41edo

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This revision was by author spt3125 and made on 2017-07-22 19:56:32 UTC.
The original revision id was 615832291.
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<span style="color: #004d25; font-family: 'Times New Roman',Times,serif; font-size: 20px;">**41 Tone Equal Temperament**</span>
<span style="display: block; text-align: right;">[[xenharmonie/41edo|Deutsch]]
</span>
[[toc|flat]]
----
=Introduction= 
The 41-tET, 41-EDO, or 41-ET, is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.268 [[cent]]s, an [[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]]// <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>[[http://en.wikipedia.org/wiki/Schismatic_temperament|Schismatic temperament]]</ref> the //[[Magic family|Magic temperament]]// <ref>[[http://en.wikipedia.org/wiki/Magic_temperament|Magic temperament]]</ref> and the superkleismic (41&26) 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%20EDO%20lists|zeta integral edo]] after 31; it is not, however, a [[The Riemann Zeta Function and Tuning#Zeta%20EDO%20lists|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|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. (In comparison, [[31edo]] is only consistent up to the 11-limit, and the intervals 12/31 and 19/31 have no 11-limit approximations).

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]] intervals requiring fine tuning +/- 1 [[http://www.h-pi.com/theory/huntsystem2.html|average JND]] from the 41-ET circle in [[205edo]].

41edo is the 13th [[prime numbers|prime]] edo, following [[37edo]] and coming before [[43edo]].

=Commas= 
41 EDO tempers out the following commas using its patent val, < 41 65 95 115 142 152 168 174 185 199 203 |.
||~ Name ||~ Monzo ||~ Ratio ||~ Cents ||
|| odiheim || | -1 2 -4 5 -2 > ||=   || 0.15 ||
|| harmonisma || | 3 -2 0 -1 3 -2 > ||= 10648/10647 || 0.16 ||
|| tridecimal schisma, Sagittal schismina || | 12 -2 -1 -1 0 -1/1 > ||= 4096/4095 || 0.42 ||
|| Lehmerisma || | -4 -3 2 -1 2 > ||= 3025/3024 || 0.57 ||
|| Breedsma || | -5 -1 -2 4 > ||= 2401/2400 || 0.72 ||
|| Eratosthenes' comma || | 6 -5 -1 0 0 0 0 1 > ||= 1216/1215 || 1.42 ||
|| schisma || | -15 8 1 > ||= 32805/32768 || 1.95 ||
|| squbema || | -3 6 0 -1 0 -1 > ||= 729/728 || 2.38 ||
|| septendecimal bridge comma || | -1 -1 1 -1 1 1 -1 > ||= 715/714 || 2.42 ||
|| Swets' comma, swetisma || | 2 3 1 -2 -1 > ||= 540/539 || 3.21 ||
|| undevicesimal comma, Boethius' comma || | -9 3 0 0 0 0 0 1 > ||= 513/512 || 3.38 ||
|| moctdel || | -2 0 3 -3 1 > ||= 1375/1372 || 3.78 ||
|| Beta 2, septimal schisma, garischisma || | 25 -14 0 -1 > ||=   || 3.80 ||
|| Werckmeister's undecimal septenarian schisma, werckisma || | -3 2 -1 2 -1 > ||= 441/440 || 3.93 ||
|| cuthbert || | 0 0 -1 1 2 -2 > ||= 847/845 || 4.09 ||
|| undecimal kleisma, keenanisma || | -7 -1 1 1 1 > ||= 385/384 || 4.50 ||
|| gentle comma || | 2 -1 0 1 -2 1 > ||= 364/363 || 4.76 ||
|| minthma || | 5 -3 0 0 1 -1 > ||= 352/351 || 4.93 ||
|| marveltwin || | -2 -4 2 0 0 1 > ||= 325/324 || 5.34 ||
|| Beta 5, Garibaldi comma, hemifamity || | 10 -6 1 -1 > ||= 5120/5103 || 5.76 ||
|| hemimage || | 5 -7 -1 3 > ||= 10976/10935 || 6.48 ||
|| septendecimal kleisma || | 8 -1 -1 0 0 0 -1 > ||= 256/255 || 6.78 ||
|| small BP diesis, mirkwai || | 0 3 4 -5 > ||= 16875/16807 || 6.99 ||
|| neutral third comma, rastma || | -1 5 0 0 -2 > ||= 243/242 || 7.14 ||
|| kestrel comma || | 2 3 0 -1 1 -2 > ||= 1188/1183 || 7.30 ||
|| septimal kleisma, marvel comma || | -5 2 2 -1 > ||= 225/224 || 7.71 ||
|| huntma || | 7 0 1 -2 0 -1 > ||= 640/637 || 8.13 ||
|| spleen comma || | 1 1 1 1 -1 0 0 -1 > ||= 210/209 || 8.26 ||
|| orgonisma || | 16 0 0 -2 -3 > ||= 65536/65219 || 8.39 ||
|| gamelan residue, gamelisma || | -10 1 0 3 > ||= 1029/1024 || 8.43 ||
|| septendecimal comma || | -7 7 0 0 0 0 -1 > ||= 2187/2176 || 8.73 ||
|| mynucuma || | 2 -1 -1 2 0 -1 > ||= 196/195 || 8.86 ||
|| quince || | -15 0 -2 7 > ||=   || 9.15 ||
|| undecimal semicomma, pentacircle (minthma * gentle) || | 7 -4 0 1 -1 > ||= 896/891 || 9.69 ||
|| 29th-partial chroma || | -4 -2 1 0 0 0 0 0 0 1 > ||= 145/144 || 11.98 ||
|| grossma || | 4 2 0 0 -1 -1 > ||= 144/143 || 12.06 ||
|| gassorma || | 0 -1 2 -1 1 -1 > ||= 275/273 || 12.64 ||
|| septimal semicomma, octagar || | 5 -4 3 -2 > ||= 4000/3969 || 13.47 ||
|| minor BP diesis, sensamagic || | 0 -5 1 2 > ||= 245/243 || 14.19 ||
|| secorian || | 12 -7 0 1 0 -1/1 > ||= 28672/28431 || 14.61 ||
|| mirwomo comma || | -15 3 2 2 > ||= 33075/32768 || 16.14 ||
|| vicesimotertial comma || | 5 -6 0 0 0 0 0 0 1 > ||= 736/729 || 16.54 ||
|| small tridecimal comma, animist || | -3 1 1 1 0 -1 > ||= 105/104 || 16.57 ||
|| hemimin || | 6 1 0 1 -3 > ||= 1344/1331 || 16.83 ||
|| Ptolemy's comma, ptolemisma || | 2 -2 2 0 -1 > ||= 100/99 || 17.40 ||
|| '41-tone' comma || | 65 -41 > ||=   || 19.84 ||
|| tolerma || | 10 -11 2 1 > ||=   || 19.95 ||
|| major BP diesis, gariboh || | 0 -2 5 -3 > ||= 3125/3087 || 21.18 ||
|| cassacot || | -1 0 1 2 -2 > ||= 245/242 || 21.33 ||
|| keema || | -5 -3 3 1 > ||= 875/864 || 21.90 ||
|| blackjackisma || | -10 7 8 -7 > ||=   || 22.41 ||
|| roda || | 20 -17 3 > ||=   || 25.71 ||
|| minimal diesis, tetracot comma || | 5 -9 4 > ||= 20000/19683 || 27.66 ||
|| small diesis, magic comma || | -10 -1 5 > ||= 3125/3072 || 29.61 ||
|| thuja comma || | 15 0 1 0 -5 > ||=   || 29.72 ||
|| Ampersand's comma || | -25 7 6 > ||=   || 31.57 ||
|| great BP diesis || | 0 -7 6 -1 > ||= 15625/15309 || 35.37 ||
|| shibboleth || | -5 -10 9 > ||=   || 57.27 ||

=Temperaments= 
[[List of edo-distinct 41et rank two temperaments]]

