intervals= | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! | | ! | | ||
! | Cents | ! | Cents | ||
! | Approximate Ratios | ! | Approximate Ratios'''*''' | ||
! colspan="3" |[[Ups_and_Downs_Notation|Ups and Downs Notation]] | ! colspan="3" |[[Ups_and_Downs_Notation|Ups and Downs Notation]] | ||
! | Andrew's | ! | Andrew's | ||
| Line 47: | Line 46: | ||
| style="text-align:center;" | 3 | | style="text-align:center;" | 3 | ||
| style="text-align:center;" | 87.805 | | style="text-align:center;" | 87.805 | ||
| | [[21/20|21/20]], [[22/21|22/21]] | | | [[21/20|21/20]], [[22/21|22/21]], [[19/18]], [[20/19]] | ||
|down-aug 1sn, minor 2nd | |down-aug 1sn, minor 2nd | ||
| style="text-align:center;" | vA1, m2 | | style="text-align:center;" | vA1, m2 | ||
| Line 55: | Line 54: | ||
| style="text-align:center;" | 4 | | style="text-align:center;" | 4 | ||
| style="text-align:center;" | 117.07 | | style="text-align:center;" | 117.07 | ||
| | [[16/15|16/15]], [[15/14|15/14]] | | | [[16/15|16/15]], [[15/14|15/14]], [[14/13]] | ||
|augmented 1sn, upminor 2nd | |augmented 1sn, upminor 2nd | ||
| style="text-align:center;" | A1, ^m2 | | style="text-align:center;" | A1, ^m2 | ||
| Line 63: | Line 62: | ||
| style="text-align:center;" | 5 | | style="text-align:center;" | 5 | ||
| style="text-align:center;" | 146.34 | | style="text-align:center;" | 146.34 | ||
| | [[12/11|12/11]] | | | [[12/11|12/11]], [[13/12]] | ||
|mid 2nd | |mid 2nd | ||
| style="text-align:center;" | ~2 | | style="text-align:center;" | ~2 | ||
| Line 71: | Line 70: | ||
| style="text-align:center;" | 6 | | style="text-align:center;" | 6 | ||
| style="text-align:center;" | 175.61 | | style="text-align:center;" | 175.61 | ||
| | [[10/9|10/9]], [[11/10|11/10]] | | | [[10/9|10/9]], [[11/10|11/10]], [[21/19]] | ||
|downmajor 2nd | |downmajor 2nd | ||
| style="text-align:center;" | vM2 | | style="text-align:center;" | vM2 | ||
| Line 87: | Line 86: | ||
| style="text-align:center;" | 8 | | style="text-align:center;" | 8 | ||
| style="text-align:center;" | 234.15 | | style="text-align:center;" | 234.15 | ||
| | [[8/7|8/7]] | | | [[8/7|8/7]], [[15/13]] | ||
|upmajor 2nd | |upmajor 2nd | ||
| style="text-align:center;" | ^M2 | | style="text-align:center;" | ^M2 | ||
| Line 95: | Line 94: | ||
| style="text-align:center;" | 9 | | style="text-align:center;" | 9 | ||
| style="text-align:center;" | 263.415 | | style="text-align:center;" | 263.415 | ||
| | [[7/6|7/6]] | | | [[7/6|7/6]], [[22/19]] | ||
|downminor 3rd | |downminor 3rd | ||
| style="text-align:center;" | vm3 | | style="text-align:center;" | vm3 | ||
| Line 103: | Line 102: | ||
| style="text-align:center;" | 10 | | style="text-align:center;" | 10 | ||
| style="text-align:center;" | 292.68 | | style="text-align:center;" | 292.68 | ||
| | [[32/27|32/27]], [[13/11]] | | | [[32/27|32/27]], [[13/11]], [[19/16]] | ||
|minor 3rd | |minor 3rd | ||
| style="text-align:center;" | m3 | | style="text-align:center;" | m3 | ||
| Line 119: | Line 118: | ||
| style="text-align:center;" | 12 | | style="text-align:center;" | 12 | ||
| style="text-align:center;" | 351.22 | | style="text-align:center;" | 351.22 | ||
| | [[11/9|11/9]], [[27/22|27/22]] | | | [[11/9|11/9]], [[27/22|27/22]], [[16/13]] | ||
|mid 3rd | |mid 3rd | ||
| style="text-align:center;" | ~3 | | style="text-align:center;" | ~3 | ||
