Xenharmonic Wiki - User contributions [en]

User:PiotrGrochowski/User:PiotrGrochowski

2019-03-19T07:03:41Z

PiotrGrochowski:

<span style="font-family: 'Consolas';"><div><big>''Do not ban this user''</big></div>

<div><big>''He did nothing wrong''</big></div>

Edos are for calculating approximate logarithms in mathematics. The right choice of an edo (such as [[53edo]] or even [[612edo]] for 5-limit numbers) would give accurate results.

<pre>Which number is the biggest?
A. 81^56
B. 9^100
C. 27^72
D. 2^224</pre>

53log₂(81^56)≈18816<br>
53log₂(9^100)≈16800<br>
53log₂(27^72)≈18144<br>
53log₂(2^224)=11872

<b>A.</b>

(calculated successfully with [[53edo]])

<pre>³√(2 7/9)÷³√(3/5) rounded to the nearest integer is
A. 1
B. 2
C. 3
D. 4</pre>

53log₂(³√(25/9)÷³√(3/5))≈39

2^(39÷53)≈5÷3

round(5÷3)=2

<b>B.</b>

(calculated successfully with [[53edo]])

<pre>Which number is the smallest?
A. 5√3
B. 4√7
C. 8√2
D. 3√11</pre>

41log₂(5√3)≈127.5<br>
41log₂(4√7)≈139.5<br>
41log₂(8√2)=143.5<br>
41log₂(3√11)≈136

<b>A.</b>

(calculated successfully with [[41edo]])

</span>

User:PiotrGrochowski/User:PiotrGrochowski

2019-03-19T06:30:33Z

PiotrGrochowski:

<span style="font-family: 'Consolas';"><div><big>''Do not ban this user''</big></div>

<div><big>''He did nothing wrong''</big></div>

Edos are for calculating approximate logarithms in mathematics. The right choice of an edo (such as [[53edo]] or even [[612edo]] for 5-limit numbers) would give accurate results.</span>

15-odd-limit

2019-03-19T06:28:31Z

PiotrGrochowski: use the proper centered tilde. in some fonts "(~15 .. 41 steps)" looks like "(˜15 .. 41 steps)" which is bad.

This is a list of 15-[[Odd limit|odd-limit]] intervals. It has proven helpful for showing JI properties of medium-sized [[EDO]] systems (∼15 .. 41 steps). To [[13-odd-limit]], it adds four additional interval pairs involving 15.

* [[16/15]], [[15/8]]
* [[15/14]], [[28/15]]
* [[14/13]], [[13/7]]
* [[13/12]], [[24/13]]
* [[12/11]], [[11/6]]
* [[11/10]], [[20/11]]
* [[10/9]], [[9/5]]
* [[9/8]], [[16/9]]
* [[8/7]], [[7/4]]
* [[15/13]], [[26/15]]
* [[7/6]], [[12/7]]
* [[13/11]], [[22/13]]
* [[6/5]], [[5/3]]
* [[11/9]], [[18/11]]
* [[16/13]], [[13/8]]
* [[5/4]], [[8/5]]
* [[14/11]], [[11/7]]
* [[9/7]], [[14/9]]
* [[13/10]], [[20/13]]
* [[4/3]], [[3/2]]
* [[15/11]], [[22/15]]
* [[11/8]], [[16/11]]
* [[18/13]], [[13/9]]
* [[7/5]], [[10/7|10/7]]

[[Category:15-limit-diamond]]
[[Category:Just interval]]

53edo

2019-03-19T06:26:15Z

PiotrGrochowski: restore a helpful section lost due to a big revert

{{interwiki
| de = 53edo
| en = 53edo
| es =
| ja =
}}__FORCETOC__
=Theory=
The famous ''53 equal division'' divides the octave into 53 equal comma-sized parts of 22.642 cents each. It is notable as a [[5-limit|5-limit]] system, a fact apparently first noted by Isaac Newton, tempering out the schisma, 32805/32768, the kleisma, 15625/15552, the amity comma, 1600000/1594323 and the semicomma, 2109375/2097152. In the 7-limit it tempers out 225/224, 1728/1715 and 3125/3087, the marvel comma, the gariboh, and the orwell comma. In the 11-limit, it tempers out 99/98 and 121/120, and is the [[Optimal_patent_val|optimal patent val]] for [[Nuwell_family|Big Brother]] temperament, which tempers out both, as well as 11-limit [[Semicomma_family|orwell temperament]], which also tempers out the 11-limit comma 176/175. In the 13-limit, it tempers out 169/168 and 245/243, and gives the optimal patent val for [[Marvel_family|athene temperament]]. It is the eighth [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta integral edo]] and the 16th [[prime_numbers|prime]] edo, following [[47edo|47edo]] and coming before [[59edo|59edo]].

53EDO has also found a certain dissemination as an EDO tuning for [[Arabic,_Turkish,_Persian|Arabic/Turkish/Persian music]].

It can also be treated as a no-elevens, no-seventeens tuning, on which it is consistent all the way up to the 21-limit.

[http://en.wikipedia.org/wiki/53_equal_temperament Wikipeda article about 53edo]

=Linear temperaments=
[[List_of_edo-distinct_53et_rank_two_temperaments|List of edo-distinct 53et rank two temperaments]]

=Just Approximation=
53edo provides excellent approximations for the classic 5-limit [[just|just]] chords and scales, such as the Ptolemy-Zarlino "just major" scale.

{| class="wikitable"
|-
! | interval
! | ratio
! | size
! | difference
|-
| | perfect fifth
| | 3/2
| style="text-align:center;" | 31
| | −0.07 cents
|-
| | major third
| | 5/4
| style="text-align:center;" | 17
| | −1.40 cents
|-
| | minor third
| | 6/5
| style="text-align:center;" | 14
| | +1.34 cents
|-
| | major tone
| | 9/8
| style="text-align:center;" | 9
| | −0.14 cents
|-
| | minor tone
| | 10/9
| style="text-align:center;" | 8
| | −1.27 cents
|-
| | diat. semitone
| | 16/15
| style="text-align:center;" | 5
| | +1.48 cents
|}

One notable property of 53EDO is that it offers good approximations for both pure and pythagorean major thirds.

The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.8 cents away from the just ratio 7/4, so 53EDO can also be used for 7-limit harmony, tempering out the [[septimal_kleisma|septimal kleisma]], 225/224.

=Intervals=

{| class="wikitable"
|-
! | degree
! | solfege
! | cents
! | approximate ratios
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs]] [[Ups_and_Downs_Notation|notation]]
! | generator for
|-
| style="text-align:center;" | 0
| style="text-align:center;" | do
| style="text-align:center;" | 0.00
| style="text-align:center;" | 1/1
| style="text-align:center;" | P1
| style="text-align:center;" | unison
| style="text-align:center;" | D
| |
|-
| style="text-align:center;" | 1
| style="text-align:center;" | di
| style="text-align:center;" | 22.64
| style="text-align:center;" | 81/80, 64/63, 50/49
| style="text-align:center;" | ^1
| style="text-align:center;" | up unison
| style="text-align:center;" | D^
| |
|-
| style="text-align:center;" | 2
| style="text-align:center;" | daw
| style="text-align:center;" | 45.28
| style="text-align:center;" | 49/48, 36/35, 33/32, 128/125
| style="text-align:center;" | ^^1,

vvm2
| style="text-align:center;" | double-up unison,

double-down minor 2nd
| style="text-align:center;" | D^^,

Ebvv
| | [[Quartonic|Quartonic]]
|-
| style="text-align:center;" | 3
| style="text-align:center;" | ro
| style="text-align:center;" | 67.92
| style="text-align:center;" | 27/26, 26/25, 25/24, 22/21
| style="text-align:center;" | vm2
| style="text-align:center;" | downminor 2nd
| style="text-align:center;" | Ebv
| |
|-
| style="text-align:center;" | 4
| style="text-align:center;" | rih
| style="text-align:center;" | 90.57
| style="text-align:center;" | 21/20, 256/243
| style="text-align:center;" | m2
| style="text-align:center;" | minor 2nd
| style="text-align:center;" | Eb
| |
|-
| style="text-align:center;" | 5
| style="text-align:center;" | ra
| style="text-align:center;" | 113.21
| style="text-align:center;" | 16/15, 15/14
| style="text-align:center;" | ^m2
| style="text-align:center;" | upminor 2nd
| style="text-align:center;" | Eb^
| |
|-
| style="text-align:center;" | 6
| style="text-align:center;" | ru
| style="text-align:center;" | 135.85
| style="text-align:center;" | 14/13, 13/12, 27/25
| style="text-align:center;" | v~2
| style="text-align:center;" | downmid 2nd
| style="text-align:center;" | Eb^^
| |
|-
| style="text-align:center;" | 7
| style="text-align:center;" | ruh
| style="text-align:center;" | 158.49
| style="text-align:center;" | 12/11, 11/10, 800/729
| style="text-align:center;" | ^~2
| style="text-align:center;" | upmid 2nd
| style="text-align:center;" | Evv
| | [[Hemikleismic|Hemikleismic]]
|-
| style="text-align:center;" | 8
| style="text-align:center;" | reh
| style="text-align:center;" | 181.13
| style="text-align:center;" | 10/9
| style="text-align:center;" | vM2
| style="text-align:center;" | downmajor 2nd
| style="text-align:center;" | Ev
| |
|-
| style="text-align:center;" | 9
| style="text-align:center;" | re
| style="text-align:center;" | 203.77
| style="text-align:center;" | 9/8
| style="text-align:center;" | M2
| style="text-align:center;" | major 2nd
| style="text-align:center;" | E
| |
|-
| style="text-align:center;" | 10
| style="text-align:center;" | ri
| style="text-align:center;" | 226.42
| style="text-align:center;" | 8/7, 256/225
| style="text-align:center;" | ^M2
| style="text-align:center;" | upmajor 2nd
| style="text-align:center;" | E^
| |
|-
| style="text-align:center;" | 11
| style="text-align:center;" | raw
| style="text-align:center;" | 249.06
| style="text-align:center;" | 15/13, 144/125
| style="text-align:center;" | ^^M2,

vvm3
| style="text-align:center;" | double-up major 2nd,

double-down minor 3rd
| style="text-align:center;" | E^^,

Fvv
| | [[Hemischis|Hemischis]]
|-
| style="text-align:center;" | 12
| style="text-align:center;" | ma
| style="text-align:center;" | 271.70
| style="text-align:center;" | 7/6, 75/64
| style="text-align:center;" | vm3
| style="text-align:center;" | downminor 3rd
| style="text-align:center;" | Fv
| | [[Orwell|Orwell]]
|-
| style="text-align:center;" | 13
| style="text-align:center;" | meh
| style="text-align:center;" | 294.34
| style="text-align:center;" | 13/11, 32/27
| style="text-align:center;" | m3
| style="text-align:center;" | minor 3rd
| style="text-align:center;" | F
| |
|-
| style="text-align:center;" | 14
| style="text-align:center;" | me
| style="text-align:center;" | 316.98
| style="text-align:center;" | 6/5
| style="text-align:center;" | ^m3
| style="text-align:center;" | upminor 3rd
| style="text-align:center;" | F^
| | [[Hanson|Hanson]]/[[catakleismic|Catakleismic]]
|-
| style="text-align:center;" | 15
| style="text-align:center;" | mu
| style="text-align:center;" | 339.62
| style="text-align:center;" | 11/9, 243/200
| style="text-align:center;" | v~3
| style="text-align:center;" | downmid 3rd
| style="text-align:center;" | F^^
| | [[Amity|Amity]]/[[Hitchcock|Hitchcock]]
|-
| style="text-align:center;" | 16
| style="text-align:center;" | muh
| style="text-align:center;" | 362.26
| style="text-align:center;" | 16/13, 100/81
| style="text-align:center;" | ^~3
| style="text-align:center;" | upmid 3rd
| style="text-align:center;" | F#vv
| |
|-
| style="text-align:center;" | 17
| style="text-align:center;" | mi
| style="text-align:center;" | 384.91
| style="text-align:center;" | 5/4
| style="text-align:center;" | vM3
| style="text-align:center;" | downmajor 3rd
| style="text-align:center;" | F#v
| |
|-
| style="text-align:center;" | 18
| style="text-align:center;" | maa
| style="text-align:center;" | 407.55
| style="text-align:center;" | 81/64
| style="text-align:center;" | M3
| style="text-align:center;" | major 3rd
| style="text-align:center;" | F#
| |
|-
| style="text-align:center;" | 19
| style="text-align:center;" | mo
| style="text-align:center;" | 430.19
| style="text-align:center;" | 9/7, 14/11
| style="text-align:center;" | ^M3
| style="text-align:center;" | upmajor 3rd
| style="text-align:center;" | F#^
| | [[Hamity|Hamity]]
|-
| style="text-align:center;" | 20
| style="text-align:center;" | maw
| style="text-align:center;" | 452.83
| style="text-align:center;" | 13/10, 125/96
| style="text-align:center;" | ^^M3,

vv4
| style="text-align:center;" | double-up major 3rd,

double-down 4th
| style="text-align:center;" | F#^^,

Gvv
| |
|-
| style="text-align:center;" | 21
| style="text-align:center;" | fe
| style="text-align:center;" | 475.47
| style="text-align:center;" | 21/16, 675/512, 320/243
| style="text-align:center;" | v4
| style="text-align:center;" | down 4th
| style="text-align:center;" | Gv
| | [[Vulture|Vulture]]/[[Buzzard|Buzzard]]
|-
| style="text-align:center;" | 22
| style="text-align:center;" | fa
| style="text-align:center;" | 498.11
| style="text-align:center;" | 4/3
| style="text-align:center;" | P4
| style="text-align:center;" | perfect 4th
| style="text-align:center;" | G
| |
|-
| style="text-align:center;" | 23
| style="text-align:center;" | fih
| style="text-align:center;" | 520.75
| style="text-align:center;" | 27/20
| style="text-align:center;" | ^4
| style="text-align:center;" | up 4th
| style="text-align:center;" | G^
| |
|-
| style="text-align:center;" | 24
| style="text-align:center;" | fu
| style="text-align:center;" | 543.40
| style="text-align:center;" | 11/8, 15/11
| style="text-align:center;" | ^^4
| style="text-align:center;" | double-up 4th
| style="text-align:center;" | G^^
| |
|-
| style="text-align:center;" | 25
| style="text-align:center;" | fuh
| style="text-align:center;" | 566.04
| style="text-align:center;" | 18/13
| style="text-align:center;" | vvA4,

vd5
| style="text-align:center;" | double-down aug 4th,

downdim 5th
| style="text-align:center;" | G#vv,

Abv
| | [[Tricot|Tricot]]
|-
| style="text-align:center;" | 26
| style="text-align:center;" | fi
| style="text-align:center;" | 588.68
| style="text-align:center;" | 7/5, 45/32
| style="text-align:center;" | vA4,

d5
| style="text-align:center;" | downaug 4th,

dim 5th
| style="text-align:center;" | G#v,

Ab
| |
|-
| style="text-align:center;" | 27
| style="text-align:center;" | se
| style="text-align:center;" | 611.32
| style="text-align:center;" | 10/7, 64/45
| style="text-align:center;" | A4,

^d5
| style="text-align:center;" | aug 4th,

updim 5th
| style="text-align:center;" | G#,

Ab^
| |
|-
| style="text-align:center;" | 28
| style="text-align:center;" | suh
| style="text-align:center;" | 633.96
| style="text-align:center;" | 13/9
| style="text-align:center;" | ^A4,

^^d5
| style="text-align:center;" | upaug 4th,

double-up dim 5th
| style="text-align:center;" | G#^,

Ab^^
| |
|-
| style="text-align:center;" | 29
| style="text-align:center;" | su
| style="text-align:center;" | 656.60
| style="text-align:center;" | 16/11, 22/15
| style="text-align:center;" | vv5
| style="text-align:center;" | double-down 5th
| style="text-align:center;" | Avv
| |
|-
| style="text-align:center;" | 30
| style="text-align:center;" | sih
| style="text-align:center;" | 679.25
| style="text-align:center;" | 40/27
| style="text-align:center;" | v5
| style="text-align:center;" | down 5th
| style="text-align:center;" | Av
| |
|-
| style="text-align:center;" | 31
| style="text-align:center;" | sol
| style="text-align:center;" | 701.89
| style="text-align:center;" | 3/2
| style="text-align:center;" | P5
| style="text-align:center;" | perfect 5th
| style="text-align:center;" | A
| | [[Helmholtz|Helmholtz]]/[[Garibaldi|Garibaldi]]
|-
| style="text-align:center;" | 32
| style="text-align:center;" | si
| style="text-align:center;" | 724.53
| style="text-align:center;" | 32/21, 243/160, 1024/675
| style="text-align:center;" | ^5
| style="text-align:center;" | up 5th
| style="text-align:center;" | A^
| |
|-
| style="text-align:center;" | 33
| style="text-align:center;" | saw
| style="text-align:center;" | 747.17
| style="text-align:center;" | 20/13, 192/125
| style="text-align:center;" | ^^5,

vvm6
| style="text-align:center;" | double-up 5th,

double-down minor 6th
| style="text-align:center;" | A^^,

Bbvv
| |
|-
| style="text-align:center;" | 34
| style="text-align:center;" | lo
| style="text-align:center;" | 769.81
| style="text-align:center;" | 14/9, 25/16, 11/7
| style="text-align:center;" | vm6
| style="text-align:center;" | downminor 6th
| style="text-align:center;" | Bbv
| |
|-
| style="text-align:center;" | 35
| style="text-align:center;" | leh
| style="text-align:center;" | 792.45
| style="text-align:center;" | 128/81
| style="text-align:center;" | m6
| style="text-align:center;" | minor 6th
| style="text-align:center;" | Bb
| |
|-
| style="text-align:center;" | 36
| style="text-align:center;" | le
| style="text-align:center;" | 815.09
| style="text-align:center;" | 8/5
| style="text-align:center;" | ^m6
| style="text-align:center;" | upminor 6th
| style="text-align:center;" | Bb^
| |
|-
| style="text-align:center;" | 37
| style="text-align:center;" | lu
| style="text-align:center;" | 837.74
| style="text-align:center;" | 13/8, 81/50
| style="text-align:center;" | v~6
| style="text-align:center;" | downmid 6th
| style="text-align:center;" | Bb^^
| |
|-
| style="text-align:center;" | 38
| style="text-align:center;" | luh
| style="text-align:center;" | 860.38
| style="text-align:center;" | 18/11, 400/243
| style="text-align:center;" | ^~6
| style="text-align:center;" | upmid 6th
| style="text-align:center;" | Bvv
| |
|-
| style="text-align:center;" | 39
| style="text-align:center;" | la
| style="text-align:center;" | 883.02
| style="text-align:center;" | 5/3
| style="text-align:center;" | vM6
| style="text-align:center;" | downmajor 6th
| style="text-align:center;" | Bv
| |
|-
| style="text-align:center;" | 40
| style="text-align:center;" | laa
| style="text-align:center;" | 905.66
| style="text-align:center;" | 22/13, 27/16
| style="text-align:center;" | M6
| style="text-align:center;" | major 6th
| style="text-align:center;" | B
| |
|-
| style="text-align:center;" | 41
| style="text-align:center;" | lo
| style="text-align:center;" | 928.30
| style="text-align:center;" | 12/7
| style="text-align:center;" | ^M6
| style="text-align:center;" | upmajor 6th
| style="text-align:center;" | B^
| |
|-
| style="text-align:center;" | 42
| style="text-align:center;" | law
| style="text-align:center;" | 950.94
| style="text-align:center;" | 26/15, 125/72
| style="text-align:center;" | ^^M6

vvm7
| style="text-align:center;" | double-up major 6th,

double-down minor 7th
| style="text-align:center;" | B^^,

Cvv
| |
|-
| style="text-align:center;" | 43
| style="text-align:center;" | ta
| style="text-align:center;" | 973.58
| style="text-align:center;" | 7/4
| style="text-align:center;" | vm7
| style="text-align:center;" | downminor 7th
| style="text-align:center;" | Cv
| |
|-
| style="text-align:center;" | 44
| style="text-align:center;" | teh
| style="text-align:center;" | 996.23
| style="text-align:center;" | 16/9
| style="text-align:center;" | m7
| style="text-align:center;" | minor 7th
| style="text-align:center;" | C
| |
|-
| style="text-align:center;" | 45
| style="text-align:center;" | te
| style="text-align:center;" | 1018.87
| style="text-align:center;" | 9/5
| style="text-align:center;" | ^m7
| style="text-align:center;" | upminor 7th
| style="text-align:center;" | C^
| |
|-
| style="text-align:center;" | 46
| style="text-align:center;" | tu
| style="text-align:center;" | 1041.51
| style="text-align:center;" | 11/6, 20/11, 729/400
| style="text-align:center;" | v~7
| style="text-align:center;" | downmid 7th
| style="text-align:center;" | C^^
| |
|-
| style="text-align:center;" | 47
| style="text-align:center;" | tuh
| style="text-align:center;" | 1064.15
| style="text-align:center;" | 13/7, 24/13, 50/27
| style="text-align:center;" | ^~7
| style="text-align:center;" | upmid 7th
| style="text-align:center;" | C#vv
| |
|-
| style="text-align:center;" | 48
| style="text-align:center;" | ti
| style="text-align:center;" | 1086.79
| style="text-align:center;" | 15/8
| style="text-align:center;" | vM7
| style="text-align:center;" | downmajor 7th
| style="text-align:center;" | C#v
| |
|-
| style="text-align:center;" | 49
| style="text-align:center;" | tih
| style="text-align:center;" | 1109.43
| style="text-align:center;" | 40/21, 243/128
| style="text-align:center;" | M7
| style="text-align:center;" | major 7th
| style="text-align:center;" | C#
| |
|-
| style="text-align:center;" | 50
| style="text-align:center;" | to
| style="text-align:center;" | 1132.08
| style="text-align:center;" | 48/25, 27/14
| style="text-align:center;" | ^M7
| style="text-align:center;" | upmajor 7th
| style="text-align:center;" | C#^
| |
|-
| style="text-align:center;" | 51
| style="text-align:center;" | taw
| style="text-align:center;" | 1154.72
| style="text-align:center;" | 125/64
| style="text-align:center;" | ^^M7,

vv8
| style="text-align:center;" | double-up major 7th,

double-down 8ve
| style="text-align:center;" | C#^^,

Dvv
| |
|-
| style="text-align:center;" | 52
| style="text-align:center;" | da
| style="text-align:center;" | 1177.36
| style="text-align:center;" | 160/81
| style="text-align:center;" | v8
| style="text-align:center;" | down 8ve
| style="text-align:center;" | Dv
| |
|-
| style="text-align:center;" | 53
| style="text-align:center;" | do
| style="text-align:center;" | 1200
| style="text-align:center;" | 2/1
| style="text-align:center;" | P8
| style="text-align:center;" | perfect 8ve
| style="text-align:center;" | D
| |
|}
Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:

{| class="wikitable"
|-
! | quality
! | [[Kite's color notation|color]]
! | monzo format
! | examples
|-
| style="text-align:center;" | downminor
| style="text-align:center;" | zo
| style="text-align:center;" | {a, b, 0, 1}
| style="text-align:center;" | 7/6, 7/4
|-
| style="text-align:center;" | minor
| style="text-align:center;" | fourthward wa
| style="text-align:center;" | {a, b}, b < -1
| style="text-align:center;" | 32/27, 16/9
|-
| style="text-align:center;" | upminor
| style="text-align:center;" | gu
| style="text-align:center;" | {a, b, -1}
| style="text-align:center;" | 6/5, 9/5
|-
| style="text-align:center;" | downmid
| style="text-align:center;" | ilo
| style="text-align:center;" | {a, b, 0, 0, 1}
| style="text-align:center;" | 11/9, 11/6
|-
| style="text-align:center;" | upmid
| style="text-align:center;" | lu
| style="text-align:center;" | {a, b, 0, 0, -1}
| style="text-align:center;" | 12/11, 18/11
|-
| style="text-align:center;" | downmajor
| style="text-align:center;" | yo
| style="text-align:center;" | {a, b, 1}
| style="text-align:center;" | 5/4, 5/3
|-
| style="text-align:center;" | major
| style="text-align:center;" | fifthward wa
| style="text-align:center;" | {a, b}, b > 1
| style="text-align:center;" | 9/8, 27/16
|-
| style="text-align:center;" | upmajor
| style="text-align:center;" | ru
| style="text-align:center;" | {a, b, 0, -1}
| style="text-align:center;" | 9/7, 12/7
|}
All 53edo chords can be named using ups and downs. Here are the zo, gu, ilo, yo and ru triads:

{| class="wikitable"
|-
! | [[Kite's color notation|color of the 3rd]]
! | JI chord
! | notes as edosteps
! | notes of C chord
! | written name
! | spoken name
|-
| style="text-align:center;" | zo
| style="text-align:center;" | 6:7:9
| style="text-align:center;" | 0-12-31
| style="text-align:center;" | C Ebv G
| style="text-align:center;" | C.vm
| style="text-align:center;" | C downminor
|-
| style="text-align:center;" | gu
| style="text-align:center;" | 10:12:15
| style="text-align:center;" | 0-14-31
| style="text-align:center;" | C Eb^ G
| style="text-align:center;" | C.^m
| style="text-align:center;" | C upminor
|-
| style="text-align:center;" | ilo
| style="text-align:center;" | 18:22:27
| style="text-align:center;" | 0-15-31
| style="text-align:center;" | C Eb^^ G
| style="text-align:center;" | C.v~
| style="text-align:center;" | C downmid
|-
| style="text-align:center;" | yo
| style="text-align:center;" | 4:5:6
| style="text-align:center;" | 0-17-31
| style="text-align:center;" | C Ev G
| style="text-align:center;" | C.v
| style="text-align:center;" | C downmajor or C dot down
|-
| style="text-align:center;" | ru
| style="text-align:center;" | 14:18:27
| style="text-align:center;" | 0-19-31
| style="text-align:center;" | C E^ G
| style="text-align:center;" | C.^
| style="text-align:center;" | C upmajor or C dot up
|}
For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].

==Selected just intervals by error==
The following table shows how [[15-odd-limit|some prominent just intervals]] are represented in 53edo (ordered by absolute error).

{| class="wikitable"
|-
| | '''Interval, complement'''
| | '''Error (abs., in [[cent]]s)'''
|-
| style="text-align:center;" | [[4/3]], [[3/2]]
| style="text-align:center;" | 0.068
|-
| style="text-align:center;" | [[9/8]], [[16/9]]
| style="text-align:center;" | 0.136
|-
| style="text-align:center;" | [[10/9]], [[9/5]]
| style="text-align:center;" | 1.272
|-
| style="text-align:center;" | [[15/13]], [[26/15]]
| style="text-align:center;" | 1.316
|-
| style="text-align:center;" | [[6/5]], [[5/3]]
| style="text-align:center;" | 1.340
|-
| style="text-align:center;" | [[13/10]], [[20/13]]
| style="text-align:center;" | 1.384
|-
| style="text-align:center;" | [[5/4]], [[8/5]]
| style="text-align:center;" | 1.408
|-
| style="text-align:center;" | [[16/15]], [[15/8]]
| style="text-align:center;" | 1.476
|-
| style="text-align:center;" | [[18/13]], [[13/9]]
| style="text-align:center;" | 2.655
|-
| style="text-align:center;" | [[13/12]], [[24/13]]
| style="text-align:center;" | 2.724
|-
| style="text-align:center;" | [[16/13]], [[13/8]]
| style="text-align:center;" | 2.792
|-
| style="text-align:center;" | [[8/7]], [[7/4]]
| style="text-align:center;" | 4.759
|-
| style="text-align:center;" | [[7/6]], [[12/7]]
| style="text-align:center;" | 4.827
|-
| style="text-align:center;" | [[9/7]], [[14/9]]
| style="text-align:center;" | 4.895
|-
| style="text-align:center;" | [[13/11]], [[22/13]]
| style="text-align:center;" | 5.130
|-
| style="text-align:center;" | [[7/5]], [[10/7]]
| style="text-align:center;" | 6.167
|-
| style="text-align:center;" | [[15/14]], [[28/15]]
| style="text-align:center;" | 6.235
|-
| style="text-align:center;" | [[15/11]], [[22/15]]
| style="text-align:center;" | 6.445
|-
| style="text-align:center;" | [[11/10]], [[20/11]]
| style="text-align:center;" | 6.514
|-
| style="text-align:center;" | [[14/13]], [[13/7]]
| style="text-align:center;" | 7.551
|-
| style="text-align:center;" | [[11/9]], [[18/11]]
| style="text-align:center;" | 7.785
|-
| style="text-align:center;" | [[12/11]], [[11/6]]
| style="text-align:center;" | 7.854
|-
| style="text-align:center;" | [[11/8]], [[16/11]]
| style="text-align:center;" | 7.922
|-
| style="text-align:center;" | [[14/11]], [[11/7]]
| style="text-align:center;" | 12.681
|}

=Compositions=
[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/prelude1-53.mp3 Bach WTC1 Prelude 1 in 53] by Bach and [[Mykhaylo_Khramov|Mykhaylo Khramov]]

[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Khramov/fugue1-53.mp3 Bach WTC1 Fugue 1 in 53] by Bach and Mykhaylo Khramov

[http://bumpermusic.blogspot.com/2007/05/whisper-song-in-53-edo-now-526-slower.html Whisper Song in 53EDO] [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Prent/sing53-c5-slow.mp3 play] by [[Prent_Rodgers|Prent Rodgers]]

[http://www.archive.org/details/TrioInOrwell Trio in Orwell] [http://www.archive.org/download/TrioInOrwell/TrioInOrwell.mp3 play] by [[Gene_Ward_Smith|Gene Ward Smith]]

[http://www.akjmusic.com/audio/desert_prayer.mp3 Desert Prayer] by [http://www.akjmusic.com/ Aaron Krister Johnson]

[http://micro.soonlabel.com/gene_ward_smith/Others/Rodgers/sing53-c5-slow.mp3 Whisper Song in 53 EDO] by [[Prent_Rodgers|Prent Rodgers]]

[http://andrewheathwaite.bandcamp.com/track/elf-dine-on-ho-ho Elf Dine on Ho Ho] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2005%20Elf%20Dine%20on%20Ho%20Ho.mp3 play] and [http://andrewheathwaite.bandcamp.com/track/spun Spun] [http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2008%20Spun.mp3 play] by [[Andrew_Heathwaite|Andrew Heathwaite]]

[http://chrisvaisvil.com/the-fallen-of-kleismic15/ The Fallen of Kleismic15][http://micro.soonlabel.com/53edo/20130903_Kleismic%5b15%5d.mp3 play] by [[Chris_Vaisvil|Chris Vaisvil]]

[https://soundcloud.com/cam-taylor-2-1/mothers mothers] by Cam Taylor

[[Category:Amity]]
[[Category:Athene]]
[[Category:Big brother]]
[[Category:Edo]]
[[Category:Hanson]]
[[Category:Kleismic]]
[[Category:Listen]]
[[Category:Marvel]]
[[Category:Orwell]]
[[Category:Prime EDO]]
[[Category:Pythagorean]]
[[Category:Schismic]]
[[Category:Semicomma]]
[[Category:Zeta]]

User talk:PiotrGrochowski

2018-10-25T14:52:49Z

PiotrGrochowski: remove spam

== Main page ==

Hi Piotr - we saw you added large text to the top of the main page about "Incomplete pages." We removed this as we want to keep the main page clean -- the whole Wiki has plenty of *lot* of incomplete pages and you should add them to the [[Wikifuture]] pages. But the pages you added aren't on the original Wiki - it looks like you made them yourself? Where did you get them from? Some of them would be good to complete (Kleisma and so on), but others seem strange to me, like [[0/0]], which I haven't heard many people try to use as a legit musical interval. [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 12:50, 19 September 2018 (UTC)

: That's like having a dirty house inside and removing the sign that it needs cleaning. This wiki needs more edo articles to progress but I can't add placeholders anymore because there are already several. And I don't know how to add additional info to the "Table of..." articles (see Category:Interval list).

:: I thought you were saying the MediaWiki import from the old wiki was incomplete or somehow screwed up. I didn't realize you were just saying the Wiki is in general a work in progress (which is true). I changed the language on the front page to make that clearer. I just didn't want a big thing like that at the top of the page, since it seemed like there was an error in the import or something wrong with the site, rather than a general call to create new pages. [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 13:14, 19 September 2018 (UTC)

::: Also, what do you mean by not being able to add info to those articles? They should be editable using either the visual editor or regular editor...

:::: The [[31edo]] page has lots of info. The [[107edo]] page is lacking in info. I just don't know what is notable or what commas it tempers out. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 16:20, 19 September 2018 (UTC)

== Hi Piotr ==

Great to see you here :) --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 09:22, 17 September 2018 (UTC)

== Why [[Coming soon]]? ==

Hi Piotr, <br/> what's the '''idea behind''' pages like this? <br/> <small>From my long-year wiki experience, it's better to start not too many pages at once, it's hard to keep them up to date.</small> <br/> Best, --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 17:07, 20 September 2018 (UTC)
: It's so the users can see what xen.wiki was in the past. I repeat: {{:Coming soon}} [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 18:23, 20 September 2018 (UTC)
:: FYI Piotr, while I am happy to give you room to do your thing, I think pages like "Coming soon" will be confusing to new users. I appreciate your sense of humor though, and think lots of your ideas are pretty funny (love calling the logo "The Xen"!), and you have been really helpful in creating new content here for the new Wiki. [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 17:09, 21 September 2018 (UTC)

== dev.xen.wiki ==
Piotr, just FYI, dev.xen.wiki is a testing page for us to do development stuff. Feel free to play around and make changes there, but just know that any changes there aren't permanent - we will periodically wiping that wiki clean and refreshing it with new stuff. Sometimes we will import the "en" wiki there, sometimes "es", etc, for us to test changes... [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 17:09, 21 September 2018 (UTC)
: You can't however test [[hard redirect]]s, as they don't work in dev.xen.wiki. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 17:20, 21 September 2018 (UTC)
:: Piotr - the thing you are calling "hard redirects" are really "interwiki redirects." If you want to link to dev, you can do [[:dev:Main_Page]], for example. If you put ":en" in a redirect on the dev wiki, it will simply redirect to the English language wiki. [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 18:04, 21 September 2018 (UTC)
::: I'm calling that "hard redirects" because it has a similar effect to "hard redirects" in <code>wikispaces</code>, see http://xenharmonic.wikispaces.com/wiki+help#x-Using%20links-Redirects. And you're wrong that :en: in dev.xen.wiki will redirect to en.xen.wiki, because [[:dev:2 1]] does not work. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 18:11, 21 September 2018 (UTC)
:::: Piotr, the "en" thing is not a redirect code, it is just a special interwiki code. For example, look at [[:purdal:Home]], or [[:purdal:Migration_FAQ]], which redirects to [[:en:Migration_FAQ]]. But you are right about dev, which is a bug! Dev should have interwiki set up, but it doesn't. I'll add it later [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 10:48, 22 September 2018 (UTC)
:::::It however has the "[[hard redirect]]" effect, and because <code>wikispaces</code> had hard redirects, they are pretty much necessary to include in [[Wiki help#Redirects]]. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 10:51, 22 September 2018 (UTC)
:I found about dev.xen.wiki by looking at description of some of your edits.<br>Text replacement - "(\[\[\:(dev|es|purdal|de|en):[^[:space:]\|]*) (.*])" to "$1|$3"<br>Text replacement - "(\[\[\:('''dev'''|es|purdal|de|en):[^[:space:]\|]*) (.*])" to "$1|$3"<br>So, I went to dev.xen.wiki for curiosity. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 09:04, 22 September 2018 (UTC)

== Vandalism ==
I saw that you added a redirect to the top of Main_Page that leads to [[Special:Block/PiotrGrochowski]]. This wiki has a very open policy, just like it did at Wikispaces. It is easy for anyone to join and contribute. If you want to experiment and try to break things, feel free to do that with your own installation of MediaWiki, or on the dev site. [[User:Tyler|Tyler]] ([[User talk:Tyler|talk]]) 10:57, 22 September 2018 (UTC)
:Some experiments however are not possible in dev. Compare [[:en:2 1]] and [[:dev:2 1]]. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 10:38, 22 September 2018 (UTC)
:: Piotr, the thing is, this website gets a lot of visitors, so when you break the Main Page, it interferes with people's ability to see the site unless you change it back really fast. Why did you have the Main Page redirect to [[Special:Block/PiotrGrochowski]]? What was your experiment there? If you are going to do an experiment, do it on dev -- if you really can't do it on dev, and it's important, then you have to try to do it in a way that doesn't interfere with others - everyone who went to xen.wiki got sent to the Block page and couldn't see the wiki while you changed it! Those kinds of experiments are dangerous. The rule is you must ''clean up after yourself'' if you do anything like that here. Thank you [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 10:59, 22 September 2018 (UTC)

== Rules for tidiness ==

Piotr, after talking internally with the admin pages, we want to use these rules: If you are going to do any HTML, MediaWiki, "crash," or redirect tests, please do it at dev. If you are going to make tuning pages, please do it at en. Your tuning pages are great! And it is ok that you are learning about MediaWiki, but please do those changes at dev. I will get prefixes like "en" working on dev later today (called "Interwiki") so you can experiment with redirects there.

If you ever want to do a MediaWiki experiment that you can't do on dev, please ask one of us admins first and we will see what can be done. But please, no matter what, do not do anything like that on en, especially to the main page. We do not want to break the wiki for everyone else.

Thank you Piotr! - [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 11:46, 22 September 2018 (UTC) [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 11:46, 22 September 2018 (UTC)

== Resetting my password on dev? ==

Why did you try to reset my password on dev? Don't play around like that - we can see [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 12:16, 22 September 2018 (UTC)
: Isn't [[:dev:]] supposed to be for testing?! [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 12:18, 22 September 2018 (UTC)
:: Testing for MediaWiki pages, yes, but don't test hacking my account! What kind of testing are you trying to do? I thought you wanted to try redirects and HTML and stuff - why try to reset my password? [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 12:23, 22 September 2018 (UTC)
:: Testing for MediaWiki pages, yes, but don't test hacking my account! What kind of testing are you trying to do? I thought you wanted to try redirects and HTML and stuff - why ''not'' try to reset my password? [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 12:23, 22 September 2018 (UTC)
Fixed that. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 12:28, 22 September 2018 (UTC)
::: The answer to "why not reset my password" is because it creates a headache and makes me do extra work if I have to reset my password. This sort of thing is usually a bannable offense - while we enjoy you being here, you cannot do things that give us more work. Again: don't play with stuff like that! We don't want to ban you, but that's really serious. [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 16:52, 22 September 2018 (UTC)

== Patent val mappings and direct mappings - using both ==

Hi Piotr, just saw you added a bunch of patent val mappings to the EDO pages. This is very good information to have. It is also a good reference to have the "direct" mappings, even if inconsistent - we use these a lot for generating subgroup mappings, so I've added both to each page. There is also the concept of the "best" mapping using generalized patent vals - I don't know if that is on the wiki yet, but it would be good to add if not. [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 18:20, 22 September 2018 (UTC)
: Do you like https://scratch.mit.edu/projects/247892491/? I can't easily reprogram it to use the closest matches instead of patent ones though, as the project calculates the errors for 3, 5, 7, 9, 11, 13 and 15 (9 and 15 from the patent val), and it's hardwired so that each fraction has been given a text fraction pair and a code, e.g. the code 15 standing for "add the error for 3, subtract the error for 11" for 12/11, 11/6 (1=3, 2=5, 3=7, 4=9, 5=11, 6=13, 7=15). [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 18:27, 22 September 2018 (UTC)

== Wiping Dev ==

Hi Piotr - we need to wipe dev soon to restart it from scratch. Are there any pages on there you want to save? [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 15:38, 23 September 2018 (UTC)
:No, everything Xen has done is testing, except for [[The Xen]] which is now present here too. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 15:43, 23 September 2018 (UTC)
::OK, we have erased it. Can you sign up there again? Thanks [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 15:49, 23 September 2018 (UTC)
:::OK, I saw your new account and saved it so you do not have to re-sign up every time we erase it. It's currently synchronizing with the English wiki, so give it 15 minutes and should be working. [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 15:54, 23 September 2018 (UTC)
::::Hello Piotr - we just reset dev. Can you confirm your email address? Let me know once you do and I will then save dev again from this point, so that every time we reset it, it doesn't reset your user. Thanks [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 07:18, 25 September 2018 (UTC)
::::: Done! [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 16:54, 25 September 2018 (UTC)
:::::: OK, we have saved it - you shouldn't have to re-confirm every time. [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 17:37, 25 September 2018 (UTC)

== 15edo-a ==

Hi Piotr - 15edo-a isn't a different tuning, Igs is just making a temporary 15edo page for now. He went to sleep and will be updating the main 15edo page with the info tomorrow. You should join us on Facebook at [https://www.facebook.com/groups/xenwiki] - that's where we've been talking about this stuff [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 05:22, 25 September 2018 (UTC)
: That's confusing; pages like [[306edo]] talking about 306c tempering out different commas not explaining what it is, etc. that confused me into thinking like this. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 05:24, 25 September 2018 (UTC)
:: That's ok, but can you please not keep making -a, -a-a, etc pages? They were temporary and part of a discussion on the Facebook group; they were always intended to be merged back into the original pages. [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 23:40, 29 September 2018 (UTC)

== hard reditrect ==

Hi Piotr, <br/>
this was new to me :-) thanks. Maybe the sidebar should probable link to the help namespace. I'm not sure if moving help stuff into the help NS gets consensus in the FB group. Are you on FB meanwhile, if not: are you planning to create an account there? <br/>
Best --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 12:54, 30 September 2018 (UTC)
:See [[redirect]] for more information about redirects. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 12:58, 30 September 2018 (UTC)
:: I see. A small usability consideration: sometimes (well, not in the current case with the help page) it's good to see that a page is the result of a redirect, I don't consider redirects ''per se'' as bad. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 13:13, 30 September 2018 (UTC)

== Editor pages ==

These pages are an (in my opinion) outdated concept, introduces in the Wikispaces era, to compensate the lack of real user pages. [[User:Carl Lumma]] and [[User:Xenwolf|I (Xenwolf)]] were the only members that made use of it, so it seems that [[Editor PiotrGrochowski]] would better redirect to your user page again. If there will be a [[Help:User pages|help page about user pages]], the [[Editor yourName]] will most likely be transformed into a redirect to that help page. But: What do you think about this topic? Best --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 16:29, 30 September 2018 (UTC)
:Editor pages are for information about the user, user pages are for possible testing. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 19:14, 30 September 2018 (UTC)
::Editor pages should still be in the User namespace then. —[[User:Keenan Pepper|Keenan Pepper]] ([[User talk:Keenan Pepper|talk]]) 19:15, 30 September 2018 (UTC)
:::Wait, maybe I misunderstood what you're saying. If you're talking about an article about the person, because they're notable in the xenharmonic community (for example [[Harry Partch]] or something), then it should just be that person's name, without the word "Editor". —[[User:Keenan Pepper|Keenan Pepper]] ([[User talk:Keenan Pepper|talk]]) 19:16, 30 September 2018 (UTC)
:::: I assume by User namespace you mean pages like [[User:PiotrGrochowski]] and NOT [[User:PiotrGrochowski/Ups and Downs Notation-a]]. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 19:18, 30 September 2018 (UTC)
:::::Both of those pages are in the User namespace. In fact, anything that begins with "User:" is. See https://www.mediawiki.org/wiki/Namespaces for more info about Mediawiki namespaces. —[[User:Keenan Pepper|Keenan Pepper]] ([[User talk:Keenan Pepper|talk]]) 19:22, 30 September 2018 (UTC)
:I agree with this. All drafts/sandboxes should be in the User namespace. —[[User:Keenan Pepper|Keenan Pepper]] ([[User talk:Keenan Pepper|talk]]) 19:12, 30 September 2018 (UTC)
:: Don't forget that there is an official [[SandBox]] page for every user to try out some wiki markup, or to discuss stuff with other users. '''@Piotr:''' the <code>Editor_</code> suffix was a surrogate for a real user namespace (Wikispaces had no namespaces). The user namespace is the user page and all subpages whereas - by convention - a users's subpages "belong" to the prefixing user (if not indicated otherwise). The talk pages are by default open to other users. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 19:29, 30 September 2018 (UTC)

== Status of [[Ups and Downs Notation]], and personal attacks in edit summaries ==

I'm going to restore [[Ups and Downs Notation]] back to the main namespace. I don't like to revert people's actions without a thorough explanation, so here goes: I moved articles like [[15edo-b]], [[Wikifuture-a]], etc. into user namespaces since I believed the '''intent''' of the people creating them was as drafts or sandboxes, and not permanent articles meant to live on the wiki indefinitely. These drafts should be stored in the User namespace since they're not meant as final, published articles, but instead eventually meant to replace (or be merged into) the main articles at [[15edo]], [[Wikifuture]]. Importantly, I believed the authors of these pages would likely not be angry that I moved them, because they can keep working on them with basically no interruption, and it aligns with what they want to do. (I can't be '''sure''' of that, but it was my sincere belief.)