=Intervals= 
||~   ||~ cents value ||~ Approximate
Ratios in the [[11-limit]] ||||~ [[xenharmonic/Ups and Downs Notation|ups and]]
[[xenharmonic/Ups and Downs Notation|downs]]
[[xenharmonic/Ups and Downs Notation|notation]] ||~ Proposed names ||~ Andrew's
solfege
syllable ||~ generator for ||~ some MOS and MODMOS Scales available ||
||= 0 ||= 0.00 ||< [[1_1|1/1]] ||= P1 ||= D || Unison || do ||   ||   ||
||= 1 ||= 29.27 ||< [[81_80|81/80]] ||= ^1 ||= D^ || Red unison || di ||   ||   ||
||= 2 ||= 58.54 ||< [[25_24|25/24]], [[28_27|28/27]],
[[33_32|33/32]] ||= vm2 ||= Ebv || Blue minor second || ro || [[Hemimiracle]] ||   ||
||= 3 ||= 87.80 ||< [[21_20|21/20]], [[22_21|22/21]] ||= m2 ||= Eb || Gray minor second || rih || 88cET (approx),
[[octacot]] ||   ||
||= 4 ||= 117.07 ||< [[16_15|16/15]], [[15_14|15/14]] ||= ^m2 ||= Eb^ || Red minor second || ra || [[Miracle]] ||   ||
||= 5 ||= 146.34 ||< [[12_11|12/11]] ||= ~2 ||= Evv || Neutral second || ru || [[Bohlen-Pierce]]/[[bohpier]] ||   ||
||= 6 ||= 175.61 ||< [[10_9|10/9]], [[11_10|11/10]] ||= vM2 ||= Ev || Blue major second || reh || [[Tetracot]]/[[bunya]]/[[monkey]] || 13-tone MOS: 1 5 1 5 1 5 1 5 5 1 5 1 5 ||
||= 7 ||= 204.88 ||< [[9_8|9/8]] ||= M2 ||= E || Gray major second || re || [[Baldy]] || 11-tone MOS: 6 1 6 6 1 6 1 6 1 6 1 ||
||= 8 ||= 234.15 ||< [[8_7|8/7]] ||= ^M2 ||= E^ || Red major second || ri || [[Rodan]]/[[guiron]] || 11-tone MOS: 7 1 7 1 7 1 7 1 1 7 1 ||
||= 9 ||= 263.41 ||< [[7_6|7/6]], [[32_25|32/25]] ||= vm3 ||= Fv || Blue minor third || ma || [[Septimin]] || 9-tone MOS: 5 4 5 5 4 5 4 5 4 ||
||= 10 ||= 292.68 ||< [[32_27|32/27]] ||= m3 ||= F || Gray minor third || meh || [[Quasitemp]] ||   ||
||= 11 ||= 321.95 ||< [[6_5|6/5]] ||= ^m3 ||= F^ || Red minor third || me || [[Superkleismic]] || 11-tone MOS: 5 3 5 3 3 5 3 3 5 3 3 ||
||= 12 ||= 351.22 ||< [[11_9|11/9]],[[27_22|27/22]] ||= ~3 ||= F^^ || Neutral third || mu || [[Hemififths]]/[[karadeniz]] || 10-tone MOS: 5 2 5 5 2 5 5 5 2 5 ||
||= 13 ||= 380.49 ||< [[5_4|5/4]] ||= vM3 ||= F#v || Blue major third || mi || [[Magic]]/[[witchcraft]] || 10-tone MOS: 2 9 2 2 9 2 2 9 2 2 ||
||= 14 ||= 409.76 ||< [[14_11|14/11]], [[81_64|81/64]] ||= M3 ||= F# || Gray major third || maa || [[Hocus]] ||   ||
||= 15 ||= 439.02 ||< [[9_7|9/7]] ||= ^M3 ||= F#^ || Red major third || mo ||   || 11-tone MOS: 4 3 4 4 4 3 4 4 3 4 4 ||
||= 16 ||= 468.29 ||< [[21_16|21/16]] ||= v4 ||= Gv || Blue fourth || fe || [[Barbad]] ||   ||
||= 17 ||= 497.56 ||< [[4_3|4/3]] ||= P4 ||= G || Perfect fourth || fa || [[Schismatic]] ([[helmholtz]], [[Garibaldi temperament|garibaldi]], [[cassandra]]) ||   ||
||= 18 ||= 526.83 ||< [[15_11|15/11]], [[27_20|27/20]] ||= ^4 ||= G^ || Red fourth || fih || [[Trismegistus]] || 9-tone MOS: 5 5 3 5 5 5 5 3 5 ||
||= 19 ||= 556.10 ||< [[11_8|11/8]] ||= ^^4 ||= G^^ || Blue minor tritone || fu ||   ||   ||
||= 20 ||= 585.37 ||< [[7_5|7/5]] ||= vA4, d5 ||= G#v,
Ab || Minor tritone / diminished fifth || fi || [[Pluto]] ||   ||
||= 21 ||= 614.63 ||< [[10_7|10/7]] ||= A4, ^d5 ||= G#,
Ab^ || Major tritone / augmented fourth || se ||   ||   ||
||= 22 ||= 643.90 ||< [[16_11|16/11]] ||= vv5 ||= Avv || Red major tritone || su ||   ||   ||
||= 23 ||= 673.17 ||< [[22_15|22/15]], [[40_27|40/27]] ||= v5 ||= Av || Blue fifth || sih ||   ||   ||
||= 24 ||= 702.44 ||< [[3_2|3/2]] ||= P5 ||= A || Perfect fifth || sol ||   ||   ||
||= 25 ||= 731.71 ||< [[32_21|32/21]] ||= ^5 ||= A^ || Red fifth || si ||   ||   ||
||= 26 ||= 760.98 ||< [[14_9|14/9]], [[25_16|25/16]] ||= vm6 ||= Bbv || Blue minor sixth || lo ||   ||   ||
||= 27 ||= 790.24 ||< [[11_7|11/7]], [[128_81|128/81]] ||= m6 ||= Bb || Gray minor sixth || leh ||   ||   ||
||= 28 ||= 819.51 ||< [[8_5|8/5]] ||= ^m6 ||= Bb^ || Red minor sixth || le ||   ||   ||
||= 29 ||= 848.78 ||< [[18_11|18/11]], [[44_27|44/27]] ||= ~6 ||= Bvv || Neutral sixth || lu ||   ||   ||
||= 30 ||= 878.05 ||< [[5_3|5/3]] ||= vM6 ||= Bv || Blue major sixth || la ||   ||   ||
||= 31 ||= 907.32 ||< [[27_16|27/16]] ||= M6 ||= B || Gray major sixth || laa ||   ||   ||
||= 32 ||= 936.59 ||< [[12_7|12/7]] ||= ^M6 ||= B^ || Red major sixth || li ||   ||   ||
||= 33 ||= 965.85 ||< [[7_4|7/4]] ||= vm7 ||= vC || Blue minor seventh || ta ||   ||   ||
||= 34 ||= 995.12 ||< [[16_9|16/9]] ||= m7 ||= C || Gray minor seventh || teh ||   ||   ||
||= 35 ||= 1024.39 ||< [[9_5|9/5]], [[20_11|20/11]] ||= ^m7 ||= C^ || Red minor seventh || te ||   ||   ||
||= 36 ||= 1053.66 ||< [[11_6|11/6]] ||= ~7 ||= C^^ || Neutral seventh || tu ||   ||   ||
||= 37 ||= 1082.93 ||< [[15_8|15/8]] ||= vM7 ||= C#v || Blue major seventh || ti ||   ||   ||
||= 38 ||= 1112.20 ||< [[40_21|40/21]], [[21_11|21/11]] ||= M7 ||= C# || Gray major seventh || taa ||   ||   ||
||= 39 ||= 1141.46 ||< [[48_25|48/25]], [[27_14|27/14]],
[[64_33|64/33]] ||= ^M7 ||= C#^ || Red major seventh || to ||   ||   ||
||= 40 ||= 1170.73 ||< [[160_81|160/81]] ||= v8 ||= Dv || Blue octave || da ||   ||   ||
||= 41 ||= 1200 ||< 2/1 ||= P8 ||= D ||   || do ||   ||   ||

41edo chord names using ups and downs:
0-10-20 = D F Ab = Ddim = "D dim"
0-10-21 = D F Ab^ = Ddim(^5) = "D dim up-five"
0-10-22 = D F Avv = Dm(vv5) = "D minor double-down five", or possibly Ddim(^^5)
0-10-23 = D F Av = Dm(v5) = "D minor down-five"
0-10-24 = D F A = Dm = "D minor"
0-11-24 = D F^ A = D.^m = "D upminor"
0-12-24 = D F^^ A = D.~ = "D mid"
0-13-24 = D F#v A = D.v = "D downmajor" or "D dot down"
0-14-24 = D F# A = D = "D" or "D major"
0-14-25 = D F# A^ = D(^5) = "D up-five"
0-14-26 = D F# A^^ = D(^^5) = "D double-up-five", or possibly Daug(vv5)
0-14-27 = D F# A#v = Daug(v5) = "D aug down-five"
0-14-28 = D F# A# is Daug = "D aug"
etc.
For a more complete list, see [[xenharmonic/Ups and Downs Notation#Chord%20names%20in%20other%20EDOs|Ups and Downs Notation - Chord names in other EDOs]].

==Selected just intervals by error== 
The following table shows how [[Just-24|some prominent just intervals]] are represented in 41edo (ordered by absolute error).
|| **Interval, complement** || **Error (abs., in [[cent|cents]])** ||
||= [[4_3|4/3]], [[3_2|3/2]] ||= 0.484 ||
||= [[9_8|9/8]], [[16_9|16/9]] ||= 0.968 ||
||= [[15_14|15/14]], [[28_15|28/15]] ||= 2.370 ||
||= [[7_5|7/5]], [[10_7|10/7]] ||= 2.854 ||
||= [[8_7|8/7]], [[7_4|7/4]] ||= 2.972 ||
||= [[7_6|7/6]], [[12_7|12/7]] ||= 3.456 ||
||= [[13_11|13/11]], [[22_13|22/13]] ||= 3.473 ||
||= [[11_9|11/9]], [[18_11|18/11]] ||= 3.812 ||
||= [[9_7|9/7]], [[14_9|14/9]] ||= 3.940 ||
||= [[12_11|12/11]], [[11_6|11/6]] ||= 4.296 ||
||= [[11_8|11/8]], [[16_11|16/11]] ||= 4.780 ||
||= [[16_15|16/15]], [[15_8|15/8]] ||= 5.342 ||
||= [[5_4|5/4]], [[8_5|8/5]] ||= 5.826 ||
||= [[6_5|6/5]], [[5_3|5/3]] ||= 6.310 ||
||= [[10_9|10/9]], [[9_5|9/5]] ||= 6.794 ||
||= [[18_13|18/13]], [[13_9|13/9]] ||= 7.285 ||
||= [[14_11|14/11]], [[11_7|11/7]] ||= 7.752 ||
||= [[13_12|13/12]], [[24_13|24/13]] ||= 7.769 ||
||= [[16_13|16/13]], [[13_8|13/8]] ||= 8.253 ||
||= [[15_11|15/11]], [[22_15|22/15]] ||= 10.122 ||
||= [[11_10|11/10]], [[20_11|20/11]] ||= 10.606 ||
||= [[14_13|14/13]], [[13_7|13/7]] ||= 11.225 ||
||= [[15_13|15/13]], [[26_15|26/15]] ||= 13.595 ||
||= [[13_10|13/10]], [[20_13|20/13]] ||= 14.079 ||

=Instruments= 
[[image:41-EDD elektrische gitaar.jpg width="560" height="745"]]
//41-EDO Electric guitar, by Gregory Sanchez.//

[[image:Ron_Sword_with_a_41ET_Guitar.jpg]]
//41-EDO Classical guitar, by Ron Sword.//

A possible system to tune keyboards in 41EDO is discussed in [[http://launch.groups.yahoo.com/group/tuning/message/74155]].

=Scales and modes=

A list of [[41edo modes]] (MOS and others).

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

|| Overtones in "Mode 8": || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 ||
|| ...as JI Ratio from 1/1: || 1/1 || 9/8 || 5/4 || 11/8 || 3/2 || 13/8 || 7/4 || 15/8 || 2/1 ||
|| ...in cents: || 0 || 203.9 || 386.3 || 551.3 || 702.0 || 840.5 || 968.8 || 1088.3 || 1200.0 ||
|| Nearest degree of 41edo: || 0 || 7 || 13 || 19 || 24 || 29 || 33 || 37 || 41 ||
|| ...in cents: || 0 || 204.9 || 380.5 || 556.1 || 702.4 || 848.8 || 965.9 || 1082.9 || 1200.0 ||

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.
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).
4\41 (117.1 cents) stands in for 14/13 (128.3 cents), 15/14 (119.4 cents), and 16/15 (111.7 cents).

The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.