| Line 127: | Line 126: | ||
| style="text-align:center;" | 13 | | style="text-align:center;" | 13 | ||
| style="text-align:center;" | 380.49 | | style="text-align:center;" | 380.49 | ||
| | [[5/4|5/4]] | | | [[5/4|5/4]], [[26/21]] | ||
|downmajor 3rd | |downmajor 3rd | ||
| style="text-align:center;" | vM3 | | style="text-align:center;" | vM3 | ||
| Line 135: | Line 134: | ||
| style="text-align:center;" | 14 | | style="text-align:center;" | 14 | ||
| style="text-align:center;" | 409.76 | | style="text-align:center;" | 409.76 | ||
| | [[14/11|14/11]], [[ | | | [[81/64]], [[14/11|14/11]], [[24/19]], [[19/15]] | ||
|major 3rd | |major 3rd | ||
| style="text-align:center;" | M3 | | style="text-align:center;" | M3 | ||
| Line 167: | Line 166: | ||
| style="text-align:center;" | 18 | | style="text-align:center;" | 18 | ||
| style="text-align:center;" | 526.83 | | style="text-align:center;" | 526.83 | ||
| | [[15/11|15/11]], [[27/20|27/20]] | | | [[15/11|15/11]], [[27/20|27/20]], [[19/14]] | ||
|up-4th | |up-4th | ||
| style="text-align:center;" | ^4 | | style="text-align:center;" | ^4 | ||
| Line 175: | Line 174: | ||
| style="text-align:center;" | 19 | | style="text-align:center;" | 19 | ||
| style="text-align:center;" | 556.1 | | style="text-align:center;" | 556.1 | ||
| | [[11/8|11/8]] | | | [[11/8|11/8]], [[18/13]], [[26/19]] | ||
|mid-4th | |mid-4th | ||
| style="text-align:center;" | ~4 | | style="text-align:center;" | ~4 | ||
| Line 199: | Line 198: | ||
| style="text-align:center;" | 22 | | style="text-align:center;" | 22 | ||
| style="text-align:center;" | 643.90 | | style="text-align:center;" | 643.90 | ||
| | [[16/11|16/11]], [[13/9]] | | | [[16/11|16/11]], [[13/9]], [[19/13]] | ||
|mid-5th | |mid-5th | ||
| style="text-align:center;" | ~5 | | style="text-align:center;" | ~5 | ||
| Line 207: | Line 206: | ||
| style="text-align:center;" | 23 | | style="text-align:center;" | 23 | ||
| style="text-align:center;" | 673.17 | | style="text-align:center;" | 673.17 | ||
| | [[22/15|22/15]], [[40/27|40/27]] | | | [[22/15|22/15]], [[40/27|40/27]], [[28/19]] | ||
|down-5th | |down-5th | ||
| style="text-align:center;" | v5 | | style="text-align:center;" | v5 | ||
| Line 239: | Line 238: | ||
| style="text-align:center;" | 27 | | style="text-align:center;" | 27 | ||
| style="text-align:center;" | 790.24 | | style="text-align:center;" | 790.24 | ||
| | [[11/7|11/7]], [[ | | | [[128/81|128/81]], [[11/7|11/7]], [[19/12]], [[30/19]] | ||
|minor 6th | |minor 6th | ||
| style="text-align:center;" | m6 | | style="text-align:center;" | m6 | ||
| Line 247: | Line 246: | ||
| style="text-align:center;" | 28 | | style="text-align:center;" | 28 | ||
| style="text-align:center;" | 819.51 | | style="text-align:center;" | 819.51 | ||
| | [[8/5|8/5]] | | | [[8/5|8/5]], [[21/13]] | ||
|upminor 6th | |upminor 6th | ||
| style="text-align:center;" | ^m6 | | style="text-align:center;" | ^m6 | ||
| Line 255: | Line 254: | ||
| style="text-align:center;" | 29 | | style="text-align:center;" | 29 | ||
| style="text-align:center;" | 848.78 | | style="text-align:center;" | 848.78 | ||
| | [[18/11|18/11]], [[44/27|44/27]] | | | [[18/11|18/11]], [[44/27|44/27]], [[13/8]] | ||
|mid 6th | |mid 6th | ||
| style="text-align:center;" | ~6 | | style="text-align:center;" | ~6 | ||
| Line 271: | Line 270: | ||
| style="text-align:center;" | 31 | | style="text-align:center;" | 31 | ||