Now, for [[Ups and Downs Notation]] the story seems quite different to me. I think it was on the old wiki for a while, and on this new wiki it's always been on the main namespace with no indication that it's any kind of draft or sandbox. Moving it to a user subpage might send the unfriendly message that that user's contributions are not welcome here, and they're going to be marginalized or outcast. Personal attacks like "K**e curse" in edit summaries certainly do nothing to help this, and I consider them uncivil.

Instead of doing things like this, I encourage you to edit articles directly. If you wanted to, I don't think it would be unwelcome to add a "Criticism" section or otherwise change the article to reflect that much of the content is not the consensus of all editors here. As long as we're civil to each other and have the common goal of improving articles, I think even people with widely differing opinions can work together to improve the wiki. —[[User:Keenan Pepper|Keenan Pepper]] ([[User talk:Keenan Pepper|talk]]) 17:26, 30 September 2018 (UTC)

== List style ==

Hi Piotr, <br/>
please don't try to push the <code><nowiki><ul><li></li></ul></nowiki></code> style into the wiki. The wiki idea is that you have to know few or - better - no markup for adding (or editing) content. This way it's really easy to see how the wiki text relates to the HTML output. You may even read the article without transforming it into HTML. Why do you prefer to embed HTML tags if it isn't necessary at all (MediaWiki transforms it into HTML automatically). Are you maybe using the visual editor? <br/> Best --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 20:31, 30 September 2018 (UTC)
: We just discussing the issue on Facebook, here I read - let me cite: '' "the proper, canonical syntax" on a wiki is wiki markup''<sup> Jonathan Bright</sup> and ''One only breaks into HTML if there is no support for the markup, and lists are supported.''<sup> Joe McMahon</sup> - all in all, there is not a single vote pro your preference. Please tell me what's your point. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 20:42, 30 September 2018 (UTC)
:I haven't ever used the visual editor on xen.wiki and I don't want to. I live my life without overcomplications of the [[list model]]. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]) 07:44, 10 October 2018 (UTC)

== I removed a partial addition ==

Hi Piotr, <br/>
you started adding a column in the page [[15edo-interval names]] which broke the table, that's why I removed it (I beg your understanding). Feel free to re-add a ''whole'' column, but for linking to background information (I'm talking about [[User:PiotrGrochowski/Extra-Diatonic Intervals|Extra-Diatonic Intervals]]), it would be better use a page in the main namespace (soon), so that also others feel invited to elaborate about your idea. <br/> Best --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 09:40, 1 October 2018 (UTC)
:Stop saying you're the best. It's annoying. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]) 06:51, 15 October 2018 (UTC)

== Pardon? ==

I'm a bit confused about [https://en.xen.wiki/index.php?title=User_talk%3AXenwolf&type=revision&diff=34318&oldid=34317 your way] to ''explain'' things. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 15:53, 1 October 2018 (UTC)
:You said you intentionally left out the link, so you are hiding some secret. I deleted my section because I can't trust you like this. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 15:56, 1 October 2018 (UTC)

== Accidental reversion on Spanish Wiki ==

Hello Piotr, just wanted to let you know that while I was updating the discussion page format on the Spanish wiki, I screwed up and we had to revert to the last SQL backup. This has had the unfortunate effect of reverting two of your edits, which I think were under the name "Piotr's Sock Puppet" or something like that. My apologies! This was unintentional; you may need to re-add these. [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 18:08, 1 October 2018 (UTC)

== Name Calling ==

Hello Piotr,

I wanted to address the name calling that you made on [[User_talk:TallKite]]'s page here: [[https://en.xen.wiki/index.php?title=User_talk:TallKite&oldid=34094]]

This is not appropriate behavior for this wiki. Namecalling is not acceptable and is generally a bannable offense. I know there is a language barrier here, and I'm not sure if you understand the meaning of the word "creepy," but it is considered an insult in English. We are giving you some slack here as we would usually ban for this right away, because we know that English is your second language and you may not 100% understand the meaning of your words. However, please refrain from such language in the future. [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 18:18, 1 October 2018 (UTC)

: Piotr, to add to this, I see you had cursed at Kite in the edit logs (K**e is so f*****g creepy). This is really unacceptable; we have temporarily blocked your account from editing the wiki for three days. You can still view the Wiki but not edit (except for this page). When the ban expires you can edit again. Please do not ever curse at users here; we want this to be a positive atmosphere for everyone. Thank you. [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 18:37, 1 October 2018 (UTC)

== The Piotr Notation system ==

Hello Piotr,

I wanted to explain the situation with Kite's notation system. Simply put, he is writing a book on his theory, and we are documenting it on the Wiki. We want this to be a place to document everyone's ideas, whether they are yours or Kite's, as though this were an encyclopedia. No notation is perfect, but we think if we have everything here, composers will be able to decide which one they like best!

It looks like you have some good ideas for a notation system. I am very interested to understand more what they are. Unfortunately, the way things are set up now, it is difficult to see what the '''differences''' are between yours and Kite's system. This isn't your fault, but it's because the Wiki format does not make it easy to see the difference between two pages.

But we all see you are interested in this stuff and would like to encourage you to contribute. I thought one of the following two options would work best:

1. Perhaps you simply have some suggested additions to Kite's system. If this is the case, then rather than just copying Kite's page, why not make a page called "Piotr's suggested alterations to Kite's Ups and Downs Notation System", where the '''differences''' are clearly listed. That is, rather than copying Kite's page and just changing stuff, simply writing your thoughts about which things should be different, and why. This is just to make it easier for us to understand your ideas, because right now your page looks exactly the same as Kite's page, with some minor differences, and everyone has to search side by side to try to find what they are.

2. Or, if you are interested in notation systems, why not go all the way and make a Piotr notation system? Instead of saying it's "Piotr's version of Kite's system," why not simply develop your very own system that can be exactly the way you want, designed exactly the way you like it? Of course you can use Kite's ideas or page as a starting point, but if you have some good thoughts for a notation system, then I think you should feel free to make it '''your''' system, not just a variant of Kite's! It can go in whatever direction you like.

Believe me, nobody's system is perfect, not Kite's, not anyone's! But I am also interested in designing a notation system, and I would be happy to work with you on this, to the extent that I have time. I created an outline for a [[KISS notation system|KISS_notation]] many years ago, but the page needs work... [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 01:06, 4 October 2018 (UTC)

:What do you mean by the wiki setup? The differences are here: <s>https://en.xen.wiki/index.php?title=Ups_and_Downs_Notation&type=revision&diff=34008&oldid=35895</s>https://en.xen.wiki/w/Special:ComparePages?page1=Ups+and+Downs+Notation&rev1=&page2=User%3APiotrGrochowski%2FUps+and+Downs+Notation-a&rev2=&action=&diffonly=&unhide=

:I made [[User:PiotrGrochowski/Ups and Downs Notation-a]] because in my opinion, just like the [[edo]]s, it's something that is a result of maths existing. It's neither Kite's, mine, yours or anyone else's. The difference is that all Kite's conventions are removed; scales now start with C and not D, augmented unison is included for 12edo, pentatonic scales removed, etc. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 14:46, 4 October 2018 (UTC)

:: Piotr - the thing is, as Kite continues to change his page, it will become difficult to see the differences between his page and yours. The link you gave above will only do a diff between the old version of his page and your current page. If you are not interested in making a new notation system, it would be helpful to make a page that emphasizes the difference between your sytem and his, so we can see what they are. [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 07:32, 8 October 2018 (UTC)
:::Stop calling it my system when it isn't! Kite minus Kite is no one. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]) 15:45, 8 October 2018 (UTC)
::::Regardless, still, it would be helpful to make a page that emphasizes the differences between what you are proposing and what he is proposing. [[User:Mike Battaglia|Mike Battaglia]] ([[User talk:Mike Battaglia|talk]]) 17:54, 8 October 2018 (UTC)

== A word about templates ==

Hello Piotr, <br/>
I understand your idea to include (some) templates into the [[:Category:Shortcuts]] but you are doing it wrong: if you place <code><nowiki>[[Category:Shortcuts]]</nowiki></code> into a template without surrounding it by <code><nowiki><noinclude></nowiki></code> and <code><nowiki></noinclude></nowiki></code>, you cause the including (or "transcluding") pages to be added to this category, what may be intended, but obviously not in this very case ([https://en.xen.wiki/index.php?title=SandBox&oldid=36275#List_model currently] [[SandBox]] is categorized as a [[:Category:Shortcuts|Shortcut]] but in fact it isn't). BTW [https://en.xen.wiki/index.php?title=Template%3AMonzo&type=revision&diff=36459&oldid=36458 I protected Template:Monzo] to help you concentrate on fixing the problems in [[Template:List]] fist. <br/>
Best regards --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 17:24, 9 October 2018 (UTC)

== broken ratios ==

Good to point on this! But I'd do it openly in your place. Maybe by typing "broken ratio!" behind the wrong numbers so that everyone can see the error. (If he/she is so smart as to understand that an 0-ending in numerator and denominator clearly indicates a rounding error). I woudn't hide the errors by commenting them out. Best --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 22:21, 13 October 2018 (UTC)

== Scratch programming ==

Hi Piotr, <br/>
you seem to be a [https://scratch.mit.edu/users/ihgfedcba/ Scratch programmer]. What are the prerequisites to run these programs? Is it possible from within modern Browsers? Doesn't is need a Flash Player? <br/>
BTW: Do you also program in other languages? <br/>
Best --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 22:01, 14 October 2018 (UTC)
:The projects run on Flash. It's possible to use Flash in Android using the Puffin Browser. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]) 05:09, 15 October 2018 (UTC)
:Also, the generator I was talking about in [[Talk:13-limit/WikispacesArchive]] was also made in Scratch, although it wasn't shared. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]) 05:10, 15 October 2018 (UTC)

== Your list modelling efforts ==

Hi Piotr, <br/>
MediaWiki has a built-in list feature that makes it easy to add and ''maintain'' lists (unordered, ordered, mixed) on wiki pages. This is for a purpose. If you are interested in doing it the hard way yourself, maybe invest your time into it. But stop ''wasting other people's time'' with it. If you wish to suggest technical improvements to the wiki community, try to discuss it first. If not on Facebook then here on talk pages or in separate project pages (in the [https://en.xen.wiki/w/Special:AllPages?from=&to=&namespace=4 project namespace]). You may as well be bold and introduce it into normal pages but be open to ''accept'' that the feedback may differ from you expectations. Don't push your ideas into the wiki by pretending they're common conventions. Believe me, there are lots of people here that are absolute familiar with HTML. Wikis were invented to ''avoid'' HTML wherever possible, I thought you knew about that. Please move your list model project into the project namespace. You may link to it from your user namespace at will. You may also link to it from talk pages if you explain why. <br/>
Best regards --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 06:43, 15 October 2018 (UTC)
:You are very biased towards that [[list model]]. The [[Conventions]] are not biased towards any particular [[list model]]. What you're describing is <code>wikispaces</code>, which is closing for a reason. Not to mention it's a pain to get newlines in many environments. Please stop being biased. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]) 06:51, 15 October 2018 (UTC)

== last warning ==

Immediately repair the damage you have done since your blocking ended. In an ''acceptable'' way. No kidding. You have one hour. If you don't, you will be '''blocked for one week''' - the deadline is '''10:35 local time (CEST)''' --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 07:35, 16 October 2018 (UTC)
:I did not do any damage. It's already repaired. You will not ban me. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]) 08:19, 16 October 2018 (UTC)
:: I see '''only one constructive edit''' in [https://en.xen.wiki/index.php?limit=50&title=Special%3AContributions&contribs=user&target=PiotrGrochowski&namespace=&tagfilter=&start=2018-10-16&end=2018-10-16 these] You are still promoting forcefully (without any regard) your pointless [[list model]]. You have '''6 minutes left.'''. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 08:29, 16 October 2018 (UTC)
::: What? All of these are constructive, no [[list model]] is pointless and you are the one promoting a specific list model. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]) 08:35, 16 October 2018 (UTC)

You have been [https://en.xen.wiki/index.php?title=Special%3ALog&type=block&user=Xenwolf&page=User%3APiotrGrochowski&year=&month=-1&tagfilter=&subtype= blocked for one <s>month</s> <u>week</u>] to carefully study what's been going on in the last ≈90 minutes. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 08:47, 16 October 2018 (UTC)

== Thanks for contributing again ==

Hello Piotr,<br/>
I ask you to do a critical review of what you are calling "[[list model]]". There will be very few people that will call it like this. And, much more important, MediaWiki will export '''valid HTML''' lists for the simplified wiki markup (<code>#</code>, <code>*</code> and <code>;</code>+<code>:</code>) as well as for list built via HTML markup, the same is true for your templates, but not for the other "options". Please have a look on the resulting HTML delivered by MediaWiki, I mean: check the delivered HTML source code (for example Ctrl+U for Firefox under Windows). <br/>
Thanks in advance! <br/>
--[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 15:10, 23 October 2018 (UTC)
:The [[list models]] need not have the same HTML output. Plaintext lists are named so because the output is plaintext, so it can be copied anywhere, including personal notes in Notepad or in the <nowiki><pre></pre></nowiki> safety zone. Defined lists intentionally override the MediaWiki list styling for a defined one. All of these are valid HTML. Do you see any HTML errors in your browser's debugger in [https://en.xen.wiki/index.php?title=SandBox&oldid=37569#List_model SandBox#List model]? [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]) 17:39, 23 October 2018 (UTC)

=== Why on earth are you insisting to re-invent the wheel?? ===
The wiki source code ''is meant'' to be the ideal fall-back version. That's why it's a bad idea to overcomplicate it. In MediaWiki it's easy to get access to the source code, even if in read-only mode (BTW: did you know [https://en.xen.wiki/w/Chroma?action=raw <code>?action=raw</code>]). Do you really believe that someone is interested in adding unreadable number pictures and hard-coded line breaks, if styling is normally based on CSS these times? Here is what W3Cschools have to say about [https://www.w3schools.com/html/html_lists.asp HTML Lists]. Please consider to further clarify your intention with the help of some sensible reference. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 18:08, 23 October 2018 (UTC)
:Not everything has to go along your or W3C's opinions. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]) 09:05, 24 October 2018 (UTC)

== Serious warning ==

You are obviously unable to present your intention in a polite and understandable manner. I assume that your intrusive changes will result in you being permanently put into read-only mode in the near future. I very much regret that you have such difficulties with self-criticism. It is still open how long your writing pause will last. I know from some dedicated Xenwiki users that they plead for a permanent ban and resent me for not having you permanently blocked already.

--[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 13:06, 24 October 2018 (UTC)

Help:Editing

2018-10-24T09:08:50Z

PiotrGrochowski: REVERT VANDALISM. That was a second hell of an opinion...

Wikis should be easy to edit, so most text you add or change will appear as you expect it. But you can do more if you want.

[https://wikiindex.org/Xenharmonic_Wiki This wiki] is currently in [https://wikiindex.org/Category:ConfirmEmail ConfirmEmail mode] after the move from [[Wikispaces]] to a MediaWiki-based installation (the edit mode may change again some time). For editing or adding information to the wiki, you have to join it. See [[How to Get Your Xenwiki Account]] for more!

== Using links ==
What you always have dreamed of (for your paper notes) - in a wiki it becomes true: fast links!

If you enclose a word in double brackets (like <code><nowiki>[[word]]</nowiki></code>), you get an internal link, hopefully to an already existing page - if not, it appears red, and you just created the opportunity for your co-authors to add the details. If you want a text differing from the link target, you can place it after a pipe symbol: <code><nowiki>[[word|different text]]</nowiki></code>.

For external links, simply place the URL into the text, certain protocols (http, https, mailto) will be automatically transformed into links. For alternate text, use single brackets, start with the URL and after a space enter the title, like so <code><nowiki>[http://acid3.acidtests.org the Acid3 test]</nowiki></code> which renders to [http://acid3.acidtests.org the Acid3 test].

You may link to sections in the same page by using the section title with the prefix <code>#</code>. MediaWiki automatically converts it into a link with the correct anchor identifier. For example to link to the beginning of this section use <code><nowiki>[[#Using links]]</nowiki></code> (rendered as [[#Using links]]) or with changed text <code><nowiki>[[#Using links|top of this section]]</nowiki></code> (rendered as [[#Using links|top of this section]]).

=== Link text ===
The link text should ideally relate to the link target, try to avoid "[[help:here-links|here]]" links. There is an interesting extension for Firefox, [https://addons.mozilla.org/de/firefox/addon/url2clipboard/ Copy URL To Clipboard], that does automatic formatting of page URL and page title in MediaWiki syntax into the clipboard.

=== Redirects ===
There are 3 types of redirects: soft redirect, regular redirect and hard redirect.

A "soft redirect" or "smart redirect" is just an ordinary page with the text "See XYZ.", where XYZ is a link to the target of the smart redirect. A smart redirect should not point to another smart redirect - if you see this you should fix it by shortcutting past the second one and making the first redirect point directly to the ultimate target. This kind of redirect were introduced in the Wikispaces era, to make redirects accessible later, for example to flip page titles.

A regular redirect is performed by <code><nowiki>#REDIRECT [[Page title]]</nowiki></code>. A double redirect should be fixed be skipping intermediate targets. The page [[Special:DoubleRedirects]] can be used to detect them. Regular redirects did not exist in ''Wikispaces''. This kind of redirect displays a small information message when it was used to get to the current page.

A hard redirect directly leads to the target page without a notice of redirect. Therefore, they should only be used in certain cases like common misspellings. The rule is that if anyone might ever want to flesh the redirect out into its own article, it should not be a hard redirect. Hard redirects can be chained with no problem, which is why they're still appropriate to use for misspellings. The code for a hard redirect in MediaWiki is <code><nowiki>#REDIRECT [[:en:Page title]]</nowiki></code>. Hard redirects worked somewhat differently in <code>wikispaces</code>; only admins could create or change them, and they could not redirect to a section.

see also

* http://groups.yahoo.com/neo/groups/tuning/conversations/topics/101227 - redirects were discussed on the [[tuning_list|tuning list]] in the [[Wikispaces]] era
* [[:Category:soft redirect|list of pages tagged as soft redirect]] - pages that contain only see <tt>[[</tt><nowiki>''some other page''</nowiki><tt>]]</tt> (be cautious with other tags on these)
* [[:Category:smart redirect|list of pages tagged as smart redirect]] - pages that have more information (a small definition, classification etc.)

== Text formatting ==

=== Emphasis ===
Some typographic markup can easy be added by including the phrase in doubled <code><nowiki>''</nowiki></code> or tripled <code><nowiki>'''</nowiki></code> (apostrophe chars):

* ''italic'' using <nowiki>''italic''</nowiki>
* '''bold''' using <nowiki>'''bold'''</nowiki>
* '''''bold italic''''' using <nowiki>'''''bold italic'''''</nowiki>

=== Lists ===
:''See [[Help:List]] for advanced use cases for lists and other [[list model]]s.''
There are tree basic types of lists in HTML that are supported in MediaWiki markup: [[#unordered|unordered]] (or "bulleted") lists, [[#numbered|numbered]] lists, [[#description|description]] lists. Each type of lists can be created within wiki source code by just indicating the items at the beginning of the line (the list is automatically built around them). Each item has to start on a new line.

==== Unordered ====
Unordered list items are marked by <code>*</code> (asterisk):

* Coffee
* Wine
* Milk

==== Numbered ====
Numbered list items are marked by <code>#</code> (number sign):

# Coffee
# Wine
# Milk

==== Description ====
Description items are marked by <code>;</code> (semicolon) for term and <code>:</code> (colon) for the description:

; Coffee
: a brewed drink prepared from roasted coffee beans. (from: ''[https://en.wikipedia.org/wiki/Coffee Coffee - Wikipedia''])
; Wine
: an alcoholic beverage made from fermented grapes. (from: ''[https://en.wikipedia.org/wiki/Wine Wine - Wikipedia]'')
; Milk
: a white liquid nutrient-rich food produced by the mammary glands of mammals. (from: ''[https://en.wikipedia.org/wiki/Milk Milk - Wikipedia]'')

==== Combined List Styles ====

# Morning
#* Coffee
#* Milk
# Evening
#* Coffee
#* Wine

=== Tables ===
:''for advanced examples see [[Help:Table]]''

You can add tables to wiki pages:
{| class="wikitable"
! what you write
|
<nowiki>
{| class="wikitable"
|+ The Title
! head A
! head B
|-
| cell A1
| cell B1
|-
| cell A2
| cell B2
|}</nowiki>
|-
! what you get
|

{| class="wikitable"
|+ The Title
! head A
! head B
|-
| cell A1
| cell B1
|-
| cell A2
| cell B2
|}

|}

=== Images ===
Inserting a JPEG, GIF, or PNG image is pretty straightforward - when using the visual page editor click the "File" button and upload the image. Or in Wikitext, use <code><nowiki>[[image:filename]]</nowiki></code> or <code><nowiki>[[file:filename]]</nowiki></code>. Also SVG images are supported (<code><nowiki>[[file:some.svg]]</nowiki></code>), they are automatically transformed into PNG images by the MediaWiki software, because not all Browsers come with native SVG support. A possible problem can arise when the image uses fonts that are not available on the wiki server. To prevent such errors, you can transform text into graphic objects before uploading the image here.

=== Maths formulas ===
...can be used [http://en.wikipedia.org/wiki/Help:Displaying_a_formula like in the Wikipedia]: start it with <code><nowiki><math></nowiki></code> and end it with <code><nowiki></math></nowiki></code>. Take care: using the math tags will break paragraphs. See for example

<math>V =\frac{4}{3} \pi r^3</math>

Here is a browser-based formula editor that may be helpful: http://www.codecogs.com/latex/eqneditor.php

=== Table of contents ===
Place <code><nowiki>__TOC__</nowiki></code> where you wish to place the table of contents, mostly this will be at the top of the page.

== Social aspects ==
In this wiki, you can use talk pages and [[user talk]] pages.

If you want to discuss the wiki itself, do it on the page [[wikifuture]].

=== Categories ===
Categories all serve different purposes: sometimes they say what something is or the context in which it belongs ([[:Category:edo]], [[:Category:interval]]). Sometimes they are used for annotation purposes ([[:Category:todo:reduce mathslang]]), etc. All these forms have in common is that they can combine wiki pages in a way that is independent of their authors.

A category may have its own content (and category)
:''see [https://en.xen.wiki/w/Special:AllPages?from=&to=&namespace=14 Pages in the namespace '''Category''']''
or not yet (despite being used to categorize other pages):
:''see [[Special:WantedCategories]]''

Some more examples:

* [[:Category:todo:add_definition|todo:add definition]] - here a term needs to be defined
* [[:Category:Todo]] - category of categories that flag articles

=== Discussion ===
Discussion pages are provided to each content page. Please keep in mind that the email notification to people who are observing the page cannot be taken back!

Sometimes you will wait forever for a reply because nobody monitors the page or topic. In this case you should find out who is most familiar with the topic and ask that person directly. You can check the history of pages for its authors. Another approach is to ask questions on further media, examples are listed on the page [[GeneralDiscussion]].

== Links ==
* [[Help:Help]]
* [[Conventions]]

[[Category:help]]
[[Category:syntax]]
[[Category:wiki help]]

Help:List

2018-10-24T09:07:39Z

PiotrGrochowski: REVERT VANDALISM. That was a hell of an opinion...

:''See [[Help:Editing #Lists]] for basic list syntax.''

Lists can get complicated, if they are nested in other lists, items get longer than just one line, for instance whole paragraphs. Items contain images and so on. The wiki markup is optimized for simple lists, for complex lists MediaWiki allows for embedding HTML all three types of lists.

== Unordered lists ==
<nowiki><ul>
<li>
One item that contains much text even if it isn't really interesting to read all of it:
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor
invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores
</li>
<li>
Another item that contains much text even if it isn't really interesting to read all of it:
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor
invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua.
At vero eos et accusam et justo duo dolores
</li>
</ul></nowiki>

<ul>
<li>
One item that contains much text even if it isn't really interesting to read all of it:
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor
invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores
</li>
<li>
Another item that contains much text even if it isn't really interesting to read all of it:
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor
invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores
</li>
</ul>

== Numbered lists ==

Numbered (or ordered) list show an ascending number for each item by default. It is possible to change this style, see sections [[#roman numbers]] or [[#latin letters]] for how.

<nowiki><ol>
<li>
One item that contains much text even if it isn't really interesting to read all of it:
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor
invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores
</li>
<li>
Another item that contains much text even if it isn't really interesting to read all of it:
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor
invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores
</li>
</ol></nowiki>

<ol>
<li>
One item that contains much text even if it isn't really interesting to read all of it:
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor
invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores
</li>
<li>
Another item that contains much text even if it isn't really interesting to read all of it:
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor
invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores
</li>
</ol>

=== roman numbers ===
<ol style="list-style-type: upper-roman;">
<li>
One item
</li>
<li>
Another item
</li>
</ol>

<ol style="list-style-type: lower-roman;">
<li>
One item
</li>
<li>
Another item
</li>
</ol>

=== latin letters ===
<ol style="list-style-type: upper-alpha;">
<li>
One item
</li>
<li>
Another item
</li>
</ol>

<ol style="list-style-type: lower-alpha;">
<li>
One item
</li>
<li>
Another item
</li>
</ol>

== Description lists ==
Here are examples for huge descriptions for terms in definition lists that contain floating images as illustrations and tables.

<nowiki><dl>
<dt>Cent</dt>
<dd>[[File:Test-120x60px.png|right|60px]] The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each. Typically, cents are used to express small intervals, or to compare the sizes of comparable intervals in different tuning systems, and in fact the interval of one cent is too small to be heard between successive notes.

Alexander J. Ellis based the measure on the acoustic logarithms decimal semitone system developed by Gaspard de Prony in the 1830s, at Robert Holford Macdowell Bosanquet's suggestion. Ellis made extensive measurements of musical instruments from around the world, using cents extensively to report and compare the scales employed,[1] and further described and employed the system in his 1875 edition of Hermann von Helmholtz's On the Sensations of Tone. It has become the standard method of representing and comparing musical pitches and intervals. (from: ''[https://en.wikipedia.org/wiki/Cent_(music) Cent (music) - Wikipedia]'') </dd>
<dt>Monzo</dt>
<dd>For example, the interval 15/8 can be thought of as having 5*3 in the numerator, and 2*2*2 in the denominator. This can be compactly represented by the expression 2^-3 * 3^1 * 5^1, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the {{Monzo| ... }} brackets, hence yielding {{Monzo|-3 1 1}}.

:''Practical hint:'' Because the pipe symbol and the greater sign have special meaning in wiki syntax and HTML, there is a helper template ([[Template:Monzo]]) that can be used like this {{Monzo|arguments}} to get the monzo brackets (<code>{{Monzo|arguments}}</code>) from it.

Here are some common 5-limit monzos, for your reference:

{| class="wikitable"
|-
! Ratio
! Monzo
|-
| style="text-align:center;" | [[3/2]]
| {{Monzo| -1 1 0 }}
|-
| style="text-align:center;" | [[5/4]]
| {{Monzo| -2 0 1 }}
|-
| style="text-align:center;" | [[9/8]]
| {{Monzo| -3 2 0 }}
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
|}
</dd>
</dl></nowiki>

<dl>
<dt>Cent</dt>
<dd>[[File:Test-120x60px.png|right|60px]] The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each. Typically, cents are used to express small intervals, or to compare the sizes of comparable intervals in different tuning systems, and in fact the interval of one cent is too small to be heard between successive notes.

Alexander J. Ellis based the measure on the acoustic logarithms decimal semitone system developed by Gaspard de Prony in the 1830s, at Robert Holford Macdowell Bosanquet's suggestion. Ellis made extensive measurements of musical instruments from around the world, using cents extensively to report and compare the scales employed,[1] and further described and employed the system in his 1875 edition of Hermann von Helmholtz's On the Sensations of Tone. It has become the standard method of representing and comparing musical pitches and intervals. (from: ''[https://en.wikipedia.org/wiki/Cent_(music) Cent (music) - Wikipedia]'')</dd>
<dt>Monzo</dt>
<dd>For example, the interval 15/8 can be thought of as having 5*3 in the numerator, and 2*2*2 in the denominator. This can be compactly represented by the expression 2^-3 * 3^1 * 5^1, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the {{Monzo| ... }} brackets, hence yielding {{Monzo|-3 1 1}}.

:''Practical hint:'' Because the pipe symbol and the greater sign have special meaning in wiki syntax and HTML, there is a helper template ([[Template:Monzo]]) that can be used like this <code><nowiki>{{Monzo|arguments}}</nowiki></code> to get the monzo brackets (<code>{{Monzo|arguments}}</code>) from it.

Here are some common 5-limit monzos, for your reference:

{| class="wikitable"
|-
! Ratio
! Monzo
|-
| style="text-align:center;" | [[3/2]]
| {{Monzo| -1 1 0 }}
|-
| style="text-align:center;" | [[5/4]]
| {{Monzo| -2 0 1 }}
|-
| style="text-align:center;" | [[9/8]]
| {{Monzo| -3 2 0 }}
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
|}
</dd>
</dl>

== List templates ==
There are two templates in this wiki for building HTML lists.

; [[Template:List]]
: for creating an unordered list with up to 50 items.
; [[Template:Numlist]]
: for creating a numbered list with up to 50 items.

Both templates were created by [[User:PiotrGrochowski]], please ask him for help and improvements.

== Plaintext and defined lists ==

It's also possible to create lists in plaintext:

<pre>•list<br>
1. list</pre>

which will output itself, minus the br tags. Colons can be used for indentation to create nested lists.

Defined lists define bullets and numbers in images instead of using text:

<pre>[[File:Listitem.png]]list<br>
[[File:1.png]] list</pre>

outputting

[[File:Listitem.png]]list<br>
[[File:1.png]] list

== See also ==
* [[Help:Editing]]
* [https://en.wikipedia.org/wiki/Help:List Help:List - Wikipedia]

[[Category:Help]]
[[Category:todo:expand|?]]

User talk:PiotrGrochowski

2018-10-24T09:05:05Z

PiotrGrochowski: off-topic

Help:Editing

2018-10-23T17:42:34Z

PiotrGrochowski: confirm email

Help:Editing

2018-10-23T17:41:27Z

PiotrGrochowski: this is the link in the proper place

Wikis should be easy to edit, so most text you add or change will appear as you expect it. But you can do more if you want.

[https://wikiindex.org/Xenharmonic_Wiki This wiki] is currently in [https://wikiindex.org/Category:LoginToEdit login-to-edit mode] after the move from [[Wikispaces]] to a MediaWiki-based installation (the edit mode may change again some time). For editing or adding information to the wiki, you have to join it. See [[How to Get Your Xenwiki Account]] for more!

== Using links ==
What you always have dreamed of (for your paper notes) - in a wiki it becomes true: fast links!

If you enclose a word in double brackets (like <code><nowiki>[[word]]</nowiki></code>), you get an internal link, hopefully to an already existing page - if not, it appears red, and you just created the opportunity for your co-authors to add the details. If you want a text differing from the link target, you can place it after a pipe symbol: <code><nowiki>[[word|different text]]</nowiki></code>.

For external links, simply place the URL into the text, certain protocols (http, https, mailto) will be automatically transformed into links. For alternate text, use single brackets, start with the URL and after a space enter the title, like so <code><nowiki>[http://acid3.acidtests.org the Acid3 test]</nowiki></code> which renders to [http://acid3.acidtests.org the Acid3 test].

You may link to sections in the same page by using the section title with the prefix <code>#</code>. MediaWiki automatically converts it into a link with the correct anchor identifier. For example to link to the beginning of this section use <code><nowiki>[[#Using links]]</nowiki></code> (rendered as [[#Using links]]) or with changed text <code><nowiki>[[#Using links|top of this section]]</nowiki></code> (rendered as [[#Using links|top of this section]]).

=== Link text ===
The link text should ideally relate to the link target, try to avoid "[[help:here-links|here]]" links. There is an interesting extension for Firefox, [https://addons.mozilla.org/de/firefox/addon/url2clipboard/ Copy URL To Clipboard], that does automatic formatting of page URL and page title in MediaWiki syntax into the clipboard.

=== Redirects ===
There are 3 types of redirects: soft redirect, regular redirect and hard redirect.

A "soft redirect" or "smart redirect" is just an ordinary page with the text "See XYZ.", where XYZ is a link to the target of the smart redirect. A smart redirect should not point to another smart redirect - if you see this you should fix it by shortcutting past the second one and making the first redirect point directly to the ultimate target. This kind of redirect were introduced in the Wikispaces era, to make redirects accessible later, for example to flip page titles.

A regular redirect is performed by <code><nowiki>#REDIRECT [[Page title]]</nowiki></code>. A double redirect should be fixed be skipping intermediate targets. The page [[Special:DoubleRedirects]] can be used to detect them. Regular redirects did not exist in ''Wikispaces''. This kind of redirect displays a small information message when it was used to get to the current page.

A hard redirect directly leads to the target page without a notice of redirect. Therefore, they should only be used in certain cases like common misspellings. The rule is that if anyone might ever want to flesh the redirect out into its own article, it should not be a hard redirect. Hard redirects can be chained with no problem, which is why they're still appropriate to use for misspellings. The code for a hard redirect in MediaWiki is <code><nowiki>#REDIRECT [[:en:Page title]]</nowiki></code>. Hard redirects worked somewhat differently in <code>wikispaces</code>; only admins could create or change them, and they could not redirect to a section.

see also

* http://groups.yahoo.com/neo/groups/tuning/conversations/topics/101227 - redirects were discussed on the [[tuning_list|tuning list]] in the [[Wikispaces]] era
* [[:Category:soft redirect|list of pages tagged as soft redirect]] - pages that contain only see <tt>[[</tt><nowiki>''some other page''</nowiki><tt>]]</tt> (be cautious with other tags on these)
* [[:Category:smart redirect|list of pages tagged as smart redirect]] - pages that have more information (a small definition, classification etc.)

== Text formatting ==

=== Emphasis ===
Some typographic markup can easy be added by including the phrase in doubled <code><nowiki>''</nowiki></code> or tripled <code><nowiki>'''</nowiki></code> (apostrophe chars):

* ''italic'' using <nowiki>''italic''</nowiki>
* '''bold''' using <nowiki>'''bold'''</nowiki>
* '''''bold italic''''' using <nowiki>'''''bold italic'''''</nowiki>

=== Lists ===
:''See [[Help:List]] for advanced use cases for lists and other [[list model]]s.''
There are tree basic types of lists in HTML that are supported in MediaWiki markup: [[#unordered|unordered]] (or "bulleted") lists, [[#numbered|numbered]] lists, [[#description|description]] lists. Each type of lists can be created within wiki source code by just indicating the items at the beginning of the line (the list is automatically built around them). Each item has to start on a new line.

==== Unordered ====
Unordered list items are marked by <code>*</code> (asterisk):

* Coffee
* Wine
* Milk

==== Numbered ====
Numbered list items are marked by <code>#</code> (number sign):

# Coffee
# Wine
# Milk

==== Description ====
Description items are marked by <code>;</code> (semicolon) for term and <code>:</code> (colon) for the description:

; Coffee
: a brewed drink prepared from roasted coffee beans. (from: ''[https://en.wikipedia.org/wiki/Coffee Coffee - Wikipedia''])
; Wine
: an alcoholic beverage made from fermented grapes. (from: ''[https://en.wikipedia.org/wiki/Wine Wine - Wikipedia]'')
; Milk
: a white liquid nutrient-rich food produced by the mammary glands of mammals. (from: ''[https://en.wikipedia.org/wiki/Milk Milk - Wikipedia]'')

==== Combined List Styles ====

# Morning
#* Coffee
#* Milk
# Evening
#* Coffee
#* Wine

=== Tables ===
:''for advanced examples see [[Help:Table]]''

You can add tables to wiki pages:
{| class="wikitable"
! what you write
|
<nowiki>
{| class="wikitable"
|+ The Title
! head A
! head B
|-
| cell A1
| cell B1
|-
| cell A2
| cell B2
|}</nowiki>
|-
! what you get
|

{| class="wikitable"
|+ The Title
! head A
! head B
|-
| cell A1
| cell B1
|-
| cell A2
| cell B2
|}

|}

=== Images ===
Inserting a JPEG, GIF, or PNG image is pretty straightforward - when using the visual page editor click the "File" button and upload the image. Or in Wikitext, use <code><nowiki>[[image:filename]]</nowiki></code> or <code><nowiki>[[file:filename]]</nowiki></code>. Also SVG images are supported (<code><nowiki>[[file:some.svg]]</nowiki></code>), they are automatically transformed into PNG images by the MediaWiki software, because not all Browsers come with native SVG support. A possible problem can arise when the image uses fonts that are not available on the wiki server. To prevent such errors, you can transform text into graphic objects before uploading the image here.

=== Maths formulas ===
...can be used [http://en.wikipedia.org/wiki/Help:Displaying_a_formula like in the Wikipedia]: start it with <code><nowiki><math></nowiki></code> and end it with <code><nowiki></math></nowiki></code>. Take care: using the math tags will break paragraphs. See for example

<math>V =\frac{4}{3} \pi r^3</math>

Here is a browser-based formula editor that may be helpful: http://www.codecogs.com/latex/eqneditor.php

=== Table of contents ===
Place <code><nowiki>__TOC__</nowiki></code> where you wish to place the table of contents, mostly this will be at the top of the page.

== Social aspects ==
In this wiki, you can use talk pages and [[user talk]] pages.

If you want to discuss the wiki itself, do it on the page [[wikifuture]].

=== Categories ===
Categories all serve different purposes: sometimes they say what something is or the context in which it belongs ([[:Category:edo]], [[:Category:interval]]). Sometimes they are used for annotation purposes ([[:Category:todo:reduce mathslang]]), etc. All these forms have in common is that they can combine wiki pages in a way that is independent of their authors.

A category may have its own content (and category)
:''see [https://en.xen.wiki/w/Special:AllPages?from=&to=&namespace=14 Pages in the namespace '''Category''']''
or not yet (despite being used to categorize other pages):
:''see [[Special:WantedCategories]]''

Some more examples:

* [[:Category:todo:add_definition|todo:add definition]] - here a term needs to be defined
* [[:Category:Todo]] - category of categories that flag articles

=== Discussion ===
Discussion pages are provided to each content page. Please keep in mind that the email notification to people who are observing the page cannot be taken back!

Sometimes you will wait forever for a reply because nobody monitors the page or topic. In this case you should find out who is most familiar with the topic and ask that person directly. You can check the history of pages for its authors. Another approach is to ask questions on further media, examples are listed on the page [[GeneralDiscussion]].