=Nonoctave Temperaments= 
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 class="wiki_link_new">[[BP|Bohlen-Pierce]]</span>[[BP| Scale]] (or the 13th root of 3). See chart:

||||||= 3 degrees of 41edo (near 88cET) ||= overlap ||||||= 5 degrees of 41edo (near BP) ||
||~ deg of 41edo ||~ deg of 88cET ||~ cents ||~ cents ||~ cents ||~ deg of BP ||~ deg of 41edo ||
||= 0 ||= 0 ||=   ||= 0 ||=   ||= 0 ||= 0 ||
||= 3 ||= 1 ||= 87.8 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 146.3 ||= 1 ||= 5 ||
||= 6 ||= 2 ||= 175.6 ||=   ||=   ||=   ||=   ||
||= 9 ||= 3 ||= 263.4 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 292.7 ||= 2 ||= 10 ||
||= 12 ||= 4 ||= 351.2 ||=   ||=   ||=   ||=   ||
||= 15 ||= 5 ||=   ||= 439.0 ||=   ||= 3 ||= 15 ||
||= 18 ||= 6 ||= 526.8 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 585.4 ||= 4 ||= 20 ||
||= 21 ||= 7 ||= 614.6 ||=   ||=   ||=   ||=   ||
||= 24 ||= 8 ||= 702.4 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 731.7 ||= 5 ||= 25 ||
||= 27 ||= 9 ||= 790.2 ||=   ||=   ||=   ||=   ||
||= 30 ||= 10 ||=   ||= 878.0 ||=   ||= 6 ||= 30 ||
||= 33 ||= 11 ||= 965.9 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 1024.4 ||= 7 ||= 35 ||
||= 36 ||= 12 ||= 1053.7 ||=   ||=   ||=   ||=   ||
||= 39 ||= 13 ||= 1141.5 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 1170.7 ||= 8 ||= 40 ||
||||||||||||||~ [ second octave ] ||
||= 1 ||= 14 ||= 29.2 ||=   ||=   ||=   ||=   ||
||= 4 ||= 15 ||=   ||= 117.1 ||=   ||= 9 ||= 4 ||
||= 7 ||= 16 ||= 204.9 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 263.4 ||= 10 ||= 9 ||
||= 10 ||= 17 ||= 292.7 ||=   ||=   ||=   ||=   ||
||= 13 ||= 18 ||= 380.5 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 409.8 ||= 11 ||= 14 ||
||= 16 ||= 19 ||= 468.3 ||=   ||=   ||=   ||=   ||
||= 19 ||= 20 ||=   ||= 556.1 ||=   ||= 12 ||= 19 ||
||= 22 ||= 21 ||= 643.9 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 702.4 ||= 13 ||= 24 ||
||= 25 ||= 22 ||= 731.7 ||=   ||=   ||=   ||=   ||
||= 28 ||= 23 ||= 819.5 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 848.8 ||= 14 ||= 29 ||
||= 31 ||= 24 ||= 907.3 ||=   ||=   ||=   ||=   ||
||= 34 ||= 25 ||=   ||= 995.1 ||=   ||= 15 ||= 34 ||
||= 37 ||= 26 ||= 1082.9 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 1141.5 ||= 16 ||= 39 ||
||= 40 ||= 27 ||= 1170.7 ||=   ||=   ||=   ||=   ||
||||||||||||||~ [ third octave ] ||
||= 2 ||= 28 ||= 58.5 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 87.8 ||= 17 ||= 3 ||
||= 5 ||= 29 ||= 146.3 ||=   ||=   ||=   ||=   ||
||= 8 ||= 30 ||=   ||= 234.1 ||=   ||= 18 ||= 8 ||
||= 11 ||= 31 ||= 322.0 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 380.5 ||= 19 ||= 13 ||
||= 14 ||= 32 ||= 409.8 ||=   ||=   ||=   ||=   ||
||= 17 ||= 33 ||= 497.6 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 526.8 ||= 20 ||= 18 ||
||= 20 ||= 34 ||= 585.3 ||=   ||=   ||=   ||=   ||
||= 23 ||= 35 ||=   ||= 673.2 ||=   ||= 21 ||= 23 ||
||= 26 ||= 36 ||= 761.0 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 819.5 ||= 22 ||= 28 ||
||= 29 ||= 37 ||= 848.8 ||=   ||=   ||=   ||=   ||
||= 32 ||= 38 ||= 936.6 ||=   ||=   ||=   ||=   ||
||=   ||=   ||=   ||=   ||= 965.9 ||= 23 ||= 33 ||
||= 35 ||= 39 ||= 1024.4 ||=   ||=   ||=   ||=   ||
||= 38 ||= 40 ||=   ||= 1112.2 ||=   ||= 24 ||= 38 ||

==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 [[xenharmonic/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.

Interval classes could also be named by analogy. The natural, colorless, or gray interval classes are the Pythagorean ones (which show up in the standard diatonic scale), while "red" and "blue" versions are one step higher or lower. Gray thirds, sixths, and sevenths are usually more dissonant than their colorful counterparts, but the reverse is true of fourths and fifths.

The step size of 41edo is small enough that the smallest interval (the "red/blue unison", seventh-tone, comma, diesis or whatever you want to call it) is actually fairly consonant with most timbres; it resembles a "noticeably out of tune unison" rather than a minor second, and has its own distinct character and appeal.


=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

=Links= 
* [[http://en.wikipedia.org/wiki/41_equal_temperament|Wikipedia article on 41edo]]
* [[Magic22 as srutis#magic22assrutis]] describes a possible use of 41edo for [[indian]] music.
* see also [[Magic family]]
* Sword, Ron. [[@http://www.ronsword.com|"Tetracontamonophonic Scales for Guitar"]]
* Taylor, Cam. [[https://drive.google.com/open?id=0B3wIGTmjY_VZYllwcHI0d3hEc3M|Intervals, Scales and Chords in 41EDO]], a work in progress using just intonation concepts and simplified Sagittal notation.

Original HTML content:

<html><head><title>41edo</title></head><body><span style="color: #004d25; font-family: 'Times New Roman',Times,serif; font-size: 20px;"><strong>41 Tone Equal Temperament</strong></span><br />
<span style="display: block; text-align: right;"><a class="wiki_link" href="http://xenharmonie.wikispaces.com/41edo">Deutsch</a><br />
</span><br />
<!-- ws:start:WikiTextTocRule:33:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><a href="#Introduction">Introduction</a><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --> | <a href="#Commas">Commas</a><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --> | <a href="#Temperaments">Temperaments</a><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --> | <a href="#Intervals">Intervals</a><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --> | <a href="#Instruments">Instruments</a><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | <a href="#Scales and modes">Scales and modes</a><!-- ws:end:WikiTextTocRule:40 --><!-- ws:start:WikiTextTocRule:41: --><!-- ws:end:WikiTextTocRule:41 --><!-- ws:start:WikiTextTocRule:42: --> | <a href="#Nonoctave Temperaments">Nonoctave Temperaments</a><!-- ws:end:WikiTextTocRule:42 --><!-- ws:start:WikiTextTocRule:43: --><!-- ws:end:WikiTextTocRule:43 --><!-- ws:start:WikiTextTocRule:44: --> | <a href="#Music">Music</a><!-- ws:end:WikiTextTocRule:44 --><!-- ws:start:WikiTextTocRule:45: --> | <a href="#Links">Links</a><!-- ws:end:WikiTextTocRule:45 --><!-- ws:start:WikiTextTocRule:46: -->
<!-- ws:end:WikiTextTocRule:46 --><hr />
<!-- ws:start:WikiTextHeadingRule:9:&lt;h1&gt; --><h1 id="toc0"><a name="Introduction"></a><!-- ws:end:WikiTextHeadingRule:9 -->Introduction</h1>
 The 41-tET, 41-EDO, or 41-ET, is the scale derived by dividing the octave into 41 equally-sized steps. Each step represents a frequency ratio of 29.268 <a class="wiki_link" href="/cent">cent</a>s, an <a class="wiki_link" href="/interval">interval</a> close in size to <a class="wiki_link" href="/64_63">64/63</a>, the <a class="wiki_link" href="/Septimal%20comma">septimal comma</a>. 41-ET can be seen as a tuning of the <em><a class="wiki_link" href="/Schismatic%20family#Garibaldi">Garibaldi temperament</a></em> <!-- ws:start:WikiTextRefRule:2:&amp;lt;ref&amp;gt;&lt;a class=&quot;wiki_link_ext&quot; href=&quot;http://x31eq.com/schismic.htm&quot; rel=&quot;nofollow&quot;&gt;&amp;quot;Schismic Temperaments&amp;quot;&lt;/a&gt; at x31eq.com the website of &lt;a class=&quot;wiki_link&quot; href=&quot;/Graham%20Breed&quot;&gt;Graham Breed&lt;/a&gt;&amp;lt;/ref&amp;gt; --><sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup><!-- ws:end:WikiTextRefRule:2 --> , <!-- ws:start:WikiTextRefRule:4:&amp;lt;ref&amp;gt;&lt;a class=&quot;wiki_link_ext&quot; href=&quot;http://x31eq.com/decimal_lattice.htm&quot; rel=&quot;nofollow&quot;&gt;&amp;quot;Lattices with Decimal Notation&amp;quot;&lt;/a&gt; at x31eq.com&amp;lt;/ref&amp;gt; --><sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup><!-- ws:end:WikiTextRefRule:4 --> , <!-- ws:start:WikiTextRefRule:6:&amp;lt;ref&amp;gt;&lt;a class=&quot;wiki_link_ext&quot; href=&quot;http://en.wikipedia.org/wiki/Schismatic_temperament&quot; rel=&quot;nofollow&quot;&gt;Schismatic temperament&lt;/a&gt;&amp;lt;/ref&amp;gt; --><sup id="cite_ref-3" class="reference"><a href="#cite_note-3">[3]</a></sup><!-- ws:end:WikiTextRefRule:6 --> the <em><a class="wiki_link" href="/Magic%20family">Magic temperament</a></em> <!-- ws:start:WikiTextRefRule:8:&amp;lt;ref&amp;gt;&lt;a class=&quot;wiki_link_ext&quot; href=&quot;http://en.wikipedia.org/wiki/Magic_temperament&quot; rel=&quot;nofollow&quot;&gt;Magic temperament&lt;/a&gt;&amp;lt;/ref&amp;gt; --><sup id="cite_ref-4" class="reference"><a href="#cite_note-4">[4]</a></sup><!-- ws:end:WikiTextRefRule:8 --> and the superkleismic (41&amp;26) temperament. It is the second smallest equal temperament (after <a class="wiki_link" href="/29edo">29edo</a>) whose perfect fifth is closer to just intonation than that of <a class="wiki_link" href="/12edo">12-ET</a>, and is the seventh <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta integral edo</a> after 31; it is not, however, a <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta%20EDO%20lists">zeta gap edo</a>. This has to do with the fact that it can deal with the <a class="wiki_link" href="/11-limit">11-limit</a> fairly well, and the <a class="wiki_link" href="/13-limit">13-limit</a> perhaps close enough for government work, though its <a class="wiki_link" href="/13_10">13/10</a> is 14 cents sharp. Various 13-limit <a class="wiki_link" href="/magic%20extensions">magic extensions</a> 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.<br />
<br />
41edo is consistent in the 15 odd limit. In fact, <em>all</em> of its intervals between 100 and 1100 cents in size are 15-odd-limit consonances. (In comparison, <a class="wiki_link" href="/31edo">31edo</a> is only consistent up to the 11-limit, and the intervals 12/31 and 19/31 have no 11-limit approximations).<br />
<br />
41-ET forms the foundation of the <a class="wiki_link_ext" href="http://www.h-pi.com/theory/huntsystem1.html" rel="nofollow">H-System</a>, which uses the scale degrees of 41-ET as the basic <a class="wiki_link" href="/13-limit">13-limit</a> intervals requiring fine tuning +/- 1 <a class="wiki_link_ext" href="http://www.h-pi.com/theory/huntsystem2.html" rel="nofollow">average JND</a> from the 41-ET circle in <a class="wiki_link" href="/205edo">205edo</a>.<br />
<br />
41edo is the 13th <a class="wiki_link" href="/prime%20numbers">prime</a> edo, following <a class="wiki_link" href="/37edo">37edo</a> and coming before <a class="wiki_link" href="/43edo">43edo</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:11:&lt;h1&gt; --><h1 id="toc1"><a name="Commas"></a><!-- ws:end:WikiTextHeadingRule:11 -->Commas</h1>
 41 EDO tempers out the following commas using its patent val, &lt; 41 65 95 115 142 152 168 174 185 199 203 |.<br />