| style="text-align:center;" | 907.32 | | style="text-align:center;" | 907.32 | ||
| | [[27/16|27/16]] | | | [[27/16|27/16]], [[22/13]], [[32/19]] | ||
|major 6th | |major 6th | ||
| style="text-align:center;" | M6 | | style="text-align:center;" | M6 | ||
| Line 279: | Line 278: | ||
| style="text-align:center;" | 32 | | style="text-align:center;" | 32 | ||
| style="text-align:center;" | 936.59 | | style="text-align:center;" | 936.59 | ||
| | [[12/7|12/7]] | | | [[12/7|12/7]], [[19/11]] | ||
|upmajor 6th | |upmajor 6th | ||
| style="text-align:center;" | ^M6 | | style="text-align:center;" | ^M6 | ||
| Line 287: | Line 286: | ||
| style="text-align:center;" | 33 | | style="text-align:center;" | 33 | ||
| style="text-align:center;" | 965.85 | | style="text-align:center;" | 965.85 | ||
| | [[7/4|7/4]] | | | [[7/4|7/4]], [[26/15]] | ||
|downminor 7th | |downminor 7th | ||
| style="text-align:center;" | vm7 | | style="text-align:center;" | vm7 | ||
| Line 303: | Line 302: | ||
| style="text-align:center;" | 35 | | style="text-align:center;" | 35 | ||
| style="text-align:center;" | 1024.39 | | style="text-align:center;" | 1024.39 | ||
| | [[9/5|9/5]], [[20/11|20/11]] | | | [[9/5|9/5]], [[20/11|20/11]], [[38/21]] | ||
|upminor 7th | |upminor 7th | ||
| style="text-align:center;" | ^m7 | | style="text-align:center;" | ^m7 | ||
| Line 311: | Line 310: | ||
| style="text-align:center;" | 36 | | style="text-align:center;" | 36 | ||
| style="text-align:center;" | 1053.66 | | style="text-align:center;" | 1053.66 | ||
| | [[11/6|11/6]] | | | [[11/6|11/6]], [[24/13]] | ||
|mid 7th | |mid 7th | ||
| style="text-align:center;" | ~7 | | style="text-align:center;" | ~7 | ||
| Line 319: | Line 318: | ||
| style="text-align:center;" | 37 | | style="text-align:center;" | 37 | ||
| style="text-align:center;" | 1082.93 | | style="text-align:center;" | 1082.93 | ||
| | [[15/8|15/8]] | | | [[15/8|15/8]], [[28/15]], [[13/7]] | ||
|downmajor 7th | |downmajor 7th | ||
| style="text-align:center;" | vM7 | | style="text-align:center;" | vM7 | ||
| Line 327: | Line 326: | ||
| style="text-align:center;" | 38 | | style="text-align:center;" | 38 | ||
| style="text-align:center;" | 1112.195 | | style="text-align:center;" | 1112.195 | ||
| | [[40/21|40/21]], [[21/11|21/11]] | | | [[40/21|40/21]], [[21/11|21/11]], [[36/19]], [[19/10]] | ||
|major 7th | |major 7th | ||
| style="text-align:center;" | M7 | | style="text-align:center;" | M7 | ||
| Line 357: | Line 356: | ||
| | do | | | do | ||
|} | |} | ||
<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 [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors: | ||
| Line 475: | Line 475: | ||
0-14-28 = D F# A# = Daug = D aug | 0-14-28 = D F# A# = Daug = D aug | ||
For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]]. | For a more complete list, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]]. | ||
==Red-Blue Notation== | == Notation == | ||
===Red-Blue Notation=== | |||
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-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: | ||
| Line 1,053: | Line 1,055: | ||
|19u1uzy | |19u1uzy | ||
| | spleen comma | | | spleen comma | ||
|- | |||
|" | |||
|361/360 | |||
|4.80 | |||
|<nowiki> | -3 -2 -1 0 0 0 0 2 </nowiki>> | |||
|Nonogu | |||
|19oog2 | |||
|go comma | |||
|- | |- | ||
|" | |" | ||