== Links ==
* [[Help:Help]]
* [[Conventions]]

[[Category:help]]
[[Category:syntax]]
[[Category:wiki help]]

User talk:PiotrGrochowski

2018-10-23T17:39:23Z

PiotrGrochowski: /* Thanks for contributing again */

User talk:PiotrGrochowski

2018-10-23T17:39:10Z

PiotrGrochowski:

Help:List

2018-10-23T14:24:59Z

PiotrGrochowski: /* Plaintext and defined lists */

Help:List

2018-10-23T14:15:04Z

PiotrGrochowski: add the other two list models

:''See [[Help:Editing #Lists]] for basic list syntax.''

Lists can get complicated, if they are nested in other lists, items get longer than just one line, for instance whole paragraphs. Items contain images and so on. The wiki markup is optimized for simple lists, for complex lists MediaWiki allows for embedding HTML all three types of lists.

== Unordered lists ==
<nowiki><ul>
<li>
One item that contains much text even if it isn't really interesting to read all of it:
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor
invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores
</li>
<li>
Another item that contains much text even if it isn't really interesting to read all of it:
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor
invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua.
At vero eos et accusam et justo duo dolores
</li>
</ul></nowiki>

<ul>
<li>
One item that contains much text even if it isn't really interesting to read all of it:
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor
invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores
</li>
<li>
Another item that contains much text even if it isn't really interesting to read all of it:
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor
invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores
</li>
</ul>

== Numbered lists ==

Numbered (or ordered) list show an ascending number for each item by default. It is possible to change this style, see sections [[#roman numbers]] or [[#latin letters]] for how.

<nowiki><ol>
<li>
One item that contains much text even if it isn't really interesting to read all of it:
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor
invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores
</li>
<li>
Another item that contains much text even if it isn't really interesting to read all of it:
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor
invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores
</li>
</ol></nowiki>

<ol>
<li>
One item that contains much text even if it isn't really interesting to read all of it:
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor
invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores
</li>
<li>
Another item that contains much text even if it isn't really interesting to read all of it:
Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam nonumy eirmod tempor
invidunt ut labore et dolore magna aliquyam erat, sed diam voluptua. At vero eos et
accusam et justo duo dolores
</li>
</ol>

=== roman numbers ===
<ol style="list-style-type: upper-roman;">
<li>
One item
</li>
<li>
Another item
</li>
</ol>

<ol style="list-style-type: lower-roman;">
<li>
One item
</li>
<li>
Another item
</li>
</ol>

=== latin letters ===
<ol style="list-style-type: upper-alpha;">
<li>
One item
</li>
<li>
Another item
</li>
</ol>

<ol style="list-style-type: lower-alpha;">
<li>
One item
</li>
<li>
Another item
</li>
</ol>

== Description lists ==
Here are examples for huge descriptions for terms in definition lists that contain floating images as illustrations and tables.

<nowiki><dl>
<dt>Cent</dt>
<dd>[[File:Test-120x60px.png|right|60px]] The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each. Typically, cents are used to express small intervals, or to compare the sizes of comparable intervals in different tuning systems, and in fact the interval of one cent is too small to be heard between successive notes.

Alexander J. Ellis based the measure on the acoustic logarithms decimal semitone system developed by Gaspard de Prony in the 1830s, at Robert Holford Macdowell Bosanquet's suggestion. Ellis made extensive measurements of musical instruments from around the world, using cents extensively to report and compare the scales employed,[1] and further described and employed the system in his 1875 edition of Hermann von Helmholtz's On the Sensations of Tone. It has become the standard method of representing and comparing musical pitches and intervals. (from: ''[https://en.wikipedia.org/wiki/Cent_(music) Cent (music) - Wikipedia]'') </dd>
<dt>Monzo</dt>
<dd>For example, the interval 15/8 can be thought of as having 5*3 in the numerator, and 2*2*2 in the denominator. This can be compactly represented by the expression 2^-3 * 3^1 * 5^1, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the {{Monzo| ... }} brackets, hence yielding {{Monzo|-3 1 1}}.

:''Practical hint:'' Because the pipe symbol and the greater sign have special meaning in wiki syntax and HTML, there is a helper template ([[Template:Monzo]]) that can be used like this {{Monzo|arguments}} to get the monzo brackets (<code>{{Monzo|arguments}}</code>) from it.

Here are some common 5-limit monzos, for your reference:

{| class="wikitable"
|-
! Ratio
! Monzo
|-
| style="text-align:center;" | [[3/2]]
| {{Monzo| -1 1 0 }}
|-
| style="text-align:center;" | [[5/4]]
| {{Monzo| -2 0 1 }}
|-
| style="text-align:center;" | [[9/8]]
| {{Monzo| -3 2 0 }}
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
|}
</dd>
</dl></nowiki>

<dl>
<dt>Cent</dt>
<dd>[[File:Test-120x60px.png|right|60px]] The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each. Typically, cents are used to express small intervals, or to compare the sizes of comparable intervals in different tuning systems, and in fact the interval of one cent is too small to be heard between successive notes.

Alexander J. Ellis based the measure on the acoustic logarithms decimal semitone system developed by Gaspard de Prony in the 1830s, at Robert Holford Macdowell Bosanquet's suggestion. Ellis made extensive measurements of musical instruments from around the world, using cents extensively to report and compare the scales employed,[1] and further described and employed the system in his 1875 edition of Hermann von Helmholtz's On the Sensations of Tone. It has become the standard method of representing and comparing musical pitches and intervals. (from: ''[https://en.wikipedia.org/wiki/Cent_(music) Cent (music) - Wikipedia]'')</dd>
<dt>Monzo</dt>
<dd>For example, the interval 15/8 can be thought of as having 5*3 in the numerator, and 2*2*2 in the denominator. This can be compactly represented by the expression 2^-3 * 3^1 * 5^1, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the {{Monzo| ... }} brackets, hence yielding {{Monzo|-3 1 1}}.

:''Practical hint:'' Because the pipe symbol and the greater sign have special meaning in wiki syntax and HTML, there is a helper template ([[Template:Monzo]]) that can be used like this <code><nowiki>{{Monzo|arguments}}</nowiki></code> to get the monzo brackets (<code>{{Monzo|arguments}}</code>) from it.

Here are some common 5-limit monzos, for your reference:

{| class="wikitable"
|-
! Ratio
! Monzo
|-
| style="text-align:center;" | [[3/2]]
| {{Monzo| -1 1 0 }}
|-
| style="text-align:center;" | [[5/4]]
| {{Monzo| -2 0 1 }}
|-
| style="text-align:center;" | [[9/8]]
| {{Monzo| -3 2 0 }}
|-
| style="text-align:center;" | [[81/80]]
| {{Monzo| -4 4 -1 }}
|}
</dd>
</dl>

== List templates ==
There are two templates in this wiki for building HTML lists.

; [[Template:List]]
: for creating an unordered list with up to 50 items.
; [[Template:Numlist]]
: for creating a numbered list with up to 50 items.

Both templates were created by [[User:PiotrGrochowski]], please ask him for help and improvements.

== Plaintext and defined lists ==

It's also possible to create lists in plaintext:

<pre>•list<br>
1. list<pre>

which will output itself, minus the br tags. Colons can be used for indentation to create nested lists.

Defined lists define bullets and numbers in images instead of using text:

<pre>[[File:Listitem.png]]list<br>
[[File:1.png]] list</pre>

outputting

[[File:Listitem.png]]list<br>
[[File:1.png]] list
== See also ==
* [[Help:Editing]]
* [https://en.wikipedia.org/wiki/Help:List Help:List - Wikipedia]

[[Category:Help]]
[[Category:todo:expand|?]]

Help:Editing

2018-10-23T14:06:58Z

PiotrGrochowski:

Wikis should be easy to edit, so most text you add or change will appear as you expect it. But you can do more if you want.

[https://wikiindex.org/Xenharmonic_Wiki This wiki] is currently in [https://wikiindex.org/Category:LoginToEdit login-to-edit mode] after the move from [[Wikispaces]] to a MediaWiki-based installation (the edit mode may change again some time). For editing or adding information to the wiki, you have to join it. See [[How to Get Your Xenwiki Account]] for more!

== Using links ==
What you always have dreamed of (for your paper notes) - in a wiki it becomes true: fast links!

If you enclose a word in double brackets (like <code><nowiki>[[word]]</nowiki></code>), you get an internal link, hopefully to an already existing page - if not, it appears red, and you just created the opportunity for your co-authors to add the details. If you want a text differing from the link target, you can place it after a pipe symbol: <code><nowiki>[[word|different text]]</nowiki></code>.

For external links, simply place the URL into the text, certain protocols (http, https, mailto) will be automatically transformed into links. For alternate text, use single brackets, start with the URL and after a space enter the title, like so <code><nowiki>[http://acid3.acidtests.org the Acid3 test]</nowiki></code> which renders to [http://acid3.acidtests.org the Acid3 test].

You may link to sections in the same page by using the section title with the prefix <code>#</code>. MediaWiki automatically converts it into a link with the correct anchor identifier. For example to link to the beginning of this section use <code><nowiki>[[#Using links]]</nowiki></code> (rendered as [[#Using links]]) or with changed text <code><nowiki>[[#Using links|top of this section]]</nowiki></code> (rendered as [[#Using links|top of this section]]).

=== Link text ===
The link text should ideally relate to the link target, try to avoid "[[help:here-links|here]]" links. There is an interesting extension for Firefox, [https://addons.mozilla.org/de/firefox/addon/url2clipboard/ Copy URL To Clipboard], that does automatic formatting of page URL and page title in MediaWiki syntax into the clipboard.

=== Redirects ===
There are 3 types of redirects: soft redirect, regular redirect and hard redirect.

A "soft redirect" or "smart redirect" is just an ordinary page with the text "See XYZ.", where XYZ is a link to the target of the smart redirect. A smart redirect should not point to another smart redirect - if you see this you should fix it by shortcutting past the second one and making the first redirect point directly to the ultimate target. This kind of redirect were introduced in the Wikispaces era, to make redirects accessible later, for example to flip page titles.

A regular redirect is performed by <code><nowiki>#REDIRECT [[Page title]]</nowiki></code>. A double redirect should be fixed be skipping intermediate targets. The page [[Special:DoubleRedirects]] can be used to detect them. Regular redirects did not exist in ''Wikispaces''. This kind of redirect displays a small information message when it was used to get to the current page.

A hard redirect directly leads to the target page without a notice of redirect. Therefore, they should only be used in certain cases like common misspellings. The rule is that if anyone might ever want to flesh the redirect out into its own article, it should not be a hard redirect. Hard redirects can be chained with no problem, which is why they're still appropriate to use for misspellings. The code for a hard redirect in MediaWiki is <code><nowiki>#REDIRECT [[:en:Page title]]</nowiki></code>. Hard redirects worked somewhat differently in <code>wikispaces</code>; only admins could create or change them, and they could not redirect to a section.

see also

* http://groups.yahoo.com/neo/groups/tuning/conversations/topics/101227 - redirects were discussed on the [[tuning_list|tuning list]] in the [[Wikispaces]] era
* [[:Category:soft redirect|list of pages tagged as soft redirect]] - pages that contain only see <tt>[[</tt><nowiki>''some other page''</nowiki><tt>]]</tt> (be cautious with other tags on these)
* [[:Category:smart redirect|list of pages tagged as smart redirect]] - pages that have more information (a small definition, classification etc.)

== Text formatting ==

=== Emphasis ===
Some typographic markup can easy be added by including the phrase in doubled <code><nowiki>''</nowiki></code> or tripled <code><nowiki>'''</nowiki></code> (apostrophe chars):

* ''italic'' using <nowiki>''italic''</nowiki>
* '''bold''' using <nowiki>'''bold'''</nowiki>
* '''''bold italic''''' using <nowiki>'''''bold italic'''''</nowiki>

=== Lists ===
:''See [[Help:List]] for advanced use cases for lists.''
There are tree basic types of lists in HTML that are supported in MediaWiki markup: [[#unordered|unordered]] (or "bulleted") lists, [[#numbered|numbered]] lists, [[#description|description]] lists. Each type of lists can be created within wiki source code by just indicating the items at the beginning of the line (the list is automatically built around them). Each item has to start on a new line. You may also use a different [[list model]] if you want.

==== Unordered ====
Unordered list items are marked by <code>*</code> (asterisk):

* Coffee
* Wine
* Milk

==== Numbered ====
Numbered list items are marked by <code>#</code> (number sign):

# Coffee
# Wine
# Milk

==== Description ====
Description items are marked by <code>;</code> (semicolon) for term and <code>:</code> (colon) for the description:

; Coffee
: a brewed drink prepared from roasted coffee beans. (from: ''[https://en.wikipedia.org/wiki/Coffee Coffee - Wikipedia''])
; Wine
: an alcoholic beverage made from fermented grapes. (from: ''[https://en.wikipedia.org/wiki/Wine Wine - Wikipedia]'')
; Milk
: a white liquid nutrient-rich food produced by the mammary glands of mammals. (from: ''[https://en.wikipedia.org/wiki/Milk Milk - Wikipedia]'')

==== Combined List Styles ====

# Morning
#* Coffee
#* Milk
# Evening
#* Coffee
#* Wine

=== Tables ===
:''for advanced examples see [[Help:Table]]''

You can add tables to wiki pages:
{| class="wikitable"
! what you write
|
<nowiki>
{| class="wikitable"
|+ The Title
! head A
! head B
|-
| cell A1
| cell B1
|-
| cell A2
| cell B2
|}</nowiki>
|-
! what you get
|

{| class="wikitable"
|+ The Title
! head A
! head B
|-
| cell A1
| cell B1
|-
| cell A2
| cell B2
|}

|}

=== Images ===
Inserting a JPEG, GIF, or PNG image is pretty straightforward - when using the visual page editor click the "File" button and upload the image. Or in Wikitext, use <code><nowiki>[[image:filename]]</nowiki></code> or <code><nowiki>[[file:filename]]</nowiki></code>. Also SVG images are supported (<code><nowiki>[[file:some.svg]]</nowiki></code>), they are automatically transformed into PNG images by the MediaWiki software, because not all Browsers come with native SVG support. A possible problem can arise when the image uses fonts that are not available on the wiki server. To prevent such errors, you can transform text into graphic objects before uploading the image here.

=== Maths formulas ===
...can be used [http://en.wikipedia.org/wiki/Help:Displaying_a_formula like in the Wikipedia]: start it with <code><nowiki><math></nowiki></code> and end it with <code><nowiki></math></nowiki></code>. Take care: using the math tags will break paragraphs. See for example

<math>V =\frac{4}{3} \pi r^3</math>

Here is a browser-based formula editor that may be helpful: http://www.codecogs.com/latex/eqneditor.php

=== Table of contents ===
Place <code><nowiki>__TOC__</nowiki></code> where you wish to place the table of contents, mostly this will be at the top of the page.

== Social aspects ==
In this wiki, you can use talk pages and [[user talk]] pages.

If you want to discuss the wiki itself, do it on the page [[wikifuture]].

=== Categories ===
Categories all serve different purposes: sometimes they say what something is or the context in which it belongs ([[:Category:edo]], [[:Category:interval]]). Sometimes they are used for annotation purposes ([[:Category:todo:reduce mathslang]]), etc. All these forms have in common is that they can combine wiki pages in a way that is independent of their authors.

A category may have its own content (and category)
:''see [https://en.xen.wiki/w/Special:AllPages?from=&to=&namespace=14 Pages in the namespace '''Category''']''
or not yet (despite being used to categorize other pages):
:''see [[Special:WantedCategories]]''

Some more examples:

* [[:Category:todo:add_definition|todo:add definition]] - here a term needs to be defined
* [[:Category:Todo]] - category of categories that flag articles

=== Discussion ===
Discussion pages are provided to each content page. Please keep in mind that the email notification to people who are observing the page cannot be taken back!

Sometimes you will wait forever for a reply because nobody monitors the page or topic. In this case you should find out who is most familiar with the topic and ask that person directly. You can check the history of pages for its authors. Another approach is to ask questions on further media, examples are listed on the page [[GeneralDiscussion]].

== Links ==
* [[Help:Help]]
* [[Conventions]]

[[Category:help]]
[[Category:syntax]]
[[Category:wiki help]]

User talk:PiotrGrochowski

2018-10-16T08:35:03Z

PiotrGrochowski: /* last warning */

User talk:PiotrGrochowski

2018-10-16T08:19:58Z

PiotrGrochowski:

User:PiotrGrochowski/User:PiotrGrochowski

2018-10-16T07:34:16Z

PiotrGrochowski:

<div><big>''Do not ban this user''</big></div>

<div><big>''He did nothing wrong''</big></div>

User:PiotrGrochowski/User:PiotrGrochowski

2018-10-16T07:33:55Z

PiotrGrochowski:

<div><big>''Do not ban this user''</big></div>

<div><big>''He did nothing wrong''</big></div>



Will write down the commas of 15–odd–limit below

{{list
|[[16/15]], [[15/8]]
|[[15/14]], [[28/15]]
|[[14/13]], [[13/7]]
|[[13/12]], [[24/13]]
|[[12/11]], [[11/6]]
|[[11/10]], [[20/11]]
|[[10/9]], [[9/5]]
|[[9/8]], [[16/9]]
|[[8/7]], [[7/4]]
|[[15/13]], [[26/15]]
|[[7/6]], [[12/7]]
|[[13/11]], [[22/13]]
|[[6/5]], [[5/3]]
|[[11/9]], [[18/11]]
|[[16/13]], [[13/8]]
|[[5/4]], [[8/5]]
|[[14/11]], [[11/7]]
|[[9/7]], [[14/9]]
|[[13/10]], [[20/13]]
|[[4/3]], [[3/2]]
|[[15/11]], [[22/15]]
|[[11/8]], [[16/11]]
|[[18/13]], [[13/9]]
|[[7/5]], [[10/7]]
}}

<pre>16/15 and 15/14 — '''225/224'''
16/15 and 14/13 — '''105/104'''
16/15 and 13/12 — '''65/64'''
16/15 and 12/11 — '''45/44'''
16/15 and 11/10 — '''33/32'''
16/15 and 10/9 — '''25/24'''
16/15 and 9/8 — '''135/128'''
15/14 and 14/13 — '''196/195'''
15/14 and 13/12 — '''91/90'''
15/14 and 12/11 — '''56/55'''
15/14 and 11/10 — '''77/75'''
15/14 and 10/9 — '''28/27'''
15/14 and 9/8 — '''21/20'''
14/13 and 13/12 — '''169/168'''
-->

Xenharmonic Wiki:Conventions

2018-10-16T07:30:07Z

PiotrGrochowski: Undo revision 37103 by Xenwolf (talk)

A few conventions have evolved in this wiki that would be good to keep consistent with. Any pages that break the conventions may have links to them removed.

(please feel free to add rules in use here, and please provide reasons for it! Thanks!)

== Decimal numbers ==
Use the English convention with decimal point, when articles are in English language (which is mostly the case). Do not misspell numbers.

{| class="wikitable"
! correct
! wrong
|-
| style="background-color: #efe" | 701.995
| style="background-color: #fee" | <s>702,995</s>
|}

==Logarithmic interval measures ==
For normal tables, we decided to stick to [[cent|cent]] values with a fixed precision, mostly 2 or 3 in the fractional part. 3 is helpful when serving as a canonical reference point for people to compare their own calculations with, which helps in the development of table creators such as [[Piotr Grochowski]]'s [https://scratch.mit.edu/projects/249278782/ one for Selected just intervals by error]. Its main purpose is to give an idea of interval sizes and relation to known intervals in the [[12edo|12edo]] system of western music. Alternative measures are acceptable if they better reflect the idea behind an interval system, scale etc. More than one logarithmic size column should be avoided.

== Links ==
Links - both wiki-internal and external ones - should have meaningful text; something like "[[Conventions|here]]" is not very useful for understanding and orientation, often the page title is a good starting point.

: ''For more info see [[Help:Here-links]]''

==[[List model]]==

Any [[list model]] may be used according to preference, however an edit consisting only of a list model change should not be done, to prevent edit wars. Do not promote any particular [[list model]] to users.

== See also ==
* [[Help:Help|Wiki help]]

[[Category:about]]
[[Category:conventions]]
[[Category:help]]

User:PiotrGrochowski/User:PiotrGrochowski

2018-10-16T07:26:09Z

PiotrGrochowski:

<div><big>''Do not ban this user''</big></div>

For information about me, see [[Editor PiotrGrochowski]]

<span style="font-family: 'Consolas';">'''DO NOT MOVE [[Editor PiotrGrochowski]] HERE. THIS IS A PERSONAL [[SandBox]].'''</span>



Will write down the commas of 15–odd–limit below

{{list
|[[16/15]], [[15/8]]
|[[15/14]], [[28/15]]
|[[14/13]], [[13/7]]
|[[13/12]], [[24/13]]
|[[12/11]], [[11/6]]
|[[11/10]], [[20/11]]
|[[10/9]], [[9/5]]
|[[9/8]], [[16/9]]
|[[8/7]], [[7/4]]
|[[15/13]], [[26/15]]
|[[7/6]], [[12/7]]
|[[13/11]], [[22/13]]
|[[6/5]], [[5/3]]
|[[11/9]], [[18/11]]
|[[16/13]], [[13/8]]
|[[5/4]], [[8/5]]
|[[14/11]], [[11/7]]
|[[9/7]], [[14/9]]
|[[13/10]], [[20/13]]
|[[4/3]], [[3/2]]
|[[15/11]], [[22/15]]
|[[11/8]], [[16/11]]
|[[18/13]], [[13/9]]
|[[7/5]], [[10/7]]
}}

<pre>16/15 and 15/14 — '''225/224'''
16/15 and 14/13 — '''105/104'''
16/15 and 13/12 — '''65/64'''
16/15 and 12/11 — '''45/44'''
16/15 and 11/10 — '''33/32'''
16/15 and 10/9 — '''25/24'''
16/15 and 9/8 — '''135/128'''
15/14 and 14/13 — '''196/195'''
15/14 and 13/12 — '''91/90'''
15/14 and 12/11 — '''56/55'''
15/14 and 11/10 — '''77/75'''
15/14 and 10/9 — '''28/27'''
15/14 and 9/8 — '''21/20'''
14/13 and 13/12 — '''169/168'''

Editor PiotrGrochowski

2018-10-16T07:25:26Z

PiotrGrochowski:

[[Editor PiotrGrochowski]] is a 13 years old boy, and lives in Poland.

Sleep time: 21:30 (local time, which is CET or CEST)<br>
Wake up time: 7:00 (local time)

Diagnosed with: nearsightedness, Asperger's syndrome

Signature:
[[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]])

According to him, unlike the organizers, PiotrGrochowski is an excellent editor.

== Controversial conventions he uses ==

<ul><li>The use of real superscripts, ¹²³, as opposed to fake ones, <sup>1</sup><sup>2</sup><sup>3</sup></li><li>The use of standard [[list model]]</li></ul>

see [[Editor yourName]]

[[Category:Wiki editor|PiotrGrochowski]]

User:PiotrGrochowski/Interval summary

2018-10-16T07:09:54Z

PiotrGrochowski: don't be evil

== [[PiotrGrochowski]]'s summary ==

{| class="wikitable
! interval
! description
|-
| unison
| [[1/1]], or 0 cents in any tuning
|-
| augmented unison
| 28/27, 25/24, 21/20, 135/128
|-
| minor second
| 16/15, 15/14, 27/25
|-
| major second
| 10/9, 9/8
|-
| fifth
| The '''perfect fifth''' is [[3/2]], or 701.955 cents.<br>1/4 comma meantone: 696.578 cents<br>[[12edo]]: 700 cents<br>[[17edo]]: 705.882 cents<br>[[19edo]]: 694.737 cents<br>[[22edo]]: 709.091 cents
|}

Below is me trying to name all the [[15-odd-limit]] intervals.

[[10/9]], 182.404 cents – lesser major second [[9/5]]

[[9/8]], 203.910 cents — greater major second [[16/9]]

[[8/7]], 231.174 cents — septimal supermajor second [[7/4]]

[[7/6]], 266.871 cents — septimal subminor third [[12/7]]

[[6/5]], 315.641 cents — perfect minor third [[5/3]]

[[5/4]], 386.314 cents — perfect major third [[8/5]]

[[9/7]], 435.084 cents — septimal supermajor third [[14/9]]

[[4/3]], 498.045 cents — perfect fourth

[[7/5]], 582.512 cents — septimal augmented fourth

[[10/7]], 617.488 cents — septimal diminished fifth

[[3/2]], 701.955 cents — perfect fifth

== Xenwolf's summary ==

== Mike Battaglia's summary ==

== [[User:Diatonic, Chromatic, Enharmonic, Subchromatic|Diatonic, Chromatic, Enharmonic, Subchromatic]]'s summary ==

<pre>0 cents unison
20 cents potato unison
40 cents xencomma
60 cents potato augmented unison
80 cents augmented unison
100 cents semitone
120 cents minor second
140 cents xeneutral second
160 cents xeneutral second II
180 cents potato major second
200 cents major second</pre>

== Dave Keenan's summary ==

== Spt3125's summary ==

== Keenan Pepper's summary ==

== Igliashon's summary ==

== TallKite's summary ==

== Gareth.hearne's summary ==

== Margo Schulter's summary ==

Talk:FKH Extended-diatonic Interval Names

2018-10-16T07:09:20Z

PiotrGrochowski: Created page with "Please, move this to your section in Interval summary. ~~~~"

Please, move this to your section in [[Interval summary]]. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]) 07:09, 16 October 2018 (UTC)

User:Lhearne/Extra-Diatonic Intervals

2018-10-16T07:07:49Z

PiotrGrochowski:

Today a small under of competing interval naming schemes exist for the description of microtonal music. More common than any particular defined standard are certain tendencies for microtonal interval naming, or names for specific intervals. While risking the creation of simply another competing standard, an effort is made to develop a scheme that is able to take the best aspects of the existing standards and apply them in a formal interval naming system built on common undefined practice. Such a system is developed, where in addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', only the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees. Using this system all intervals in three fifths of all [[Equal division of the octave|edo]]<nowiki/>s up to 50 can be named such that 'S' and 's' correspond to a displacement of an interval up or down a single interval of the edo, respectively. Many commonly used [[MOS scale|MOS scales]] may also be described using this scheme such that these scales' interval names are consistent expression in any tuning that supports them. The resultant scheme can also be easily mapped to any of the current naming standards, and may even facilitate translation between. The resulting scheme should improve pedagogy and communication in microtonal music.

== Background ==

=== The origin of diatonic interval names ===
[[File:Mesopotamian interval names table.jpg|thumb|500x500px|Mesopotamian interval names, from http://www.historyofmusictheory.com/?page_id=130, accessed October 7, 2018.]]
Music theory describing the use of heptatonic-diatonic scales, including interval names, has been traced back as far as 2000BC, deciphered from a Sumerian cuneiform tablet from Nippur by Kilmer (1986). From Kummel (1970) we know that 'the names given to the seven tunings/scales were derived from the specific intervals on which the tuning procedure started' (Kilmer, 1986). This formed the basis of their musical notation ([http://www.jstor.org/stable/985853. Kilmer, 2016]). The table to the right following table displays the Ancient Mesopotamian interval names accompanied by their modern names.

Kilmer also writes that 'the ancient Mesopotamian musicians/“musicologists” knew what we call today the Pythagorean series of fifths, and that the series could be accomplished within a single octave by means of “inversion.” '. The Mesopotamian's music and theory was passed down through the Babylonians and the Assyrians to the Ancient Greeks, as well as their mathematics, particularly concerning musical and acoustical sound [[ratios]] (Ibid, [http://math-cs.aut.ac.ir/~shamsi/HoM/Hodgkin%20-%20A%20History%20of%20Mathematics%20From%20Mesopotamia%20to%20Modernity.pdf Hodgekin, 2005]).

Such mathematical and musical ideas are attributed to Pythagoras, who undoubtedly made them popular., although many scholars suggest he may have learned these ideas from his Babylonian and Egyptian mentors. None the less, Pythagoras' idea that that by dividing the length of a string into ratios of halves, thirds, quarters and fifths created the musical intervals of an octave, a perfect fifth, an octave again, and a major third form the basis of Ancient Greek music theory (http://www.historyofmusictheory.com/?page_id=20). His tuning of the diatonic scale by only octaves and perfect fifths ([[Pythagorean tuning]]) is influential through to today.

=== Ancient Greek interval names ===
Intervals in Ancient Greek music were written either as string length ratios, after Pythagoras, or as positions in a [[tetrachord]].

[[2/1]], the [[octave]], was named ''diapason'' meaning ''<nowiki/>'''through all [strings]'

[[3/2]], the [[perfect fifth]] was labelled ''diapente,'' meaning 'through 5 [strings]'

[[4/3]], the [[perfect fourth]], was labelled ''diatessaron'', meaning 'through 4 [strings]'

''dieses,'' 'sending through', refers to any interval smaller than about 1/3 of a perfect fourth

''tonos'' referred both to the interval of a whole tone, and something more akin to [[mode]] or key in the modern sense ([http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf Chalmers, 1993])

''ditone'' referred to the interval made by stacking two [[9/8]] whole tones, resulting in [[81/64]], the Pythagorean major third. ([[Joe Monzo|Monzo]], http://www.tonalsoft.com)

[[256/243]] - the ''limma'', which is the ratio between left over after subtracting two 9/8 tones (together making a ditone) a perfect fourth, the ''diatonic semitone'' of the Pythagorean diatonic scale

[[2187/2048]] - the ''apotome'', which is the ratio between the tone and the limma, the ''chromatic semitone'' of the Pythagorean diatonic scale

The Ancient greek term ''diatonon'', meaning 'through tones', refers to the genus with two whole tones and a semitone, or any genus in which no interval is greater than one half of the fourth (Chalmers, 1993). The Pythaogrean diatonic scale is the scale that may be built from one two Pythagorean tetrachords, and the left over interval of 9/8.

=== Zarlino and Meantone ===
[[File:Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1.png|thumb|566x573px|''Le institutioni harmoniche,'' Zarlino, 1558, Cap. 15: Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1, pg. 25.]]
Intervals were referred to by the Ancient Greek names through the the 18th century, as Latin names. By the Renaissance it had been discovered that a Pythagorean diminished fourth sounded sweet, and approximated the string length ratio 5/4. This just tuning for the major third was sought after, along with the complementary 6/5 tuning for the minor third, and octave complements to both - 8/5 for the minor sixth and 5/3 for the major sixth. Influential Italian music theorist and composer Gioseffo Zarlino put forth that choirs tuned the diatonic scale to the tuning built from this tetrachord, the ''intense diatonic scale'', also known as the ''syntonic or syntonus diatonic scale'' or the ''Ptolemaic sequence'':

1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

Included in this scale, however, were ''wolf intervals:'' imperfect consonances that occurred as tunings of the same interval as perfect consonances. For example, between 1/1 and 3/2, 4/3 and 1/1, 5/3 and 5/4; and 5/4 and 15/8 occurs the perfect fifth, 3/2, whereas between 9/8 and 5/3 occurs the wolf fifth, 40/27, flat of 3/2 by 81/80. This was also the interval by which four 3/2 fifths missed 5/1 (the interval two octaves above 5/4). It was named the ''syntonic comma'' after Ptolemy's ''syntonus'' or ''intense diatonic tetrachord'' which consists of the intervals 9/8, 10/9 and 16/15, where 9/8 and 10/9 differ by this interval. By making the syntonic comma a unison the wolf fifth could be made a perfect fifth. It was discovered that this could be achieved by flattening (tempering) the perfect fifth by some fraction of this comma such that four of these fifths less two octaves gave an approximation of 5/4. Where two fifths less an octave give 9/8, the next two add another 10/9 to result in the 5/4. 9/8 and 10/9 were referred to as the ''major tone'' (''tunono maggiore'') and ''minor tone'' (''tunono minore''), respectively, and where this tuning led to them being equated, it was referred to as Meantone temperament, which is said to 'temper out' the syntonic comma. Zarlino advocated the flattening of the fifth by 2/7 of a comma, leading to 2/7-comma Meantone, but also described 1/3-comma and 1/4-comma Meantone as usable (Zarlino, 1558).

The diagram on the right, from Zarlino's 1558 treatise ''Le istitutioni harmoniche'' associates many intervals with their tuning as perfect consonances. The perfect tuning for the ditone was considered then to be 5/4, rather than 81/64. The interval for which 6/5 is considered a perfect tuning was referred to as a ''semiditone'' (labelled also in ''Le istitutioni harmoniche'' by as ''Trihemituono)''. This may seem odd to us now, but in Latin 'semi' referred not to 'half', but to 'smaller', so 'semiditone' translated to something like 'smaller ditone'. Additionally 'semitone' referred to the interval smaller than the 'tone'. Like the tone, this interval possessed two alternative perfect tunings: 16/15, the difference between 15/8 and 2/1, or 5/4 and 4/3, and 25/24, the difference between 6/5 and 5/4. 16/15 was referred to as the ''major semitone'' (''semituono maggiore'') and 25/24 as the ''minor semitone (semituono maggiore'').

In addition to the Latin interval names, derived from the Ancient Greek interval names, we see on the diagram a single interval name in Italian: ''Essachordo maggiore'', referring to the ratio 5/3, which we are tempted to translate to 'major sixth'. Chapter 16, ''Quel che sia Consonanze semplice, e Composta; & che nel Senario si ritouano le sorme di tutte le somplici consonanze; & onde habbia origine l'Essachordo minore'', puts forward that the ''Essachordo minore,'' or perhaps 'minor sixth' be tuned to 8/5.

In the 1691 ''Lettre de Monsieur Huygens à l'Auteur [Henri Basnage de Beauval] touchant le Cycle Harmonique,'' theorist Christiaan Huygens gave names and ratios to common intervals and mapped them to 31-tET, which very closely approximates 1/4-comma Meantone. Translated from French, 3/2 was labelled a Fifth, 4/3 a Fourth, 5/4 a major Third, 6/5 and minor Third, 5/3 a major Sixth and 8/5 a minor Sixth. Here we really begin to see today's interval names.

=== English interval names in the Baroque ===
[[File:Harmonics, or The Philosophy of Musical Sounds, Section 2 figure 3.png|thumb|517x548px|''Harmonics, or The Philosophy of Musical Sounds'', Edition 2, Smith, 1759, Section 2: On the Names and Notation of consonance and their intervals, Fig. 2 & 3 , pg. 10]]
After English superseded Latin as the the main language of scholarship, the Latin interval names were rejected and the convention we saw in Zarlino's Italian for naming the smaller of a pair of sizes of an interval 'minor' and the larger 'major' was further applied.

Music theorist and mathematician Robert Smith provides the diagram and table on the right in his 1749 ''Harmonics, or, The Philosophy of Musical Sounds,'' with the description: <blockquote>'Fig. 2. If a musical string ''CO'' and it's parts ''DO'', ''EO'', ''FO'', ''GO'', ''AO'', ''BO'', ''cO'', be in proportion to one another as the numbers 1, 8/9, 4/5, 3/4, 2/3, 3/5, 8/15, 1/2, their vibrations will exhibit the system of 8 sounds which musicians donate by the letters ''C'', ''D'', ''E'', ''F'', ''G'', ''A'', ''B'', ''c''.</blockquote><blockquote>Fig. 3. And supposing those strings to be ranged like ordinates to a right line ''Cc'', and their distances ''CD'', ''DE'', ''EF'', ''FG'', ''GA'', ''AB'', ''BC'', not to be the differences of their lengths, as in fig. 2. but to be the magnitudes proportional to the intervals of their sounds, the received Names of these intervals are shewn in the following Table; and are taken from the numbers of the strings or sounds in each interval inclusively; as a Second, Third, Fourth, Fifth, &c, with the epithet of ''major'' or ''minor'', according as the name or number belongs to a greater of smaller total interval; the difference of which results chiefly from the different magnitudes of the major and minor second, called the Tone and Hemitone.'</blockquote>We note that Smith uses the tuning of the diatonic scale that Zarlino put forward: the Ptolemaic Sequence.

We may be surprised to see 4:3 here labelled as a minor Fourth, and 3/2 as a major Fifth, but it is obvious that this naming is more consistent than today's. Smith adds that <blockquote>'Any one of the ratios in the third column of the foregoing Table, except 80 to 81, or any one of them compounded once of oftener with the ratio 2 to 1 or 1 to 2, is called a Perfect ratio when reduced to it's least terms. And when the times of the single vibrations of any two sounds have a perfect ratio, the consonance and it's interval too after called Perfect; and is called Imperfect or Tempered when that perfect ratio and interval is a little increased or decreased.'</blockquote><blockquote>'Any small increment of decrement of a perfect interval is called respectively the Sharp or Flat Temperament of the imperfect consonance, and is measured most conveniently by the proportion it bears to the comma'</blockquote>Therefore in this system 3/2 is the ''Perfect major Fifth'' and 5/4 the ''Perfect major Third''. 81/64 might be labelled a ''comma sharp major Third'', 32/27 a ''comma flat minor Third'', and the 1/4-comma Meantone fifth a ''1/4-comma flat major Fifth''. The interval naming scheme Smith describes may be immediately applied to 5-limit microtonal systems. There is an inconsistency, however, where it seems that 9/8 should be called a ''Perfect major Second,'' but that, while 9/5 be named a ''comma sharp minor Seventh'', it's inverse, 10/9, is a ''Perfect minor Tone.''

In ''An Elementary Treatise on Musical Intervals and Temperament,'' published in 1876'','' R. H. M Bosanquet refers to 5/4 as the ''perfect third'', and 81/64 as the P''ythagorean third''. Bosanquet also labels other intervals of the Pythagorean diatonic scale similarly, i.e. 256/243, the limma, is labelled the ''Pythagorean semitone'', and 27/16 the ''Pythagorean sixth''. 81/80 is labelled the ''ordinary comma'', or simple the ''comma,'' and the Pythagorean comma is defined as the difference between twelve fifths and seven octaves. The apotome of 2187/2048 is referred to as Apatomè Pythagoria. The following relationships are then described: (<span style="color:#FF0000">'''<big>the image is too inappropriate to show, see pg 28 for rules regarding posting images to the Internet</span>'''</big>)

=== Helmholtz and Ellis ===
[[File:Helmholtz consonances table.png|thumb|617x469px|Table describing the influence of the different consonances on one another, up to the 9th partial, from ''On the Sensations of Tone as a Phycological Basis for Theory of Music'', Helmholtz, 1863, Translation by Ellis, 1875, pg. 187|]]
Through the investigations of Galileo (1638), Newton, Euler (1729), and Bernouilli (1771), theorist Hermann von Helmholtz was aware that ratios governing the lengths of strings existed also for the vibrations of the tones they produced. His investigation of the harmonic series associated with these ratios of vibration led him to the consideration of ratios above the 5-limit. In his seminal ''On the Sensations of Tone as a Physiological Basis for the Theory of Music'', published in German in 1863 and translated into English in 1875 by Alexander Ellis, he listed just intervals as show in the table to the right. It is interesting to note that 8:9 is labelled a 'Second' rather than a 'major Second'. The minor Seventh is shown as 5:9 rather than as 9:16 seemingly because of the 9 partial limit imposed on the table. It is also worth noting that 5:7 is labelled a subminor Fifth. 'Super', indicated in notation with a '+', raises an interval by 35:36, the septimal quarter tone, and 'sub', indicated with a '-' lowers by the same interval with the exception of the Supersecond 7:8, which lies 63:64, the septimal comma above the Second. The subminor fifth is not included in this as no minor Fifth is shown. If we assume that 'sub' lowers an interval 35:36, then the minor Fifth would be 18:25, 80:81 above Smith's 45:64 minor Fifth, however in table 2 below, Ellis labels 18:25 a ''superfluous Fourth'', and it's inverse, 25:32, an ''acute diminished Fifth'', whilst 64:45 is labelled a ''diminished Fifth'' and its inverse 32:45 a ''false Fourth or Tritone.'' If we label 9:10 as a 'major Second', and 7:8 as a 'supermajor Second' then they differ by 35:36, the major Second is the inverse of the minor Seventh, and the supermajor Second is the octave inverse of the subminor Seventh. Perhaps 'major' has been left off name of the major Fifth, and minor off the name of the minor Fourth since the time of Smith. We can add to this table the remaining octave inversions as well as the super Fourth and sub Fifth.
{| class="wikitable"
|+Table 1. Additional Intervals
!Intervals
!Notation
!Ratio
!Cents in the interval
|-
|Major Second
|D
|9:10
|182
|-
|Sub Fourth
|F-
|16:21
|471
|-
|Supermajor Fourth
|F#+
|7:10
|617
|-
|Super Fifth
|G+
|21:32
|729
|-
|Subminor Sixth
|A♭-
|9:14
|765
|-
|Supermajor Sixth
|A+
|7:12
|933
|}
Helmholtz defined the ''perfect consonances'' as the Octave, Twelfth and Double Octave as well as Fourth and Fifth. The major Sixth and major Third are next called ''medial consonances'', considered to in the era of Pythagorean tuning to be ''imperfect consonances'', which Helmholtz defined instead to be the minor Third and the minor Sixth.

Regarding the tuning of the intervals however, those corresponding to simple ratios of vibration are, as in Smith, referred to as perfect, however hey are also described as 'justly-intoned', or by Ellis as 'just'. The perfect tuning for the semitone is listed as 16/15, or 182c. The perfect tunings are compared to the Pythagorean tunings, where the Pythagorean tuning of the major Third and sixth are described as 81/80 above the perfect tunings, and of the minor Third, minor Sixth and semitone to be 81/80 below the perfect tunings. Helmholtz notes that the Pythagorean tunings are closer to the equal tempered tunings than the perfect tunings. Helmholtz also describes the Pythagorean Tritone as of 612c.