<table class="wiki_table">
    <tr>
        <th>Name<br />
</th>
        <th>Monzo<br />
</th>
        <th>Ratio<br />
</th>
        <th>Cents<br />
</th>
    </tr>
    <tr>
        <td>odiheim<br />
</td>
        <td>| -1 2 -4 5 -2 &gt;<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td>0.15<br />
</td>
    </tr>
    <tr>
        <td>harmonisma<br />
</td>
        <td>| 3 -2 0 -1 3 -2 &gt;<br />
</td>
        <td style="text-align: center;">10648/10647<br />
</td>
        <td>0.16<br />
</td>
    </tr>
    <tr>
        <td>tridecimal schisma, Sagittal schismina<br />
</td>
        <td>| 12 -2 -1 -1 0 -1/1 &gt;<br />
</td>
        <td style="text-align: center;">4096/4095<br />
</td>
        <td>0.42<br />
</td>
    </tr>
    <tr>
        <td>Lehmerisma<br />
</td>
        <td>| -4 -3 2 -1 2 &gt;<br />
</td>
        <td style="text-align: center;">3025/3024<br />
</td>
        <td>0.57<br />
</td>
    </tr>
    <tr>
        <td>Breedsma<br />
</td>
        <td>| -5 -1 -2 4 &gt;<br />
</td>
        <td style="text-align: center;">2401/2400<br />
</td>
        <td>0.72<br />
</td>
    </tr>
    <tr>
        <td>Eratosthenes' comma<br />
</td>
        <td>| 6 -5 -1 0 0 0 0 1 &gt;<br />
</td>
        <td style="text-align: center;">1216/1215<br />
</td>
        <td>1.42<br />
</td>
    </tr>
    <tr>
        <td>schisma<br />
</td>
        <td>| -15 8 1 &gt;<br />
</td>
        <td style="text-align: center;">32805/32768<br />
</td>
        <td>1.95<br />
</td>
    </tr>
    <tr>
        <td>squbema<br />
</td>
        <td>| -3 6 0 -1 0 -1 &gt;<br />
</td>
        <td style="text-align: center;">729/728<br />
</td>
        <td>2.38<br />
</td>
    </tr>
    <tr>
        <td>septendecimal bridge comma<br />
</td>
        <td>| -1 -1 1 -1 1 1 -1 &gt;<br />
</td>
        <td style="text-align: center;">715/714<br />
</td>
        <td>2.42<br />
</td>
    </tr>
    <tr>
        <td>Swets' comma, swetisma<br />
</td>
        <td>| 2 3 1 -2 -1 &gt;<br />
</td>
        <td style="text-align: center;">540/539<br />
</td>
        <td>3.21<br />
</td>
    </tr>
    <tr>
        <td>undevicesimal comma, Boethius' comma<br />
</td>
        <td>| -9 3 0 0 0 0 0 1 &gt;<br />
</td>
        <td style="text-align: center;">513/512<br />
</td>
        <td>3.38<br />
</td>
    </tr>
    <tr>
        <td>moctdel<br />
</td>
        <td>| -2 0 3 -3 1 &gt;<br />
</td>
        <td style="text-align: center;">1375/1372<br />
</td>
        <td>3.78<br />
</td>
    </tr>
    <tr>
        <td>Beta 2, septimal schisma, garischisma<br />
</td>
        <td>| 25 -14 0 -1 &gt;<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td>3.80<br />
</td>
    </tr>
    <tr>
        <td>Werckmeister's undecimal septenarian schisma, werckisma<br />
</td>
        <td>| -3 2 -1 2 -1 &gt;<br />
</td>
        <td style="text-align: center;">441/440<br />
</td>
        <td>3.93<br />
</td>
    </tr>
    <tr>
        <td>cuthbert<br />
</td>
        <td>| 0 0 -1 1 2 -2 &gt;<br />
</td>
        <td style="text-align: center;">847/845<br />
</td>
        <td>4.09<br />
</td>
    </tr>
    <tr>
        <td>undecimal kleisma, keenanisma<br />
</td>
        <td>| -7 -1 1 1 1 &gt;<br />
</td>
        <td style="text-align: center;">385/384<br />
</td>
        <td>4.50<br />
</td>
    </tr>
    <tr>
        <td>gentle comma<br />
</td>
        <td>| 2 -1 0 1 -2 1 &gt;<br />
</td>
        <td style="text-align: center;">364/363<br />
</td>
        <td>4.76<br />
</td>
    </tr>
    <tr>
        <td>minthma<br />
</td>
        <td>| 5 -3 0 0 1 -1 &gt;<br />
</td>
        <td style="text-align: center;">352/351<br />
</td>
        <td>4.93<br />
</td>
    </tr>
    <tr>
        <td>marveltwin<br />
</td>
        <td>| -2 -4 2 0 0 1 &gt;<br />
</td>
        <td style="text-align: center;">325/324<br />
</td>
        <td>5.34<br />
</td>
    </tr>
    <tr>
        <td>Beta 5, Garibaldi comma, hemifamity<br />
</td>
        <td>| 10 -6 1 -1 &gt;<br />
</td>
        <td style="text-align: center;">5120/5103<br />
</td>
        <td>5.76<br />
</td>
    </tr>
    <tr>
        <td>hemimage<br />
</td>
        <td>| 5 -7 -1 3 &gt;<br />
</td>
        <td style="text-align: center;">10976/10935<br />
</td>
        <td>6.48<br />
</td>
    </tr>
    <tr>
        <td>septendecimal kleisma<br />
</td>
        <td>| 8 -1 -1 0 0 0 -1 &gt;<br />
</td>
        <td style="text-align: center;">256/255<br />
</td>
        <td>6.78<br />
</td>
    </tr>
    <tr>
        <td>small BP diesis, mirkwai<br />
</td>
        <td>| 0 3 4 -5 &gt;<br />
</td>
        <td style="text-align: center;">16875/16807<br />
</td>
        <td>6.99<br />
</td>
    </tr>
    <tr>
        <td>neutral third comma, rastma<br />
</td>
        <td>| -1 5 0 0 -2 &gt;<br />
</td>
        <td style="text-align: center;">243/242<br />
</td>
        <td>7.14<br />
</td>
    </tr>
    <tr>
        <td>kestrel comma<br />
</td>
        <td>| 2 3 0 -1 1 -2 &gt;<br />
</td>
        <td style="text-align: center;">1188/1183<br />
</td>
        <td>7.30<br />
</td>
    </tr>
    <tr>
        <td>septimal kleisma, marvel comma<br />
</td>
        <td>| -5 2 2 -1 &gt;<br />
</td>
        <td style="text-align: center;">225/224<br />
</td>
        <td>7.71<br />
</td>
    </tr>
    <tr>
        <td>huntma<br />
</td>
        <td>| 7 0 1 -2 0 -1 &gt;<br />
</td>
        <td style="text-align: center;">640/637<br />
</td>
        <td>8.13<br />
</td>
    </tr>
    <tr>
        <td>spleen comma<br />
</td>
        <td>| 1 1 1 1 -1 0 0 -1 &gt;<br />
</td>
        <td style="text-align: center;">210/209<br />
</td>
        <td>8.26<br />
</td>
    </tr>
    <tr>
        <td>orgonisma<br />
</td>
        <td>| 16 0 0 -2 -3 &gt;<br />
</td>
        <td style="text-align: center;">65536/65219<br />
</td>
        <td>8.39<br />
</td>
    </tr>
    <tr>
        <td>gamelan residue, gamelisma<br />
</td>
        <td>| -10 1 0 3 &gt;<br />
</td>
        <td style="text-align: center;">1029/1024<br />
</td>
        <td>8.43<br />
</td>
    </tr>
    <tr>
        <td>septendecimal comma<br />
</td>
        <td>| -7 7 0 0 0 0 -1 &gt;<br />
</td>
        <td style="text-align: center;">2187/2176<br />
</td>
        <td>8.73<br />
</td>
    </tr>
    <tr>
        <td>mynucuma<br />
</td>
        <td>| 2 -1 -1 2 0 -1 &gt;<br />
</td>
        <td style="text-align: center;">196/195<br />
</td>
        <td>8.86<br />
</td>
    </tr>
    <tr>
        <td>quince<br />
</td>
        <td>| -15 0 -2 7 &gt;<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td>9.15<br />
</td>
    </tr>
    <tr>
        <td>undecimal semicomma, pentacircle (minthma * gentle)<br />
</td>
        <td>| 7 -4 0 1 -1 &gt;<br />
</td>
        <td style="text-align: center;">896/891<br />
</td>
        <td>9.69<br />
</td>
    </tr>
    <tr>
        <td>29th-partial chroma<br />
</td>
        <td>| -4 -2 1 0 0 0 0 0 0 1 &gt;<br />
</td>
        <td style="text-align: center;">145/144<br />
</td>
        <td>11.98<br />
</td>
    </tr>
    <tr>
        <td>grossma<br />
</td>
        <td>| 4 2 0 0 -1 -1 &gt;<br />
</td>
        <td style="text-align: center;">144/143<br />
</td>
        <td>12.06<br />
</td>
    </tr>
    <tr>
        <td>gassorma<br />
</td>
        <td>| 0 -1 2 -1 1 -1 &gt;<br />
</td>
        <td style="text-align: center;">275/273<br />
</td>
        <td>12.64<br />
</td>
    </tr>
    <tr>
        <td>septimal semicomma, octagar<br />
</td>
        <td>| 5 -4 3 -2 &gt;<br />
</td>
        <td style="text-align: center;">4000/3969<br />
</td>
        <td>13.47<br />
</td>
    </tr>
    <tr>
        <td>minor BP diesis, sensamagic<br />
</td>
        <td>| 0 -5 1 2 &gt;<br />
</td>
        <td style="text-align: center;">245/243<br />
</td>
        <td>14.19<br />
</td>
    </tr>
    <tr>
        <td>secorian<br />
</td>
        <td>| 12 -7 0 1 0 -1/1 &gt;<br />
</td>
        <td style="text-align: center;">28672/28431<br />
</td>
        <td>14.61<br />
</td>
    </tr>
    <tr>
        <td>mirwomo comma<br />
</td>
        <td>| -15 3 2 2 &gt;<br />
</td>
        <td style="text-align: center;">33075/32768<br />
</td>
        <td>16.14<br />
</td>
    </tr>
    <tr>
        <td>vicesimotertial comma<br />
</td>
        <td>| 5 -6 0 0 0 0 0 0 1 &gt;<br />
</td>
        <td style="text-align: center;">736/729<br />
</td>
        <td>16.54<br />
</td>
    </tr>
    <tr>
        <td>small tridecimal comma, animist<br />
</td>
        <td>| -3 1 1 1 0 -1 &gt;<br />
</td>
        <td style="text-align: center;">105/104<br />
</td>
        <td>16.57<br />
</td>
    </tr>
    <tr>
        <td>hemimin<br />
</td>
        <td>| 6 1 0 1 -3 &gt;<br />
</td>
        <td style="text-align: center;">1344/1331<br />
</td>
        <td>16.83<br />
</td>
    </tr>
    <tr>
        <td>Ptolemy's comma, ptolemisma<br />
</td>
        <td>| 2 -2 2 0 -1 &gt;<br />
</td>
        <td style="text-align: center;">100/99<br />
</td>
        <td>17.40<br />
</td>
    </tr>
    <tr>
        <td>'41-tone' comma<br />
</td>
        <td>| 65 -41 &gt;<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td>19.84<br />
</td>
    </tr>
    <tr>
        <td>tolerma<br />
</td>
        <td>| 10 -11 2 1 &gt;<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td>19.95<br />
</td>
    </tr>
    <tr>
        <td>major BP diesis, gariboh<br />
</td>
        <td>| 0 -2 5 -3 &gt;<br />
</td>
        <td style="text-align: center;">3125/3087<br />
</td>
        <td>21.18<br />
</td>
    </tr>
    <tr>
        <td>cassacot<br />
</td>
        <td>| -1 0 1 2 -2 &gt;<br />
</td>
        <td style="text-align: center;">245/242<br />
</td>
        <td>21.33<br />
</td>
    </tr>
    <tr>
        <td>keema<br />
</td>
        <td>| -5 -3 3 1 &gt;<br />
</td>
        <td style="text-align: center;">875/864<br />
</td>
        <td>21.90<br />
</td>
    </tr>
    <tr>
        <td>blackjackisma<br />
</td>
        <td>| -10 7 8 -7 &gt;<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td>22.41<br />
</td>
    </tr>
    <tr>
        <td>roda<br />
</td>
        <td>| 20 -17 3 &gt;<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td>25.71<br />
</td>
    </tr>
    <tr>
        <td>minimal diesis, tetracot comma<br />
</td>
        <td>| 5 -9 4 &gt;<br />
</td>
        <td style="text-align: center;">20000/19683<br />
</td>
        <td>27.66<br />
</td>
    </tr>
    <tr>
        <td>small diesis, magic comma<br />
</td>
        <td>| -10 -1 5 &gt;<br />
</td>
        <td style="text-align: center;">3125/3072<br />
</td>
        <td>29.61<br />
</td>
    </tr>
    <tr>
        <td>thuja comma<br />
</td>
        <td>| 15 0 1 0 -5 &gt;<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td>29.72<br />
</td>
    </tr>
    <tr>
        <td>Ampersand's comma<br />
</td>
        <td>| -25 7 6 &gt;<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td>31.57<br />
</td>
    </tr>
    <tr>
        <td>great BP diesis<br />
</td>
        <td>| 0 -7 6 -1 &gt;<br />
</td>
        <td style="text-align: center;">15625/15309<br />
</td>
        <td>35.37<br />
</td>
    </tr>
    <tr>
        <td>shibboleth<br />
</td>
        <td>| -5 -10 9 &gt;<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td>57.27<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:13:&lt;h1&gt; --><h1 id="toc2"><a name="Temperaments"></a><!-- ws:end:WikiTextHeadingRule:13 -->Temperaments</h1>
 <a class="wiki_link" href="/List%20of%20edo-distinct%2041et%20rank%20two%20temperaments">List of edo-distinct 41et rank two temperaments</a><br />
<br />
<!-- ws:start:WikiTextHeadingRule:15:&lt;h1&gt; --><h1 id="toc3"><a name="Intervals"></a><!-- ws:end:WikiTextHeadingRule:15 -->Intervals</h1>
 