Ellis includes an [[Ellis' interval table|additional table]] providing names for many different just and tempered intervals, perfect and imperfect. The interval names do not appear to follow any sort of consistent naming system, rather intervals seem to be named case-by-case. They also largely do not correspond to the interval names used by Helmholtz.

In ''Notes on the Observations of Musical Beats'', Proc. R. Soc. Lond. 1880, Ellis named many just intervals of the 7-limit (including 3 and 5-limit intervals):<blockquote>‘Fifth 3:2, Fourth 4:3, Major Third 5:4, Minor Third 6:5, Major Sixth 5:3, Sub-Fifth 7:5, Super-Fourth 10:7, Super-major Third 9:7, Sub-minor Sixth 14:9, Sub-minor Third 7:6, Super-major Sixth 12:7, Sub-minor or Harmonic Seventh 7:4, Super-major Second 8:7, Major Tone 9:8, Minor Tone 10:9, Small Major Seventh 9:5 and Diatomic (sic.) Semitone 16:15’</blockquote>He later lists ‘The Major Sevenths 16:9 and 15:8’. The labeling of 16:9 as a Major Seventh and 9:5 as a Small Major Seventh is interesting and at odds with Smith's interval names. Given that 9:5 is larger than 16:9, and no Minor Seventh is mentioned, we can assume 16:9 was mislabeled as a Major Seventh and was understood to be a Minor Seventh, as referred to by Smith.

There is an inconsistency associated with the labeling of 9:5 as a Small Major Seventh also, as it lies a 3:2 Fifth above the 6:5 Minor Third, and we know a fifth and a minor third when added together to give a minor, rather than major seventh.

Ellis here uses the name for 8:7 we suggested above, Super-major Second, and includes our suggested Sub-minor Sixth and Super-major Sixth, however rather than Subminor Fifth and Supermajor Fourth, 7:5 and 10:7 are labelled Sub-Fifth and Super-Fourth, where in this instance sub and super are seen to raise and lower by 21:20 instead of by 36:35.

Ellis, in a footnote to his translation of Helmholtz,'s treatise also provides names for a single 11-limit interval. The interval 22:27, of 355c, introduced by Zalzal, says Ellis was termed a ''neutral Third'' by Herr J. P. N. Land originally in ''Over de Toonladders der Arabische Musiek'' (On the Scales of Arabic Music) in 1880. An interval a fourth higher than this is mentioned, but a ratio is not given, and it is not named. We can ourselves however find it's ratio as 11:18, and guess it's name to be a ''neutral Sixth'', given that it lies a perfect Fourth above the neutral Third. Following a similar process as in our completion of Helmholtz table above, and assuming that the octave inverse of a neutral Third should be a neutral Sixth we may introduce the following 11-limit intervals that see common use among music theorists and microtonal musicians through to today:
{| class="wikitable"
|+Table 2. 11-limit intervals
!Intervals
!Notation
!Ratio
!Cents in the interval
|-
|Neutral Second
|D♭^
|81:88
|143
|-
|Neutral Second
|Dv
|11:12
|151
|-
|Neutral Third
|E♭^
|9:11
|347
|-
|Neutral Third
|Ev
|22:27
|355
|-
|Neutral Sixth
|A♭^
|27:44
|845
|-
|Neutral Sixth
|Av
|11:18
|853
|-
|Neutral Seventh
|B♭^
|6:11
|1049
|-
|Neutral Seventh
|Bv
|44:81
|1057
|}
Each interval name has two sizes that differ by the comma 242:243. The notation included in the table is from HEWM notation, developed as an extension to the Helmholtz-Ellis use of '+' and '-' by Joe Monzo (http://www.tonalsoft.com/enc/h/hewm.aspx<nowiki/>).'^' indicates raising 'v' a lowered of 33/32. In HEWM notation '+' and '-' are refined to mean raising and lowering of 81/80 respectively and '>' and '<' are added instead to indicate raising and lowering of 64/63. Letter names correspond instead of the the Ptolemaic sequence, as in Smith's and Helmholtz' descriptions, but to a Pythagorean tuning of the diatonic scale, where '#' and '♭' and respectively raise and lower the apotome, 2187/2048. HEWM notation is not accompanied by an interval naming system.

=== Common interval names today ===
These interval names are used by theorists and microtonal musicians today, though 7:5 and 10:7 are given many different names, today also considered to be an augmented fourth and diminished fifth, lesser septimal tritone and greater septimal tritone, or simply as tritones. The fourth and fifth are today called perfect fourth and perfect fifth, and Smith's major Fourth and minor Fifth referred to as augmented fourth and diminished fifth respectively. As can be seen in Tchaikovsky's ''A Guide to the Practical Study of Harmony,'' by the beginning of end of the 19th century the familiar conventions for the naming of intervals were set, wherein
* Seconds, thirds, sixths and sevenths appear in the diatonic in two sizes, the larger labelled 'major' and the smaller, 'minor'.
* Major, when raised by a semitone, becomes 'augmented', and minor, lowered by a semitone, 'diminished'.
* The smaller of the two sizes of fourth and the larger of the two sizes of fifth are labelled 'perfect', along with the unison and octave.
* A perfect interval, when raised a semitone is labelled 'augmented', and when lowered a semitone, 'diminished'.

=== Fokker/Keenan Extended-diatonic interval-names ===

Considering the 11-limit otonal chord 4:5:6:7:9:11 a chain of thirds, in addition to the familiar major, minor, subminor, supermajor and neutral thirds, Dave Kennan labelled 5:7 a sub-diminished fifth and 7:11 an augmented fifth. 7:10, the inversion of 5:7, is labelled a diminished. 5:7, therefore, is also an augmented fourth. In terms of sevenths, 4:7 is subminor, 5:9 is minor and 11:6 is neutral.

From this, Keenan defines a consistent interval naming system, meaning one which obeys diatonic interval arithmetic (In each column, the parenthesised prefix is the one that is implied when there is no prefix). When adding intervals the indexes are added together to give the index of the resulting interval. Keenan also adds corrections for each interval class to the indexes in order to account for inconsistencies that occur within diatonic interval arithmetic when concerning intervals greater than an octave, so that his system, unlike regular diatonic interval names, may be completely consistent.

As can be seen above, sub, super, augmented and diminished have also carried inconsistent meaning historically, where in Keenan's system they always alter intervals by the same amount.
{| class="wikitable"
|+
Table 3. Fokker/Keenan Extended-diatonic interval-names indexes
!Index
!Prefix for unisons, fourths, fifths, octaves
!Prefix for seconds, thirds, sixths, sevenths, ninths
|-
| -4
|double diminished
|subdiminished
|-
| -3
|subdiminished
|diminished
|-
| -2
|diminished
|subminor
|-
| -1
|sub
|minor
|-
|0
|(perfect)
|neutral
|-
|1
|super
|(major)
|-
|2
|augmented
|supermajor
|-
|3
|superaugmented
|augmented
|-
|4
|double augmented
|superaugmented
|}
The index values correspond most directly to degrees of 31-tET, whose interval names by this method are given in the following table:
{| class="wikitable"
|+Table 4. Fokker/Keenan Extended-diatonic interval-names in 31-tET
!31-tET degree
!Ratios
!Names
!
|-
|0
|1:1
|
|unison
|-
|1
|48:49 44:45 36:35 33:32
|(diminished second)
|super unison
|-
|2
|27:28 24:25 20:21
|subminor second
|(augmented unison)
|-
|3
|15:16 14:15
|minor second
|
|-
|4
|11:12 10:11
|neutral second
|
|-
|5
|9:10 8:9
|major second
|
|-
|6
|7:8
|supermajor second
|(diminished third)
|-
|7
|6:7
|(augmented second)
|subminor third
|-
|8
|5:6
|
|minor third
|-
|9
|9:11
|
|neutral third
|-
|10
|4:5
|(subdiminished fourth)
|major third
|-
|11
|11:14 7:9
|(diminished fourth)
|supermajor third
|-
|12
|16:21
|subfourth
|(augmented third)
|-
|13
|3:4
|perfect fourth
|
|-
|14
|8:11
|super fourth
|
|-
|15
|5:7
|augmented fourth
|(subdiminished fifth)
|-
|16
|7:10
|(superaugmented fourth)
|diminished fifth
|-
|17
|11:16
|
|sub fifth
|-
|18
|2:3
|
|perfect fifth
|-
|19
|21:32
|(diminished sixth)
|super fifth
|-
|20
|9:14 7:11
|subminor sixth
|(augmented fifth)
|-
|21
|5:8
|minor sixth
|(superaugmented fifth)
|-
|22
|11:18
|neutral sixth
|
|-
|23
|3:5
|major sixth
|
|-
|24
|7:12
|supermajor sixth
|(diminished seventh)
|-
|25
|4:7
|(augmented sixth)
|subminor seventh
|-
|26
|9:16 5:9
|
|minor seventh
|-
|27
|11:20 6:11
|
|neutral seventh
|-
|28
|8:15
|
|major seventh
|-
|29
|14:27
|(diminished octave)
|supermajor seventh
|-
|30
|18:35
|sub octave
|(augmented seventh)
|-
|31
|1:2
|octave
|
|}
The interval names shown in brackets could be said to be 'secondary', the others, 'primary'.

After releasing his system Keenan was informed that is was identical to the extended-diatonic interval-naming scheme of Adriaan Fokker but for the acknowledgment of more 11-limit ratios.This system depends on the tempering out of 81/80, where the diatonic major third, from four stacked fifths, approximates the just major third, 5/4. It also depends on the existence of neutral intervals, i.e., that the perfect fifth or equivalently, the chromatic semitone, subtends an even number of degrees of the ET. To simply to our familiar naming scheme for 12-tET, we observe that it applies to 24-tET equally as directly as in 31-tET, where the prefixes correspond to degrees of the edo. Exactly the same is also true for 38-tET, twice 19-tET, a meantone which very closely approximates 1/3-comma meantone. Meantone temperament wherein the fifth is divided into two equally sized neutral thirds is referred to as neutral temperament. Whereas meantone temperament is generated by the fifth, iin neutral temperament the generator is half this interval, the neutral third. Where it was seen above that there are two neutral thirds, 9:11 and 22:27 that differ by 243/242, neutral temperament is at its most simple the temperament defined by this equivalence: the tempering out of 243/242, as meantone is defined by the tempering out of 81/80. The temperament that tempers out both 81/80 and 243/242 is called Mohajira, upon which Keenan's scheme can be said to be based. As well as 24-tET, 31-tET and 38-tET, Mohajira is supported by 7-tET and 17-tET.

The primary interval names resulting in this system's application to these ETs is now show for easy comparison, where 'M', 'm', 'P', 'N', 'A', 'd', 'S' and 's' are shorthand for major, minor, perfect, neutral, augmented, diminished, super and sub, respectively:

7-tET: P1 N2 N3 P4 P5 N6 N7 P8

17-tET: P1 m2 N2 M2 m3 N3 M3 P4 S4 s5 P5 m6 N6 M6 m7 N7 M7 P8

24-tET: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 S4 A4/d5 s5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8

31-tET: P1 S1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 S4 A4 d5 s5 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 s8 P8

38-tET: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 S4 A4 SA4/sd5 d5 s5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8

to Meantone tunings that are not Mohajira tunings, the regular diatonic interval names can be applied, but with the addition of double augmented and double diminished from Fokker/Keenan's system.

19-tET: P1 A1 m2 M2 A2/d3 m3 M3 d4 P4 A4 d5 P5 A5 m6 M6 A6/d7 m7 d8 P8 (every second step of 38edo)

26-tET: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 AA4/AA5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8

Keenan adds further that if it is desired to distinguish between ratios that are in 31-tET approximated by the same number of steps, an addition prefix be added to describe the prime limit of the approximated interval. For 3-limit intervals, the obvious choice is 'Pythagorean', for 5-limit Keenan chooses 'classic', for 7, 'septimal, 11, 'undecimal' and 13, 'tridecimal'. When the highest prime is the same, Keenan suggests adding 'small' and 'large' as final prefixes for this purpose.

In non-Meantone tunings, the two definitions of major third - 4:5 and 64:81, the just (or classic) and Pythagorean major thirds no-longer correspond. If intervals are to receive unique names then to one or both of these major thirds must be added a prefix. Keenan has been involved with the development of both types of systems. Only when the major is defined by it's mapping as fourth fifths, i.e. 81/64, can conserve interval arithmetic.

Keenan later describes how the scheme can be extended to also cover 72-tET and 41-tET. In 31-tET the fifth may divided into six minor seconds. This temperament is called Miracle, and is also supported by 41-tET and 72-tET. The first six generators of Miracle give the following intervals: P1 m2 SM2 m3 S4 P5, as can be seen in the table above. 31-tET may be covered by 15 generators downwards and 15 generators upwards from 1:1. In 72-tET, either side of the intervals that that result from these 31 notes, called Miracle[31] 15|15, lie unnamed intervals that may be found first at either 31 or 41 generators further upwards or downwards and in 41-tET, either at 10 or 31 generators. If, one degree of 41 or 72-tET above an interval or Miracle[31] 15|15 lies an unnamed interval that can be first found by an additional 31 generators upwards, it is given the same name as the interval directly below it, with the addition of the prefix 'n', for 'narrow'. Similarly, 'W' for 'wide' prefixes an unnamed interval one degree, 31 generators below.

41-tET: P1 S1 nsm2 sm2 m2 N2 nM2 M2 SM2 sm3 nm3 m3 N3 M3 nSM3 SM3 s4 P4 nS4 S4 A4 d5 s5 Ws5 P5 S5 sm6 Wsm6 m6 N6 M6 Wm6 SM6 sm7 m7 Wm7 N7 M7 SM7 WSM7 s8 P8

In 41-tET, fourth fifths make a wide major third, rather than a major third, and interval arithmetic is no longer conserved. The same is true for 72-tET, so we have still yet to a system able to conserve interval arithmetic in non-meantone ETs. Though many edos can be covered, many still cannot, including the Superpythagorean edos, where the fifth is sharper than just, and four fifths give an approximation to 7:9, the super major third, tempering out the septimal comma, 63:64.

=== [[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak] ===
One system which in it's naming of meantone and non-meantone edos is able to conserve interval arithmetic, sagispeak, was developed by [[Dave Keenan]] and others as an interval naming system that maps 1-1 with the Sagittal microtonal music notation system. Sagittal notation was developed as a generalised diatonic-based notation system applicable equally to [[just intonation]], [[Equal Temperaments|equal tunings]] and rank-''n'' [[temperaments]]. Dozens of different accidentals can be used on a regular diatonic [[Staff notation|staff]] to notate up to extremely fine divisions, however in most cases only a handful are needed. In sagispeak, each accidental is presented by a prefix, made up of a single letter, in most cases, followed by either 'ai' if the accidental raises a note, or 'ao' if it lowers a note. As in HEWM notation, Pythagorean intonation is assumed as a basis. Then the prefixes depart from Pythaogrean intonation, altering by commas and introducing other primes. In place of the prefixes 'sub' and 'super', generally signifying an alteration of 36/35 from 5-limit intervals or 64/63 for 3-limit, Sagittal features an accidental of [[64/63]], which may be used to take a Pythagorean major interval to a supermajor, minor to subminor, or perfect to super or sub. The prefix 'tao' indicates a decrease of 64/63 and and the prefix 'tai' an increase. Whereas in previous interval naming schemes 'major' and 'minor' were synonymous with the 5-limit tunings, in sagispeak they map instead to Pythagorean. A prefix is needed then to take a Pythagorean intoned interval to a 5-limit tuning. Where 5/4 is 81/80 below the the Pythagorean third, the prefixes 'pai' and 'pao' (where 'p' is for 'pental', as in, involving prime 5), which raise or lower a note by [[81/80]] respectively. Similarly, 'vai' and 'vao', which raise or lower a note by [[33/32]] respectively, leading to ratios of 11.

Because it is built off of the diatonic scale, sagispeak conserves diatonic interval arithmetic, i.e. familiar relations in the diatonic scale, i.e. M2 + m3 = P4. As in Fokker/Keenan Extended-diatonic Interval-names, diatonic interval arithmetic is also extended, where, for example, tai-major 2 + tao-minor 3 = P4 (8/7 + 7/6 = 4/3), where opposite alterations cancel each other out, and diatonic interval arithmetic is conserved, a very useful property for a microtonal interval naming system to possess. Another helpful property of sagispeak is its generalised applicability to edos, just intonation and other tunings, where the same intervals maintain their spelling across different tunings. Despite these benefits however, many see Sagittal and Sagispeak as overly complex (even though the entire extended system need hardly ever be applied), and requiring too many new terms to be learnt.

We have seen that there are two competing definitions of a major third, the ratio '5/4' or the interval built from fourth stacked fifths, that may or may not correspond to 5/4. In meantone systems, those we are used to, they correspond, but in most edos they do not. Interval naming systems wherein the major third is defined as an approximation to 5/4 rather than as four fifths minus two octaves may benefit from a familiar name for 5/4, but they are unable to conserve diatonic interval arithmetic.

For comparison, 31-tET, is shown below in sagispeak:

31-tET: P1 tai-1/vai-1 tao-m2 m2 vai-2m/vao-M2 M2 tai-M2 tao-m3 m3 vai-m3/vao-M3 M3 tai-M3 tao-4 P4 vai-4 A4 d5 vao-5 P5 tai-5 tao-m6 m6 vai-m6/vao-M6 M6 tai-M6 tao-m7 m7 vai-m7.vao-M7 M7 tai-M7 vao-8/tao-8 P8

=== Dave Keenan's most recent system ===
In 2016 Dave Keenan proposed an alternative generalised [http://dkeenan.com/Music/EdoIntervalNames.pdf microtonal interval naming system for edos]. In what might be understood as a generalisation of his extended-diatonic interval-naming system described above onto any equal tuning. Employing as prefixes the familiar 'sub', 'super', and 'neutral'. His scheme is based on the diatonic scale, however the diatonic interval names are not defined by their position in a cycle of fifths like is Sagispeak. In Keenan's system the ET's best 3/2 is first labelled P5, and the fourth P4. The interval half-way between the tonic and fifth is labelled the neutral third, or 'N3', and halfway between the fourth and the octave N6. Then the interval a perfect fifth larger than N3 is labelled N7, and the interval a fifth smaller than N6 labelled N2. The neutral intervals then lie either at a step of the ET, or between two steps. After this the remaining interval names are decided based on the distance they lie in pitch from the 7 labelled intervals, which make up the ''Neutral scale'', P1 N2 N3 P4 P5 N6 N7, which, like the diatonic, is an MOS scale, which may be labelled [[Neutral7|Neutral[7]]] 3|3 using [[Modal UDP Notation|Modal UPD notation]]. To name an interval in an ET, the number of steps of 72-tET that most closely approximate the size of the interval difference from a note of the neutral scale is first found. Then the prefix corresponding to that number of steps of 72-tET is applied to the interval name. The following chart details this process (can't load the chart :( ). An interval just smaller than a major third in Keenan's system is labelled a ''narrow major third'', and an interval just wider than a [[6/5]] minor third a ''wide minor third'', however he notes that 'narrow' and 'wide' are only necessary in edos greater than 31. This system is equivalent to the Fokker/Keenan Extended-diatonic interval-naming system when applied to any of the ETs it was able to cover.

Keenan's system is an elegant way to keep the 'major 3rd' label for 5/4, where labels depend on the size of the best fifth, however it suffers from it's applicability only to ETs, and that it does not conserve interval arithmetic. Another potentially undesirable result of the system is that the major second approximates 10/9, and a ''wide major second'' 9/8, where as 9/8 is almost always considered a major second, and [[10/9]] often a narrow or small major second. One such system that considers 10/9 a narrow major second is that of Aaron Hunt.

=== Size-based systems ===
Microtonal theorist [[Aaron Andrew Hunt]] devised [http://musictheory.zentral.zone/huntsystem4.html the Hunt system], which includes interval name assignments for JI (just intonation) and edos based on [[41edo]]. Compared to Keenan's 72 interval names, Aaron's system includes 41. His system is based directly on 41edo, and unlike Keenan's system, interval are given the name of the closest step of 41edo, and no account is taken of the size of the edos fifth. In 41edo, Major, minor, augmented and diminished intervals are those obtained through the approximately Pythagorean cycle of fifths. Intervals one step of 41edo above these are given the prefix 'small', one step larger are given the prefix 'large', two steps smaller the prefix 'narrow' and two larger the prefix 'wide'. As a result, 5/4 is labelled a 'small major 3rd', or SM3 (not to be confused with a super major third, a label that does not exist in this system).

Neo-medieval musicians and early music historian and theorist [[Margo Schulter]] described her own [http://www.bestii.com/~mschulter/IntervalSpectrumRegions.txt interval naming scheme] built on approximations to JI intervals. Each interval names corresponds to an approximate size, and no particular edo is referenced. In her scheme middle major thirds range in size from 400-423 cents, and small major thirds from 372-400c. 5/4 is labelled a small major third, 81/64 a middle major third and 9/7 a large major third. Margo's scheme includes small, middle and large varieties of major, minor and neutral 2nds, 3rds, 6ths, 7ths; perfect fourth and fifths; and tritones, as well as a sub fifth and super fourth a dieses and comma and an octave less dieses and comma and ''interseptimals'', which correspond to intermediates, her name referencing the fact that they may each approximate two ratios of 7.

In Hunt's system when used in 41edo or JI diatonic interval arithmetic is conserved, but in other tunings it may not be, and Margo's system may not conserve diatonic interval arithmetic either. Both systems may be applied to arbitrary tunings, but the same intervals (defined, perhaps by a MOS scale) may not be given the same interval names across different tunings.

<ul><li>[[User:PiotrGrochowski/Extra-Diatonic Intervals]] gives each 43edo interval a name, then maps each desired interval to a 43edo interval. [[User:PiotrGrochowski/Extra-Diatonic Intervals — 50edo]] does this with 50edo. '''It has a very amazingly excellent solution to the 5/4 and 81/64 problem''': The 5/4 is named major third and this notation is split in half, while 81/64 is named high major third. 10/9 and 9/8 are both named major second. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]). </li></ul>Size-based systems are completely generalisable, but do not conserve interval arithmetic.

=== Ups and Downs ===
One final interval naming system, associated with the [[Ups and Downs Notation|Ups and Downs Notation]] system, belonging to microtonal theorist and musicians [[KiteGiedraitis|Kite Giedraitis]], like Sagittal is based on deviations from the diatonic scale. In this system however, deviations (from major, minor, perfect, augmented and diminished) are notated simply by the addition of up or down arrows: '^' or 'v', corresponding to raising or lowering of a single step of an edo. In some tunings ([[12edo]], [[19edo]] or [[31edo]] for example) 5/4 may be a M3, and in others a vM3 (downmajor 3rd) (e.g. [[15edo]], [[22edo]], 41edo, 72edo), or even an up-major 3rd (e.g. [[21edo]]). Ups and downs also includes neutrals, which lay exactly in-between major and minor intervals of the same degree, labelled '~' (mid). 'Up' and 'down' prefixes may be used before mid also, i.e. 'v~ 3). P1, P4, P5 and P8 are simply labelled '1', '4', '5' and '8'. This system benefits from it's simplicity as well as it's conservation of interval arithmetic. It can be used for some MOS scales where one of the generators is a perfect fifth or a fraction of a perfect fifth, but not all of these (e.g. Diminished[8]), and not all MOS scales (if such scales are to be described, an additional pair of accidentals/qualifiers is used. Although the scales then are described, their intervals still are not given the same names in Ups and Downs' edo names). Another criticism of Kite's system that does not apply to the others is the fact that when an edo is doubled or multiplied by some simple fraction, and the best fifth is constant across the two edos, the same intervals may be be given different names.

[[Igliashon Jones]] is a supporter of this system, but for the relabeling of 'down' as 'sub' and 'up' as 'super' and 'mid' as 'neutral', so that more common names are used, wherein 'super' infers a raise of 1 step of the edo, and 'sub' a lowering of one step. In this 'Extra-diatonic' system 'super' and 'sub' may be doubly applied, as in Ups and Downs, but they may not be applied before 'neutral' where in Ups and Downs they may be applied before 'mid'. The author's own extra-diatonic system is developed as a departure with caveat that 'S' and 's' prefixes are defined not as alterations by a single step of the edo, but by comma alterations as in Sagittal, in order that interval of MOS scales may be represented consistently across different tunings. Throughout the rest of the article the development is detailed, and the system defined.

31edo in Ups and Downs:

1 ^1 vm2 m2 ~2 M2 ^M2 vm3 m3 ~3 M3 ^M3 v4 4 ^4 A4 d5 v5 5 ^5 vm6 m6 ~6 M6 ^M6 vm7 m7 ~7 M7 ^M7 v8 8

In Jones' relabeling 31edo appears as in Fokker/Keenan Extended-diatonic Interval-names.

== Premise: ==
Extra-diatonic names should be simple, generalisable, widely applicable, backwards compatible with standard diatonic notation and reflecting current informal practice as closely as possible. Extra-diatonic interval names are fifth based; extended from the familiar major, minor and perfect interval names so that diatonic interval arithmetic is conserved. ‘M’, ‘m’, and ‘P’ remain the short-hand for major, minor and perfect. ‘A’ and ‘d’ for Augmented and diminished may also be used in the familiar way. In cases where the chroma (the chromatic semitone, or augmented unison) is represented by multiple steps in the tuning the prefix ‘super’ raises major and perfect intervals by a single step while ‘sub’ lowers minor and perfect intervals, with short-hand ‘S’ and ‘s’. ‘S’ and ‘s’ may also be used to raise minor and lower major intervals respectively, reflecting occasion practice. In this case ‘S’ is short-hand for ‘supra’, and 's' for 'small'. They may also be used to raise or lower diminished and augmented intervals similarly. In this way this scheme is equivalent thus far to Ups and Downs notation, where ‘^’ or ‘up’ corresponds to ‘S’, ‘super’ or ‘supra’ and ‘v’ or ‘down’ to ‘sub’ or 'small' .

== Additions and examples: ==
''Neutrals'' and ''intermediates'' are also included, where neutrals occur between opposing sizes of a single generic interval the intermediates between each generic interval and the next.

Interval names for equal tunings are ranked in eight tiers.
# Perfect
# Neutral
# Major, minor, A4 and d5 and, if the chroma is subtended by a single (positive) step of the edo, other augmented and diminished intervals
# Super, sub, supra and small prefixes to major, minor, perfect interval and to A4 and d5
# Intermediates
# Augmented and diminished intervals (for when the chroma is subtended by more than a single (positive) step of the edo)
# Super, sub, supra and small prefixes to augmented and diminished intervals
# Intervals augmented and diminished more than singularly
When more than one interval name corresponds to a specific interval, the names are privileged in order of the tiers. By this ordering, the first available name is the ‘primary’ for that interval, the second available ‘secondary’ and third 'tertiary'.

I initially intended for Intermediates to lay in the second tier with neutrals, but that led to primary intervals results that were not preferred by my colleagues over other options and I was persuaded to avoid using intermediates unless they are necessary.

=== Neutrals ===
N2, N3, N6 and N7, i.e. neutral 2nds, 3rds, 6ths and 7ths, falling exactly in-between the major and minor intervals of the same interval class, add native support for neutral-thirds temperament, where the N3 divides the P5 in exact halves and N2 divides the m3 is exact halves. In ups and downs neutrals indicated with '~' and said 'mid'.

Extending this familiar application to provide support for larger neutral scales, we add that neutrals occur also between P4 and A4; P5 and d5; P1 and A1; and P8 and d8.

Then [[Neutral7|Neutral[7]]] 3|3 can then be written:

P1 N2 N3 P4 P5 N6 N7 P8.

The same names give the primary interval names for [[7edo]], whose secondary intervals names are:

N1 m2/M2 m3/M3 N4 N5 m6/M6 m7/M7 N8.

The secondary interval names show that the chroma is equivalent to a unison in 7edo.

[[Neutral10|Neutral[10]]] 5|4 may then be written as

P1 N2 M2 N3 P4 N4 P5 N6 m7 N7 P8

[[Neutral17|Neutral[17]]] 8|8 may be written as

P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8,

which is equivalent to the primary interval names of [[17edo]].

=== Intermediates ===
To provide native support for [[The Archipelago|Barbados]] and Diminishes temperaments, intermediates are also added to the system. It should be noted immediately that intermediates are not as common to common microtonal interval naming as neutrals and though are a useful addition to this scheme, may be left out if desired. The appendix includes the MOS scales and edos from 'lists of all edos and MOS Scales', but without any intermediates.

‘2-3’ lies exactly half-way between M2 and m3 and divides the P4 in half. It may be read ‘second-third’ or ‘serd’. ‘6-7’, it’s octave-inverse lies exactly half-way between M6 and m7 and may be read ‘sixth-seventh’ or ‘sinth’.

‘1-2’ lies exactly half-way between P1 and m2, dividing the m2 in half. It may be read ‘unison-second’ or ‘unicond’. Its octave-inverse, ‘7-8’, lies exactly half-way between M7 and P8 and may be read ‘seventh-octave’ or ‘sevtave’.


‘3-4’ lies exactly half-way between M3 and P4, dividing the M6 in half. It may be read ‘third-fourth’ or ‘thourth’. It’s octave-inverse, ‘5-6’, lies exactly half-way between P5 and m6 and may be read ‘fifth-sixth’ or ‘fixth’. 

[[5edo]] can be spelled with the list of only these intermediates:

1-2 2-3 3-4 5-6 6-7 7-8.

The primary interval names for 5edo do not include them however, as they may be described by diatonic intervals:

P1 M2/m3 P4 P5 M6/m7 P8

Barbados[5] 2|2 may be described as

P1 2-3 P4 P5 6-7 P8,

and Barbados[9] 4|4 as

P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8.

The final intermediate, ‘4-5’ lies exactly between P4 and P5 and divides the octave in half. It may be read ‘fourth-fifth’ or ‘firth’. It is necessary for [[Diminished]] temperament, where the half-octave cannot be represented as any alteration of A4 or d5. Diminished temperament has a period of 1/4 of an octave, an approximation of 6/5. Therefore, the difference between four 6/5's and 2, 648/625, is tempered out.

Diminished[8] 1|0 (4) can be written as P1 sM2 Sm3 P4 4-5 Sm6 sM6 sM7 P8, and

[[Diminished12|Diminished[12]]] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 4-5 P5 Sm6 sM6 Sm7 sM7 P8,

== [[12edo]] and a problem ==
The primary interval names for 12edo are as we are familiar:

P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8,

but the interval labeled A4/d5 has the secondary name '4-5'.

We know that 12edo supports Diminished temperament, but we can't see that 12edo as equivalent to Diminished[12]: If we follow the rules from our premise, the notes of Diminished[12] in 12edo would give: P1 M2 m2 M3 m3 P4 A4/d5 P5 M6 m6 M7 m7 P8, with the majors and minor flipped. To be equivalent, sM must equal M and Sm must equal m.

=== [[10edo]], Pajara and a problem ===
10edo may be written as only neutrals and intermediates:

N1/1-2 N2 2-3 N3 3-4 N4/4-5/N5 5-6 N6 6-7 N7 7-8/N8

However it much more sense to write is using the primary interval name set:

P1 N2 M2/m3 N3 P4 N4/N5 P5 N6 M6/m7 N7 P8

It's secondary intervals as follows:

m2 Sm2/sM2 2-3 Sm3/sM3 M3 S4/s5 m6 Sm6/sM6 6-7 Sm7/sM7 M7

We can see that 10edo supports [[Neutral third scales]], given that we can make the interval names for Neutral[10] using the primary interval names for 10edo.

We know 10edo and 12edo both support Pajara temperament. Pajara[10] 2|2 (2) consists of:

P1 Sm2 M2 sM3 P4 sA4/Sd5 P5 Sm6 m7 sM7 P8,

and [[Pajara12|Pajara[12]]] 3|2 (2) of

P1 Sm2 M2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 m7 sM7 P8.

We can see Pajara[10] in 10edo, but in 12edo, wouldn’t sM3 be m3?

This is where my system diverges from Igliashon Jones’. We have to break our first rule here, or at least add some conditions to it.

== Comma association ==
To address this problem of consistency, we now state that when 81/80 is tempered out, M=sM and m=Sm, and when 64/63 is tempered out, M=SM and m=sm. In the case of sm and SM, ‘S’ and ‘s’ raise and lower by 64/63, and in the case of Sm and sM, ‘S’ and ‘s’ raise and lower by 81/80. In this way extra-diatonic interval names are equivalent to [http://forum.sagittal.org/viewforum.php?f=9 Sagispeak] interval names, where for sm and SM ‘S’ and ‘s’ are equivalent to ‘tai’ and ‘pao’ and for Sm and sM ‘S’ and ‘s’ are equivalent to ‘pai’ and ‘pao’.

It is important to note that given this change, 'S' and 's' may alter an interval by a different number of steps in an edo depending on which interval names they prefix. This may seem confusing, but it seems to reflect existing informal practice.

In 12edo, which represents the union between the two, where both 64/63 and 81/80 are tempered out, ‘S’ and ‘s’ do not raise or lower intervals at all. We can now easily see that 12edo supports Pajara, where simply removing all the ‘s’s and ‘S’s from Pajara[12] gives us our primary interval names of 12edo.

=== [[14edo]] and Injera ===
Like 10edo, 14edo may be written using all the neutrals and intermediates, but without any intervals described both as a neutral and as an intermediate:

N1 1-2 N2 2-3 N3 3-4 N4 4-5 N5 5-6 N6 6-7 N7 7-8 N8.

The primary interval names for 14edo are as follows:

P1 sm2 N2 SM2/sm3 N3 SM3/s4 P4 SA4/sd5 P5 S5/sm6 N6 SM6/sm7 N7 SM7 P1

We know that 12edo and 14edo support Injera, where Injera[12] 3|2 (2) may be labelled

P1 sm2 M2 sm3 SM3 P4 SA4/sd5 P5 sm6 M6 m7 SM7 P8,

and Injera[14] 3|3 (2) labelled

P1 sm2 M2 sm3 m3 SM3 P4 SA4/sd5 P5 sm6 M6 SM6 m7 SM7 P8.

We can see both in 14edo, and to get 12edo from Injera[12], as with Pajara, we remove all the ‘s’ and ‘S’ prefixes.

=== [[Blacksmith]] and further extension ===
10edo also supports Blacksmith temperament, and we may think to write Blacksmith[10] 1|0 (5) as:

P1 sM2 M2/m3 sM3 P4 s5 P5 sM6 M6/m7 sM7 P8

But we have now added mappings, but are yet to define the use of ‘S’ and ‘s’ for perfect intervals. In Blacksmith, the interval we might call ‘s5’ is 81/80 below P5, however, more commonly ‘s5’ is used to refer to [[16/11]], and S4 [[11/8]]. Since these intervals have above been labelled N4 and N5 above however, we do not need to worry about that, and can add that s5, a 'small 5th', is 81/80 below 3/2, and S4, a 'supra 4th' lies 81/80 above 4/3. where ‘s4’ has been typically been used to refer to [[21/16]], and ‘S5’ to [[32/21]], we add that s4 is lower than P4 by 64/63 and that S5 is higher than P5 by 64/63.

The primary intervals names of Blacksmith[15] 1|1 (5) then are as follows:

P1 Sm2 sM2 M2/m3 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 M6/m7 Sm7 sM7 P8,

these being identical to the the primary interval names of 15edo.

We may further add that ‘S’ (supra) and ‘s’ (small) may raise diminished, and lower augmented intervals by 81/80 as they do to minor and major respectively and that when ‘S’ (super) raised an augmented interval, or ‘s’ (sub) lowers it, the change is by 64/63.

In edos where 81/80 or 64/63 are represented by a single step, or when the apotome is represented by a single step, Ups and Downs with 'S', 's', and 'N' can name the intervals equivalently to this system. This can be seen in add edos considered thus far, as well as in those listed directly below. What this system adds is that it may also describe the intervals of MOS scales (as well as JI scales), such that these interval can be identically named in edos that approximate the scales they belong to. This is true for all MOS scales mentioned so far, as well as those listed below.

=== Further application in edos ===
The primary interval names are shown below for some larger edos.

[[22edo]]: P1 m2 Sm2 sM2 M2 m3 Sm3 sM3 M3 P4 S4 4-5 s5 P5 m6 Sm6 sM6 M6 m7 Sm7 sM7 M7 P8

[[24edo]]: P1 N1 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 N4 A4/d5 N5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 N8 P8

[[26edo]]: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 SA4/sd5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8

[[27edo]]: P1 m2 Sm2 N2 sM2 M2 m3 Sm3 N3 sM3 M3 P4 N4 d6 A3 N5 P5 m6 Sm6 N6 sM6 M6 m7 Sm7 N7 sM7 M7 P8

[[29edo]]: P1 sm2 m2 Sm2 sM2 M2 SM2/sm3 m3 Sm3 sM3 M3 SM3/s4 P4 S4 d5 A4 s5 P5 sm6 m6 Sm6 sM6 M6 SM6/sm7 m7 Sm7 sM7 M7 SM7 P8

[[31edo]]: P1 N1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 N4 A4 d5 N4 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 N8 P8

[[34edo]]: P1 1-2 m2 Sm2 N2 sM2 M2 2-3 m3 Sm3 N3 sM3 M3 3-4 P4 S4 N4 4-5 N5 s5 P5 5-6 m6 Sm6 N6 sM6 M6 6-7 m7 Sm7 N7 sM7 M7 7-8 P8

[[38edo]]: P1 S1 1-2 sm2 m2 N2 M2 SM2 2-3 sm3 m3 N3 M3 SM3 3-4 s4 P4 N4 A4 SA4/sd5 d5 N5 P5 S5 5-6 sm6 m6 N6 M6 SM6 6-7 sm7 m7 N7 M7 SM7 7-8 s8 P8

[[41edo]]: P1 N1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 s4 P4 S4 N4 d5 A4 N5 sM5 P5 S5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 N8 P8

[[43edo]]: P1 S1 1-2 A1/sm2 m2 dd3 AA1 M2 SM2/d3 2-3 A2/sm3 m3 dd4 AA2 M3 SM3/d4 3-4 A3/s4 P4 dd5 AA3 A4 d5 dd6 AA4 P5 S5/A6 5-6 A5/sm6 m6 dd7 AA5 M6 SM6/d7 6-7 A6/sm7 m7 dd8 AA6 M7 SM7/d8 7-8 s8 P8

[[46edo]]: P1 S1 sm2 m2 Sm2 A1 d3 sM2 M2 SM2 sm3 m3 Sm3 A2 d4 sM3 M3 SM3 s4 P4 S4 sd5 d5 sA4/Sd5 A4 SA4 s5 P5 S5 sm6 m6 Sm6 A4 d7 sM6 M6 SM6 sm7 m7 Sm7 A6 d8 sM7 M7 SM7 s8 P8

In 43edo we encounter the first time we have to use double Augmented and diminished intervals. 43edo marks the first instance in which Jones' alternative ups and downs interval names do not match those from this system. In his system, for example, AA2 would simply be sM3, but in this system since sM3 implies an approximation to 5/4 and the M3 already represents 5/4, and therefore is equivalent to sM3, we cannot do this. This tells us however that no simple ratio is approximated by the interval, and perhaps it is better understood as an AA2. Larger edos contain unlabeled intervals (without resorting to extended diatonic interval names). The association of 'super' and 'sub' with 64/63 and with 'supra' and 'small' with 81/80 may effect the assignment of primary interval names, but for all of these edos, as well as all those mentioned before, when 'S' and 's' are used, they still signify a raising or lowering by a single step of the edo, and thus appear equivalent to the Ups and Downs version. The comma associations add that, though use of enharmonic equivalences and secondary interval names may be necessary, intervals from MOS scales may be spelled in a consistent way across tuning to different edos.
=== Other rank-2 temperaments' MOS scales ===
On top of those discussed thus far, other temperaments generated by the P5 or a fraction of it are also supported to some extent, where their MOS scales may be represented, including [[Semaphore]], [[Augmented]], [[Porcupine]], [[Diminished]], [[Negri]], [[Tetracot]] and [[Slendric]].

Semaphore[5] 2|2: P1 SM2/sm3 P4 P5 SM6/sm7 P8

Semaphore[9] 4|4: P1 M2 SM2/sm3 SM3/s4 P4 P5 S5/sm6 SM6/sm7 m7 P8

Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8

Augmented[9] 1|1 (3): P1 Sm2 Sm3 sM3 P4 P5 Sm6 SM6 sM7 P8

Augmented[12] 2|1 (3): P1 Sm2 M2 Sm3 sM3 P4 sA4 P5 Sm6 Sm7 SM6 sM7 P8

Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8

Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8

Porcupine[8] 4|3: P1 sM2 Sm3 P4 s5 P5 sM6 Sm7 P8

Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8

Negri[9] 4|4: P1 Sm2 SM2/sm3 sM3 P4 P5 Sm6 SM6/sm7 sM7 P8

Negri[10] 4|4: P1 Sm2 SM2/sm3 sM3 P4 A4 P5 Sm6 SM6/sm7 sM7 P8

Tetracot[6] 3|2: P1 sM2 N3 S4 N6 Sm7 P8

Tetracot[7] 4|2: P1 sM2 N3 S4 P5 N6 Sm7 P8

Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 P8

[[Slendric5|Slendric[5]]] 2|3: P1 SM2 s4 S5 sm7 P8

[[Slendric6|Slendric[6]]] 3|2: P1 SM2 s4 P5 S5 sm7 P8

[[Slendric11|Slendric[11]]] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8

We may also write temperaments with a 9/8 but no 3/2. The most well known of these is [[Machine]]:

[[Machine5|Machine[5]]] 2|2: P1 M2 M3 m6 m7 P8

[[Machine6|Machine[6]]] 3|2: P1 M2 M3 A4 m6 m7 P8

[[Machine11|Machine[11]]] 5|5: P1 d3 M2 d4 M3 d5 A4 m6 A5 m7 A6 P8
=== Formal summary ===
The resultant system may be formally summarised as follows:

'''Definition 1a.''' M and m label the two sizes of 2nd, 3rd, 6th and 7th in the Pythagorean diatonic scale.

'''Definition 1b.''' The smaller 4th and larger 5th are labelled P.

'''Definition 1c.''' The single size of 1 and 8 is labelled P.

'''Definition 2.''' A chroma above M or P is A and below m or P is d.

'''Definition 3a.''' Within generic interval classes 2, 3, 6 and 7, half way between M and m is N.

'''Definition 3b'''. Within generic interval classes 1 and 4, half way between P and A is N.

'''Definition 3c.''' Within generic interval classes 5 and 8, half way between P and d is N.

'''Corollary:''' 7edo can be written P1 N2 N3 P4 P5 N6 N7 N8.

'''Definition 4.''' Half way between adjacent generic interval classes lie the intermediates.

'''Corollary:''' 5edo can be written 1-2 2-3 3-4 5-6 6-7 7-8.

'''Definition 5a.''' 'S' when applied before M, and 's' when applied before m, raise or lower by 64/63 respectively.

'''Definition 5b.''' 'S' when applied before m, and 's' when applied before M, raise or lower by 81/80 respectively.

'''Definition 5c.''' 'S' when applied to P5 (leading to S5) and 's' when applied to P4 (leading to s4) raise and lower by 64/63 respectively.

'''Definition 5d.''' 'S' when applied to P4 (leading to S4) and 's' when applied to P5 (leading to s5) raise and lower by 81/80 respectively.

'''Definition 5e.''' 'S' when applied to P1 and 's' when applied to P8 may raise and lower by 64/63, or by 81/80.

'''Definition 6a.''' When 'S' and 's' imply alterations of 64/63, they have long-form 'super' and 'sub' respectively.

'''Definition 6b.''' When 'S' and 's' imply alterations of 81/80, they have long-form 'supra' and 'small' respectively.

'''Definition 7.''' If alterations of 81/80 and of 64/63 need to be distinguished from one another in short-form, alterations of 81/80 can be written 'SR' and 'sl'.

== edos with extreme fifths ==
Thus far the best fifth of all edos described lies in the 'diatonic range', between 4 steps of 7edo and 3 steps of 5edo. The best fifths of some edos lies outside this range, in either directly. Whereas in edos where the best fifth lies between 7 steps of 12edo and 4 steps of 7edo, 81/80, the meantone comma is tempered out, wherein four fifths approximate the fifth harmonic (a sM3 raised two octave), in edos with fifths flatter than this, 135/128, a meantone chromatic semitone is instead tempered out, resulting in the four fifths instead approximate the Sm3, 6/5 (raised two ocaves). This system is called Mavila temperament. In the other direction, whereas the best fifth lies between 7 steps of 12edo and 3 steps of 5edo, 64/63, the septimal or Archytas comma is tempered out, wherein four fifths approximate a SM3, 9/7, raised two octaves, in edos who's best fifth is sharper than this, 9/7, as well as 5/4 are approximated by the perfect fourth, tempering out 16/15 and 28/27. This system is called Father temperament. The application of this system to edos of both of these fifth sizes is addressed below.

=== [[Mavila]] ===
In Mavila, the perfect 5th is flatter than in 7edo, so major intervals are below minor and augmented below major. In Mavila the major third approximates 6/5 and the minor third 5/4, tempering out [[135/128]]. This presents no problem to the scheme however, and the rules are applied just the same. The small major third, 81/80 below 6/5 or 81/64 comes to [[32/27]], the minor 3rd, and the sub minor 3rd remains 7/6.

Mavila[7] 3|3 can be written

P1 M2 m3 P4 P5 M6 m7 P8,

the same as the diatonic scale.

Mavila[9] 4|4 can be written

P1 M2 M3 m3 P4 P5 M6 m6 m7 P8,

which is also the primary interval name set for 9edo.

If we don't want to have major being below minor, we can hide it with some secondary interval names:

P1 M2 Sm3 sM3 P4 P5 Sm6 sM6 m7 P8, arriving at Augmented[9].

We say that our first Mavila[9] interval names are not ''well-ordered'', where for an interval name set to be well-ordered, for each degree major must be above minor. By extension we defined a well-ordered interval names set as one in which ... ≤ dd ≤ d ≤ m ≤ M ≤ A ≤ AA ≤ ... or ... ≤ dd ≤ d ≤ P ≤ A ≤ AA ≤ ..., and where s_ ≤ _ ≤ S_ (where '_' represents any of ... dd, d, m, (P), M, A, AA ...).

Using the fact that 64/63 is represented by a single step in 9edo, it is left as an exercise for the reader to prove that 9edo supports Negri temperament (by replacing some names with sub and super prefixed names to arrive at Negri[9] 4|4).

The primary interval names for Augmented[9] 1|1 (3) are well-ordered.

Mavila[16] 8|7 can be writtten

P1 d1 M2 m2 M3 m3 A4 P4 A5 P5 d5 M6 m6 M7 m7 A8 P8.

The enharmonic equivalences for 16edo can be generated by: M=Sm=sm or m=sM=SM

This leads us to our primary well-ordered interval name set for 16edo:

P1 S1 Sm2 sM2 Sm3 sM3 s4 P4 4-5 P5 S5 Sm6 sM6 Sm7 sM7 s8 P8,

wherein we can see that it supports Diminished temperament.

Mavila[23]11|11 can be written as:

P1 d1 A2 M2 m2 A3 M3 m3 d3 A4 P4 d4 A5 P5 d5 A6 M6 m6 d6 M7 m7 d7 A8 P8

And the primary well-ordered interval names for [[23edo]] are appear as:

P1 S1 1-2 M2 m2 2-3 M3 m3 3-4 A4 P4 d4 A5 P5 d5 5-6 M6 m6 6-7 M7 m7 7-8 d8 P8

=== [[Father]] ===
In the other direction, the fifths of Father temperament are sharper than those of 5edo, leading to minor second going backwards. In Father temperament, 5/4 and 4/3 are tempered to a unison, along with 9/7. As 64/63 is tempered out, alterations of 64/63 act as identity alterations.The M2, larger than m3 is also a Sm3. sM2, then, returns to m3.

The primary interval names for Father[5] 2|2,

P1 M2 P4 P5 m7 P8,

present no problems.

In the primary interval names for Father[8] 4|3, however:

P1 m3 M2 P4 M3 P5 m7 M6 P8,

which are the same as the primary intervals for [[8edo]], but with M3 rather than 4-5, we see our diatonic interval names begin to cross over. We will add to our definition of well-ordered interval names that no interval names from interval-class ''n''-1 may be subtended by a larger number of steps that any interval names from interval-class ''n.'' As above, we can may use some secondary interval names to address it, leading to

P1 sM2 M2 P4 S4 P5 m7 Sm7 P8,

as the primary well-ordered interval name set.

If we also re-write M2 and m7 as Sm3 and sM6, we get Porcupine[8] 4|3.

Using some secondary interval names to 'fix' the order leads us to

P1 sM2 M2 P4 S4 P5 m7 M6 P8

The primary interval names for Father[13] 6|6:

P1 M7 m3 M2 d5 P4 M3 m6 P5 A4 m7 M6 m2 P8,

look very unruly.

We will fix up the ordering again with secondary interval names:

P1 S1 sM2 M2 Sm3 P4 S4 s5 P5 sM6 m7 Sm7 s8 P8,.

The primary interval names for [[13edo]] are similar:

P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8.

Minimally fixing the order leads us to

P1 N2 sM2 M2 N3 P4 S4 s5 P5 N6 m7 Sm7 N7 P8,

which we have seen before as Tetracot[13] 6|6.

=== [[Miracle]], [[11edo]] and [[21edo]] ===
All the scales discusses to this point use either smalls and supras or subs and supers, so in the rare instance that we see a S1 or s8, we can infer whether or not it's small/supra or sub/super, probably without even thinking too much about it. Rarely do MOS scales in this scheme require alterations of both 81/80 and 64/63. One important temperament that includes such scales is Miracle. We do not encounter either S1 in the 10 and 11-note MOS:

[[Miracle 10|Miracle[10]]] 5|4: P1 Sm2 SM2 N3 s4 sA4 S5 N6 sm7 sM7 P8

Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8

We first see S1 and s8 in the 21-note MOS:

[[Blackjack|Miracle[21]]] 10|10: P1 S1 Sm2 N2 SM2 sm3 N3 sM3 s4 P4 Sd5 sA4 P5 S5 Sm6 N6 SM6 sm7 N7 sM7 s8 P8,

the largest MOS scale we have attempted yet to write. S1 in this scale is 64/63. We could maybe guess this from the presence of S5, but it is not obvious. If we need to make it clear when we are referring to small/supra, we can write their short-hand instead as 'sl' for 'small' and 'Sa' for supra. Miracle[21] would then be re-written:

P1 S1 Sam2 N2 SM2 sm3 N3 slM3 s4 P4 Sad5 slA4 P5 S5 Sam6 N6 SM6 sm7 N7 slM7 s8 P8, if complete clarity is needed, otherwise if long-form names are provided there is no need. It looks unwieldy to me and so I would avoid it, but it is there as a possibility, if the intervals of Mavila[21] need be written with only 4 letters allowed for each interval name.

Having encountered 11 and 21-note scales now, and haven't not described 11 and 21edo, I will add these here.

The fifth in 11edo is too flat even for it to be considered to support Mavila. Let's see what happens:

P1 M2 M3 m2 m3 P4 P5 M6 M7 m6 m7 P8.

Adding neutrals gives us our primary interval names for 11edo:

P1 M2 M3 N3 m3 P4 P5 M6 N6 m6 m7 P8.

This is clearly not Mavila, so we don't know what's tempered out, such that we might add our alterations to arrive at a well-ordered interval name set. Let's review the 11-note scales we have encountered above:

Slendric[11] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8

Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8

From these union of these scales we can see from P4=Sd5 that 135/128, the Mavila comma is tempered out. We apply our Mavila re-spellings to arrive at:

P1 Sm2 Sm3 N3 sM3 P4 P5 Sm6 N6 sM6 sM7 P8 as a well-ordered interval name set.

We should not expect our Machine[11] scale to be represented in this spelling of 11edo: A spelling of 11edo that shows that it supports Machine uses a different mapping, using the 9/8 from two 22edo P5s. We could spell 11edo as every other note of 22edo if we wish to see how it supports Machine.

21edo can be written as three 7edos as it's best fifth is that of 7edo. Since 81/80 is -1 steps in 21edo, we use 64/63 alterations:

P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8.

Using the available conversion sm = Sm (and therefor SM = sM), we can confirm that 21edo supports Miracle and Whitewood temperaments. This is left as an exercise for the the inspired reader.

=== [[6edo]] ===
Though 6edo is normally only ever considered as a subset of 12edo, given that we have encountered 6-note MOS I'll give it a red hot go.

Our P5 and P4 in 6edo is our half-octave, 4-5, so 9/8 is tempered out and our chromatic scale only covers 2edo: P1/M2/M3/A4 m2/m3/P4/4-5/P5/M6/M7 m6/m7/P8. If we want to write 6edo is a well-ordered way, we might choose:

P1 SM2 sm3 P4/4-5/P5 SM6 sm7 P8.

Writing s4 and S5 instead of sm3 and SM6 would give us Slendric[6] 3|2.

What of Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8?

This tells us that in 6edo 81/80 is mapped to -2 steps of 6edo. This is not a problem, as we can use alterations of 64/63, mapped to 1 step, though I don't see why anyone would want to think of 6edo in this way.

Similarly to Machine in 11edo, Machine in 6edo uses a different (much better) mapping of 9/8: That of 12edo. 6edo is much better spelled as a subset of 12edo, where we can see if supports Machine.

The primary interval names for the remaining trivial edos are trivially derived and are given along with all those described so far in the section below.

=== [[28edo]] ===
We encounter a new problem with 28edo. 28edo's best fifth is that of 7edo. It's primary intervals names are as such:

P1 sA1 1-2 Sm2 N2 sM2 2-3 Sm3 N3 sM3 3-4 S4 P4 sA4 4-5 Sd5 P5 s5 5-6 Sm6 N6 sM6 6-7 Sm7 N7 sM7 7-8 Sd8 P8

in 28edo 81/80 is represented by -1 steps. We en-devour to maintain as best we can in our primary interval names the original premise behind the notation - that the prefix 's' takes an interval down a single step of the edo and 'S' a single step up. In our primary interval names for 28edo we have S4 below P4 and d5 above P5. We can avoid this confusion however by using the secondary interval names for P4 and P5 in 28edo - N4 and N5. In our list of edos below this change is made.

== How to ==
Though we have derived our interval names through many examples thus far, the process has not been explained as yet such that the reader may immediately apply them. This will be addressed here, with a step-by-step derivation guide. This guide gives all possible labels to each interval.

For ''n-''edo:
# Label P1 = 1, P8 = ''n''
# Find the best approximations of 3/2, 5/4 and 7/4, which are to be labelled P5, sM3 and sm7 (This is equivalent to finding the [[7-limit]] [[patent val]]).
#* P5 = round(ln(3/2)/ln(''n'')) steps
#* sM3 = round(ln(5/4)/ln(''n'')) steps
#* sm7 = round(ln(7/4)/ln(''n'')) steps
# Using the best fifth, label the diatonic intervals. i.e.
#* M2 = (2*P5) mod ''n''
#* M6 = (3*P5) mod ''n''
#* M3 = (4*P5) mod ''n''
#* M7 = (5*P5) mod ''n''
#* (P4, m7, m3, m6, m2) = ''n'' - (P4, M2, M6, M3, M7)
# Determine the size of the apotome: A = M-m (for any of degree 2, 3, 6 or 7).
# The initial chromatic intervals may be labelled:
#* A1 = A
#* d8 = P8 - A
#* A4 = P4 + A
#* d5 = P5 - A
# If A is even, then intervals half way between M and m within a degree may be labelled N.
# Determine the size of meantone and septimal comma alterations (steps 7-12 may be skipped if A = 1)
#* meantone comma = M3 - sM3. If equal to 0 then 81/80 is tempered out, the edo is said to support Meantone temperament and alterations of 81/80 are not needed.
#* septimal comma = m7 - sm7. If equal to 0 then 64/63 is tempered out, the edo is said to support Superpyth temperament, and alterations of 64/63 are not needed.
# If sub/super < 0, alterations of 64/63 should not be applied.
# If small/supra < 0, alterations of 81/80 my be applied only if A = 0, wherein all diatonic intervals are given the label 'N'.
# Acknowledging points 7, 8 and 9, label 's' and 'S' alterations to diatonic intervals, where:
#* sM (small major) = M - meantone comma
#* Sm (supra minor) = m + meantone comma
#* SM (super major) = M + septimal comma
#* sm (sub minor) = m - septimal comma
# Acknowledging points 7, 8 and 9, label 's' and 'S' alterations to P4 and P5 where:
#* S4 (supra 4th) = P4 + meantone comma
#* s5 (small 5th) = P5 - meantone comma
#* s4 (sub 4th) = P4 - septimal comma
#* S5 (super 5th) = P5 + septimal comma
# If meantone comma alterations are used, S1 (supra unison) may be labelled at P1 + meantone comma and d8 (small octave) at P8 - meantone comma. If septimal comma alterations are used, S1 (super unison) may be labelled at P1 + septimal comma and s8 (sub octave) may be labelled at P8 - septimal comma. If both are needed and not equal, and after steps 13-16 are completed as desired the interval corresponding to a super or supra unison (equivalently, sub or small octave) is yet otherwise unlabeled then to ascertain whether S1 is a supra or super unison (or equivalently s8 a small or sub octave), for all degrees, the short hand for 'small' is to be 'sl' rather than 's', and for 'supra' is to be 'SR' rather than 'S'.
# Remaining augmented (M + A) and diminished (m - A) intervals may be labelled 'A' and 'd'.
# If desired, augmentation and diminution may be iterated through the continued adding or subtracting of A, labelled by additional 'A' or 'd' prefixes respectively.
# 'S' and 's' may be further applied as if 'A' were 'M' or 'd' were 'm'.
# If desired and if ''n'' is even, the interval subtended by ''n''/2 steps may be labelled 4-5.
# If desired and if m2 is subtended by an even number of steps, the remaining intermediates may be labelled: (1-2, 2-3, 3-4, 5-6, 6-7, 7-8) = (P1, M2, M3, P5, M6, M7) + m2/2.
#

== Lists of edos and MOS scales ==

=== Primary well-ordered (unless otherwise noted) interval names for edos ===
2edo: P1 P4/P5 P8

3edo: P1 P4 P5 P8

4edo: P1 SM2/sm3 P4/P5 SM6/sm7 P8

5edo: P1/m2 M2/m3 M3/P4 P5/m6 M6/m7 M7/P8

6edo: P1 SM2 sm3/s4 P4/P5 S5/sM6 sm7 P8

7edo: P1 N2 N3 P4 P5 N6 N7 P8

8edo: P1 sM2 M2 P4 S4/s5 P5 m7 Sm7 P8

9edo: P1 M2 M3 m3 P4 P5 M6 m6 m7 P8

10edo: P1/m2 N2 M2/m3 N3 M3/P4 S4/s5 P5/m6 N6 M6/m7 N7 M7/P8

11edo: P1 Sm2 Sm3 N3 sM3 P4 P5 Sm6 N6 sM6 sM7 P8

12edo: P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8

13edo: P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8

14edo: P1 S1/sm2 N2 SM2/sm3 N3 SM3/s4 P4 SA4/sd5 P5 S5/sm6 N6 SM6/sm7 N7 SM7/s8 P8

15edo: P1/m2 S1/Sm2 sM2 M2/m3 Sm3 sM3 M3/P4 S4 s5 P5/m6 Sm6 sM6 M6/m7 Sm7 sM7/s8 M7/P8

16edo: P1 1-2 Sm2 sM2 Sm3 sM3 3-4 P4 4-5 P5 5-6 Sm6 sM6 Sm7 sM7 7-8 P8

17edo: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8

19edo: P1 S1/sm2 m2 M2 SM2/sm3 m3 M3 SM3/s4 P4 A4 d5 P5 S5/sm6 m6 M6 SM6/sm7 m7 M7/s8 P8

21edo: P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8

22edo: P1 m2 Sm2 sM2 M2 m3 Sm3 sM3 M3 P4 S4 4-5 s5 P5 m6 Sm6 sM6 M6 m7 Sm7 sM7 M7 P8

23edo: P1 S1 1-2 Sm2 sM2 2-3 Sm3 sM3 3-4 A4 P4 S4 s5 P5 d5 5-6 Sm6 sM6 6-7 Sm7 sM7 7-8 s8 P8 (can't quite get it well-ordered)

24edo: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 N4 A4/d5 N5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8

26edo: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 4-5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8

27edo: P1 m2 Sm2 N2 sM2 M2 m3 Sm3 N3 sM3 M3 P4 N4 d6 A3 N5 P5 m6 Sm6 N6 sM6 M6 m7 Sm7 N7 sM7 M7 P8

28edo: P1 sA1 1-2 Sm2 N2 sM2 2-3 Sm3 N3 sM3 3-4 S4 N4 sA4 4-5 Sd5 N5 s5 5-6 Sm6 N6 sM6 6-7 Sm7 N7 sM7 7-8 Sd8 P8

29edo: P1 S1 m2 Sm2 sM2 M2 SM2/sm3 m3 Sm3 sM3 M3 SM3/s4 P4 S4 d5 A4 s5 P5 S5/sm6 m6 Sm6 sM6 M6 SM6/sm7 m7 Sm7 sM7 M7 s8 P8

31edo: P1 N1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 N4 A4 d5 N4 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 N8 P8

34edo: P1 S1 m2 Sm2 N2 sM2 M2 2-3 m3 Sm3 N3 sM3 M3 3-4 P4 S4 N4 4-5 N5 s5 P5 5-6 m6 Sm6 N6 sM6 M6 6-7 m7 Sm7 N7 sM7 M7 s8 P8

38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 N4 A4 SA4/sd5 d5 N5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8

41edo: P1 N1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 s4 P4 S4 N4 sA4 Sd5 N5 sM5 P5 S5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 N8 P8

43edo: P1 S1 1-2 A1/sm2 m2 dd3 AA1 M2 SM2/d3 2-3 A2/sm3 m3 dd4 AA2 M3 SM3/d4 3-4 A3/s4 P4 dd5 AA3 A4 d5 dd6 AA4 P5 S5/A6 5-6 A5/sm6 m6 dd7 AA5 M6 SM6/d7 6-7 A6/sm7 m7 dd8 AA6 M7 SM7/d8 7-8 s8 P8 (not possible to find a well-ordered interval name set)

46edo: P1 S1 sm2 m2 Sm2 A1 d3 sM2 M2 SM2 sm3 m3 Sm3 A2 d4 sM3 M3 SM3 s4 P4 S4 sd5 d5 4-5 A4 SA4 s5 P5 S5 sm6 m6 Sm6 A4 d7 sM6 M6 SM6 sm7 m7 Sm7 A6 d8 sM7 M7 SM7 s8 P8 (not possible to find a well-ordered interval name set)

=== Interval names for MOS scales ===
Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8

Augmented[9] 1|1 (3): P1 Sm2 Sm3 sM3 P4 P5 Sm6 SM6 sM7 P8

Augmented[12] 2|1 (3): P1 Sm2 M2 Sm3 sM3 P4 sA4 P5 Sm6 Sm7 SM6 sM7 P8

Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8

Barbados[5] 2|2: P1 2-3 P4 P5 6-7 P8

Barbados[9] 4|4: P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8

Blackwood[10] 1|0 (5): P1 sM2 M2/m3 sM3 P4 s5 P5 sM6 M6/m7 sM7/s8 P8

Blackwood[15] 1|1 (5): P1 S1/Sm2 sM2 M2/m3 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 M6/m7 Sm7 sM7/s8 P8

Diminished[8] 1|0 (4): P1 sM2 Sm3 P4 sA4/Sd5 Sm6 sM6 sM7 P8

Diminished[12] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 Sm7 sM7 P8

Injera[12] 3|2 (2): P1 sm2 M2 sm3 SM3 P4 SA4/sd5 P5 sm6 M6 m7 SM7 P8

Injera[14] 3|3 (2): P1 sm2 M2 sm3 m3 SM3 P4 SA4/sd5 P5 sm6 M6 SM6 m7 SM7 P8

Machine[5] 2|2: P1 M2 M3 m6 m7 P8

Machine[6] 3|2: P1 M2 M3 A4 m6 m7 P8

Machine[11] 5|5: P1 d3 M2 d4 M3 d5 A4 m6 A5 m7 A6 P8

Mavila[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8

Mavila[9] 4|4: P1 M2 M3 m3 P4 P5 M6 m6 m7 P8

Mavila[16] 8|7: P1 d1 M2 m2 M3 m3 A4 P4 A5 P5 d5 M6 m6 M7 m7 A8 P8

Meantone[5] 2|2: P1 M2 P4 P5 m7 P8

Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8

Meantone[12] 6|5: P1 m2 M2 m3 M3 P4 A4 P5 m6 M6 m7 M7 P8

Meantone[19] 9|9: P1 A1 m2 M2 A2 m3 M3 d4 P4 A4 d5 P5 A5 m6 M6 d7 m7 M7 d8 P8

Miracle[10] 5|4: P1 Sm2 SM2 N3 s4 sA4 S5 N6 sm7 sM7 P8

Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8

Negri[9] 4|4: P1 Sm2 SM2/sm3 sM3 P4 P5 Sm6 SM6/sm7 sM7 P8

Negri[10] 4|4: P1 Sm2 SM2/sm3 sM3 P4 A4 P5 Sm6 SM6/sm7 sM7 P8

Neutral[7] 3|3: P1 N2 N3 P4 P5 N6 N7 P8

Neutral[10] 5|4: P1 N2 M2 N3 P4 N4 P5 N6 m7 N7 P8

Neutral[17] 8|8: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8

Pajara[10] 2|2 (2): P1 Sm2 M2 sM3 P4 sA4/Sd5 P5 Sm6 m7 sM7 P8

Pajara[12] 3|2 (2): P1 Sm2 M2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 m7 sM7 P8

Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8

Porcupine[8] 4|3: P1 sM2 Sm3 P4 s5 P5 sM6 Sm7 P8

Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8

Semaphore[5] 2|2: P1 SM2/sm3 P4 P5 SM6/sm7 P8

Semaphore[9] 4|4: P1 M2 SM2/sm3 SM3/s4 P4 P5 S5/sm6 SM6/sm7 m7 P8

Slendric[5] 2|3: P1 SM2 s4 S5 sm7 P8

Slendric[6] 3|2: P1 SM2 s4 P5 S5 sm7 P8

Slendric[11] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8

Superpyth[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8

Superpyth[12] 6|5: P1 m2 M2 m3 M3 P4 A4 P5 m6 M6 m7 M7 P8

Superpyth[17] 8|8: P1 m2 A1 M2 m3 d4 M3 P4 d5 A4 P5 m6 A5 M6 m7 d8 M7 P8

Tetracot[6] 3|2: P1 sM2 N3 S4 N6 Sm7 P8

Tetracot[7] 4|2: P1 sM2 N3 S4 P5 N6 Sm7 P8

Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 P8

== Conclusion ==
Using only 'S' and 's' as qualifiers to M, m, P, A and d, along with N and intermediate degrees, a system is developed wherein for most edos below 50 intervals can be systematically named such that for primary interval names, 'S' and 's' raise and lower, respectively, by 1 step of an edo , whilst the intervals of many MOS scales may be consistently named across different tunings, taking the best of both Igliashon Jones' extra diatonic interval names and Sagispeak. One does not need to understand the comma associations to make use of the interval names. While the intervals of some MOS scales may hold consistent names in edos in an Ups and Downs based scheme, there are many common scales that do not in such a system, that do in this one, such as scales of Diminished and Augmented temperament. What's more, the use of neutrals and intermediates leads to quicker recognition of MOS scales that may be supported in edos.

Help:Introduction

2018-10-16T07:05:43Z

PiotrGrochowski:

; [[Xenharmonic Wiki:Conventions|Conventions]]: a few thoughts about a few conventions currently in use
; [[Help:How to Get Your Xenwiki Account|How to Get Your Xenwiki Account]]: an account is necessary to edit articles, it provides also a user talk page
; Start your [[Help:User pages|user page]] (was before [[Editor yourName]]): say a few words about you as a wiki editor (user pages also serve as the temporary "host" (root) for subpages where complicated edits are carried out)

If you have any (seemingly) unsolvable problem, look or ask for a solution [[{{TALKPAGENAME}}|on the talk page]], Thanks :-)

__FORCETOC__

Wikis should be easy to edit, so most text you add or change will appear as you expect it. But you can do more if you want.

[https://wikiindex.org/Xenharmonic_Wiki This wiki] is currently in [https://wikiindex.org/Category:LoginToEdit login-to-edit mode] after the move from [[Wikispaces]] to MediaWiki-based installation (the edit mode may change again some time). For editing or adding information to the wiki, you have to join the wiki. See [[How to Get Your Xenwiki Account]] for more!

== Using links ==
What you always have dreamed of (for your paper notes) - in a wiki it becomes true: fast links!

If you enclose a word in double brackets (like <code><nowiki>[[word]]</nowiki></code>), you get an internal link, hopefully to an already existing page - if not, it appears red, and you just created the opportunity for your co-authors to add the details. If you want a text differing from the link target, you can place it after a pipe symbol: <code><nowiki>[[word|different text]]</nowiki></code>.

For external links, simply place the URL into the text, certain protocols (http, https, mailto) will be automatically transformed into links. For alternate text, use single brackets, start with the URL and after a space enter the title, like so <code><nowiki>[http://acid3.acidtests.org the Acid3 test]</nowiki></code> which renders to [http://acid3.acidtests.org the Acid3 test].

=== Link text ===
''tbd''

The link text should relate to the link target, don't use "[[help:here-links|here]]".

=== Redirects ===
There are 3 types of redirects: soft redirect, regular redirect and hard redirect.

A "soft redirect" or "smart redirect" is just an ordinary page with the text "See XYZ.", where XYZ is a link to the target of the smart redirect. A smart redirect should not point to another smart redirect - if you see this you should fix it by shortcutting past the second one and making the first redirect point directly to the ultimate target.

A regular redirect is performed by <code><nowiki>#REDIRECT [[Page title]]</nowiki></code>. A double redirect should be fixed by the same method as above, or by replacing it with a hard redirect. Regular redirects did not exist in <code>wikispaces</code>. This kind of redirect displays a small information message when it was used to display the current page.

Real or "hard" redirects directly go to the target page without a notice of redirect. Therefore, they should only be used in the case of simple misspellings or alternate spellings. The rule is that if anyone might ever want to flesh the redirect out into its own article, it should not be a hard redirect. Hard redirects can be chained with no problem, which is why they're still appropriate to use for misspellings. The code for a hard redirect in MediaWiki is <code><nowiki>#REDIRECT [[:en:Page title]]</nowiki></code>. Hard redirects worked somewhat differently in <code>wikispaces</code>; only admins could create or change them, and they could not redirect to a section.

see also

* http://groups.yahoo.com/neo/groups/tuning/conversations/topics/101227 - redirects were discussed on the [[tuning_list|tuning list]] in the [[Wikispaces]] era
* [[:Category:soft redirect|list of pages tagged as soft redirect]] - pages that contain only see <tt>[[</tt><nowiki>''some other page''</nowiki><tt>]]</tt> (be cautious with other tags on these)
* [[:Category:smart redirect|list of pages tagged as smart redirect]] - pages that have more information (a small definition, classification etc.)

== Text formatting ==

=== Emphasis ===
Some typographic markup can easy be added by including the phrase in doubled <code>''</code> or tripled <code>'''</code> (apostrophe chars):

* ''italic'' using <nowiki>''italic''</nowiki>
* '''bold''' using <nowiki>'''bold'''</nowiki>
* '''''bold italic''''' using <nowiki>'''''bold italic'''''</nowiki>

=== Lists ===

See [[list model]] for ways to write a list.

=== Tables ===
You can add tables to wiki pages:
{| class="wikitable"
! what you write
|
<nowiki>
{| class="wikitable"
|+ The Title
! column A
! column B
|-
| cell A1
| cell B1
|-
| cell A2
| cell B2
|}</nowiki>
|-
! what you get
|
{| class="wikitable"
|+ The Title
! column A
! column B
|-
| cell A1
| cell B1
|-
| cell A2
| cell B2
|}
|}

==== Adjusting Text Alignment ====
The table cells are left aligned by default, the heading cells are center aligned. You may change this via inline CSS style definitions:

{| class="wikitable"
! what you want for the cell
! what you do to get it
|-
| left aligned text
| nothing to do
|-
| style="text-align:center;" | centered text
| place <code><nowiki>style="text-align:center;" |</nowiki></code> after the opening pipe
|-
| style="text-align:right;" | right aligned
| place <code><nowiki>style="text-align:center;" |</nowiki></code> after the opening pipe
|-
! headline style
| exclamation mark instead of the pipe symbol starts the cell
|}

If you want to override the default text alignment for all cells, place the inline CSS (for example <code>style="text-align:center;"</code>) directly at the beginning of the table, right after the <code>class="wikitable"</code> statement.

==== External helper tools ====
If you have to construct really complex tables with colspans and rowspans, you may use this online table generator:

[http://www.tablesgenerator.com/mediawiki_tables Generate tables in MediaWiki format - TablesGenerator.com]

For example, this wiki markup
<nowiki>
{| class="wikitable"
! colspan="3" | 1
! 2
! 3
|-
| rowspan="2" | A
| colspan="3" | B
| C
|-
| rowspan="2" colspan="2" | D
| colspan="2" | E
|-
| F
| G
| H
|}
</nowiki>

will be rendered into this table:

{| class="wikitable"
! colspan="3" | 1
! 2
! 3
|-
| rowspan="2" | A
| colspan="3" | B
| C
|-
| rowspan="2" colspan="2" | D
| colspan="2" | E
|-
| F
| G
| H
|}

=== Images ===
Inserting a JPEG or PNG image is pretty straightforward - when using the visual page editor click the "File" button and upload the image. Or in Wikitext, use <code><nowiki>[[image:filename]]</nowiki></code>. SVG images are possible but require a bit more work; see [[SVG_images_on_wiki_pages|SVG images on wiki pages]] for details.

=== Maths formulas ===
...can be used [http://en.wikipedia.org/wiki/Help:Displaying_a_formula like in the Wikipedia]: start it with <code><nowiki><math></nowiki></code> and end it with <code><nowiki></math></nowiki></code>. Take care: using the math tags will break paragraphs. See for example

<math>V =\frac{4}{3} \pi r^3</math>

Here is a browser-based formula editor that may be helpful: http://www.codecogs.com/latex/eqneditor.php

=== Table of contents ===
Place <code><nowiki>__TOC__</nowiki></code> where you wish to place the table of contents, mostly this will be at the top of the page.

== Social aspects ==
In this wiki, you can use talk pages and user talk pages.

If you want to discuss the wiki itself, do it on the page [[wikifuture]].

=== Categories ===
Categories all serve different purposes: sometimes they say what something is or the context in which it belongs ([[:Category:edo]], [[:Category:interval]]). Sometimes they are used for annotation purposes ([[:Category:todo:reduce mathslang]]), etc. All these forms have in common is that they can combine wiki pages in a way that is independent of their authors.

A category may have its own content (and category)
:''see [https://en.xen.wiki/w/Special:AllPages?from=&to=&namespace=14 Pages in the namespace '''Category''']''
or not yet (despite being used to categorize other pages):
:''see [[Special:WantedCategories]]''

Some more examples:

* [[:Category:todo:add_definition|todo:add definition]] - here a term needs to be defined
* [[:Category:Todo]] - category of categories that flag articles

=== Discussion ===
Discussion pages are provided to each content page. Please keep in mind that the email notification to people who are observing the page cannot be taken back!

Sometimes you will wait forever for a reply because nobody monitors the page or topic. In this case you should find out who is most familiar with the topic and ask that person directly. You can check the history of pages for its authors. Another approach is to ask questions on further media, examples are listed on the page [[GeneralDiscussion]].

== Links ==
* [[Help:Help]]
* [[Conventions]]

[[Category:help]]
[[Category:syntax]]
[[Category:wiki help]]
[[Category:help]]

Redirect

2018-10-16T07:04:43Z

PiotrGrochowski: Changed redirect target from Help:Wiki help#Redirects to Help:Help#Redirects

#REDIRECT [[Help:Help#Redirects]]

Hard redirect

2018-10-16T07:04:26Z

PiotrGrochowski: Changed redirect target from en:Help:Wiki help#Redirects to en:Help:Help#Redirects

#REDIRECT [[:en:Help:Help#Redirects]]

Soft redirect

2018-10-16T07:04:02Z

PiotrGrochowski: Changed redirect target from Help:Wiki help#Redirects to Help:Help#Redirects

#redirect [[Help:Help#Redirects]]

Wiki help

2018-10-16T07:03:15Z

PiotrGrochowski: revert vandalism

#REDIRECT [[Help:Help]]

Help:User pages

2018-10-16T07:02:28Z

PiotrGrochowski:

The wiki currently requires a user to be logged in to edit its content. Each wiki edit is linked to a user page in the history. This user page may be used for telling a few words about the editor: This may be strengths, weaknesses, to-do lists and so on. This page is often a good starting point for the editor, containing todo-lists, project descriptions, and any other purpose more or less related to the wiki work.

A user page gets added to the [[:category:wiki editor|category wiki editor]] if the line <code><nowiki>[[Category:Wiki editor]]</nowiki></code> is added in the wiki markup.

The talk page of the user page can be used to directly get into contact with the editor. If someone changes this page, the editor gets a notification bar displayed if she/he does the next navigates step within the wiki.

== See also ==
* [[Help:Help]]
* [[:category:wiki editor]]

[[category:help]]

User talk:Xenwolf

2018-10-16T07:01:03Z

PiotrGrochowski:

When in the MonoBook skin, the Search bar is down, so it requires scrolling. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 11:47, 17 September 2018 (UTC)
: I suggest not to use a user-defined skin. After Wikipedia switched to Vector, MonoBook seems to be no longer maintained by the MediaWiki team. Maybe you can still use MonoBook and change the search bar position via user-defined CSS and/or user-defined JavaScript, but I'm currently not able to help you with this issue. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 13:27, 17 September 2018 (UTC)
:: That's in [[Special:Preferences]] by the way. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 14:01, 17 September 2018 (UTC)

A nice advantage of MediaWiki is that broken links are immediately visible, which is helpful in pages like [[EDO#100...199]]. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 14:01, 17 September 2018 (UTC)
: Absolutely. I even more like the MediaWiki tables, look here for a [[SandBox#Table_Syntax|little table demo]] :) --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 14:09, 17 September 2018 (UTC)

Why is there a site named dev.xen.wiki that copies the xenharmonic articles and uses Timeless as the default skin? What's the point of that site? [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 13:10, 18 September 2018 (UTC)
: It's a test page as it seems, there was a [https://www.facebook.com/groups/xenwiki/permalink/2119836524949007/ poll on Facebook which skin to prefer], it's still open, but a majority for Default (=Vector) is on the horizon. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 14:28, 18 September 2018 (UTC)

== Why did you ban me?! ==

Allowing any [[list model]] is the exact opposite of a personal opinion! You are so ironic! [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 07:04, 15 October 2018 (UTC)
: Nobody disallows your opinion. But it's extremely disturbing that you tell people working on this wikis for years that they are childish if they use the simplest thing possible to write lists. Maybe it's a misconception to think that MediaWiki is about coding, it's about writing and communication. ''Did you read'' the help pages in [[wikipedia:Help:List|Wikipedia]] or [[mediawikiwiki:Help:List|MediaWiki]]? What do you think about their expertise? They start with simple lists, later they explain HTML-like features, but they try to keep the wiki text as readable as possible and usable in diffs. Most (if not all lists we used in this wiki were simple enough for ''basic lists''). Show me the actual problem that can be solved via [[template:list]] that cannot be solved via * and #. I mean: ''find'' an existing one, not ''construct'' one. I see a problem in very long lines containing a lot of <code><nowiki></li><li></nowiki></code> markup: they are not easy to change and they have a lower signal-to-noise ratio. And this can be solved by basic lists. <br/> And as to answer your initial question: You try to invent Conventions for the wiki with a lot of effort and when I tried to correct you about this, you show how fast you can revert my justified changes and tell me that I'm vandalizing the wiki? I have not enough time to play this ''I'm better than you'' game. I have the feeling that you don't read carefully what other people write. <br/> Best regards --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 13:19, 15 October 2018 (UTC)
:: It's true that you are worse than me. You are the one trying to push a personal opinion. All [[list model]]s are equally acceptable and a specific one should not be taught in [[:en:Help:Wiki help]], no matter what advantages it might have. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 14:42, 15 October 2018 (UTC)
::: I suggest to set up an own page "Help:Links" or "Help:Link" to contain the options <code><nowiki><ul><li></li></ul></nowiki></code> and <code><nowiki><ol><li></li></ol></nowiki></code> as well as your list template(s), provided that they are introduced with realistic use cases not as "just another opinion". Pages with style A ("my pages") and pages with style B ("your pages") without good reason will cause endless future wars, won't it? As for your claim that preferring * and # would show a "personal opinion": you remember your "unofficial poll"? I hope this helps you understand why a help page should start to teach the simplest way first. And when you claim ''"All [[list model]]s are equally acceptable"'', this is wrong. A list template will have limits, and <code><nowiki><br/></nowiki></code> doesn't build a valid list, right? The HTML list tags are acceptable if they are well structured even in the source code since basically ''everyone'' should be able to edit the wiki not only those who are coding in HTML. Best regards --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 15:37, 15 October 2018 (UTC)
:::: Realistic use cases don't matter. Every single list model is subjectively best, so it's very wrong to include only one in [[:en:Help:Wiki help]]. And all [[list model]]s really are equally acceptable; it's racist to say that a plaintext or defined list model results in an invalid list. And the convention that an edit consisting only of a list model change should not be done is supposed to prevent the edit war bug. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 16:11, 15 October 2018 (UTC)
::::Maybe [[:en:Help:Wiki help#Lists]] should soft redirect to [[:en:List model]], like [[:dev:Help:Wiki help#Lists]] does? Whatever the final solution may be, it definitely must show all [[list model]]s as possibilities that users may use, with no subjective recommendation or anything. The objective advantages and disadvantages are too specific to be shown in [[:en:Help:Wiki help|the xen.wiki school]], and they can always be found out by users later on, so it's best to state all [[list model]]s neutrally. Can't believe you are complaining while doing worse in [[:en:Help:Wiki help]]. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 05:17, 16 October 2018 (UTC)

Help:Introduction

2018-10-15T06:58:03Z

PiotrGrochowski: might as well merge

; [[Xenharmonic Wiki:Conventions|Conventions]]: a few thoughts about a few conventions currently in use
; [[Help:How to Get Your Xenwiki Account|How to Get Your Xenwiki Account]]: an account is necessary to edit articles, it provides also a user talk page
; Start your [[Help:User pages|user page]] (was before [[Editor yourName]]): say a few words about you as a wiki editor (user pages also serve as the temporary "host" (root) for subpages where complicated edits are carried out)

If you have any (seemingly) unsolvable problem, look or ask for a solution [[{{TALKPAGENAME}}|on the talk page]], Thanks :-)

__FORCETOC__

Wikis should be easy to edit, so most text you add or change will appear as you expect it. But you can do more if you want.

[https://wikiindex.org/Xenharmonic_Wiki This wiki] is currently in [https://wikiindex.org/Category:LoginToEdit login-to-edit mode] after the move from [[Wikispaces]] to MediaWiki-based installation (the edit mode may change again some time). For editing or adding information to the wiki, you have to join the wiki. See [[How to Get Your Xenwiki Account]] for more!