<table class="wiki_table">
    <tr>
        <th><br />
</th>
        <th>cents value<br />
</th>
        <th>Approximate<br />
Ratios in the <a class="wiki_link" href="/11-limit">11-limit</a><br />
</th>
        <th colspan="2"><a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">ups and</a><br />
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">downs</a><br />
<a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation">notation</a><br />
</th>
        <th>Proposed names<br />
</th>
        <th>Andrew's<br />
solfege<br />
syllable<br />
</th>
        <th>generator for<br />
</th>
        <th>some MOS and MODMOS Scales available<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: center;">0.00<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/1_1">1/1</a><br />
</td>
        <td style="text-align: center;">P1<br />
</td>
        <td style="text-align: center;">D<br />
</td>
        <td>Unison<br />
</td>
        <td>do<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">1<br />
</td>
        <td style="text-align: center;">29.27<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/81_80">81/80</a><br />
</td>
        <td style="text-align: center;">^1<br />
</td>
        <td style="text-align: center;">D^<br />
</td>
        <td>Red unison<br />
</td>
        <td>di<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">2<br />
</td>
        <td style="text-align: center;">58.54<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/25_24">25/24</a>, <a class="wiki_link" href="/28_27">28/27</a>,<br />
<a class="wiki_link" href="/33_32">33/32</a><br />
</td>
        <td style="text-align: center;">vm2<br />
</td>
        <td style="text-align: center;">Ebv<br />
</td>
        <td>Blue minor second<br />
</td>
        <td>ro<br />
</td>
        <td><a class="wiki_link" href="/Hemimiracle">Hemimiracle</a><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3<br />
</td>
        <td style="text-align: center;">87.80<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/21_20">21/20</a>, <a class="wiki_link" href="/22_21">22/21</a><br />
</td>
        <td style="text-align: center;">m2<br />
</td>
        <td style="text-align: center;">Eb<br />
</td>
        <td>Gray minor second<br />
</td>
        <td>rih<br />
</td>
        <td>88cET (approx),<br />
<a class="wiki_link" href="/octacot">octacot</a><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4<br />
</td>
        <td style="text-align: center;">117.07<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/16_15">16/15</a>, <a class="wiki_link" href="/15_14">15/14</a><br />
</td>
        <td style="text-align: center;">^m2<br />
</td>
        <td style="text-align: center;">Eb^<br />
</td>
        <td>Red minor second<br />
</td>
        <td>ra<br />
</td>
        <td><a class="wiki_link" href="/Miracle">Miracle</a><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">5<br />
</td>
        <td style="text-align: center;">146.34<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/12_11">12/11</a><br />
</td>
        <td style="text-align: center;">~2<br />
</td>
        <td style="text-align: center;">Evv<br />
</td>
        <td>Neutral second<br />
</td>
        <td>ru<br />
</td>
        <td><a class="wiki_link" href="/Bohlen-Pierce">Bohlen-Pierce</a>/<a class="wiki_link" href="/bohpier">bohpier</a><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">6<br />
</td>
        <td style="text-align: center;">175.61<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/10_9">10/9</a>, <a class="wiki_link" href="/11_10">11/10</a><br />
</td>
        <td style="text-align: center;">vM2<br />
</td>
        <td style="text-align: center;">Ev<br />
</td>
        <td>Blue major second<br />
</td>
        <td>reh<br />
</td>
        <td><a class="wiki_link" href="/Tetracot">Tetracot</a>/<a class="wiki_link" href="/bunya">bunya</a>/<a class="wiki_link" href="/monkey">monkey</a><br />
</td>
        <td>13-tone MOS: 1 5 1 5 1 5 1 5 5 1 5 1 5<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">7<br />
</td>
        <td style="text-align: center;">204.88<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/9_8">9/8</a><br />
</td>
        <td style="text-align: center;">M2<br />
</td>
        <td style="text-align: center;">E<br />
</td>
        <td>Gray major second<br />
</td>
        <td>re<br />
</td>
        <td><a class="wiki_link" href="/Baldy">Baldy</a><br />
</td>
        <td>11-tone MOS: 6 1 6 6 1 6 1 6 1 6 1<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">8<br />
</td>
        <td style="text-align: center;">234.15<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/8_7">8/7</a><br />
</td>
        <td style="text-align: center;">^M2<br />
</td>
        <td style="text-align: center;">E^<br />
</td>
        <td>Red major second<br />
</td>
        <td>ri<br />
</td>
        <td><a class="wiki_link" href="/Rodan">Rodan</a>/<a class="wiki_link" href="/guiron">guiron</a><br />
</td>
        <td>11-tone MOS: 7 1 7 1 7 1 7 1 1 7 1<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">9<br />
</td>
        <td style="text-align: center;">263.41<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/7_6">7/6</a>, <a class="wiki_link" href="/32_25">32/25</a><br />
</td>
        <td style="text-align: center;">vm3<br />
</td>
        <td style="text-align: center;">Fv<br />
</td>
        <td>Blue minor third<br />
</td>
        <td>ma<br />
</td>
        <td><a class="wiki_link" href="/Septimin">Septimin</a><br />
</td>
        <td>9-tone MOS: 5 4 5 5 4 5 4 5 4<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">10<br />
</td>
        <td style="text-align: center;">292.68<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/32_27">32/27</a><br />
</td>
        <td style="text-align: center;">m3<br />
</td>
        <td style="text-align: center;">F<br />
</td>
        <td>Gray minor third<br />
</td>
        <td>meh<br />
</td>
        <td><a class="wiki_link" href="/Quasitemp">Quasitemp</a><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">11<br />
</td>
        <td style="text-align: center;">321.95<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/6_5">6/5</a><br />
</td>
        <td style="text-align: center;">^m3<br />
</td>
        <td style="text-align: center;">F^<br />
</td>
        <td>Red minor third<br />
</td>
        <td>me<br />
</td>
        <td><a class="wiki_link" href="/Superkleismic">Superkleismic</a><br />
</td>
        <td>11-tone MOS: 5 3 5 3 3 5 3 3 5 3 3<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">12<br />
</td>
        <td style="text-align: center;">351.22<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/11_9">11/9</a>,<a class="wiki_link" href="/27_22">27/22</a><br />
</td>
        <td style="text-align: center;">~3<br />
</td>
        <td style="text-align: center;">F^^<br />
</td>
        <td>Neutral third<br />
</td>
        <td>mu<br />
</td>
        <td><a class="wiki_link" href="/Hemififths">Hemififths</a>/<a class="wiki_link" href="/karadeniz">karadeniz</a><br />
</td>
        <td>10-tone MOS: 5 2 5 5 2 5 5 5 2 5<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">13<br />
</td>
        <td style="text-align: center;">380.49<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/5_4">5/4</a><br />
</td>
        <td style="text-align: center;">vM3<br />
</td>
        <td style="text-align: center;">F#v<br />
</td>
        <td>Blue major third<br />
</td>
        <td>mi<br />
</td>
        <td><a class="wiki_link" href="/Magic">Magic</a>/<a class="wiki_link" href="/witchcraft">witchcraft</a><br />
</td>
        <td>10-tone MOS: 2 9 2 2 9 2 2 9 2 2<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">14<br />
</td>
        <td style="text-align: center;">409.76<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/14_11">14/11</a>, <a class="wiki_link" href="/81_64">81/64</a><br />
</td>
        <td style="text-align: center;">M3<br />
</td>
        <td style="text-align: center;">F#<br />
</td>
        <td>Gray major third<br />
</td>
        <td>maa<br />
</td>
        <td><a class="wiki_link" href="/Hocus">Hocus</a><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">15<br />
</td>
        <td style="text-align: center;">439.02<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/9_7">9/7</a><br />
</td>
        <td style="text-align: center;">^M3<br />
</td>
        <td style="text-align: center;">F#^<br />
</td>
        <td>Red major third<br />
</td>
        <td>mo<br />
</td>
        <td><br />
</td>
        <td>11-tone MOS: 4 3 4 4 4 3 4 4 3 4 4<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">16<br />
</td>
        <td style="text-align: center;">468.29<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/21_16">21/16</a><br />
</td>
        <td style="text-align: center;">v4<br />
</td>
        <td style="text-align: center;">Gv<br />
</td>
        <td>Blue fourth<br />
</td>
        <td>fe<br />
</td>
        <td><a class="wiki_link" href="/Barbad">Barbad</a><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">17<br />
</td>
        <td style="text-align: center;">497.56<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/4_3">4/3</a><br />
</td>
        <td style="text-align: center;">P4<br />
</td>
        <td style="text-align: center;">G<br />
</td>
        <td>Perfect fourth<br />
</td>
        <td>fa<br />
</td>
        <td><a class="wiki_link" href="/Schismatic">Schismatic</a> (<a class="wiki_link" href="/helmholtz">helmholtz</a>, <a class="wiki_link" href="/Garibaldi%20temperament">garibaldi</a>, <a class="wiki_link" href="/cassandra">cassandra</a>)<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">18<br />
</td>
        <td style="text-align: center;">526.83<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/15_11">15/11</a>, <a class="wiki_link" href="/27_20">27/20</a><br />
</td>
        <td style="text-align: center;">^4<br />
</td>
        <td style="text-align: center;">G^<br />
</td>
        <td>Red fourth<br />
</td>
        <td>fih<br />
</td>
        <td><a class="wiki_link" href="/Trismegistus">Trismegistus</a><br />
</td>
        <td>9-tone MOS: 5 5 3 5 5 5 5 3 5<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">19<br />
</td>
        <td style="text-align: center;">556.10<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/11_8">11/8</a><br />
</td>
        <td style="text-align: center;">^^4<br />
</td>
        <td style="text-align: center;">G^^<br />
</td>
        <td>Blue minor tritone<br />
</td>
        <td>fu<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">20<br />
</td>
        <td style="text-align: center;">585.37<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/7_5">7/5</a><br />
</td>
        <td style="text-align: center;">vA4, d5<br />
</td>
        <td style="text-align: center;">G#v,<br />
Ab<br />
</td>
        <td>Minor tritone / diminished fifth<br />
</td>
        <td>fi<br />
</td>
        <td><a class="wiki_link" href="/Pluto">Pluto</a><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">21<br />
</td>
        <td style="text-align: center;">614.63<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/10_7">10/7</a><br />
</td>
        <td style="text-align: center;">A4, ^d5<br />
</td>
        <td style="text-align: center;">G#,<br />
Ab^<br />
</td>
        <td>Major tritone / augmented fourth<br />
</td>
        <td>se<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">22<br />
</td>
        <td style="text-align: center;">643.90<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/16_11">16/11</a><br />
</td>
        <td style="text-align: center;">vv5<br />
</td>
        <td style="text-align: center;">Avv<br />
</td>
        <td>Red major tritone<br />
</td>
        <td>su<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">23<br />
</td>
        <td style="text-align: center;">673.17<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/22_15">22/15</a>, <a class="wiki_link" href="/40_27">40/27</a><br />
</td>
        <td style="text-align: center;">v5<br />
</td>
        <td style="text-align: center;">Av<br />
</td>
        <td>Blue fifth<br />
</td>
        <td>sih<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">24<br />
</td>
        <td style="text-align: center;">702.44<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/3_2">3/2</a><br />
</td>
        <td style="text-align: center;">P5<br />
</td>
        <td style="text-align: center;">A<br />
</td>
        <td>Perfect fifth<br />
</td>
        <td>sol<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">25<br />
</td>
        <td style="text-align: center;">731.71<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/32_21">32/21</a><br />
</td>
        <td style="text-align: center;">^5<br />
</td>
        <td style="text-align: center;">A^<br />
</td>
        <td>Red fifth<br />
</td>
        <td>si<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">26<br />
</td>
        <td style="text-align: center;">760.98<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/14_9">14/9</a>, <a class="wiki_link" href="/25_16">25/16</a><br />
</td>
        <td style="text-align: center;">vm6<br />
</td>
        <td style="text-align: center;">Bbv<br />
</td>
        <td>Blue minor sixth<br />
</td>
        <td>lo<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">27<br />
</td>
        <td style="text-align: center;">790.24<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/11_7">11/7</a>, <a class="wiki_link" href="/128_81">128/81</a><br />
</td>
        <td style="text-align: center;">m6<br />
</td>
        <td style="text-align: center;">Bb<br />
</td>
        <td>Gray minor sixth<br />
</td>
        <td>leh<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">28<br />
</td>
        <td style="text-align: center;">819.51<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/8_5">8/5</a><br />
</td>
        <td style="text-align: center;">^m6<br />
</td>
        <td style="text-align: center;">Bb^<br />
</td>
        <td>Red minor sixth<br />
</td>
        <td>le<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">29<br />
</td>
        <td style="text-align: center;">848.78<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/18_11">18/11</a>, <a class="wiki_link" href="/44_27">44/27</a><br />
</td>
        <td style="text-align: center;">~6<br />
</td>
        <td style="text-align: center;">Bvv<br />
</td>
        <td>Neutral sixth<br />
</td>
        <td>lu<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">30<br />
</td>
        <td style="text-align: center;">878.05<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/5_3">5/3</a><br />
</td>
        <td style="text-align: center;">vM6<br />
</td>
        <td style="text-align: center;">Bv<br />
</td>
        <td>Blue major sixth<br />
</td>
        <td>la<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">31<br />
</td>
        <td style="text-align: center;">907.32<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/27_16">27/16</a><br />
</td>
        <td style="text-align: center;">M6<br />
</td>
        <td style="text-align: center;">B<br />
</td>
        <td>Gray major sixth<br />
</td>
        <td>laa<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">32<br />
</td>
        <td style="text-align: center;">936.59<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/12_7">12/7</a><br />
</td>
        <td style="text-align: center;">^M6<br />
</td>
        <td style="text-align: center;">B^<br />
</td>
        <td>Red major sixth<br />
</td>
        <td>li<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">33<br />
</td>
        <td style="text-align: center;">965.85<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/7_4">7/4</a><br />
</td>
        <td style="text-align: center;">vm7<br />
</td>
        <td style="text-align: center;">vC<br />
</td>
        <td>Blue minor seventh<br />
</td>
        <td>ta<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">34<br />
</td>
        <td style="text-align: center;">995.12<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/16_9">16/9</a><br />
</td>
        <td style="text-align: center;">m7<br />
</td>
        <td style="text-align: center;">C<br />
</td>
        <td>Gray minor seventh<br />
</td>
        <td>teh<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">35<br />
</td>
        <td style="text-align: center;">1024.39<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/9_5">9/5</a>, <a class="wiki_link" href="/20_11">20/11</a><br />
</td>
        <td style="text-align: center;">^m7<br />
</td>
        <td style="text-align: center;">C^<br />
</td>
        <td>Red minor seventh<br />
</td>
        <td>te<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">36<br />
</td>
        <td style="text-align: center;">1053.66<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/11_6">11/6</a><br />
</td>
        <td style="text-align: center;">~7<br />
</td>
        <td style="text-align: center;">C^^<br />
</td>
        <td>Neutral seventh<br />
</td>
        <td>tu<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">37<br />
</td>
        <td style="text-align: center;">1082.93<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/15_8">15/8</a><br />
</td>
        <td style="text-align: center;">vM7<br />
</td>
        <td style="text-align: center;">C#v<br />
</td>
        <td>Blue major seventh<br />
</td>
        <td>ti<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">38<br />
</td>
        <td style="text-align: center;">1112.20<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/40_21">40/21</a>, <a class="wiki_link" href="/21_11">21/11</a><br />
</td>
        <td style="text-align: center;">M7<br />
</td>
        <td style="text-align: center;">C#<br />
</td>
        <td>Gray major seventh<br />
</td>
        <td>taa<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">39<br />
</td>
        <td style="text-align: center;">1141.46<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/48_25">48/25</a>, <a class="wiki_link" href="/27_14">27/14</a>,<br />
<a class="wiki_link" href="/64_33">64/33</a><br />
</td>
        <td style="text-align: center;">^M7<br />
</td>
        <td style="text-align: center;">C#^<br />
</td>
        <td>Red major seventh<br />
</td>
        <td>to<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">40<br />
</td>
        <td style="text-align: center;">1170.73<br />
</td>
        <td style="text-align: left;"><a class="wiki_link" href="/160_81">160/81</a><br />
</td>
        <td style="text-align: center;">v8<br />
</td>
        <td style="text-align: center;">Dv<br />
</td>
        <td>Blue octave<br />
</td>
        <td>da<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">41<br />
</td>
        <td style="text-align: center;">1200<br />
</td>
        <td style="text-align: left;">2/1<br />
</td>
        <td style="text-align: center;">P8<br />
</td>
        <td style="text-align: center;">D<br />
</td>
        <td><br />
</td>
        <td>do<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
</table>