== Using links ==
What you always have dreamed of (for your paper notes) - in a wiki it becomes true: fast links!

If you enclose a word in double brackets (like <code><nowiki>[[word]]</nowiki></code>), you get an internal link, hopefully to an already existing page - if not, it appears red, and you just created the opportunity for your co-authors to add the details. If you want a text differing from the link target, you can place it after a pipe symbol: <code><nowiki>[[word|different text]]</nowiki></code>.

For external links, simply place the URL into the text, certain protocols (http, https, mailto) will be automatically transformed into links. For alternate text, use single brackets, start with the URL and after a space enter the title, like so <code><nowiki>[http://acid3.acidtests.org the Acid3 test]</nowiki></code> which renders to [http://acid3.acidtests.org the Acid3 test].

=== Link text ===
''tbd''

The link text should relate to the link target, don't use "[[help:here-links|here]]".

=== Redirects ===
There are 3 types of redirects: soft redirect, regular redirect and hard redirect.

A "soft redirect" or "smart redirect" is just an ordinary page with the text "See XYZ.", where XYZ is a link to the target of the smart redirect. A smart redirect should not point to another smart redirect - if you see this you should fix it by shortcutting past the second one and making the first redirect point directly to the ultimate target.

A regular redirect is performed by <code><nowiki>#REDIRECT [[Page title]]</nowiki></code>. A double redirect should be fixed by the same method as above, or by replacing it with a hard redirect. Regular redirects did not exist in <code>wikispaces</code>. This kind of redirect displays a small information message when it was used to display the current page.

Real or "hard" redirects directly go to the target page without a notice of redirect. Therefore, they should only be used in the case of simple misspellings or alternate spellings. The rule is that if anyone might ever want to flesh the redirect out into its own article, it should not be a hard redirect. Hard redirects can be chained with no problem, which is why they're still appropriate to use for misspellings. The code for a hard redirect in MediaWiki is <code><nowiki>#REDIRECT [[:en:Page title]]</nowiki></code>. Hard redirects worked somewhat differently in <code>wikispaces</code>; only admins could create or change them, and they could not redirect to a section.

see also

* http://groups.yahoo.com/neo/groups/tuning/conversations/topics/101227 - redirects were discussed on the [[tuning_list|tuning list]] in the [[Wikispaces]] era
* [[:Category:soft redirect|list of pages tagged as soft redirect]] - pages that contain only see <tt>[[</tt><nowiki>''some other page''</nowiki><tt>]]</tt> (be cautious with other tags on these)
* [[:Category:smart redirect|list of pages tagged as smart redirect]] - pages that have more information (a small definition, classification etc.)

== Text formatting ==

=== Emphasis ===
Some typographic markup can easy be added by including the phrase in doubled <code>''</code> or tripled <code>'''</code> (apostrophe chars):

* ''italic'' using <nowiki>''italic''</nowiki>
* '''bold''' using <nowiki>'''bold'''</nowiki>
* '''''bold italic''''' using <nowiki>'''''bold italic'''''</nowiki>

=== Lists ===

You can create lists with one of the [[list model]]s:

* first level
** second level
** second level
* first level

# first level
## second level
## second level
# first level

...and you can of course mix both, if you know what you're doing ;)

# first level
#* second level
## second level
* first level

=== Tables ===
You can add tables to wiki pages:
{| class="wikitable"
! what you write
|
<nowiki>
{| class="wikitable"
|+ The Title
! column A
! column B
|-
| cell A1
| cell B1
|-
| cell A2
| cell B2
|}</nowiki>
|-
! what you get
|
{| class="wikitable"
|+ The Title
! column A
! column B
|-
| cell A1
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|-
| cell A2
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|}
|}

==== Adjusting Text Alignment ====
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| left aligned text
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|-
| style="text-align:center;" | centered text
| place <code><nowiki>style="text-align:center;" |</nowiki></code> after the opening pipe
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==== External helper tools ====
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[http://www.tablesgenerator.com/mediawiki_tables Generate tables in MediaWiki format - TablesGenerator.com]

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{| class="wikitable"
! colspan="3" | 1
! 2
! 3
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| rowspan="2" | A
| colspan="3" | B
| C
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| colspan="2" | E
|-
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| H
|}
</nowiki>

will be rendered into this table:

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! 2
! 3
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[[Category:help]]
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Wiki help

2018-10-15T06:57:06Z

PiotrGrochowski: That page is blatantly wrong

#REDIRECT [[Help:Help]]

Help:Introduction

2018-10-15T06:56:21Z

PiotrGrochowski: NO

Soft redirect

2018-10-15T06:56:00Z

PiotrGrochowski:

See [[Main Page]].

[[Category:Soft redirect| ]]

Redirect

2018-10-15T06:55:38Z

PiotrGrochowski: Undo revision 37088 by Xenwolf (talk)

#REDIRECT [[Main Page]]

Hard redirect

2018-10-15T06:54:38Z

PiotrGrochowski: Undo revision 37087 by Xenwolf (talk)

#REDIRECT [[:en:]]

Help:User pages

2018-10-15T06:53:34Z

PiotrGrochowski: STOP USING MISCONCEPTIONS

User pages may be used for any purpose.

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[[category:help]]

Xenharmonic Wiki:Conventions

2018-10-15T06:52:59Z

PiotrGrochowski: revert vandalism; list model is not allowed to be biased. STOP!

A few conventions have evolved in this wiki that would be good to keep consistent with. Any pages that break the conventions may have links to them removed.

(please feel free to add rules in use here, and please provide reasons for it! Thanks!)

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{| class="wikitable"
! correct
! wrong
|-
| style="background-color: #efe" | 701.995
| style="background-color: #fee" | <s>702,995</s>
|}

==Logarithmic interval measures ==
For normal tables, we decided to stick to [[cent|cent]] values with a fixed precision, mostly 2 or 3 in the fractional part. 3 is helpful when serving as a canonical reference point for people to compare their own calculations with, which helps in the development of table creators such as [[Piotr Grochowski]]'s [https://scratch.mit.edu/projects/249278782/ one for Selected just intervals by error]. Its main purpose is to give an idea of interval sizes and relation to known intervals in the [[12edo|12edo]] system of western music. Alternative measures are acceptable if they better reflect the idea behind an interval system, scale etc. More than one logarithmic size column should be avoided.

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[[Category:about]]
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[[Category:help]]

User talk:PiotrGrochowski

2018-10-15T06:51:18Z

PiotrGrochowski:

User:Lhearne/Extra-Diatonic Intervals

2018-10-15T06:39:57Z

PiotrGrochowski:

Today a small under of competing interval naming schemes exist for the description of microtonal music. More common than any particular defined standard are certain tendencies for microtonal interval naming, or names for specific intervals. While risking the creation of simply another competing standard, an effort is made to develop a scheme that is able to take the best aspects of the existing standards and apply them in a formal interval naming system built on common undefined practice. Such a system is developed, where in addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', only the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees. Using this system all intervals in three fifths of all [[Equal division of the octave|edo]]<nowiki/>s up to 50 can be named such that 'S' and 's' correspond to a displacement of an interval up or down a single interval of the edo, respectively. Many commonly used [[MOS scale|MOS scales]] may also be described using this scheme such that these scales' interval names are consistent expression in any tuning that supports them. The resultant scheme can also be easily mapped to any of the current naming standards, and may even facilitate translation between. The resulting scheme should improve pedagogy and communication in microtonal music.

== Background ==

=== The origin of diatonic interval names ===
[[File:Mesopotamian interval names table.jpg|thumb|500x500px|Mesopotamian interval names, from http://www.historyofmusictheory.com/?page_id=130, accessed October 7, 2018.]]
Music theory describing the use of heptatonic-diatonic scales, including interval names, has been traced back as far as 2000BC, deciphered from a Sumerian cuneiform tablet from Nippur by Kilmer (1986). From Kummel (1970) we know that 'the names given to the seven tunings/scales were derived from the specific intervals on which the tuning procedure started' (Kilmer, 1986). This formed the basis of their musical notation ([http://www.jstor.org/stable/985853. Kilmer, 2016]). The table to the right following table displays the Ancient Mesopotamian interval names accompanied by their modern names.

Kilmer also writes that 'the ancient Mesopotamian musicians/“musicologists” knew what we call today the Pythagorean series of fifths, and that the series could be accomplished within a single octave by means of “inversion.” '. The Mesopotamian's music and theory was passed down through the Babylonians and the Assyrians to the Ancient Greeks, as well as their mathematics, particularly concerning musical and acoustical sound [[ratios]] (Ibid, [http://math-cs.aut.ac.ir/~shamsi/HoM/Hodgkin%20-%20A%20History%20of%20Mathematics%20From%20Mesopotamia%20to%20Modernity.pdf Hodgekin, 2005]).

Such mathematical and musical ideas are attributed to Pythagoras, who undoubtedly made them popular., although many scholars suggest he may have learned these ideas from his Babylonian and Egyptian mentors. None the less, Pythagoras' idea that that by dividing the length of a string into ratios of halves, thirds, quarters and fifths created the musical intervals of an octave, a perfect fifth, an octave again, and a major third form the basis of Ancient Greek music theory (http://www.historyofmusictheory.com/?page_id=20). His tuning of the diatonic scale by only octaves and perfect fifths ([[Pythagorean tuning]]) is influential through to today.

=== Ancient Greek interval names ===
Intervals in Ancient Greek music were written either as string length ratios, after Pythagoras, or as positions in a [[tetrachord]].

[[2/1]], the [[octave]], was named ''diapason'' meaning ''<nowiki/>'''through all [strings]'

[[3/2]], the [[perfect fifth]] was labelled ''diapente,'' meaning 'through 5 [strings]'

[[4/3]], the [[perfect fourth]], was labelled ''diatessaron'', meaning 'through 4 [strings]'

''dieses,'' 'sending through', refers to any interval smaller than about 1/3 of a perfect fourth

''tonos'' referred both to the interval of a whole tone, and something more akin to [[mode]] or key in the modern sense ([http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf Chalmers, 1993])

''ditone'' referred to the interval made by stacking two [[9/8]] whole tones, resulting in [[81/64]], the Pythagorean major third. ([[Joe Monzo|Monzo]], http://www.tonalsoft.com)

[[256/243]] - the ''limma'', which is the ratio between left over after subtracting two 9/8 tones (together making a ditone) a perfect fourth, the ''diatonic semitone'' of the Pythagorean diatonic scale

[[2187/2048]] - the ''apotome'', which is the ratio between the tone and the limma, the ''chromatic semitone'' of the Pythagorean diatonic scale

The Ancient greek term ''diatonon'', meaning 'through tones', refers to the genus with two whole tones and a semitone, or any genus in which no interval is greater than one half of the fourth (Chalmers, 1993). The Pythaogrean diatonic scale is the scale that may be built from one two Pythagorean tetrachords, and the left over interval of 9/8.

=== Zarlino and Meantone ===
[[File:Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1.png|thumb|566x573px|''Le institutioni harmoniche,'' Zarlino, 1558, Cap. 15: Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1, pg. 25.]]
Intervals were referred to by the Ancient Greek names through the the 18th century, as Latin names. By the Renaissance it had been discovered that a Pythagorean diminished fourth sounded sweet, and approximated the string length ratio 5/4. This just tuning for the major third was sought after, along with the complementary 6/5 tuning for the minor third, and octave complements to both - 8/5 for the minor sixth and 5/3 for the major sixth. Influential Italian music theorist and composer Gioseffo Zarlino put forth that choirs tuned the diatonic scale to the tuning built from this tetrachord, the ''intense diatonic scale'', also known as the ''syntonic or syntonus diatonic scale'' or the ''Ptolemaic sequence'':

1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

Included in this scale, however, were ''wolf intervals:'' imperfect consonances that occurred as tunings of the same interval as perfect consonances. For example, between 1/1 and 3/2, 4/3 and 1/1, 5/3 and 5/4; and 5/4 and 15/8 occurs the perfect fifth, 3/2, whereas between 9/8 and 5/3 occurs the wolf fifth, 40/27, flat of 3/2 by 81/80. This was also the interval by which four 3/2 fifths missed 5/1 (the interval two octaves above 5/4). It was named the ''syntonic comma'' after Ptolemy's ''syntonus'' or ''intense diatonic tetrachord'' which consists of the intervals 9/8, 10/9 and 16/15, where 9/8 and 10/9 differ by this interval. By making the syntonic comma a unison the wolf fifth could be made a perfect fifth. It was discovered that this could be achieved by flattening (tempering) the perfect fifth by some fraction of this comma such that four of these fifths less two octaves gave an approximation of 5/4. Where two fifths less an octave give 9/8, the next two add another 10/9 to result in the 5/4. 9/8 and 10/9 were referred to as the ''major tone'' (''tunono maggiore'') and ''minor tone'' (''tunono minore''), respectively, and where this tuning led to them being equated, it was referred to as Meantone temperament, which is said to 'temper out' the syntonic comma. Zarlino advocated the flattening of the fifth by 2/7 of a comma, leading to 2/7-comma Meantone, but also described 1/3-comma and 1/4-comma Meantone as usable (Zarlino, 1558).

The diagram on the right, from Zarlino's 1558 treatise ''Le istitutioni harmoniche'' associates many intervals with their tuning as perfect consonances. The perfect tuning for the ditone was considered then to be 5/4, rather than 81/64. The interval for which 6/5 is considered a perfect tuning was referred to as a ''semiditone'' (labelled also in ''Le istitutioni harmoniche'' by as ''Trihemituono)''. This may seem odd to us now, but in Latin 'semi' referred not to 'half', but to 'smaller', so 'semiditone' translated to something like 'smaller ditone'. Additionally 'semitone' referred to the interval smaller than the 'tone'. Like the tone, this interval possessed two alternative perfect tunings: 16/15, the difference between 15/8 and 2/1, or 5/4 and 4/3, and 25/24, the difference between 6/5 and 5/4. 16/15 was referred to as the ''major semitone'' (''semituono maggiore'') and 25/24 as the ''minor semitone (semituono maggiore'').

In addition to the Latin interval names, derived from the Ancient Greek interval names, we see on the diagram a single interval name in Italian: ''Essachordo maggiore'', referring to the ratio 5/3, which we are tempted to translate to 'major sixth'. Chapter 16, ''Quel che sia Consonanze semplice, e Composta; & che nel Senario si ritouano le sorme di tutte le somplici consonanze; & onde habbia origine l'Essachordo minore'', puts forward that the ''Essachordo minore,'' or perhaps 'minor sixth' be tuned to 8/5.

In the 1691 ''Lettre de Monsieur Huygens à l'Auteur [Henri Basnage de Beauval] touchant le Cycle Harmonique,'' theorist Christiaan Huygens gave names and ratios to common intervals and mapped them to 31-tET, which very closely approximates 1/4-comma Meantone. Translated from French, 3/2 was labelled a Fifth, 4/3 a Fourth, 5/4 a major Third, 6/5 and minor Third, 5/3 a major Sixth and 8/5 a minor Sixth. Here we really begin to see today's interval names.

=== English interval names in the Baroque ===
[[File:Harmonics, or The Philosophy of Musical Sounds, Section 2 figure 3.png|thumb|517x548px|''Harmonics, or The Philosophy of Musical Sounds'', Edition 2, Smith, 1759, Section 2: On the Names and Notation of consonance and their intervals, Fig. 2 & 3 , pg. 10]]
After English superseded Latin as the the main language of scholarship, the Latin interval names were rejected and the convention we saw in Zarlino's Italian for naming the smaller of a pair of sizes of an interval 'minor' and the larger 'major' was further applied.

Music theorist and mathematician Robert Smith provides the diagram and table on the right in his 1749 ''Harmonics, or, The Philosophy of Musical Sounds,'' with the description: <blockquote>'Fig. 2. If a musical string ''CO'' and it's parts ''DO'', ''EO'', ''FO'', ''GO'', ''AO'', ''BO'', ''cO'', be in proportion to one another as the numbers 1, 8/9, 4/5, 3/4, 2/3, 3/5, 8/15, 1/2, their vibrations will exhibit the system of 8 sounds which musicians donate by the letters ''C'', ''D'', ''E'', ''F'', ''G'', ''A'', ''B'', ''c''.</blockquote><blockquote>Fig. 3. And supposing those strings to be ranged like ordinates to a right line ''Cc'', and their distances ''CD'', ''DE'', ''EF'', ''FG'', ''GA'', ''AB'', ''BC'', not to be the differences of their lengths, as in fig. 2. but to be the magnitudes proportional to the intervals of their sounds, the received Names of these intervals are shewn in the following Table; and are taken from the numbers of the strings or sounds in each interval inclusively; as a Second, Third, Fourth, Fifth, &c, with the epithet of ''major'' or ''minor'', according as the name or number belongs to a greater of smaller total interval; the difference of which results chiefly from the different magnitudes of the major and minor second, called the Tone and Hemitone.'</blockquote>We note that Smith uses the tuning of the diatonic scale that Zarlino put forward: the Ptolemaic Sequence.

We may be surprised to see 4:3 here labelled as a minor Fourth, and 3/2 as a major Fifth, but it is obvious that this naming is more consistent than today's. Smith adds that <blockquote>'Any one of the ratios in the third column of the foregoing Table, except 80 to 81, or any one of them compounded once of oftener with the ratio 2 to 1 or 1 to 2, is called a Perfect ratio when reduced to it's least terms. And when the times of the single vibrations of any two sounds have a perfect ratio, the consonance and it's interval too after called Perfect; and is called Imperfect or Tempered when that perfect ratio and interval is a little increased or decreased.'</blockquote><blockquote>'Any small increment of decrement of a perfect interval is called respectively the Sharp or Flat Temperament of the imperfect consonance, and is measured most conveniently by the proportion it bears to the comma'</blockquote>Therefore in this system 3/2 is the ''Perfect major Fifth'' and 5/4 the ''Perfect major Third''. 81/64 might be labelled a ''comma sharp major Third'', 32/27 a ''comma flat minor Third'', and the 1/4-comma Meantone fifth a ''1/4-comma flat major Fifth''. The interval naming scheme Smith describes may be immediately applied to 5-limit microtonal systems. There is an inconsistency, however, where it seems that 9/8 should be called a ''Perfect major Second,'' but that, while 9/5 be named a ''comma sharp minor Seventh'', it's inverse, 10/9, is a ''Perfect minor Tone.''

In ''An Elementary Treatise on Musical Intervals and Temperament,'' published in 1876'','' R. H. M Bosanquet refers to 5/4 as the ''perfect third'', and 81/64 as the P''ythagorean third''. Bosanquet also labels other intervals of the Pythagorean diatonic scale similarly, i.e. 256/243, the limma, is labelled the ''Pythagorean semitone'', and 27/16 the ''Pythagorean sixth''. 81/80 is labelled the ''ordinary comma'', or simple the ''comma,'' and the Pythagorean comma is defined as the difference between twelve fifths and seven octaves. The apotome of 2187/2048 is referred to as Apatomè Pythagoria. The following relationships are then described: (<span style="color:#FF0000">'''<big>the image is too inappropriate to show, see pg 28 for rules regarding posting images to the Internet</span>'''</big>)

=== Helmholtz and Ellis ===
[[File:Helmholtz consonances table.png|thumb|617x469px|Table describing the influence of the different consonances on one another, up to the 9th partial, from ''On the Sensations of Tone as a Phycological Basis for Theory of Music'', Helmholtz, 1863, Translation by Ellis, 1875, pg. 187|]]
Through the investigations of Galileo (1638), Newton, Euler (1729), and Bernouilli (1771), theorist Hermann von Helmholtz was aware that ratios governing the lengths of strings existed also for the vibrations of the tones they produced. His investigation of the harmonic series associated with these ratios of vibration led him to the consideration of ratios above the 5-limit. In his seminal ''On the Sensations of Tone as a Physiological Basis for the Theory of Music'', published in German in 1863 and translated into English in 1875 by Alexander Ellis, he listed just intervals as show in the table to the right. It is interesting to note that 8:9 is labelled a 'Second' rather than a 'major Second'. The minor Seventh is shown as 5:9 rather than as 9:16 seemingly because of the 9 partial limit imposed on the table. It is also worth noting that 5:7 is labelled a subminor Fifth. 'Super', indicated in notation with a '+', raises an interval by 35:36, the septimal quarter tone, and 'sub', indicated with a '-' lowers by the same interval with the exception of the Supersecond 7:8, which lies 63:64, the septimal comma above the Second. The subminor fifth is not included in this as no minor Fifth is shown. If we assume that 'sub' lowers an interval 35:36, then the minor Fifth would be 18:25, 80:81 above Smith's 45:64 minor Fifth, however in table 2 below, Ellis labels 18:25 a ''superfluous Fourth'', and it's inverse, 25:32, an ''acute diminished Fifth'', whilst 64:45 is labelled a ''diminished Fifth'' and its inverse 32:45 a ''false Fourth or Tritone.'' If we label 9:10 as a 'major Second', and 7:8 as a 'supermajor Second' then they differ by 35:36, the major Second is the inverse of the minor Seventh, and the supermajor Second is the octave inverse of the subminor Seventh. Perhaps 'major' has been left off name of the major Fifth, and minor off the name of the minor Fourth since the time of Smith. We can add to this table the remaining octave inversions as well as the super Fourth and sub Fifth.
{| class="wikitable"
|+Table 1. Additional Intervals
!Intervals
!Notation
!Ratio
!Cents in the interval
|-
|Major Second
|D
|9:10
|182
|-
|Sub Fourth
|F-
|16:21
|471
|-
|Supermajor Fourth
|F#+
|7:10
|617
|-
|Super Fifth
|G+
|21:32
|729
|-
|Subminor Sixth
|A♭-
|9:14
|765
|-
|Supermajor Sixth
|A+
|7:12
|933
|}
Helmholtz defined the ''perfect consonances'' as the Octave, Twelfth and Double Octave as well as Fourth and Fifth. The major Sixth and major Third are next called ''medial consonances'', considered to in the era of Pythagorean tuning to be ''imperfect consonances'', which Helmholtz defined instead to be the minor Third and the minor Sixth.

Regarding the tuning of the intervals however, those corresponding to simple ratios of vibration are, as in Smith, referred to as perfect, however hey are also described as 'justly-intoned', or by Ellis as 'just'. The perfect tuning for the semitone is listed as 16/15, or 182c. The perfect tunings are compared to the Pythagorean tunings, where the Pythagorean tuning of the major Third and sixth are described as 81/80 above the perfect tunings, and of the minor Third, minor Sixth and semitone to be 81/80 below the perfect tunings. Helmholtz notes that the Pythagorean tunings are closer to the equal tempered tunings than the perfect tunings. Helmholtz also describes the Pythagorean Tritone as of 612c.

Ellis includes an [[Ellis' interval table|additional table]] providing names for many different just and tempered intervals, perfect and imperfect. The interval names do not appear to follow any sort of consistent naming system, rather intervals seem to be named case-by-case. They also largely do not correspond to the interval names used by Helmholtz.

In ''Notes on the Observations of Musical Beats'', Proc. R. Soc. Lond. 1880, Ellis named many just intervals of the 7-limit (including 3 and 5-limit intervals):<blockquote>‘Fifth 3:2, Fourth 4:3, Major Third 5:4, Minor Third 6:5, Major Sixth 5:3, Sub-Fifth 7:5, Super-Fourth 10:7, Super-major Third 9:7, Sub-minor Sixth 14:9, Sub-minor Third 7:6, Super-major Sixth 12:7, Sub-minor or Harmonic Seventh 7:4, Super-major Second 8:7, Major Tone 9:8, Minor Tone 10:9, Small Major Seventh 9:5 and Diatomic (sic.) Semitone 16:15’</blockquote>He later lists ‘The Major Sevenths 16:9 and 15:8’. The labeling of 16:9 as a Major Seventh and 9:5 as a Small Major Seventh is interesting and at odds with Smith's interval names. Given that 9:5 is larger than 16:9, and no Minor Seventh is mentioned, we can assume 16:9 was mislabeled as a Major Seventh and was understood to be a Minor Seventh, as referred to by Smith.

There is an inconsistency associated with the labeling of 9:5 as a Small Major Seventh also, as it lies a 3:2 Fifth above the 6:5 Minor Third, and we know a fifth and a minor third when added together to give a minor, rather than major seventh.

Ellis here uses the name for 8:7 we suggested above, Super-major Second, and includes our suggested Sub-minor Sixth and Super-major Sixth, however rather than Subminor Fifth and Supermajor Fourth, 7:5 and 10:7 are labelled Sub-Fifth and Super-Fourth, where in this instance sub and super are seen to raise and lower by 21:20 instead of by 36:35.

Ellis, in a footnote to his translation of Helmholtz,'s treatise also provides names for a single 11-limit interval. The interval 22:27, of 355c, introduced by Zalzal, says Ellis was termed a ''neutral Third'' by Herr J. P. N. Land originally in ''Over de Toonladders der Arabische Musiek'' (On the Scales of Arabic Music) in 1880. An interval a fourth higher than this is mentioned, but a ratio is not given, and it is not named. We can ourselves however find it's ratio as 11:18, and guess it's name to be a ''neutral Sixth'', given that it lies a perfect Fourth above the neutral Third. Following a similar process as in our completion of Helmholtz table above, and assuming that the octave inverse of a neutral Third should be a neutral Sixth we may introduce the following 11-limit intervals that see common use among music theorists and microtonal musicians through to today:
{| class="wikitable"
|+Table 3. 11-limit intervals
!Intervals
!Notation
!Ratio
!Cents in the interval
|-
|Neutral Second
|D♭^
|81:88
|143
|-
|Neutral Second
|Dv
|11:12
|151
|-
|Neutral Third
|E♭^
|9:11
|347
|-
|Neutral Third
|Ev
|22:27
|355
|-
|Neutral Sixth
|A♭^
|27:44
|845
|-
|Neutral Sixth
|Av
|11:18
|853
|-
|Neutral Seventh
|B♭^
|6:11
|1049
|-
|Neutral Seventh
|Bv
|44:81
|1057
|}
Each interval name has two sizes that differ by the comma 242:243. The notation included in the table is from HEWM notation, developed as an extension to the Helmholtz-Ellis use of '+' and '-' by Joe Monzo (http://www.tonalsoft.com/enc/h/hewm.aspx<nowiki/>).'^' indicates raising 'v' a lowered of 33/32. In HEWM notation '+' and '-' are refined to mean raising and lowering of 81/80 respectively and '>' and '<' are added instead to indicate raising and lowering of 64/63. Letter names correspond instead of the the Ptolemaic sequence, as in Smith's and Helmholtz' descriptions, but to a Pythagorean tuning of the diatonic scale, where '#' and '♭' and respectively raise and lower the apotome, 2187/2048. HEWM notation is not accompanied by an interval naming system.

=== Common interval names today ===
These interval names are used by theorists and microtonal musicians today, though 7:5 and 10:7 are given many different names, today also considered to be an augmented fourth and diminished fifth, lesser septimal tritone and greater septimal tritone, or simply as tritones. The fourth and fifth are today called perfect fourth and perfect fifth, and Smith's major Fourth and minor Fifth referred to as augmented fourth and diminished fifth respectively. As can be seen in Tchaikovsky's ''A Guide to the Practical Study of Harmony,'' by the beginning of end of the 19th century the familiar conventions for the naming of intervals were set, wherein
* Seconds, thirds, sixths and sevenths appear in the diatonic in two sizes, the larger labelled 'major' and the smaller, 'minor'.
* Major, when raised by a semitone, becomes 'augmented', and minor, lowered by a semitone, 'diminished'.
* The smaller of the two sizes of fourth and the larger of the two sizes of fifth are labelled 'perfect', along with the unison and octave.
* A perfect interval, when raised a semitone is labelled 'augmented', and when lowered a semitone, 'diminished'.

=== Fokker/Keenan Extended-diatonic interval-names ===

Considering the 11-limit otonal chord 4:5:6:7:9:11 a chain of thirds, in addition to the familiar major, minor, subminor, supermajor and neutral thirds, Dave Kennan labelled 5:7 a sub-diminished fifth and 7:11 an augmented fifth. 7:10, the inversion of 5:7, is labelled a diminished. 5:7, therefore, is also an augmented fourth. In terms of sevenths, 4:7 is subminor, 5:9 is minor and 11:6 is neutral.

Super and sub were further generalised by some theorists and musicians such that an interval a bit smaller than a major is referred to as a ''subminor third'', and an interval a bit larger than a minor third as a ''supraminor third''.

From this, Keenan defines a consistent interval naming system, meaning one which obeys diatonic interval arithmetic (In each column, the parenthesised prefix is the one that is implied when there is no prefix). When adding intervals the indexes are added together to give the index of the resulting interval. Keenan also adds corrections for each interval class to the indexes in order to account for inconsistencies that occur within diatonic interval arithmetic when concerning intervals greater than an octave, so that his system, unlike regular diatonic interval names, may be completely consistent.

As can be seen above, sub, super, augmented and diminished have also carried inconsistent meaning historically, where in Keenan's system they always alter intervals by the same amount.
{| class="wikitable"
|+
!Index
!Prefix for unisons, fourths, fifths, octaves
!Prefix for seconds, thirds, sixths, sevenths, ninths
|-
| -4
|double diminished
|subdiminished
|-
| -3
|subdiminished
|diminished
|-
| -2
|diminished
|subminor
|-
| -1
|sub
|minor
|-
|0
|(perfect)
|neutral
|-
|1
|super
|(major)
|-
|2
|augmented
|supermajor
|-
|3
|superaugmented
|augmented
|-
|4
|double augmented
|superaugmented
|}
The index values correspond most directly to degrees of 31edo, whose interval names by this method are given in the following table:
{| class="wikitable"
|+Extended-diatonic interval names in 31-tET
!31-tET degree
!Ratios
!Names
!
|-
|0
|1:1
|
|unison
|-
|1
|48:49 44:45 36:35 33:32
|(diminished second)
|super unison
|-
|2
|27:28 24:25 20:21
|subminor second
|(augmented unison)
|-
|3
|15:16 14:15
|minor second
|
|-
|4
|11:12 10:11
|neutral second
|
|-
|5
|9:10 8:9
|major second
|
|-
|6
|7:8
|supermajor second
|(diminished third)
|-
|7
|6:7
|(augmented second)
|subminor third
|-
|8
|5:6
|
|minor third
|-
|9
|9:11
|
|neutral third
|-
|10
|4:5
|(subdiminished fourth)
|major third
|-
|11
|11:14 7:9
|(diminished fourth)
|supermajor third
|-
|12
|16:21
|subfourth
|(augmented third)
|-
|13
|3:4
|perfect fourth
|
|-
|14
|8:11
|super fourth
|
|-
|15
|5:7
|augmented fourth
|(subdiminished fifth)
|-
|16
|7:10
|(superaugmented fourth)
|diminished fifth
|-
|17
|11:16
|
|sub fifth
|-
|18
|2:3
|
|perfect fifth
|-
|19
|21:32
|(diminished sixth)
|super fifth
|-
|20
|9:14 7:11
|subminor sixth
|(augmented fifth)
|-
|21
|5:8
|minor sixth
|(superaugmented fifth)
|-
|22
|11:18
|neutral sixth
|
|-
|23
|3:5
|major sixth
|
|-
|24
|7:12
|supermajor sixth
|(diminished seventh)
|-
|25
|4:7
|(augmented sixth)
|subminor seventh
|-
|26
|9:16 5:9
|
|minor seventh
|-
|27
|11:20 6:11
|
|neutral seventh
|-
|28
|8:15
|
|major seventh
|-
|29
|14:27
|(diminished octave)
|supermajor seventh
|-
|30
|18:35
|sub octave
|(augmented seventh)
|-
|31
|1:2
|octave
|
|}
The interval names shown in brackets could be said to be 'secondary', the others, 'primary'.

After releasing his system Keenan was informed that is was identical to the extended-diatonic interval-naming scheme of Adriaan Fokker but for the acknowledgment of more 11-limit ratios.This system depends on the tempering out of 81/80, where the diatonic major third, from four stacked fifths, approximates the just major third, 5/4. It also depends on the existence of neutral intervals, i.e., that the perfect fifth or equivalently, the chromatic semitone, subtends an even number of degrees of the ET. To simply to our familiar naming scheme for 12-tET, we observe that it applies to 24edo equally as directly as in 31-tET, where the prefixes correspond to degrees of the edo. Exactly the same is also true for 38-tET, twice 19-tET, a meantone which very closely approximates 1/3-comma meantone. Meantone temperament wherein the fifth is divided into two equally sized neutral thirds is referred to as neutral temperament. Whereas meantone temperament is generated by the fifth, iin neutral temperament the generator is half this interval, the neutral third. Where it was seen above that there are two neutral thirds, 9:11 and 22:27 that differ by 243/242, neutral temperament is at its most simple the temperament defined by this equivalence: the tempering out of 243/242, as meantone is defined by the tempering out of 81/80. The temperament that tempers out both 81/80 and 243/242 is called Mohajira, upon which Keenan's scheme can be said to be based.

The primary interval names resulting in this system's application to 24-tET and 38-tET is now show, along with 31-tET again for easy comparison, where 'M', 'm', 'P', 'N', 'A', 'd', 'S' and 's' are shorthand for major, minor, perfect, neutral, augmented, diminished, super and sub, respectively:

24edo: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 S4 A4/d5 s5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8

31edo: P1 S1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 S4 A4 d5 s5 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 s8 P8

38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 S4 A4 SA4/sd5 d5 s5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8

to Meantone tunings that are not Mohajira tunings, the regular diatonic interval names can be applied, but with the addition of double augmented and double diminished:

19edo: P1 A1 m2 M2 A2/d3 m3 M3 d4 P4 A4 d5 P5 A5 m6 M6 A6/d7 m7 d8 P8

26edo: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 AA4/AA5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8

Keenan adds further that if it is desired to distinguish between ratios that are in 31-tET approximated by the same number of steps, an addition prefix be added to describe the prime limit of the approximated interval. For 3-limit intervals, the obvious choice is 'Pythagorean', for 5-limit Keenan chooses 'classic', for 7, 'septimal, 11, 'undecimal' and 13, 'tridecimal'. When the highest prime is the same, Keenan suggests adding 'small' and 'large' as final prefixes for this purpose.

In non-Meantone tunings, the two definitions of major third - 4:5 and 64:81, the just (or classic) and Pythagorean major thirds no-longer correspond. If intervals are to receive unique names then to one or both of these major thirds must be added a prefix. Keenan has been involved with the development of both types of systems. Only when the major is defined by it's mapping as fourth fifths, i.e. 81/64, can conserve interval arithmetic.

=== [[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak] ===
One system in which 81/64 is the major third, sagispeak, was developed by [[Dave Keenan]] and others as an interval naming system that maps 1-1 with the Sagittal microtonal music notation system. Sagittal notation was developed as a generalised diatonic-based notation system applicable equally to [[just intonation]], [[Equal Temperaments|equal tunings]] and rank-''n'' [[temperaments]]. Dozens of different accidentals can be used on a regular diatonic [[Staff notation|staff]] to notate up to extremely fine divisions, however in most cases only a handful are needed. In sagispeak, each accidental is presented by a prefix, made up of a single letter, in most cases, followed by either 'ai' if the accidental raises a note, or 'ao' if it lowers a note. As in HEWM notation, Pythagorean intonation is assumed as a basis. Then the prefixes depart from Pythaogrean intonation, altering by commas and introducing other primes. In place of the prefixes 'sub' and 'super', generally signifying an alteration of 36/35 from 5-limit intervals or 64/63 for 3-limit, Sagittal features an accidental of [[64/63]], which may be used to take a Pythagorean major interval to a supermajor, minor to subminor, or perfect to super or sub. The prefix 'tao' indicates a decrease of 64/63 and and the prefix 'tai' an increase. Whereas in previous interval naming schemes 'major' and 'minor' were synonymous with the 5-limit tunings, in sagispeak they map instead to Pythagorean. A prefix is needed then to take a Pythagorean intoned interval to a 5-limit tuning. Where 5/4 is 81/80 below the the Pythagorean third, the prefixes 'pai' and 'pao' (where 'p' is for 'pental', as in, involving prime 5), which raise or lower a note by [[81/80]]<nowiki/>respectively. Similarly, 'vai' and 'vao', which raise or lower a note by [[33/32]] respectively, leading to ratios of 11.

Because it is built off of the diatonic scale, sagispeak conserves diatonic interval arithmetic, i.e. familiar relations in the diatonic scale, i.e. M2 + m3 = P4. As in Fokker/Keenan Extended-diatonic Interval-names, diatonic interval arithmetic is also extended, where, for example, tai-major 2 + tao-minor 3 = P4 (8/7 + 7/6 = 4/3), where opposite alterations cancel each other out, and diatonic interval arithmetic is conserved, a very useful property for a microtonal interval naming system to possess. Another helpful property of sagispeak is its generalised applicability to edos, just intonation and other tunings, where the same intervals maintain their spelling across different tunings. Despite these benefits however, many see Sagittal and Sagispeak as overly complex (even though the entire extended system need hardly ever be applied), and requiring too many new terms to be learnt.

We have seen that there are two competing definitions of a major third, the ratio '5/4' or the interval built from fourth stacked fifths, that may or may not correspond to 5/4. In meantone systems, those we are used to, they correspond, but in most edos they do not. Interval naming systems wherein the major third is defined as an approximation to 5/4 rather than as four fifths minus two octaves may benefit from a familiar name for 5/4, but they are unable to conserve diatonic interval arithmetic.

For comparison, 31edo is shown below in sagispeak:

31edo: P1 tai-1/vai-1 tao-m2 m2 vai-2m/vao-M2 M2 tai-M2 tao-m3 m3 vai-m3/vao-M3 M3 tai-M3 tao-4 P4 vai-4 A4 d5 vao-5 P5 tai-5 tao-m6 m6 vai-m6/vao-M6 M6 tai-M6 tao-m7 m7 vai-m7.vao-M7 M7 tai-M7 vao-8/tao-8 P8

=== Dave Keenan's most recent system ===
In 2016 Dave Keenan proposed an alternative generalised [http://dkeenan.com/Music/EdoIntervalNames.pdf microtonal interval naming system for edos]. In what might be understood as a generalisation of his extended-diatonic interval-naming system described above onto any equal tuning. Employing as prefixes the familiar 'sub', 'super', and 'neutral'. His scheme is based on the diatonic scale, however the diatonic interval names are not defined by their position in a cycle of fifths like is Sagispeak. In Keenan's system the ET's best 3/2 is first labelled P5, and the fourth P4. The interval half-way between the tonic and fifth is labelled the neutral third, or 'N3', and halfway between the fourth and the octave N6. Then the interval a perfect fifth larger than N3 is labelled N7, and the interval a fifth smaller than N6 labelled N2. The neutral intervals then lie either at a step of the ET, or between two steps. After this the remaining interval names are decided based on the distance they lie in pitch from the 7 labelled intervals, which make up the ''Neutral scale'', P1 N2 N3 P4 P5 N6 N7, which, like the diatonic, is an MOS scale, which may be labelled [[Neutral7|Neutral[7]]] 3|3 using [[Modal UDP Notation|Modal UPD notation]]. To name an interval in an edo, the number of steps of [[72edo]] that most closely approximate the size of the interval difference from a note of the neutral scale is first found. Then the prefix corresponding to that number of steps of 72edo is applied to the interval name. The following chart details this process (can't load the chart :( ). An interval just smaller than a major third in Keenan's system is labelled a ''narrow major third'', and an interval just wider than a [[6/5]] minor third a ''wide minor third'', however he notes that 'narrow' and 'wide' are only necessary in edos greater than 31. Note that the interval just wide of a minor third is labelled a 'supraminor third' rather than a 'superminor third'. This reflect recent tendencies among microtonal musicians and theorists.

Keenan's system is an elegant way to keep the 'major 3rd' label for 5/4, where labels depend on the size of the best fifth, however it suffers from it's applicability only to edos, and that it does not conserve interval arithmetic. Another potentially undesirable result of the system is that the major second approximates 10/9, and a ''wide major second'' 9/8, where as 9/8 is almost always considered a major second, and [[10/9]] often a narrow or small major second. One such system that considers 10/9 a narrow major second is that of Aaron Hunt.

=== Size-based systems ===
Microtonal theorist [[Aaron Andrew Hunt]] devised [http://musictheory.zentral.zone/huntsystem4.html the Hunt system], which includes interval name assignments for JI (just intonation) and edos based on [[41edo]]. Compared to Keenan's 72 interval names, Aaron's system includes 41. His system is based directly on 41edo, and unlike Keenan's system, interval are given the name of the closest step of 41edo, and no account is taken of the size of the edos fifth. In 41edo, Major, minor, augmented and diminished intervals are those obtained through the approximately Pythagorean cycle of fifths. Intervals one step of 41edo above these are given the prefix 'small', one step larger are given the prefix 'large', two steps smaller the prefix 'narrow' and two larger the prefix 'wide'. As a result, 5/4 is labelled a 'small major 3rd', or SM3 (not to be confused with a super major third, a label that does not exist in this system).

Neo-medieval musicians and early music historian and theorist [[Margo Schulter]] described her own [http://www.bestii.com/~mschulter/IntervalSpectrumRegions.txt interval naming scheme] built on approximations to JI intervals. Each interval names corresponds to an approximate size, and no particular edo is referenced. In her scheme middle major thirds range in size from 400-423 cents, and small major thirds from 372-400c. 5/4 is labelled a small major third, 81/64 a middle major third and 9/7 a large major third. Margo's scheme includes small, middle and large varieties of major, minor and neutral 2nds, 3rds, 6ths, 7ths; perfect fourth and fifths; and tritones, as well as a sub fifth and super fourth a dieses and comma and an octave less dieses and comma and ''interseptimals'', which correspond to intermediates, her name referencing the fact that they may each approximate two ratios of 7.