<br />
41edo chord names using ups and downs:<br />
0-10-20 = D F Ab = Ddim = &quot;D dim&quot;<br />
0-10-21 = D F Ab^ = Ddim(^5) = &quot;D dim up-five&quot;<br />
0-10-22 = D F Avv = Dm(vv5) = &quot;D minor double-down five&quot;, or possibly Ddim(^^5)<br />
0-10-23 = D F Av = Dm(v5) = &quot;D minor down-five&quot;<br />
0-10-24 = D F A = Dm = &quot;D minor&quot;<br />
0-11-24 = D F^ A = D.^m = &quot;D upminor&quot;<br />
0-12-24 = D F^^ A = D.~ = &quot;D mid&quot;<br />
0-13-24 = D F#v A = D.v = &quot;D downmajor&quot; or &quot;D dot down&quot;<br />
0-14-24 = D F# A = D = &quot;D&quot; or &quot;D major&quot;<br />
0-14-25 = D F# A^ = D(^5) = &quot;D up-five&quot;<br />
0-14-26 = D F# A^^ = D(^^5) = &quot;D double-up-five&quot;, or possibly Daug(vv5)<br />
0-14-27 = D F# A#v = Daug(v5) = &quot;D aug down-five&quot;<br />
0-14-28 = D F# A# is Daug = &quot;D aug&quot;<br />
etc.<br />
For a more complete list, see <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Chord%20names%20in%20other%20EDOs">Ups and Downs Notation - Chord names in other EDOs</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:17:&lt;h2&gt; --><h2 id="toc4"><a name="Intervals-Selected just intervals by error"></a><!-- ws:end:WikiTextHeadingRule:17 -->Selected just intervals by error</h2>
 The following table shows how <a class="wiki_link" href="/Just-24">some prominent just intervals</a> are represented in 41edo (ordered by absolute error).<br />


<table class="wiki_table">
    <tr>
        <td><strong>Interval, complement</strong><br />
</td>
        <td><strong>Error (abs., in <a class="wiki_link" href="/cent">cents</a>)</strong><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/4_3">4/3</a>, <a class="wiki_link" href="/3_2">3/2</a><br />
</td>
        <td style="text-align: center;">0.484<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/9_8">9/8</a>, <a class="wiki_link" href="/16_9">16/9</a><br />
</td>
        <td style="text-align: center;">0.968<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/15_14">15/14</a>, <a class="wiki_link" href="/28_15">28/15</a><br />
</td>
        <td style="text-align: center;">2.370<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/7_5">7/5</a>, <a class="wiki_link" href="/10_7">10/7</a><br />
</td>
        <td style="text-align: center;">2.854<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/8_7">8/7</a>, <a class="wiki_link" href="/7_4">7/4</a><br />
</td>
        <td style="text-align: center;">2.972<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/7_6">7/6</a>, <a class="wiki_link" href="/12_7">12/7</a><br />
</td>
        <td style="text-align: center;">3.456<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/13_11">13/11</a>, <a class="wiki_link" href="/22_13">22/13</a><br />
</td>
        <td style="text-align: center;">3.473<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/11_9">11/9</a>, <a class="wiki_link" href="/18_11">18/11</a><br />
</td>
        <td style="text-align: center;">3.812<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/9_7">9/7</a>, <a class="wiki_link" href="/14_9">14/9</a><br />
</td>
        <td style="text-align: center;">3.940<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/12_11">12/11</a>, <a class="wiki_link" href="/11_6">11/6</a><br />
</td>
        <td style="text-align: center;">4.296<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/11_8">11/8</a>, <a class="wiki_link" href="/16_11">16/11</a><br />
</td>
        <td style="text-align: center;">4.780<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/16_15">16/15</a>, <a class="wiki_link" href="/15_8">15/8</a><br />
</td>
        <td style="text-align: center;">5.342<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/5_4">5/4</a>, <a class="wiki_link" href="/8_5">8/5</a><br />
</td>
        <td style="text-align: center;">5.826<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/6_5">6/5</a>, <a class="wiki_link" href="/5_3">5/3</a><br />
</td>
        <td style="text-align: center;">6.310<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/10_9">10/9</a>, <a class="wiki_link" href="/9_5">9/5</a><br />
</td>
        <td style="text-align: center;">6.794<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/18_13">18/13</a>, <a class="wiki_link" href="/13_9">13/9</a><br />
</td>
        <td style="text-align: center;">7.285<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/14_11">14/11</a>, <a class="wiki_link" href="/11_7">11/7</a><br />
</td>
        <td style="text-align: center;">7.752<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/13_12">13/12</a>, <a class="wiki_link" href="/24_13">24/13</a><br />
</td>
        <td style="text-align: center;">7.769<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/16_13">16/13</a>, <a class="wiki_link" href="/13_8">13/8</a><br />
</td>
        <td style="text-align: center;">8.253<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/15_11">15/11</a>, <a class="wiki_link" href="/22_15">22/15</a><br />
</td>
        <td style="text-align: center;">10.122<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/11_10">11/10</a>, <a class="wiki_link" href="/20_11">20/11</a><br />
</td>
        <td style="text-align: center;">10.606<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/14_13">14/13</a>, <a class="wiki_link" href="/13_7">13/7</a><br />
</td>
        <td style="text-align: center;">11.225<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/15_13">15/13</a>, <a class="wiki_link" href="/26_15">26/15</a><br />
</td>
        <td style="text-align: center;">13.595<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><a class="wiki_link" href="/13_10">13/10</a>, <a class="wiki_link" href="/20_13">20/13</a><br />
</td>
        <td style="text-align: center;">14.079<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:19:&lt;h1&gt; --><h1 id="toc5"><a name="Instruments"></a><!-- ws:end:WikiTextHeadingRule:19 -->Instruments</h1>
 <!-- ws:start:WikiTextLocalImageRule:2722:&lt;img src=&quot;/file/view/41-EDD%20elektrische%20gitaar.jpg/610818537/560x745/41-EDD%20elektrische%20gitaar.jpg&quot; alt=&quot;&quot; title=&quot;&quot; style=&quot;height: 745px; width: 560px;&quot; /&gt; --><img src="/file/view/41-EDD%20elektrische%20gitaar.jpg/610818537/560x745/41-EDD%20elektrische%20gitaar.jpg" alt="41-EDD elektrische gitaar.jpg" title="41-EDD elektrische gitaar.jpg" style="height: 745px; width: 560px;" /><!-- ws:end:WikiTextLocalImageRule:2722 --><br />
<em>41-EDO Electric guitar, by Gregory Sanchez.</em><br />
<br />
<!-- ws:start:WikiTextLocalImageRule:2723:&lt;img src=&quot;/file/view/Ron_Sword_with_a_41ET_Guitar.jpg/221056094/Ron_Sword_with_a_41ET_Guitar.jpg&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/Ron_Sword_with_a_41ET_Guitar.jpg/221056094/Ron_Sword_with_a_41ET_Guitar.jpg" alt="Ron_Sword_with_a_41ET_Guitar.jpg" title="Ron_Sword_with_a_41ET_Guitar.jpg" /><!-- ws:end:WikiTextLocalImageRule:2723 --><br />
<em>41-EDO Classical guitar, by Ron Sword.</em><br />
<br />
A possible system to tune keyboards in 41EDO is discussed in <a class="wiki_link_ext" href="http://launch.groups.yahoo.com/group/tuning/message/74155" rel="nofollow">http://launch.groups.yahoo.com/group/tuning/message/74155</a>.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:21:&lt;h1&gt; --><h1 id="toc6"><a name="Scales and modes"></a><!-- ws:end:WikiTextHeadingRule:21 -->Scales and modes</h1>
<br />
A list of <a class="wiki_link" href="/41edo%20modes">41edo modes</a> (MOS and others).<br />
<br />
<!-- ws:start:WikiTextHeadingRule:23:&lt;h3&gt; --><h3 id="toc7"><a name="Scales and modes--Harmonic Scale"></a><!-- ws:end:WikiTextHeadingRule:23 -->Harmonic Scale</h3>
41edo is the first edo to do some justice to Mode 8 of the <a class="wiki_link" href="/OverToneSeries">harmonic series</a>, which Dante Rosati calls the &quot;<a class="wiki_link" href="/overtone%20scales">Diatonic Harmonic Series Scale</a>,&quot; consisting of overtones 8 through 16 (sometimes made to repeat at the octave).<br />
<br />


<table class="wiki_table">
    <tr>
        <td>Overtones in &quot;Mode 8&quot;:<br />
</td>
        <td>8<br />
</td>
        <td>9<br />
</td>
        <td>10<br />
</td>
        <td>11<br />
</td>
        <td>12<br />
</td>
        <td>13<br />
</td>
        <td>14<br />
</td>
        <td>15<br />
</td>
        <td>16<br />
</td>
    </tr>
    <tr>
        <td>...as JI Ratio from 1/1:<br />
</td>
        <td>1/1<br />
</td>
        <td>9/8<br />
</td>
        <td>5/4<br />
</td>
        <td>11/8<br />
</td>
        <td>3/2<br />
</td>
        <td>13/8<br />
</td>
        <td>7/4<br />
</td>
        <td>15/8<br />
</td>
        <td>2/1<br />
</td>
    </tr>
    <tr>
        <td>...in cents:<br />
</td>
        <td>0<br />
</td>
        <td>203.9<br />
</td>
        <td>386.3<br />
</td>
        <td>551.3<br />
</td>
        <td>702.0<br />
</td>
        <td>840.5<br />
</td>
        <td>968.8<br />
</td>
        <td>1088.3<br />
</td>
        <td>1200.0<br />
</td>
    </tr>
    <tr>
        <td>Nearest degree of 41edo:<br />
</td>
        <td>0<br />
</td>
        <td>7<br />
</td>
        <td>13<br />
</td>
        <td>19<br />
</td>
        <td>24<br />
</td>
        <td>29<br />
</td>
        <td>33<br />
</td>
        <td>37<br />
</td>
        <td>41<br />
</td>
    </tr>
    <tr>
        <td>...in cents:<br />
</td>
        <td>0<br />
</td>
        <td>204.9<br />
</td>
        <td>380.5<br />
</td>
        <td>556.1<br />
</td>
        <td>702.4<br />
</td>
        <td>848.8<br />
</td>
        <td>965.9<br />
</td>
        <td>1082.9<br />
</td>
        <td>1200.0<br />
</td>
    </tr>
</table>

<br />
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.)<br />
<br />
7\41 (7 degrees of 41edo) (204.9 cents) stands in for just ratio 9/8 (203.9 cents) -- a close match.<br />
6\41 (175.6 cents) stands in for both 10/9 (182.4 cents) and 11/10 (165.0 cents).<br />
5\41 (146.3 cents) stands in for both 12/11 (150.6 cents) and 13/12 (138.6 cents).<br />
4\41 (117.1 cents) stands in for 14/13 (128.3 cents), 15/14 (119.4 cents), and 16/15 (111.7 cents).<br />
<br />
The scale in 41, as adjacent steps, thus goes: 7 6 6 5 5 4 4 4.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:25:&lt;h1&gt; --><h1 id="toc8"><a name="Nonoctave Temperaments"></a><!-- ws:end:WikiTextHeadingRule:25 -->Nonoctave Temperaments</h1>
 Taking every third degree of 41edo produces a scale extremely close to <a class="wiki_link" href="/88cET">88cET</a> 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 class="wiki_link_new"><a class="wiki_link" href="/BP">Bohlen-Pierce</a></span><a class="wiki_link" href="/BP"> Scale</a> (or the 13th root of 3). See chart:<br />
<br />