In Hunt's system when used in 41edo or JI diatonic interval arithmetic is conserved, but in other tunings it may not be, and Margo's system may not conserve diatonic interval arithmetic either. Both systems may be applied to arbitrary tunings, but the same intervals (defined, perhaps by a MOS scale) may not be given the same interval names across different tunings.

<ul><li>[[User:PiotrGrochowski/Extra-Diatonic Intervals]] gives each 43edo interval a name, then maps each desired interval to a 43edo interval. [[User:PiotrGrochowski/Extra-Diatonic Intervals — 50edo]] does this with 50edo. '''It has a very amazingly excellent solution to the 5/4 and 81/64 problem''': The 5/4 is named major third and this notation is split in half, while 81/64 is named high major third. 10/9 and 9/8 are both named major second. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]). </li></ul>Size-based systems are completely generalisable, but do not conserve interval arithmetic.

=== Ups and Downs ===
One final interval naming system, associated with the [[Ups and Downs Notation|Ups and Downs Notation]] system, belonging to microtonal theorist and musicians [[KiteGiedraitis|Kite Giedraitis]], like Sagittal is based on deviations from the diatonic scale. In this system however, deviations (from major, minor, perfect, augmented and diminished) are notated simply by the addition of up or down arrows: '^' or 'v', corresponding to raising or lowering of a single step of an edo. In some tunings ([[12edo]], [[19edo]] or [[31edo]] for example) 5/4 may be a M3, and in others a vM3 (downmajor 3rd) (e.g. [[15edo]], [[22edo]], 41edo, 72edo), or even an up-major 3rd (e.g. [[21edo]]). Ups and downs also includes neutrals, which lay exactly in-between major and minor intervals of the same degree, labelled '~' (mid). 'Up' and 'down' prefixes may be used before mid also, i.e. 'v~ 3). This system benefits from it's simplicity as well as it's conservation of interval arithmetic. It can be used for some MOS scales where one of the generators is a perfect fifth or a fraction of a perfect fifth, but not all of these (e.g. Diminished[8]), and not all MOS scales (if such scales are to be described, an additional pair of accidentals/qualifiers is used. Although the scales then are described, their intervals still are not given the same names in Ups and Downs' edo names). Another criticism of Kite's system that does not apply to the others is the fact that when an edo is doubled or multiplied by some simple fraction, and the best fifth is constant across the two edos, the same intervals may be be given different names.

[[Igliashon Jones]] is a supporter of this system, but for the relabeling of 'down' as 'sub' and 'up' as 'super' (or supra) and 'mid' as 'neutral', so that more common names are used, wherein 'super' infers a raise of 1 step of the edo, and 'sub' a lowering of one step. In this 'Extra-diatonic' system 'super' and 'sub' may be doubly applied, as in Ups and Downs, but they may not be applied before 'neutral' where in Ups and Downs they may be applied before 'mid'. The author's own extra-diatonic system is developed as a departure with caveat that 'S' and 's' prefixes are defined not as alterations by a single step of the edo, but by comma alterations as in Sagittal, in order that interval of MOS scales may be represented consistently across different tunings. Throughout the rest of the article the development is detailed, and the system defined.

== Premise: ==
Extra-diatonic names should be simple, generalisable, widely applicable, backwards compatible with standard diatonic notation and reflecting current informal practice as closely as possible. Extra-diatonic interval names are fifth based; extended from the familiar major, minor and perfect interval names so that diatonic interval arithmetic is conserved. ‘M’, ‘m’, and ‘P’ remain the short-hand for major, minor and perfect. ‘A’ and ‘d’ for Augmented and diminished may also be used in the familiar way. In cases where the chroma (the chromatic semitone, or augmented unison) is represented by multiple steps in the tuning the prefix ‘super’ raises major and perfect intervals by a single step while ‘sub’ lowers minor and perfect intervals, with short-hand ‘S’ and ‘s’. ‘S’ and ‘s’ may also be used to raise minor and lower major intervals respectively, reflecting occasion practice. In this case ‘S’ is short-hand for ‘supra’, and 's' for 'small'. They may also be used to raise or lower diminished and augmented intervals similarly. In this way this scheme is equivalent thus far to Ups and Downs notation, where ‘^’ or ‘up’ corresponds to ‘S’, ‘super’ or ‘supra’ and ‘v’ or ‘down’ to ‘sub’ or 'small' .

== Additions and examples: ==
''Neutrals'' and ''intermediates'' are also included, where neutrals occur between opposing sizes of a single generic interval the intermediates between each generic interval and the next.

Interval names for equal tunings are ranked in five tiers.

Interval names for equal tunings are ranked in eight tiers.
# Perfect
# Neutral
# Major, minor, A4 and d5 and, if the chroma is subtended by a single (positive) step of the edo, other augmented and diminished intervals
# Super, sub, supra and small prefixes to major, minor, perfect interval and to A4 and d5
# Intermediates
# Augmented and diminished intervals (for when the chroma is subtended by more than a single (positive) step of the edo)
# Super, sub, supra and small prefixes to augmented and diminished intervals
# Intervals augmented and diminished more than singularly
When more than one interval name corresponds to a specific interval, the names are privileged in order of the tiers. By this ordering, the first available name is the ‘primary’ for that interval, the second available ‘secondary’ and third 'tertiary'.

I initially intended for Intermediates to lay in the second tier with neutrals, but that led to primary intervals results that were not preferred by my colleagues over other options and I was persuaded to avoid using intermediates unless they are necessary.

=== Neutrals ===
N2, N3, N6 and N7, i.e. neutral 2nds, 3rds, 6ths and 7ths, falling exactly in-between the major and minor intervals of the same interval class, add native support for neutral-thirds temperament, where the N3 divides the P5 in exact halves and N2 divides the m3 is exact halves. In ups and downs neutrals indicated with '~' and said 'mid'.

Extending this familiar application to provide support for larger neutral scales, we add that neutrals occur also between P4 and A4; P5 and d5; P1 and A1; and P8 and d8.

Then [[Neutral7|Neutral[7]]] 3|3 can then be written:

P1 N2 N3 P4 P5 N6 N7 P8.

The same names give the primary interval names for [[7edo]], whose secondary intervals names are:

N1 m2/M2 m3/M3 N4 N5 m6/M6 m7/M7 N8.

The secondary interval names show that the chroma is equivalent to a unison in 7edo.

[[Neutral10|Neutral[10]]] 5|4 may then be written as

P1 N2 M2 N3 P4 N4 P5 N6 m7 N7 P8

[[Neutral17|Neutral[17]]] 8|8 may be written as

P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8,

which is equivalent to the primary interval names of [[17edo]].

=== Intermediates ===
To provide native support for [[The Archipelago|Barbados]] and Diminishes temperaments, intermediates are also added to the system. It should be noted immediately that intermediates are not as common to common microtonal interval naming as neutrals and though are a useful addition to this scheme, may be left out if desired. The appendix includes the MOS scales and edos from 'lists of all edos and MOS Scales', but without any intermediates.

‘2-3’ lies exactly half-way between M2 and m3 and divides the P4 in half. It may be read ‘second-third’ or ‘serd’. ‘6-7’, it’s octave-inverse lies exactly half-way between M6 and m7 and may be read ‘sixth-seventh’ or ‘sinth’.

‘1-2’ lies exactly half-way between P1 and m2, dividing the m2 in half. It may be read ‘unison-second’ or ‘unicond’. Its octave-inverse, ‘7-8’, lies exactly half-way between M7 and P8 and may be read ‘seventh-octave’ or ‘sevtave’.


‘3-4’ lies exactly half-way between M3 and P4, dividing the M6 in half. It may be read ‘third-fourth’ or ‘thourth’. It’s octave-inverse, ‘5-6’, lies exactly half-way between P5 and m6 and may be read ‘fifth-sixth’ or ‘fixth’. 

[[5edo]] can be spelled with the list of only these intermediates:

1-2 2-3 3-4 5-6 6-7 7-8.

The primary interval names for 5edo do not include them however, as they may be described by diatonic intervals:

P1 M2/m3 P4 P5 M6/m7 P8

Barbados[5] 2|2 may be described as

P1 2-3 P4 P5 6-7 P8,

and Barbados[9] 4|4 as

P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8.

The final intermediate, ‘4-5’ lies exactly between P4 and P5 and divides the octave in half. It may be read ‘fourth-fifth’ or ‘firth’. It is necessary for [[Diminished]] temperament, where the half-octave cannot be represented as any alteration of A4 or d5. Diminished temperament has a period of 1/4 of an octave, an approximation of 6/5. Therefore, the difference between four 6/5's and 2, 648/625, is tempered out.

Diminished[8] 1|0 (4) can be written as P1 sM2 Sm3 P4 4-5 Sm6 sM6 sM7 P8, and

[[Diminished12|Diminished[12]]] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 4-5 P5 Sm6 sM6 Sm7 sM7 P8,

== [[12edo]] and a problem ==
The primary interval names for 12edo are as we are familiar:

P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8,

but the interval labeled A4/d5 has the secondary name '4-5'.

We know that 12edo supports Diminished temperament, but we can't see that 12edo as equivalent to Diminished[12]: If we follow the rules from our premise, the notes of Diminished[12] in 12edo would give: P1 M2 m2 M3 m3 P4 A4/d5 P5 M6 m6 M7 m7 P8, with the majors and minor flipped. To be equivalent, sM must equal M and Sm must equal m.

=== [[10edo]], Pajara and a problem ===
10edo may be written as only neutrals and intermediates:

N1/1-2 N2 2-3 N3 3-4 N4/4-5/N5 5-6 N6 6-7 N7 7-8/N8

However it much more sense to write is using the primary interval name set:

P1 N2 M2/m3 N3 P4 N4/N5 P5 N6 M6/m7 N7 P8

It's secondary intervals as follows:

m2 Sm2/sM2 2-3 Sm3/sM3 M3 S4/s5 m6 Sm6/sM6 6-7 Sm7/sM7 M7

We can see that 10edo supports [[Neutral third scales]], given that we can make the interval names for Neutral[10] using the primary interval names for 10edo.

We know 10edo and 12edo both support Pajara temperament. Pajara[10] 2|2 (2) consists of:

P1 Sm2 M2 sM3 P4 sA4/Sd5 P5 Sm6 m7 sM7 P8,

and [[Pajara12|Pajara[12]]] 3|2 (2) of

P1 Sm2 M2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 m7 sM7 P8.

We can see Pajara[10] in 10edo, but in 12edo, wouldn’t sM3 be m3?

This is where my system diverges from Igliashon Jones’. We have to break our first rule here, or at least add some conditions to it.

== Comma association ==
To address this problem of consistency, we now state that when 81/80 is tempered out, M=sM and m=Sm, and when 64/63 is tempered out, M=SM and m=sm. In the case of sm and SM, ‘S’ and ‘s’ raise and lower by 64/63, and in the case of Sm and sM, ‘S’ and ‘s’ raise and lower by 81/80. In this way extra-diatonic interval names are equivalent to [http://forum.sagittal.org/viewforum.php?f=9 Sagispeak] interval names, where for sm and SM ‘S’ and ‘s’ are equivalent to ‘tai’ and ‘pao’ and for Sm and sM ‘S’ and ‘s’ are equivalent to ‘pai’ and ‘pao’.

It is important to note that given this change, 'S' and 's' may alter an interval by a different number of steps in an edo depending on which interval names they prefix. This may seem confusing, but it seems to reflect existing informal practice.

In 12edo, which represents the union between the two, where both 64/63 and 81/80 are tempered out, ‘S’ and ‘s’ do not raise or lower intervals at all. We can now easily see that 12edo supports Pajara, where simply removing all the ‘s’s and ‘S’s from Pajara[12] gives us our primary interval names of 12edo.

=== [[14edo]] and Injera ===
Like 10edo, 14edo may be written using all the neutrals and intermediates, but without any intervals described both as a neutral and as an intermediate:

N1 1-2 N2 2-3 N3 3-4 N4 4-5 N5 5-6 N6 6-7 N7 7-8 N8.

The primary interval names for 14edo are as follows:

P1 sm2 N2 SM2/sm3 N3 SM3/s4 P4 SA4/sd5 P5 S5/sm6 N6 SM6/sm7 N7 SM7 P1

We know that 12edo and 14edo support Injera, where Injera[12] 3|2 (2) may be labelled

P1 sm2 M2 sm3 SM3 P4 SA4/sd5 P5 sm6 M6 m7 SM7 P8,

and Injera[14] 3|3 (2) labelled

P1 sm2 M2 sm3 m3 SM3 P4 SA4/sd5 P5 sm6 M6 SM6 m7 SM7 P8.

We can see both in 14edo, and to get 12edo from Injera[12], as with Pajara, we remove all the ‘s’ and ‘S’ prefixes.

=== [[Blacksmith]] and further extension ===
10edo also supports Blacksmith temperament, and we may think to write Blacksmith[10] 1|0 (5) as:

P1 sM2 M2/m3 sM3 P4 s5 P5 sM6 M6/m7 sM7 P8

But we have now added mappings, but are yet to define the use of ‘S’ and ‘s’ for perfect intervals. In Blacksmith, the interval we might call ‘s5’ is 81/80 below P5, however, more commonly ‘s5’ is used to refer to [[16/11]], and S4 [[11/8]]. Since these intervals have above been labelled N4 and N5 above however, we do not need to worry about that, and can add that s5, a 'small 5th', is 81/80 below 3/2, and S4, a 'supra 4th' lies 81/80 above 4/3. where ‘s4’ has been typically been used to refer to [[21/16]], and ‘S5’ to [[32/21]], we add that s4 is lower than P4 by 64/63 and that S5 is higher than P5 by 64/63.

The primary intervals names of Blacksmith[15] 1|1 (5) then are as follows:

P1 Sm2 sM2 M2/m3 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 M6/m7 Sm7 sM7 P8,

these being identical to the the primary interval names of 15edo.

We may further add that ‘S’ (supra) and ‘s’ (small) may raise diminished, and lower augmented intervals by 81/80 as they do to minor and major respectively and that when ‘S’ (super) raised an augmented interval, or ‘s’ (sub) lowers it, the change is by 64/63.

In edos where 81/80 or 64/63 are represented by a single step, or when the apotome is represented by a single step, Ups and Downs with 'S', 's', and 'N' can name the intervals equivalently to this system. This can be seen in add edos considered thus far, as well as in those listed directly below. What this system adds is that it may also describe the intervals of MOS scales (as well as JI scales), such that these interval can be identically named in edos that approximate the scales they belong to. This is true for all MOS scales mentioned so far, as well as those listed below.

=== Further application in edos ===
The primary interval names are shown below for some larger edos.

[[22edo]]: P1 m2 Sm2 sM2 M2 m3 Sm3 sM3 M3 P4 S4 4-5 s5 P5 m6 Sm6 sM6 M6 m7 Sm7 sM7 M7 P8

[[24edo]]: P1 N1 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 N4 A4/d5 N5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 N8 P8

[[26edo]]: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 SA4/sd5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8

[[27edo]]: P1 m2 Sm2 N2 sM2 M2 m3 Sm3 N3 sM3 M3 P4 N4 d6 A3 N5 P5 m6 Sm6 N6 sM6 M6 m7 Sm7 N7 sM7 M7 P8

[[29edo]]: P1 sm2 m2 Sm2 sM2 M2 SM2/sm3 m3 Sm3 sM3 M3 SM3/s4 P4 S4 d5 A4 s5 P5 sm6 m6 Sm6 sM6 M6 SM6/sm7 m7 Sm7 sM7 M7 SM7 P8

[[31edo]]: P1 N1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 N4 A4 d5 N4 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 N8 P8

[[34edo]]: P1 1-2 m2 Sm2 N2 sM2 M2 2-3 m3 Sm3 N3 sM3 M3 3-4 P4 S4 N4 4-5 N5 s5 P5 5-6 m6 Sm6 N6 sM6 M6 6-7 m7 Sm7 N7 sM7 M7 7-8 P8

[[38edo]]: P1 S1 1-2 sm2 m2 N2 M2 SM2 2-3 sm3 m3 N3 M3 SM3 3-4 s4 P4 N4 A4 SA4/sd5 d5 N5 P5 S5 5-6 sm6 m6 N6 M6 SM6 6-7 sm7 m7 N7 M7 SM7 7-8 s8 P8

[[41edo]]: P1 N1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 s4 P4 S4 N4 d5 A4 N5 sM5 P5 S5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 N8 P8

[[43edo]]: P1 S1 1-2 A1/sm2 m2 dd3 AA1 M2 SM2/d3 2-3 A2/sm3 m3 dd4 AA2 M3 SM3/d4 3-4 A3/s4 P4 dd5 AA3 A4 d5 dd6 AA4 P5 S5/A6 5-6 A5/sm6 m6 dd7 AA5 M6 SM6/d7 6-7 A6/sm7 m7 dd8 AA6 M7 SM7/d8 7-8 s8 P8

[[46edo]]: P1 S1 sm2 m2 Sm2 A1 d3 sM2 M2 SM2 sm3 m3 Sm3 A2 d4 sM3 M3 SM3 s4 P4 S4 sd5 d5 sA4/Sd5 A4 SA4 s5 P5 S5 sm6 m6 Sm6 A4 d7 sM6 M6 SM6 sm7 m7 Sm7 A6 d8 sM7 M7 SM7 s8 P8

In 43edo we encounter the first time we have to use double Augmented and diminished intervals. 43edo marks the first instance in which Jones' alternative ups and downs interval names do not match those from this system. In his system, for example, AA2 would simply be sM3, but in this system since sM3 implies an approximation to 5/4 and the M3 already represents 5/4, and therefore is equivalent to sM3, we cannot do this. This tells us however that no simple ratio is approximated by the interval, and perhaps it is better understood as an AA2. Larger edos contain unlabeled intervals (without resorting to extended diatonic interval names). The association of 'super' and 'sub' with 64/63 and with 'supra' and 'small' with 81/80 may effect the assignment of primary interval names, but for all of these edos, as well as all those mentioned before, when 'S' and 's' are used, they still signify a raising or lowering by a single step of the edo, and thus appear equivalent to the Ups and Downs version. The comma associations add that, though use of enharmonic equivalences and secondary interval names may be necessary, intervals from MOS scales may be spelled in a consistent way across tuning to different edos.
=== Other rank-2 temperaments' MOS scales ===
On top of those discussed thus far, other temperaments generated by the P5 or a fraction of it are also supported to some extent, where their MOS scales may be represented, including [[Semaphore]], [[Augmented]], [[Porcupine]], [[Diminished]], [[Negri]], [[Tetracot]] and [[Slendric]].

Semaphore[5] 2|2: P1 SM2/sm3 P4 P5 SM6/sm7 P8

Semaphore[9] 4|4: P1 M2 SM2/sm3 SM3/s4 P4 P5 S5/sm6 SM6/sm7 m7 P8

Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8

Augmented[9] 1|1 (3): P1 Sm2 Sm3 sM3 P4 P5 Sm6 SM6 sM7 P8

Augmented[12] 2|1 (3): P1 Sm2 M2 Sm3 sM3 P4 sA4 P5 Sm6 Sm7 SM6 sM7 P8

Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8

Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8

Porcupine[8] 4|3: P1 sM2 Sm3 P4 s5 P5 sM6 Sm7 P8

Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8

Negri[9] 4|4: P1 Sm2 SM2/sm3 sM3 P4 P5 Sm6 SM6/sm7 sM7 P8

Negri[10] 4|4: P1 Sm2 SM2/sm3 sM3 P4 A4 P5 Sm6 SM6/sm7 sM7 P8

Tetracot[6] 3|2: P1 sM2 N3 S4 N6 Sm7 P8

Tetracot[7] 4|2: P1 sM2 N3 S4 P5 N6 Sm7 P8

Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 P8

[[Slendric5|Slendric[5]]] 2|3: P1 SM2 s4 S5 sm7 P8

[[Slendric6|Slendric[6]]] 3|2: P1 SM2 s4 P5 S5 sm7 P8

[[Slendric11|Slendric[11]]] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8

We may also write temperaments with a 9/8 but no 3/2. The most well known of these is [[Machine]]:

[[Machine5|Machine[5]]] 2|2: P1 M2 M3 m6 m7 P8

[[Machine6|Machine[6]]] 3|2: P1 M2 M3 A4 m6 m7 P8

[[Machine11|Machine[11]]] 5|5: P1 d3 M2 d4 M3 d5 A4 m6 A5 m7 A6 P8
=== Formal summary ===
The resultant system may be formally summarised as follows:

'''Definition 1a.''' M and m label the two sizes of 2nd, 3rd, 6th and 7th in the Pythagorean diatonic scale.

'''Definition 1b.''' The smaller 4th and larger 5th are labelled P.

'''Definition 1c.''' The single size of 1 and 8 is labelled P.

'''Definition 2.''' A chroma above M or P is A and below m or P is d.

'''Definition 3a.''' Within generic interval classes 2, 3, 6 and 7, half way between M and m is N.

'''Definition 3b'''. Within generic interval classes 1 and 4, half way between P and A is N.

'''Definition 3c.''' Within generic interval classes 5 and 8, half way between P and d is N.

'''Corollary:''' 7edo can be written P1 N2 N3 P4 P5 N6 N7 N8.

'''Definition 4.''' Half way between adjacent generic interval classes lie the intermediates.

'''Corollary:''' 5edo can be written 1-2 2-3 3-4 5-6 6-7 7-8.

'''Definition 5a.''' 'S' when applied before M, and 's' when applied before m, raise or lower by 64/63 respectively.

'''Definition 5b.''' 'S' when applied before m, and 's' when applied before M, raise or lower by 81/80 respectively.

'''Definition 5c.''' 'S' when applied to P5 (leading to S5) and 's' when applied to P4 (leading to s4) raise and lower by 64/63 respectively.

'''Definition 5d.''' 'S' when applied to P4 (leading to S4) and 's' when applied to P5 (leading to s5) raise and lower by 81/80 respectively.

'''Definition 5e.''' 'S' when applied to P1 and 's' when applied to P8 may raise and lower by 64/63, or by 81/80.

'''Definition 6a.''' When 'S' and 's' imply alterations of 64/63, they have long-form 'super' and 'sub' respectively.

'''Definition 6b.''' When 'S' and 's' imply alterations of 81/80, they have long-form 'supra' and 'small' respectively.

'''Definition 7.''' If alterations of 81/80 and of 64/63 need to be distinguished from one another in short-form, alterations of 81/80 can be written 'SR' and 'sl'.

== edos with extreme fifths ==
Thus far the best fifth of all edos described lies in the 'diatonic range', between 4 steps of 7edo and 3 steps of 5edo. The best fifths of some edos lies outside this range, in either directly. Whereas in edos where the best fifth lies between 7 steps of 12edo and 4 steps of 7edo, 81/80, the meantone comma is tempered out, wherein four fifths approximate the fifth harmonic (a sM3 raised two octave), in edos with fifths flatter than this, 135/128, a meantone chromatic semitone is instead tempered out, resulting in the four fifths instead approximate the Sm3, 6/5 (raised two ocaves). This system is called Mavila temperament. In the other direction, whereas the best fifth lies between 7 steps of 12edo and 3 steps of 5edo, 64/63, the septimal or Archytas comma is tempered out, wherein four fifths approximate a SM3, 9/7, raised two octaves, in edos who's best fifth is sharper than this, 9/7, as well as 5/4 are approximated by the perfect fourth, tempering out 16/15 and 28/27. This system is called Father temperament. The application of this system to edos of both of these fifth sizes is addressed below.

=== [[Mavila]] ===
In Mavila, the perfect 5th is flatter than in 7edo, so major intervals are below minor and augmented below major. In Mavila the major third approximates 6/5 and the minor third 5/4, tempering out [[135/128]]. This presents no problem to the scheme however, and the rules are applied just the same. The small major third, 81/80 below 6/5 or 81/64 comes to [[32/27]], the minor 3rd, and the sub minor 3rd remains 7/6.

Mavila[7] 3|3 can be written

P1 M2 m3 P4 P5 M6 m7 P8,

the same as the diatonic scale.

Mavila[9] 4|4 can be written

P1 M2 M3 m3 P4 P5 M6 m6 m7 P8,

which is also the primary interval name set for 9edo.

If we don't want to have major being below minor, we can hide it with some secondary interval names:

P1 M2 Sm3 sM3 P4 P5 Sm6 sM6 m7 P8, arriving at Augmented[9].

We say that our first Mavila[9] interval names are not ''well-ordered'', where for an interval name set to be well-ordered, for each degree major must be above minor. By extension we defined a well-ordered interval names set as one in which ... ≤ dd ≤ d ≤ m ≤ M ≤ A ≤ AA ≤ ... or ... ≤ dd ≤ d ≤ P ≤ A ≤ AA ≤ ..., and where s_ ≤ _ ≤ S_ (where '_' represents any of ... dd, d, m, (P), M, A, AA ...).

Using the fact that 64/63 is represented by a single step in 9edo, it is left as an exercise for the reader to prove that 9edo supports Negri temperament (by replacing some names with sub and super prefixed names to arrive at Negri[9] 4|4).

The primary interval names for Augmented[9] 1|1 (3) are well-ordered.

Mavila[16] 8|7 can be writtten

P1 d1 M2 m2 M3 m3 A4 P4 A5 P5 d5 M6 m6 M7 m7 A8 P8.

The enharmonic equivalences for 16edo can be generated by: M=Sm=sm or m=sM=SM

This leads us to our primary well-ordered interval name set for 16edo:

P1 S1 Sm2 sM2 Sm3 sM3 s4 P4 4-5 P5 S5 Sm6 sM6 Sm7 sM7 s8 P8,

wherein we can see that it supports Diminished temperament.

Mavila[23]11|11 can be written as:

P1 d1 A2 M2 m2 A3 M3 m3 d3 A4 P4 d4 A5 P5 d5 A6 M6 m6 d6 M7 m7 d7 A8 P8

And the primary well-ordered interval names for [[23edo]] are appear as:

P1 S1 1-2 M2 m2 2-3 M3 m3 3-4 A4 P4 d4 A5 P5 d5 5-6 M6 m6 6-7 M7 m7 7-8 d8 P8

=== [[Father]] ===
In the other direction, the fifths of Father temperament are sharper than those of 5edo, leading to minor second going backwards. In Father temperament, 5/4 and 4/3 are tempered to a unison, along with 9/7. As 64/63 is tempered out, alterations of 64/63 act as identity alterations.The M2, larger than m3 is also a Sm3. sM2, then, returns to m3.

The primary interval names for Father[5] 2|2,

P1 M2 P4 P5 m7 P8,

present no problems.

In the primary interval names for Father[8] 4|3, however:

P1 m3 M2 P4 M3 P5 m7 M6 P8,

which are the same as the primary intervals for [[8edo]], but with M3 rather than 4-5, we see our diatonic interval names begin to cross over. We will add to our definition of well-ordered interval names that no interval names from interval-class ''n''-1 may be subtended by a larger number of steps that any interval names from interval-class ''n.'' As above, we can may use some secondary interval names to address it, leading to

P1 sM2 M2 P4 S4 P5 m7 Sm7 P8,

as the primary well-ordered interval name set.

If we also re-write M2 and m7 as Sm3 and sM6, we get Porcupine[8] 4|3.

Using some secondary interval names to 'fix' the order leads us to

P1 sM2 M2 P4 S4 P5 m7 M6 P8

The primary interval names for Father[13] 6|6:

P1 M7 m3 M2 d5 P4 M3 m6 P5 A4 m7 M6 m2 P8,

look very unruly.

We will fix up the ordering again with secondary interval names:

P1 S1 sM2 M2 Sm3 P4 S4 s5 P5 sM6 m7 Sm7 s8 P8,.

The primary interval names for [[13edo]] are similar:

P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8.

Minimally fixing the order leads us to

P1 N2 sM2 M2 N3 P4 S4 s5 P5 N6 m7 Sm7 N7 P8,

which we have seen before as Tetracot[13] 6|6.

=== [[Miracle]], [[11edo]] and [[21edo]] ===
All the scales discusses to this point use either smalls and supras or subs and supers, so in the rare instance that we see a S1 or s8, we can infer whether or not it's small/supra or sub/super, probably without even thinking too much about it. Rarely do MOS scales in this scheme require alterations of both 81/80 and 64/63. One important temperament that includes such scales is Miracle. We do not encounter either S1 in the 10 and 11-note MOS:

[[Miracle 10|Miracle[10]]] 5|4: P1 Sm2 SM2 N3 s4 sA4 S5 N6 sm7 sM7 P8

Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8

We first see S1 and s8 in the 21-note MOS:

[[Blackjack|Miracle[21]]] 10|10: P1 S1 Sm2 N2 SM2 sm3 N3 sM3 s4 P4 Sd5 sA4 P5 S5 Sm6 N6 SM6 sm7 N7 sM7 s8 P8,

the largest MOS scale we have attempted yet to write. S1 in this scale is 64/63. We could maybe guess this from the presence of S5, but it is not obvious. If we need to make it clear when we are referring to small/supra, we can write their short-hand instead as 'sl' for 'small' and 'Sa' for supra. Miracle[21] would then be re-written:

P1 S1 Sam2 N2 SM2 sm3 N3 slM3 s4 P4 Sad5 slA4 P5 S5 Sam6 N6 SM6 sm7 N7 slM7 s8 P8, if complete clarity is needed, otherwise if long-form names are provided there is no need. It looks unwieldy to me and so I would avoid it, but it is there as a possibility, if the intervals of Mavila[21] need be written with only 4 letters allowed for each interval name.

Having encountered 11 and 21-note scales now, and haven't not described 11 and 21edo, I will add these here.

The fifth in 11edo is too flat even for it to be considered to support Mavila. Let's see what happens:

P1 M2 M3 m2 m3 P4 P5 M6 M7 m6 m7 P8.

Adding neutrals gives us our primary interval names for 11edo:

P1 M2 M3 N3 m3 P4 P5 M6 N6 m6 m7 P8.

This is clearly not Mavila, so we don't know what's tempered out, such that we might add our alterations to arrive at a well-ordered interval name set. Let's review the 11-note scales we have encountered above:

Slendric[11] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8

Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8

From these union of these scales we can see from P4=Sd5 that 135/128, the Mavila comma is tempered out. We apply our Mavila re-spellings to arrive at:

P1 Sm2 Sm3 N3 sM3 P4 P5 Sm6 N6 sM6 sM7 P8 as a well-ordered interval name set.

We should not expect our Machine[11] scale to be represented in this spelling of 11edo: A spelling of 11edo that shows that it supports Machine uses a different mapping, using the 9/8 from two 22edo P5s. We could spell 11edo as every other note of 22edo if we wish to see how it supports Machine.

21edo can be written as three 7edos as it's best fifth is that of 7edo. Since 81/80 is -1 steps in 21edo, we use 64/63 alterations:

P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8.

Using the available conversion sm = Sm (and therefor SM = sM), we can confirm that 21edo supports Miracle and Whitewood temperaments. This is left as an exercise for the the inspired reader.

=== [[6edo]] ===
Though 6edo is normally only ever considered as a subset of 12edo, given that we have encountered 6-note MOS I'll give it a red hot go.

Our P5 and P4 in 6edo is our half-octave, 4-5, so 9/8 is tempered out and our chromatic scale only covers 2edo: P1/M2/M3/A4 m2/m3/P4/4-5/P5/M6/M7 m6/m7/P8. If we want to write 6edo is a well-ordered way, we might choose:

P1 SM2 sm3 P4/4-5/P5 SM6 sm7 P8.

Writing s4 and S5 instead of sm3 and SM6 would give us Slendric[6] 3|2.

What of Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8?

This tells us that in 6edo 81/80 is mapped to -2 steps of 6edo. This is not a problem, as we can use alterations of 64/63, mapped to 1 step, though I don't see why anyone would want to think of 6edo in this way.

Similarly to Machine in 11edo, Machine in 6edo uses a different (much better) mapping of 9/8: That of 12edo. 6edo is much better spelled as a subset of 12edo, where we can see if supports Machine.

The primary interval names for the remaining trivial edos are trivially derived and are given along with all those described so far in the section below.

=== [[28edo]] ===
We encounter a new problem with 28edo. 28edo's best fifth is that of 7edo. It's primary intervals names are as such:

P1 sA1 1-2 Sm2 N2 sM2 2-3 Sm3 N3 sM3 3-4 S4 P4 sA4 4-5 Sd5 P5 s5 5-6 Sm6 N6 sM6 6-7 Sm7 N7 sM7 7-8 Sd8 P8

in 28edo 81/80 is represented by -1 steps. We en-devour to maintain as best we can in our primary interval names the original premise behind the notation - that the prefix 's' takes an interval down a single step of the edo and 'S' a single step up. In our primary interval names for 28edo we have S4 below P4 and d5 above P5. We can avoid this confusion however by using the secondary interval names for P4 and P5 in 28edo - N4 and N5. In our list of edos below this change is made.

== How to ==
Though we have derived our interval names through many examples thus far, the process has not been explained as yet such that the reader may immediately apply them. This will be addressed here, with a step-by-step derivation guide. This guide gives all possible labels to each interval.

For ''n-''edo:
# Label P1 = 1, P8 = ''n''
# Find the best approximations of 3/2, 5/4 and 7/4, which are to be labelled P5, sM3 and sm7 (This is equivalent to finding the [[7-limit]] [[patent val]]).
#* P5 = round(ln(3/2)/ln(''n'')) steps
#* sM3 = round(ln(5/4)/ln(''n'')) steps
#* sm7 = round(ln(7/4)/ln(''n'')) steps
# Using the best fifth, label the diatonic intervals. i.e.
#* M2 = (2*P5) mod ''n''
#* M6 = (3*P5) mod ''n''
#* M3 = (4*P5) mod ''n''
#* M7 = (5*P5) mod ''n''
#* (P4, m7, m3, m6, m2) = ''n'' - (P4, M2, M6, M3, M7)
# Determine the size of the apotome: A = M-m (for any of degree 2, 3, 6 or 7).
# The initial chromatic intervals may be labelled:
#* A1 = A
#* d8 = P8 - A
#* A4 = P4 + A
#* d5 = P5 - A
# If A is even, then intervals half way between M and m within a degree may be labelled N.
# Determine the size of meantone and septimal comma alterations (steps 7-12 may be skipped if A = 1)
#* meantone comma = M3 - sM3. If equal to 0 then 81/80 is tempered out, the edo is said to support Meantone temperament and alterations of 81/80 are not needed.
#* septimal comma = m7 - sm7. If equal to 0 then 64/63 is tempered out, the edo is said to support Superpyth temperament, and alterations of 64/63 are not needed.
# If sub/super < 0, alterations of 64/63 should not be applied.
# If small/supra < 0, alterations of 81/80 my be applied only if A = 0, wherein all diatonic intervals are given the label 'N'.
# Acknowledging points 7, 8 and 9, label 's' and 'S' alterations to diatonic intervals, where:
#* sM (small major) = M - meantone comma
#* Sm (supra minor) = m + meantone comma
#* SM (super major) = M + septimal comma
#* sm (sub minor) = m - septimal comma
# Acknowledging points 7, 8 and 9, label 's' and 'S' alterations to P4 and P5 where:
#* S4 (supra 4th) = P4 + meantone comma
#* s5 (small 5th) = P5 - meantone comma
#* s4 (sub 4th) = P4 - septimal comma
#* S5 (super 5th) = P5 + septimal comma
# If meantone comma alterations are used, S1 (supra unison) may be labelled at P1 + meantone comma and d8 (small octave) at P8 - meantone comma. If septimal comma alterations are used, S1 (super unison) may be labelled at P1 + septimal comma and s8 (sub octave) may be labelled at P8 - septimal comma. If both are needed and not equal, and after steps 13-16 are completed as desired the interval corresponding to a super or supra unison (equivalently, sub or small octave) is yet otherwise unlabeled then to ascertain whether S1 is a supra or super unison (or equivalently s8 a small or sub octave), for all degrees, the short hand for 'small' is to be 'sl' rather than 's', and for 'supra' is to be 'SR' rather than 'S'.
# Remaining augmented (M + A) and diminished (m - A) intervals may be labelled 'A' and 'd'.
# If desired, augmentation and diminution may be iterated through the continued adding or subtracting of A, labelled by additional 'A' or 'd' prefixes respectively.
# 'S' and 's' may be further applied as if 'A' were 'M' or 'd' were 'm'.
# If desired and if ''n'' is even, the interval subtended by ''n''/2 steps may be labelled 4-5.
# If desired and if m2 is subtended by an even number of steps, the remaining intermediates may be labelled: (1-2, 2-3, 3-4, 5-6, 6-7, 7-8) = (P1, M2, M3, P5, M6, M7) + m2/2.
#

== Lists of edos and MOS scales ==

=== Primary well-ordered (unless otherwise noted) interval names for edos ===
2edo: P1 P4/P5 P8

3edo: P1 P4 P5 P8

4edo: P1 SM2/sm3 P4/P5 SM6/sm7 P8

5edo: P1/m2 M2/m3 M3/P4 P5/m6 M6/m7 M7/P8

6edo: P1 SM2 sm3/s4 P4/P5 S5/sM6 sm7 P8

7edo: P1 N2 N3 P4 P5 N6 N7 P8

8edo: P1 sM2 M2 P4 S4/s5 P5 m7 Sm7 P8

9edo: P1 M2 M3 m3 P4 P5 M6 m6 m7 P8

10edo: P1/m2 N2 M2/m3 N3 M3/P4 S4/s5 P5/m6 N6 M6/m7 N7 M7/P8

11edo: P1 Sm2 Sm3 N3 sM3 P4 P5 Sm6 N6 sM6 sM7 P8

12edo: P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8

13edo: P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8

14edo: P1 S1/sm2 N2 SM2/sm3 N3 SM3/s4 P4 SA4/sd5 P5 S5/sm6 N6 SM6/sm7 N7 SM7/s8 P8

15edo: P1/m2 S1/Sm2 sM2 M2/m3 Sm3 sM3 M3/P4 S4 s5 P5/m6 Sm6 sM6 M6/m7 Sm7 sM7/s8 M7/P8

16edo: P1 1-2 Sm2 sM2 Sm3 sM3 3-4 P4 4-5 P5 5-6 Sm6 sM6 Sm7 sM7 7-8 P8

17edo: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8

19edo: P1 S1/sm2 m2 M2 SM2/sm3 m3 M3 SM3/s4 P4 A4 d5 P5 S5/sm6 m6 M6 SM6/sm7 m7 M7/s8 P8

21edo: P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8

22edo: P1 m2 Sm2 sM2 M2 m3 Sm3 sM3 M3 P4 S4 4-5 s5 P5 m6 Sm6 sM6 M6 m7 Sm7 sM7 M7 P8

23edo: P1 S1 1-2 Sm2 sM2 2-3 Sm3 sM3 3-4 A4 P4 S4 s5 P5 d5 5-6 Sm6 sM6 6-7 Sm7 sM7 7-8 s8 P8 (can't quite get it well-ordered)

24edo: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 N4 A4/d5 N5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8

26edo: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 4-5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8

27edo: P1 m2 Sm2 N2 sM2 M2 m3 Sm3 N3 sM3 M3 P4 N4 d6 A3 N5 P5 m6 Sm6 N6 sM6 M6 m7 Sm7 N7 sM7 M7 P8

28edo: P1 sA1 1-2 Sm2 N2 sM2 2-3 Sm3 N3 sM3 3-4 S4 N4 sA4 4-5 Sd5 N5 s5 5-6 Sm6 N6 sM6 6-7 Sm7 N7 sM7 7-8 Sd8 P8

29edo: P1 S1 m2 Sm2 sM2 M2 SM2/sm3 m3 Sm3 sM3 M3 SM3/s4 P4 S4 d5 A4 s5 P5 S5/sm6 m6 Sm6 sM6 M6 SM6/sm7 m7 Sm7 sM7 M7 s8 P8

31edo: P1 N1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 N4 A4 d5 N4 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 N8 P8

34edo: P1 S1 m2 Sm2 N2 sM2 M2 2-3 m3 Sm3 N3 sM3 M3 3-4 P4 S4 N4 4-5 N5 s5 P5 5-6 m6 Sm6 N6 sM6 M6 6-7 m7 Sm7 N7 sM7 M7 s8 P8

38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 N4 A4 SA4/sd5 d5 N5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8

41edo: P1 N1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 s4 P4 S4 N4 sA4 Sd5 N5 sM5 P5 S5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 N8 P8

43edo: P1 S1 1-2 A1/sm2 m2 dd3 AA1 M2 SM2/d3 2-3 A2/sm3 m3 dd4 AA2 M3 SM3/d4 3-4 A3/s4 P4 dd5 AA3 A4 d5 dd6 AA4 P5 S5/A6 5-6 A5/sm6 m6 dd7 AA5 M6 SM6/d7 6-7 A6/sm7 m7 dd8 AA6 M7 SM7/d8 7-8 s8 P8 (not possible to find a well-ordered interval name set)

46edo: P1 S1 sm2 m2 Sm2 A1 d3 sM2 M2 SM2 sm3 m3 Sm3 A2 d4 sM3 M3 SM3 s4 P4 S4 sd5 d5 4-5 A4 SA4 s5 P5 S5 sm6 m6 Sm6 A4 d7 sM6 M6 SM6 sm7 m7 Sm7 A6 d8 sM7 M7 SM7 s8 P8 (not possible to find a well-ordered interval name set)

=== Interval names for MOS scales ===
Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8

Augmented[9] 1|1 (3): P1 Sm2 Sm3 sM3 P4 P5 Sm6 SM6 sM7 P8

Augmented[12] 2|1 (3): P1 Sm2 M2 Sm3 sM3 P4 sA4 P5 Sm6 Sm7 SM6 sM7 P8

Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8

Barbados[5] 2|2: P1 2-3 P4 P5 6-7 P8

Barbados[9] 4|4: P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8

Blackwood[10] 1|0 (5): P1 sM2 M2/m3 sM3 P4 s5 P5 sM6 M6/m7 sM7/s8 P8

Blackwood[15] 1|1 (5): P1 S1/Sm2 sM2 M2/m3 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 M6/m7 Sm7 sM7/s8 P8

Diminished[8] 1|0 (4): P1 sM2 Sm3 P4 sA4/Sd5 Sm6 sM6 sM7 P8

Diminished[12] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 Sm7 sM7 P8

Injera[12] 3|2 (2): P1 sm2 M2 sm3 SM3 P4 SA4/sd5 P5 sm6 M6 m7 SM7 P8

Injera[14] 3|3 (2): P1 sm2 M2 sm3 m3 SM3 P4 SA4/sd5 P5 sm6 M6 SM6 m7 SM7 P8

Machine[5] 2|2: P1 M2 M3 m6 m7 P8

Machine[6] 3|2: P1 M2 M3 A4 m6 m7 P8

Machine[11] 5|5: P1 d3 M2 d4 M3 d5 A4 m6 A5 m7 A6 P8

Mavila[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8

Mavila[9] 4|4: P1 M2 M3 m3 P4 P5 M6 m6 m7 P8

Mavila[16] 8|7: P1 d1 M2 m2 M3 m3 A4 P4 A5 P5 d5 M6 m6 M7 m7 A8 P8

Meantone[5] 2|2: P1 M2 P4 P5 m7 P8

Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8

Meantone[12] 6|5: P1 m2 M2 m3 M3 P4 A4 P5 m6 M6 m7 M7 P8

Meantone[19] 9|9: P1 A1 m2 M2 A2 m3 M3 d4 P4 A4 d5 P5 A5 m6 M6 d7 m7 M7 d8 P8

Miracle[10] 5|4: P1 Sm2 SM2 N3 s4 sA4 S5 N6 sm7 sM7 P8

Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8

Negri[9] 4|4: P1 Sm2 SM2/sm3 sM3 P4 P5 Sm6 SM6/sm7 sM7 P8

Negri[10] 4|4: P1 Sm2 SM2/sm3 sM3 P4 A4 P5 Sm6 SM6/sm7 sM7 P8

Neutral[7] 3|3: P1 N2 N3 P4 P5 N6 N7 P8

Neutral[10] 5|4: P1 N2 M2 N3 P4 N4 P5 N6 m7 N7 P8

Neutral[17] 8|8: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8

Pajara[10] 2|2 (2): P1 Sm2 M2 sM3 P4 sA4/Sd5 P5 Sm6 m7 sM7 P8

Pajara[12] 3|2 (2): P1 Sm2 M2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 m7 sM7 P8

Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8

Porcupine[8] 4|3: P1 sM2 Sm3 P4 s5 P5 sM6 Sm7 P8

Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8

Semaphore[5] 2|2: P1 SM2/sm3 P4 P5 SM6/sm7 P8

Semaphore[9] 4|4: P1 M2 SM2/sm3 SM3/s4 P4 P5 S5/sm6 SM6/sm7 m7 P8

Slendric[5] 2|3: P1 SM2 s4 S5 sm7 P8

Slendric[6] 3|2: P1 SM2 s4 P5 S5 sm7 P8

Slendric[11] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8

Superpyth[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8

Superpyth[12] 6|5: P1 m2 M2 m3 M3 P4 A4 P5 m6 M6 m7 M7 P8

Superpyth[17] 8|8: P1 m2 A1 M2 m3 d4 M3 P4 d5 A4 P5 m6 A5 M6 m7 d8 M7 P8

Tetracot[6] 3|2: P1 sM2 N3 S4 N6 Sm7 P8

Tetracot[7] 4|2: P1 sM2 N3 S4 P5 N6 Sm7 P8

Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 P8

== Conclusion ==
Using only 'S' and 's' as qualifiers to M, m, P, A and d, along with N and intermediate degrees, a system is developed wherein for most edos below 50 intervals can be systematically named such that for primary interval names, 'S' and 's' raise and lower, respectively, by 1 step of an edo , whilst the intervals of many MOS scales may be consistently named across different tunings, taking the best of both Igliashon Jones' extra diatonic interval names and Sagispeak. One does not need to understand the comma associations to make use of the interval names. While the intervals of some MOS scales may hold consistent names in edos in an Ups and Downs based scheme, there are many common scales that do not in such a system, that do in this one, such as scales of Diminished and Augmented temperament. What's more, the use of neutrals and intermediates leads to quicker recognition of MOS scales that may be supported in edos.