<table class="wiki_table">
    <tr>
        <td colspan="3" style="text-align: center;">3 degrees of 41edo (near 88cET)<br />
</td>
        <td style="text-align: center;">overlap<br />
</td>
        <td colspan="3" style="text-align: center;">5 degrees of 41edo (near BP)<br />
</td>
    </tr>
    <tr>
        <th>deg of 41edo<br />
</th>
        <th>deg of 88cET<br />
</th>
        <th>cents<br />
</th>
        <th>cents<br />
</th>
        <th>cents<br />
</th>
        <th>deg of BP<br />
</th>
        <th>deg of 41edo<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">0<br />
</td>
        <td style="text-align: center;">0<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">3<br />
</td>
        <td style="text-align: center;">1<br />
</td>
        <td style="text-align: center;">87.8<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">146.3<br />
</td>
        <td style="text-align: center;">1<br />
</td>
        <td style="text-align: center;">5<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">6<br />
</td>
        <td style="text-align: center;">2<br />
</td>
        <td style="text-align: center;">175.6<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">9<br />
</td>
        <td style="text-align: center;">3<br />
</td>
        <td style="text-align: center;">263.4<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">292.7<br />
</td>
        <td style="text-align: center;">2<br />
</td>
        <td style="text-align: center;">10<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">12<br />
</td>
        <td style="text-align: center;">4<br />
</td>
        <td style="text-align: center;">351.2<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">15<br />
</td>
        <td style="text-align: center;">5<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">439.0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">3<br />
</td>
        <td style="text-align: center;">15<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">18<br />
</td>
        <td style="text-align: center;">6<br />
</td>
        <td style="text-align: center;">526.8<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">585.4<br />
</td>
        <td style="text-align: center;">4<br />
</td>
        <td style="text-align: center;">20<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">21<br />
</td>
        <td style="text-align: center;">7<br />
</td>
        <td style="text-align: center;">614.6<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">24<br />
</td>
        <td style="text-align: center;">8<br />
</td>
        <td style="text-align: center;">702.4<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">731.7<br />
</td>
        <td style="text-align: center;">5<br />
</td>
        <td style="text-align: center;">25<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">27<br />
</td>
        <td style="text-align: center;">9<br />
</td>
        <td style="text-align: center;">790.2<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">30<br />
</td>
        <td style="text-align: center;">10<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">878.0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">6<br />
</td>
        <td style="text-align: center;">30<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">33<br />
</td>
        <td style="text-align: center;">11<br />
</td>
        <td style="text-align: center;">965.9<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">1024.4<br />
</td>
        <td style="text-align: center;">7<br />
</td>
        <td style="text-align: center;">35<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">36<br />
</td>
        <td style="text-align: center;">12<br />
</td>
        <td style="text-align: center;">1053.7<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">39<br />
</td>
        <td style="text-align: center;">13<br />
</td>
        <td style="text-align: center;">1141.5<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">1170.7<br />
</td>
        <td style="text-align: center;">8<br />
</td>
        <td style="text-align: center;">40<br />
</td>
    </tr>
    <tr>
        <th colspan="7">[ second octave ]<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">1<br />
</td>
        <td style="text-align: center;">14<br />
</td>
        <td style="text-align: center;">29.2<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">4<br />
</td>
        <td style="text-align: center;">15<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">117.1<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">9<br />
</td>
        <td style="text-align: center;">4<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">7<br />
</td>
        <td style="text-align: center;">16<br />
</td>
        <td style="text-align: center;">204.9<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">263.4<br />
</td>
        <td style="text-align: center;">10<br />
</td>
        <td style="text-align: center;">9<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">10<br />
</td>
        <td style="text-align: center;">17<br />
</td>
        <td style="text-align: center;">292.7<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">13<br />
</td>
        <td style="text-align: center;">18<br />
</td>
        <td style="text-align: center;">380.5<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">409.8<br />
</td>
        <td style="text-align: center;">11<br />
</td>
        <td style="text-align: center;">14<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">16<br />
</td>
        <td style="text-align: center;">19<br />
</td>
        <td style="text-align: center;">468.3<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">19<br />
</td>
        <td style="text-align: center;">20<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">556.1<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">12<br />
</td>
        <td style="text-align: center;">19<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">22<br />
</td>
        <td style="text-align: center;">21<br />
</td>
        <td style="text-align: center;">643.9<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">702.4<br />
</td>
        <td style="text-align: center;">13<br />
</td>
        <td style="text-align: center;">24<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">25<br />
</td>
        <td style="text-align: center;">22<br />
</td>
        <td style="text-align: center;">731.7<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">28<br />
</td>
        <td style="text-align: center;">23<br />
</td>
        <td style="text-align: center;">819.5<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">848.8<br />
</td>
        <td style="text-align: center;">14<br />
</td>
        <td style="text-align: center;">29<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">31<br />
</td>
        <td style="text-align: center;">24<br />
</td>
        <td style="text-align: center;">907.3<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">34<br />
</td>
        <td style="text-align: center;">25<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">995.1<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">15<br />
</td>
        <td style="text-align: center;">34<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">37<br />
</td>
        <td style="text-align: center;">26<br />
</td>
        <td style="text-align: center;">1082.9<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">1141.5<br />
</td>
        <td style="text-align: center;">16<br />
</td>
        <td style="text-align: center;">39<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">40<br />
</td>
        <td style="text-align: center;">27<br />
</td>
        <td style="text-align: center;">1170.7<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <th colspan="7">[ third octave ]<br />
</th>
    </tr>
    <tr>
        <td style="text-align: center;">2<br />
</td>
        <td style="text-align: center;">28<br />
</td>
        <td style="text-align: center;">58.5<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">87.8<br />
</td>
        <td style="text-align: center;">17<br />
</td>
        <td style="text-align: center;">3<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">5<br />
</td>
        <td style="text-align: center;">29<br />
</td>
        <td style="text-align: center;">146.3<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">8<br />
</td>
        <td style="text-align: center;">30<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">234.1<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">18<br />
</td>
        <td style="text-align: center;">8<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">11<br />
</td>
        <td style="text-align: center;">31<br />
</td>
        <td style="text-align: center;">322.0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">380.5<br />
</td>
        <td style="text-align: center;">19<br />
</td>
        <td style="text-align: center;">13<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">14<br />
</td>
        <td style="text-align: center;">32<br />
</td>
        <td style="text-align: center;">409.8<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">17<br />
</td>
        <td style="text-align: center;">33<br />
</td>
        <td style="text-align: center;">497.6<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">526.8<br />
</td>
        <td style="text-align: center;">20<br />
</td>
        <td style="text-align: center;">18<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">20<br />
</td>
        <td style="text-align: center;">34<br />
</td>
        <td style="text-align: center;">585.3<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">23<br />
</td>
        <td style="text-align: center;">35<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">673.2<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">21<br />
</td>
        <td style="text-align: center;">23<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">26<br />
</td>
        <td style="text-align: center;">36<br />
</td>
        <td style="text-align: center;">761.0<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">819.5<br />
</td>
        <td style="text-align: center;">22<br />
</td>
        <td style="text-align: center;">28<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">29<br />
</td>
        <td style="text-align: center;">37<br />
</td>
        <td style="text-align: center;">848.8<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">32<br />
</td>
        <td style="text-align: center;">38<br />
</td>
        <td style="text-align: center;">936.6<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">965.9<br />
</td>
        <td style="text-align: center;">23<br />
</td>
        <td style="text-align: center;">33<br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">35<br />
</td>
        <td style="text-align: center;">39<br />
</td>
        <td style="text-align: center;">1024.4<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td style="text-align: center;">38<br />
</td>
        <td style="text-align: center;">40<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">1112.2<br />
</td>
        <td style="text-align: center;"><br />
</td>
        <td style="text-align: center;">24<br />
</td>
        <td style="text-align: center;">38<br />
</td>
    </tr>
</table>

<br />
<!-- ws:start:WikiTextHeadingRule:27:&lt;h2&gt; --><h2 id="toc9"><a name="Nonoctave Temperaments-Notation"></a><!-- ws:end:WikiTextHeadingRule:27 -->Notation</h2>
 <br />
A red-note/blue-note system, similar to the one proposed for <a class="wiki_link" href="/36edo">36edo</a>, is one option for notating 41edo. (This is separate from and not compatible with Kite's <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Kite%27s%20color%20notation">color notation</a>.) We have the &quot;white key&quot; albitonic notes A-G (7 in total), the &quot;black key&quot; sharps and flats (10 in total), a &quot;red&quot; and &quot;blue&quot; version of each albitonic note (14 in total), a &quot;red&quot; (dark red?) version of each sharp and a &quot;blue&quot; (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:<br />
<br />
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.<br />
<br />
Interval classes could also be named by analogy. The natural, colorless, or gray interval classes are the Pythagorean ones (which show up in the standard diatonic scale), while &quot;red&quot; and &quot;blue&quot; versions are one step higher or lower. Gray thirds, sixths, and sevenths are usually more dissonant than their colorful counterparts, but the reverse is true of fourths and fifths.<br />
<br />
The step size of 41edo is small enough that the smallest interval (the &quot;red/blue unison&quot;, seventh-tone, comma, diesis or whatever you want to call it) is actually fairly consonant with most timbres; it resembles a &quot;noticeably out of tune unison&quot; rather than a minor second, and has its own distinct character and appeal.<br />
<br />
<br />
<!-- ws:start:WikiTextHeadingRule:29:&lt;h1&gt; --><h1 id="toc10"><a name="Music"></a><!-- ws:end:WikiTextHeadingRule:29 -->Music</h1>
 <a class="wiki_link_ext" href="http://soundcloud.com/cameron-bobro/eveninghorizon-cbobro" rel="nofollow">EveningHorizon</a> <a class="wiki_link_ext" href="http://micro.soonlabel.com/gene_ward_smith/Others/Bobro/EveningHorizon_CBobro.mp3" rel="nofollow">play</a> by Cameron Bobro<br />
<br />
<!-- ws:start:WikiTextHeadingRule:31:&lt;h1&gt; --><h1 id="toc11"><a name="Links"></a><!-- ws:end:WikiTextHeadingRule:31 -->Links</h1>
 <ul><li><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/41_equal_temperament" rel="nofollow">Wikipedia article on 41edo</a></li><li><a class="wiki_link" href="/Magic22%20as%20srutis#magic22assrutis">Magic22 as srutis</a> describes a possible use of 41edo for <a class="wiki_link" href="/indian">indian</a> music.</li><li>see also <a class="wiki_link" href="/Magic%20family">Magic family</a></li><li>Sword, Ron. <a class="wiki_link_ext" href="http://www.ronsword.com" rel="nofollow" target="_blank">&quot;Tetracontamonophonic Scales for Guitar&quot;</a></li><li>Taylor, Cam. <a class="wiki_link_ext" href="https://drive.google.com/open?id=0B3wIGTmjY_VZYllwcHI0d3hEc3M" rel="nofollow">Intervals, Scales and Chords in 41EDO</a>, a work in progress using just intonation concepts and simplified Sagittal notation.</li></ul><!-- ws:start:WikiTextReferencesRule:4142: --><hr class="references" /><ol class="references">
<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a class="wiki_link_ext" href="http://x31eq.com/schismic.htm" rel="nofollow">&quot;Schismic Temperaments&quot;</a> at x31eq.com the website of <a class="wiki_link" href="/Graham%20Breed">Graham Breed</a></li>
<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a class="wiki_link_ext" href="http://x31eq.com/decimal_lattice.htm" rel="nofollow">&quot;Lattices with Decimal Notation&quot;</a> at x31eq.com</li>
<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Schismatic_temperament" rel="nofollow">Schismatic temperament</a></li>
<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Magic_temperament" rel="nofollow">Magic temperament</a></li>
</ol><!-- ws:end:WikiTextReferencesRule:4142 --></body></html>
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