Xenharmonic Wiki:Conventions

2018-10-15T06:36:22Z

PiotrGrochowski:

A few conventions have evolved in this wiki that would be good to keep consistent with. Any pages that break the conventions may have links to them removed.

(please feel free to add rules in use here, and please provide reasons for it! Thanks!)

== Decimal numbers ==
Use the English convention with decimal point, when articles are in English language (which is mostly the case). Do not misspell numbers.

{| class="wikitable"
! correct
! wrong
|-
| style="background-color: #efe" | 701.995
| style="background-color: #fee" | <s>702,995</s>
|}

==Logarithmic interval measures ==
For normal tables, we decided to stick to [[cent|cent]] values with a fixed precision, mostly 2 or 3 in the fractional part. 3 is helpful when serving as a canonical reference point for people to compare their own calculations with, which helps in the development of table creators such as [[Piotr Grochowski]]'s [https://scratch.mit.edu/projects/249278782/ one for Selected just intervals by error]. Its main purpose is to give an idea of interval sizes and relation to known intervals in the [[12edo|12edo]] system of western music. Alternative measures are acceptable if they better reflect the idea behind an interval system, scale etc. More than one logarithmic size column should be avoided.

== Links ==
Links - both wiki-internal and external ones - should have meaningful text; something like "[[Conventions|here]]" is not acceptable, often the page title is a good starting point.

: ''For more info see [[Help:Here-links]]''

==[[List model]]==

Any [[list model]] may be used according to preference, however an edit consisting only of a list model change should not be done.

== See also ==
* [[Help:Help|Help]]

[[Category:about]]
[[Category:conventions]]
[[Category:help]]

User:Lhearne/Extra-Diatonic Intervals

2018-10-15T06:35:01Z

PiotrGrochowski: aaagaggsggghhhh

Today a small under of competing interval naming schemes exist for the description of microtonal music. More common than any particular defined standard are certain tendencies for microtonal interval naming, or names for specific intervals. While risking the creation of simply another competing standard, an effort is made to develop a scheme that is able to take the best aspects of the existing standards and apply them in a formal interval naming system built on common undefined practice. Such a system is developed, where in addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', only the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees. Using this system all intervals in three fifths of all [[Equal division of the octave|edo]]<nowiki/>s up to 50 can be named such that 'S' and 's' correspond to a displacement of an interval up or down a single interval of the edo, respectively. Many commonly used [[MOS scale|MOS scales]] may also be described using this scheme such that these scales' interval names are consistent expression in any tuning that supports them. The resultant scheme can also be easily mapped to any of the current naming standards, and may even facilitate translation between. The resulting scheme should improve pedagogy and communication in microtonal music.

== Background ==

=== The origin of diatonic interval names ===
[[File:Mesopotamian interval names table.jpg|thumb|500x500px|Mesopotamian interval names, from http://www.historyofmusictheory.com/?page_id=130, accessed October 7, 2018.]]
Music theory describing the use of heptatonic-diatonic scales, including interval names, has been traced back as far as 2000BC, deciphered from a Sumerian cuneiform tablet from Nippur by Kilmer (1986). From Kummel (1970) we know that 'the names given to the seven tunings/scales were derived from the specific intervals on which the tuning procedure started' (Kilmer, 1986). This formed the basis of their musical notation ([http://www.jstor.org/stable/985853. Kilmer, 2016]). The table to the right following table displays the Ancient Mesopotamian interval names accompanied by their modern names.

Kilmer also writes that 'the ancient Mesopotamian musicians/“musicologists” knew what we call today the Pythagorean series of fifths, and that the series could be accomplished within a single octave by means of “inversion.” '. The Mesopotamian's music and theory was passed down through the Babylonians and the Assyrians to the Ancient Greeks, as well as their mathematics, particularly concerning musical and acoustical sound [[ratios]] (Ibid, [http://math-cs.aut.ac.ir/~shamsi/HoM/Hodgkin%20-%20A%20History%20of%20Mathematics%20From%20Mesopotamia%20to%20Modernity.pdf Hodgekin, 2005]).

Such mathematical and musical ideas are attributed to Pythagoras, who undoubtedly made them popular., although many scholars suggest he may have learned these ideas from his Babylonian and Egyptian mentors. None the less, Pythagoras' idea that that by dividing the length of a string into ratios of halves, thirds, quarters and fifths created the musical intervals of an octave, a perfect fifth, an octave again, and a major third form the basis of Ancient Greek music theory (http://www.historyofmusictheory.com/?page_id=20). His tuning of the diatonic scale by only octaves and perfect fifths ([[Pythagorean tuning]]) is influential through to today.

=== Ancient Greek interval names ===
Intervals in Ancient Greek music were written either as string length ratios, after Pythagoras, or as positions in a [[tetrachord]].

[[2/1]], the [[octave]], was named ''diapason'' meaning ''<nowiki/>'''through all [strings]'

[[3/2]], the [[perfect fifth]] was labelled ''diapente,'' meaning 'through 5 [strings]'

[[4/3]], the [[perfect fourth]], was labelled ''diatessaron'', meaning 'through 4 [strings]'

''dieses,'' 'sending through', refers to any interval smaller than about 1/3 of a perfect fourth

''tonos'' referred both to the interval of a whole tone, and something more akin to [[mode]] or key in the modern sense ([http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf Chalmers, 1993])

''ditone'' referred to the interval made by stacking two [[9/8]] whole tones, resulting in [[81/64]], the Pythagorean major third. ([[Joe Monzo|Monzo]], http://www.tonalsoft.com)

[[256/243]] - the ''limma'', which is the ratio between left over after subtracting two 9/8 tones (together making a ditone) a perfect fourth, the ''diatonic semitone'' of the Pythagorean diatonic scale

[[2187/2048]] - the ''apotome'', which is the ratio between the tone and the limma, the ''chromatic semitone'' of the Pythagorean diatonic scale

The Ancient greek term ''diatonon'', meaning 'through tones', refers to the genus with two whole tones and a semitone, or any genus in which no interval is greater than one half of the fourth (Chalmers, 1993). The Pythaogrean diatonic scale is the scale that may be built from one two Pythagorean tetrachords, and the left over interval of 9/8.

=== Zarlino and Meantone ===
[[File:Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1.png|thumb|566x573px|''Le institutioni harmoniche,'' Zarlino, 1558, Cap. 15: Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1, pg. 25.]]
Intervals were referred to by the Ancient Greek names through the the 18th century, as Latin names. By the Renaissance it had been discovered that a Pythagorean diminished fourth sounded sweet, and approximated the string length ratio 5/4. This just tuning for the major third was sought after, along with the complementary 6/5 tuning for the minor third, and octave complements to both - 8/5 for the minor sixth and 5/3 for the major sixth. Influential Italian music theorist and composer Gioseffo Zarlino put forth that choirs tuned the diatonic scale to the tuning built from this tetrachord, the ''intense diatonic scale'', also known as the ''syntonic or syntonus diatonic scale'' or the ''Ptolemaic sequence'':

1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

Included in this scale, however, were ''wolf intervals:'' imperfect consonances that occurred as tunings of the same interval as perfect consonances. For example, between 1/1 and 3/2, 4/3 and 1/1, 5/3 and 5/4; and 5/4 and 15/8 occurs the perfect fifth, 3/2, whereas between 9/8 and 5/3 occurs the wolf fifth, 40/27, flat of 3/2 by 81/80. This was also the interval by which four 3/2 fifths missed 5/1 (the interval two octaves above 5/4). It was named the ''syntonic comma'' after Ptolemy's ''syntonus'' or ''intense diatonic tetrachord'' which consists of the intervals 9/8, 10/9 and 16/15, where 9/8 and 10/9 differ by this interval. By making the syntonic comma a unison the wolf fifth could be made a perfect fifth. It was discovered that this could be achieved by flattening (tempering) the perfect fifth by some fraction of this comma such that four of these fifths less two octaves gave an approximation of 5/4. Where two fifths less an octave give 9/8, the next two add another 10/9 to result in the 5/4. 9/8 and 10/9 were referred to as the ''major tone'' (''tunono maggiore'') and ''minor tone'' (''tunono minore''), respectively, and where this tuning led to them being equated, it was referred to as Meantone temperament, which is said to 'temper out' the syntonic comma. Zarlino advocated the flattening of the fifth by 2/7 of a comma, leading to 2/7-comma Meantone, but also described 1/3-comma and 1/4-comma Meantone as usable (Zarlino, 1558).

The diagram on the right, from Zarlino's 1558 treatise ''Le istitutioni harmoniche'' associates many intervals with their tuning as perfect consonances. The perfect tuning for the ditone was considered then to be 5/4, rather than 81/64. The interval for which 6/5 is considered a perfect tuning was referred to as a ''semiditone'' (labelled also in ''Le istitutioni harmoniche'' by as ''Trihemituono)''. This may seem odd to us now, but in Latin 'semi' referred not to 'half', but to 'smaller', so 'semiditone' translated to something like 'smaller ditone'. Additionally 'semitone' referred to the interval smaller than the 'tone'. Like the tone, this interval possessed two alternative perfect tunings: 16/15, the difference between 15/8 and 2/1, or 5/4 and 4/3, and 25/24, the difference between 6/5 and 5/4. 16/15 was referred to as the ''major semitone'' (''semituono maggiore'') and 25/24 as the ''minor semitone (semituono maggiore'').

In addition to the Latin interval names, derived from the Ancient Greek interval names, we see on the diagram a single interval name in Italian: ''Essachordo maggiore'', referring to the ratio 5/3, which we are tempted to translate to 'major sixth'. Chapter 16, ''Quel che sia Consonanze semplice, e Composta; & che nel Senario si ritouano le sorme di tutte le somplici consonanze; & onde habbia origine l'Essachordo minore'', puts forward that the ''Essachordo minore,'' or perhaps 'minor sixth' be tuned to 8/5.

In the 1691 ''Lettre de Monsieur Huygens à l'Auteur [Henri Basnage de Beauval] touchant le Cycle Harmonique,'' theorist Christiaan Huygens gave names and ratios to common intervals and mapped them to 31-tET, which very closely approximates 1/4-comma Meantone. Translated from French, 3/2 was labelled a Fifth, 4/3 a Fourth, 5/4 a major Third, 6/5 and minor Third, 5/3 a major Sixth and 8/5 a minor Sixth. Here we really begin to see today's interval names.

=== English interval names in the Baroque ===
[[File:Harmonics, or The Philosophy of Musical Sounds, Section 2 figure 3.png|thumb|517x548px|''Harmonics, or The Philosophy of Musical Sounds'', Edition 2, Smith, 1759, Section 2: On the Names and Notation of consonance and their intervals, Fig. 2 & 3 , pg. 10|547.986x547.986px]]
After English superseded Latin as the the main language of scholarship, the Latin interval names were rejected and the convention we saw in Zarlino's Italian for naming the smaller of a pair of sizes of an interval 'minor' and the larger 'major' was further applied.

Music theorist and mathematician Robert Smith provides the diagram and table on the right in his 1749 ''Harmonics, or, The Philosophy of Musical Sounds,'' with the description: <blockquote>'Fig. 2. If a musical string ''CO'' and it's parts ''DO'', ''EO'', ''FO'', ''GO'', ''AO'', ''BO'', ''cO'', be in proportion to one another as the numbers 1, 8/9, 4/5, 3/4, 2/3, 3/5, 8/15, 1/2, their vibrations will exhibit the system of 8 sounds which musicians donate by the letters ''C'', ''D'', ''E'', ''F'', ''G'', ''A'', ''B'', ''c''.</blockquote><blockquote>Fig. 3. And supposing those strings to be ranged like ordinates to a right line ''Cc'', and their distances ''CD'', ''DE'', ''EF'', ''FG'', ''GA'', ''AB'', ''BC'', not to be the differences of their lengths, as in fig. 2. but to be the magnitudes proportional to the intervals of their sounds, the received Names of these intervals are shewn in the following Table; and are taken from the numbers of the strings or sounds in each interval inclusively; as a Second, Third, Fourth, Fifth, &c, with the epithet of ''major'' or ''minor'', according as the name or number belongs to a greater of smaller total interval; the difference of which results chiefly from the different magnitudes of the major and minor second, called the Tone and Hemitone.'</blockquote>We note that Smith uses the tuning of the diatonic scale that Zarlino put forward: the Ptolemaic Sequence.

We may be surprised to see 4:3 here labelled as a minor Fourth, and 3/2 as a major Fifth, but it is obvious that this naming is more consistent than today's. Smith adds that <blockquote>'Any one of the ratios in the third column of the foregoing Table, except 80 to 81, or any one of them compounded once of oftener with the ratio 2 to 1 or 1 to 2, is called a Perfect ratio when reduced to it's least terms. And when the times of the single vibrations of any two sounds have a perfect ratio, the consonance and it's interval too after called Perfect; and is called Imperfect or Tempered when that perfect ratio and interval is a little increased or decreased.'</blockquote><blockquote>'Any small increment of decrement of a perfect interval is called respectively the Sharp or Flat Temperament of the imperfect consonance, and is measured most conveniently by the proportion it bears to the comma'</blockquote>Therefore in this system 3/2 is the ''Perfect major Fifth'' and 5/4 the ''Perfect major Third''. 81/64 might be labelled a ''comma sharp major Third'', 32/27 a ''comma flat minor Third'', and the 1/4-comma Meantone fifth a ''1/4-comma flat major Fifth''. The interval naming scheme Smith describes may be immediately applied to 5-limit microtonal systems. There is an inconsistency, however, where it seems that 9/8 should be called a ''Perfect major Second,'' but that, while 9/5 be named a ''comma sharp minor Seventh'', it's inverse, 10/9, is a ''Perfect minor Tone.''

In ''An Elementary Treatise on Musical Intervals and Temperament,'' published in 1876'','' R. H. M Bosanquet refers to 5/4 as the ''perfect third'', and 81/64 as the P''ythagorean third''. Bosanquet also labels other intervals of the Pythagorean diatonic scale similarly, i.e. 256/243, the limma, is labelled the ''Pythagorean semitone'', and 27/16 the ''Pythagorean sixth''. 81/80 is labelled the ''ordinary comma'', or simple the ''comma,'' and the Pythagorean comma is defined as the difference between twelve fifths and seven octaves. The apotome of 2187/2048 is referred to as Apatomè Pythagoria. The following relationships are then described: (<span style="color:#FF0000">the image is too inappropriate to show, see pg 28 for rules regarding posting images to the Internet</span>)

[[File:Helmholtz consonances table.png|thumb|Table describing the influence of the different consonances on one another, up to the 9th partial, from ''On the Sensations of Tone as a Phycological Basis for Theory of Music'', Helmholtz, 1863, Translation by Ellis, 1875, pg. 187]]

=== Helmholtz and Ellis ===
Through the investigations of Galileo (1638), Newton, Euler (1729), and Bernouilli (1771), theorist Hermann von Helmholtz was aware that ratios governing the lengths of strings existed also for the vibrations of the tones they produced. His investigation of the harmonic series associated with these ratios of vibration led him to the consideration of ratios above the 5-limit. In his seminal ''On the Sensations of Tone as a Physiological Basis for the Theory of Music'', published in German in 1863 and translated into English in 1875 by Alexander Ellis, he listed just intervals as show in the table to the right. It is interesting to note that 8:9 is labelled a 'Second' rather than a 'major Second'. The minor Seventh is shown as 5:9 rather than as 9:16 seemingly because of the 9 partial limit imposed on the table. It is also worth noting that 5:7 is labelled a subminor Fifth. 'Super', indicated in notation with a '+', raises an interval by 35:36, the septimal quarter tone, and 'sub', indicated with a '-' lowers by the same interval with the exception of the Supersecond 7:8, which lies 63:64, the septimal comma above the Second. The subminor fifth is not included in this as no minor Fifth is shown. If we assume that 'sub' lowers an interval 35:36, then the minor Fifth would be 18:25, 80:81 above Smith's 45:64 minor Fifth, however in table 2 below, Ellis labels 18:25 a ''superfluous Fourth'', and it's inverse, 25:32, an ''acute diminished Fifth'', whilst 64:45 is labelled a ''diminished Fifth'' and its inverse 32:45 a ''false Fourth or Tritone.'' If we label 9:10 as a 'major Second', and 7:8 as a 'supermajor Second' then they differ by 35:36, the major Second is the inverse of the minor Seventh, and the supermajor Second is the octave inverse of the subminor Seventh. Perhaps 'major' has been left off name of the major Fifth, and minor off the name of the minor Fourth since the time of Smith. We can add to this table the remaining octave inversions as well as the super Fourth and sub Fifth.
{| class="wikitable"
|+Table 1. Additional Intervals
!Intervals
!Notation
!Ratio
!Cents in the interval
|-
|Major Second
|D
|9:10
|182
|-
|Sub Fourth
|F-
|16:21
|471
|-
|Supermajor Fourth
|F#+
|7:10
|617
|-
|Super Fifth
|G+
|21:32
|729
|-
|Subminor Sixth
|A♭-
|9:14
|765
|-
|Supermajor Sixth
|A+
|7:12
|933
|}
Helmholtz defined the ''perfect consonances'' as the Octave, Twelfth and Double Octave as well as Fourth and Fifth. The major Sixth and major Third are next called ''medial consonances'', considered to in the era of Pythagorean tuning to be ''imperfect consonances'', which Helmholtz defined instead to be the minor Third and the minor Sixth.

Regarding the tuning of the intervals however, those corresponding to simple ratios of vibration are, as in Smith, referred to as perfect, however hey are also described as 'justly-intoned', or by Ellis as 'just'. The perfect tuning for the semitone is listed as 16/15, or 182c. The perfect tunings are compared to the Pythagorean tunings, where the Pythagorean tuning of the major Third and sixth are described as 81/80 above the perfect tunings, and of the minor Third, minor Sixth and semitone to be 81/80 below the perfect tunings. Helmholtz notes that the Pythagorean tunings are closer to the equal tempered tunings than the perfect tunings. Helmholtz also describes the Pythagorean Tritone as of 612c.

Ellis includes an [[Ellis' interval table|additional table]] providing names for many different just and tempered intervals, perfect and imperfect. The interval names do not appear to follow any sort of consistent naming system, rather intervals seem to be named case-by-case. They also largely do not correspond to the interval names used by Helmholtz.

In ''Notes on the Observations of Musical Beats'', Proc. R. Soc. Lond. 1880, Ellis named many just intervals of the 7-limit (including 3 and 5-limit intervals):<blockquote>‘Fifth 3:2, Fourth 4:3, Major Third 5:4, Minor Third 6:5, Major Sixth 5:3, Sub-Fifth 7:5, Super-Fourth 10:7, Super-major Third 9:7, Sub-minor Sixth 14:9, Sub-minor Third 7:6, Super-major Sixth 12:7, Sub-minor or Harmonic Seventh 7:4, Super-major Second 8:7, Major Tone 9:8, Minor Tone 10:9, Small Major Seventh 9:5 and Diatomic (sic.) Semitone 16:15’</blockquote>He later lists ‘The Major Sevenths 16:9 and 15:8’. The labeling of 16:9 as a Major Seventh and 9:5 as a Small Major Seventh is interesting and at odds with Smith's interval names. Given that 9:5 is larger than 16:9, and no Minor Seventh is mentioned, we can assume 16:9 was mislabeled as a Major Seventh and was understood to be a Minor Seventh, as referred to by Smith.

There is an inconsistency associated with the labeling of 9:5 as a Small Major Seventh also, as it lies a 3:2 Fifth above the 6:5 Minor Third, and we know a fifth and a minor third when added together to give a minor, rather than major seventh.

Ellis here uses the name for 8:7 we suggested above, Super-major Second, and includes our suggested Sub-minor Sixth and Super-major Sixth, however rather than Subminor Fifth and Supermajor Fourth, 7:5 and 10:7 are labelled Sub-Fifth and Super-Fourth, where in this instance sub and super are seen to raise and lower by 21:20 instead of by 36:35.

Ellis, in a footnote to his translation of Helmholtz,'s treatise also provides names for a single 11-limit interval. The interval 22:27, of 355c, introduced by Zalzal, says Ellis was termed a ''neutral Third'' by Herr J. P. N. Land originally in ''Over de Toonladders der Arabische Musiek'' (On the Scales of Arabic Music) in 1880. An interval a fourth higher than this is mentioned, but a ratio is not given, and it is not named. We can ourselves however find it's ratio as 11:18, and guess it's name to be a ''neutral Sixth'', given that it lies a perfect Fourth above the neutral Third. Following a similar process as in our completion of Helmholtz table above, and assuming that the octave inverse of a neutral Third should be a neutral Sixth we may introduce the following 11-limit intervals that see common use among music theorists and microtonal musicians through to today:
{| class="wikitable"
|+Table 3. 11-limit intervals
!Intervals
!Notation
!Ratio
!Cents in the interval
|-
|Neutral Second
|D♭^
|81:88
|143
|-
|Neutral Second
|Dv
|11:12
|151
|-
|Neutral Third
|E♭^
|9:11
|347
|-
|Neutral Third
|Ev
|22:27
|355
|-
|Neutral Sixth
|A♭^
|27:44
|845
|-
|Neutral Sixth
|Av
|11:18
|853
|-
|Neutral Seventh
|B♭^
|6:11
|1049
|-
|Neutral Seventh
|Bv
|44:81
|1057
|}
Each interval name has two sizes that differ by the comma 242:243. The notation included in the table is from HEWM notation, developed as an extension to the Helmholtz-Ellis use of '+' and '-' by Joe Monzo (http://www.tonalsoft.com/enc/h/hewm.aspx<nowiki/>).'^' indicates raising 'v' a lowered of 33/32. In HEWM notation '+' and '-' are refined to mean raising and lowering of 81/80 respectively and '>' and '<' are added instead to indicate raising and lowering of 64/63. Letter names correspond instead of the the Ptolemaic sequence, as in Smith's and Helmholtz' descriptions, but to a Pythagorean tuning of the diatonic scale, where '#' and '♭' and respectively raise and lower the apotome, 2187/2048. HEWM notation is not accompanied by an interval naming system.

=== Common interval names today ===
These interval names are used by theorists and microtonal musicians today, though 7:5 and 10:7 are given many different names, today also considered to be an augmented fourth and diminished fifth, lesser septimal tritone and greater septimal tritone, or simply as tritones. The fourth and fifth are today called perfect fourth and perfect fifth, and Smith's major Fourth and minor Fifth referred to as augmented fourth and diminished fifth respectively. As can be seen in Tchaikovsky's ''A Guide to the Practical Study of Harmony,'' by the beginning of end of the 19th century the familiar conventions for the naming of intervals were set, wherein
* Seconds, thirds, sixths and sevenths appear in the diatonic in two sizes, the larger labelled 'major' and the smaller, 'minor'.
* Major, when raised by a semitone, becomes 'augmented', and minor, lowered by a semitone, 'diminished'.
* The smaller of the two sizes of fourth and the larger of the two sizes of fifth are labelled 'perfect', along with the unison and octave.
* A perfect interval, when raised a semitone is labelled 'augmented', and when lowered a semitone, 'diminished'.

=== Fokker/Keenan Extended-diatonic interval-names ===

Considering the 11-limit otonal chord 4:5:6:7:9:11 a chain of thirds, in addition to the familiar major, minor, subminor, supermajor and neutral thirds, Dave Kennan labelled 5:7 a sub-diminished fifth and 7:11 an augmented fifth. 7:10, the inversion of 5:7, is labelled a diminished. 5:7, therefore, is also an augmented fourth. In terms of sevenths, 4:7 is subminor, 5:9 is minor and 11:6 is neutral.

Super and sub were further generalised by some theorists and musicians such that an interval a bit smaller than a major is referred to as a ''subminor third'', and an interval a bit larger than a minor third as a ''supraminor third''.

From this, Keenan defines a consistent interval naming system, meaning one which obeys diatonic interval arithmetic (In each column, the parenthesised prefix is the one that is implied when there is no prefix). When adding intervals the indexes are added together to give the index of the resulting interval. Keenan also adds corrections for each interval class to the indexes in order to account for inconsistencies that occur within diatonic interval arithmetic when concerning intervals greater than an octave, so that his system, unlike regular diatonic interval names, may be completely consistent.

As can be seen above, sub, super, augmented and diminished have also carried inconsistent meaning historically, where in Keenan's system they always alter intervals by the same amount.
{| class="wikitable"
|+
!Index
!Prefix for unisons, fourths, fifths, octaves
!Prefix for seconds, thirds, sixths, sevenths, ninths
|-
| -4
|double diminished
|subdiminished
|-
| -3
|subdiminished
|diminished
|-
| -2
|diminished
|subminor
|-
| -1
|sub
|minor
|-
|0
|(perfect)
|neutral
|-
|1
|super
|(major)
|-
|2
|augmented
|supermajor
|-
|3
|superaugmented
|augmented
|-
|4
|double augmented
|superaugmented
|}
The index values correspond most directly to degrees of 31edo, whose interval names by this method are given in the following table:
{| class="wikitable"
|+Extended-diatonic interval names in 31-tET
!31-tET degree
!Ratios
!Names
!
|-
|0
|1:1
|
|unison
|-
|1
|48:49 44:45 36:35 33:32
|(diminished second)
|super unison
|-
|2
|27:28 24:25 20:21
|subminor second
|(augmented unison)
|-
|3
|15:16 14:15
|minor second
|
|-
|4
|11:12 10:11
|neutral second
|
|-
|5
|9:10 8:9
|major second
|
|-
|6
|7:8
|supermajor second
|(diminished third)
|-
|7
|6:7
|(augmented second)
|subminor third
|-
|8
|5:6
|
|minor third
|-
|9
|9:11
|
|neutral third
|-
|10
|4:5
|(subdiminished fourth)
|major third
|-
|11
|11:14 7:9
|(diminished fourth)
|supermajor third
|-
|12
|16:21
|subfourth
|(augmented third)
|-
|13
|3:4
|perfect fourth
|
|-
|14
|8:11
|super fourth
|
|-
|15
|5:7
|augmented fourth
|(subdiminished fifth)
|-
|16
|7:10
|(superaugmented fourth)
|diminished fifth
|-
|17
|11:16
|
|sub fifth
|-
|18
|2:3
|
|perfect fifth
|-
|19
|21:32
|(diminished sixth)
|super fifth
|-
|20
|9:14 7:11
|subminor sixth
|(augmented fifth)
|-
|21
|5:8
|minor sixth
|(superaugmented fifth)
|-
|22
|11:18
|neutral sixth
|
|-
|23
|3:5
|major sixth
|
|-
|24
|7:12
|supermajor sixth
|(diminished seventh)
|-
|25
|4:7
|(augmented sixth)
|subminor seventh
|-
|26
|9:16 5:9
|
|minor seventh
|-
|27
|11:20 6:11
|
|neutral seventh
|-
|28
|8:15
|
|major seventh
|-
|29
|14:27
|(diminished octave)
|supermajor seventh
|-
|30
|18:35
|sub octave
|(augmented seventh)
|-
|31
|1:2
|octave
|
|}
The interval names shown in brackets could be said to be 'secondary', the others, 'primary'.

After releasing his system Keenan was informed that is was identical to the extended-diatonic interval-naming scheme of Adriaan Fokker but for the acknowledgment of more 11-limit ratios.This system depends on the tempering out of 81/80, where the diatonic major third, from four stacked fifths, approximates the just major third, 5/4. It also depends on the existence of neutral intervals, i.e., that the perfect fifth or equivalently, the chromatic semitone, subtends an even number of degrees of the ET. To simply to our familiar naming scheme for 12-tET, we observe that it applies to 24edo equally as directly as in 31-tET, where the prefixes correspond to degrees of the edo. Exactly the same is also true for 38-tET, twice 19-tET, a meantone which very closely approximates 1/3-comma meantone. Meantone temperament wherein the fifth is divided into two equally sized neutral thirds is referred to as neutral temperament. Whereas meantone temperament is generated by the fifth, iin neutral temperament the generator is half this interval, the neutral third. Where it was seen above that there are two neutral thirds, 9:11 and 22:27 that differ by 243/242, neutral temperament is at its most simple the temperament defined by this equivalence: the tempering out of 243/242, as meantone is defined by the tempering out of 81/80. The temperament that tempers out both 81/80 and 243/242 is called Mohajira, upon which Keenan's scheme can be said to be based.

The primary interval names resulting in this system's application to 24-tET and 38-tET is now show, along with 31-tET again for easy comparison, where 'M', 'm', 'P', 'N', 'A', 'd', 'S' and 's' are shorthand for major, minor, perfect, neutral, augmented, diminished, super and sub, respectively:

24edo: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 S4 A4/d5 s5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8

31edo: P1 S1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 S4 A4 d5 s5 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 s8 P8

38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 S4 A4 SA4/sd5 d5 s5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8

to Meantone tunings that are not Mohajira tunings, the regular diatonic interval names can be applied, but with the addition of double augmented and double diminished:

19edo: P1 A1 m2 M2 A2/d3 m3 M3 d4 P4 A4 d5 P5 A5 m6 M6 A6/d7 m7 d8 P8

26edo: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 AA4/AA5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8

Keenan adds further that if it is desired to distinguish between ratios that are in 31-tET approximated by the same number of steps, an addition prefix be added to describe the prime limit of the approximated interval. For 3-limit intervals, the obvious choice is 'Pythagorean', for 5-limit Keenan chooses 'classic', for 7, 'septimal, 11, 'undecimal' and 13, 'tridecimal'. When the highest prime is the same, Keenan suggests adding 'small' and 'large' as final prefixes for this purpose.

In non-Meantone tunings, the two definitions of major third - 4:5 and 64:81, the just (or classic) and Pythagorean major thirds no-longer correspond. If intervals are to receive unique names then to one or both of these major thirds must be added a prefix. Keenan has been involved with the development of both types of systems. Only when the major is defined by it's mapping as fourth fifths, i.e. 81/64, can conserve interval arithmetic.

=== [[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak] ===
One system in which 81/64 is the major third, sagispeak, was developed by [[Dave Keenan]] and others as an interval naming system that maps 1-1 with the Sagittal microtonal music notation system. Sagittal notation was developed as a generalised diatonic-based notation system applicable equally to [[just intonation]], [[Equal Temperaments|equal tunings]] and rank-''n'' [[temperaments]]. Dozens of different accidentals can be used on a regular diatonic [[Staff notation|staff]] to notate up to extremely fine divisions, however in most cases only a handful are needed. In sagispeak, each accidental is presented by a prefix, made up of a single letter, in most cases, followed by either 'ai' if the accidental raises a note, or 'ao' if it lowers a note. As in HEWM notation, Pythagorean intonation is assumed as a basis. Then the prefixes depart from Pythaogrean intonation, altering by commas and introducing other primes. In place of the prefixes 'sub' and 'super', generally signifying an alteration of 36/35 from 5-limit intervals or 64/63 for 3-limit, Sagittal features an accidental of [[64/63]], which may be used to take a Pythagorean major interval to a supermajor, minor to subminor, or perfect to super or sub. The prefix 'tao' indicates a decrease of 64/63 and and the prefix 'tai' an increase. Whereas in previous interval naming schemes 'major' and 'minor' were synonymous with the 5-limit tunings, in sagispeak they map instead to Pythagorean. A prefix is needed then to take a Pythagorean intoned interval to a 5-limit tuning. Where 5/4 is 81/80 below the the Pythagorean third, the prefixes 'pai' and 'pao' (where 'p' is for 'pental', as in, involving prime 5), which raise or lower a note by [[81/80]]<nowiki/>respectively. Similarly, 'vai' and 'vao', which raise or lower a note by [[33/32]] respectively, leading to ratios of 11.

Because it is built off of the diatonic scale, sagispeak conserves diatonic interval arithmetic, i.e. familiar relations in the diatonic scale, i.e. M2 + m3 = P4. As in Fokker/Keenan Extended-diatonic Interval-names, diatonic interval arithmetic is also extended, where, for example, tai-major 2 + tao-minor 3 = P4 (8/7 + 7/6 = 4/3), where opposite alterations cancel each other out, and diatonic interval arithmetic is conserved, a very useful property for a microtonal interval naming system to possess. Another helpful property of sagispeak is its generalised applicability to edos, just intonation and other tunings, where the same intervals maintain their spelling across different tunings. Despite these benefits however, many see Sagittal and Sagispeak as overly complex (even though the entire extended system need hardly ever be applied), and requiring too many new terms to be learnt.

We have seen that there are two competing definitions of a major third, the ratio '5/4' or the interval built from fourth stacked fifths, that may or may not correspond to 5/4. In meantone systems, those we are used to, they correspond, but in most edos they do not. Interval naming systems wherein the major third is defined as an approximation to 5/4 rather than as four fifths minus two octaves may benefit from a familiar name for 5/4, but they are unable to conserve diatonic interval arithmetic.

For comparison, 31edo is shown below in sagispeak:

31edo: P1 tai-1/vai-1 tao-m2 m2 vai-2m/vao-M2 M2 tai-M2 tao-m3 m3 vai-m3/vao-M3 M3 tai-M3 tao-4 P4 vai-4 A4 d5 vao-5 P5 tai-5 tao-m6 m6 vai-m6/vao-M6 M6 tai-M6 tao-m7 m7 vai-m7.vao-M7 M7 tai-M7 vao-8/tao-8 P8

=== Dave Keenan's most recent system ===
In 2016 Dave Keenan proposed an alternative generalised [http://dkeenan.com/Music/EdoIntervalNames.pdf microtonal interval naming system for edos]. In what might be understood as a generalisation of his extended-diatonic interval-naming system described above onto any equal tuning. Employing as prefixes the familiar 'sub', 'super', and 'neutral'. His scheme is based on the diatonic scale, however the diatonic interval names are not defined by their position in a cycle of fifths like is Sagispeak. In Keenan's system the ET's best 3/2 is first labelled P5, and the fourth P4. The interval half-way between the tonic and fifth is labelled the neutral third, or 'N3', and halfway between the fourth and the octave N6. Then the interval a perfect fifth larger than N3 is labelled N7, and the interval a fifth smaller than N6 labelled N2. The neutral intervals then lie either at a step of the ET, or between two steps. After this the remaining interval names are decided based on the distance they lie in pitch from the 7 labelled intervals, which make up the ''Neutral scale'', P1 N2 N3 P4 P5 N6 N7, which, like the diatonic, is an MOS scale, which may be labelled [[Neutral7|Neutral[7]]] 3|3 using [[Modal UDP Notation|Modal UPD notation]]. To name an interval in an edo, the number of steps of [[72edo]] that most closely approximate the size of the interval difference from a note of the neutral scale is first found. Then the prefix corresponding to that number of steps of 72edo is applied to the interval name. The following chart details this process (can't load the chart :( ). An interval just smaller than a major third in Keenan's system is labelled a ''narrow major third'', and an interval just wider than a [[6/5]] minor third a ''wide minor third'', however he notes that 'narrow' and 'wide' are only necessary in edos greater than 31. Note that the interval just wide of a minor third is labelled a 'supraminor third' rather than a 'superminor third'. This reflect recent tendencies among microtonal musicians and theorists.

Keenan's system is an elegant way to keep the 'major 3rd' label for 5/4, where labels depend on the size of the best fifth, however it suffers from it's applicability only to edos, and that it does not conserve interval arithmetic. Another potentially undesirable result of the system is that the major second approximates 10/9, and a ''wide major second'' 9/8, where as 9/8 is almost always considered a major second, and [[10/9]] often a narrow or small major second. One such system that considers 10/9 a narrow major second is that of Aaron Hunt.

=== Size-based systems ===
Microtonal theorist [[Aaron Andrew Hunt]] devised [http://musictheory.zentral.zone/huntsystem4.html the Hunt system], which includes interval name assignments for JI (just intonation) and edos based on [[41edo]]. Compared to Keenan's 72 interval names, Aaron's system includes 41. His system is based directly on 41edo, and unlike Keenan's system, interval are given the name of the closest step of 41edo, and no account is taken of the size of the edos fifth. In 41edo, Major, minor, augmented and diminished intervals are those obtained through the approximately Pythagorean cycle of fifths. Intervals one step of 41edo above these are given the prefix 'small', one step larger are given the prefix 'large', two steps smaller the prefix 'narrow' and two larger the prefix 'wide'. As a result, 5/4 is labelled a 'small major 3rd', or SM3 (not to be confused with a super major third, a label that does not exist in this system).

Neo-medieval musicians and early music historian and theorist [[Margo Schulter]] described her own [http://www.bestii.com/~mschulter/IntervalSpectrumRegions.txt interval naming scheme] built on approximations to JI intervals. Each interval names corresponds to an approximate size, and no particular edo is referenced. In her scheme middle major thirds range in size from 400-423 cents, and small major thirds from 372-400c. 5/4 is labelled a small major third, 81/64 a middle major third and 9/7 a large major third. Margo's scheme includes small, middle and large varieties of major, minor and neutral 2nds, 3rds, 6ths, 7ths; perfect fourth and fifths; and tritones, as well as a sub fifth and super fourth a dieses and comma and an octave less dieses and comma and ''interseptimals'', which correspond to intermediates, her name referencing the fact that they may each approximate two ratios of 7.

In Hunt's system when used in 41edo or JI diatonic interval arithmetic is conserved, but in other tunings it may not be, and Margo's system may not conserve diatonic interval arithmetic either. Both systems may be applied to arbitrary tunings, but the same intervals (defined, perhaps by a MOS scale) may not be given the same interval names across different tunings.

<ul><li>[[User:PiotrGrochowski/Extra-Diatonic Intervals]] gives each 43edo interval a name, then maps each desired interval to a 43edo interval. [[User:PiotrGrochowski/Extra-Diatonic Intervals — 50edo]] does this with 50edo. '''It has a very amazingly excellent solution to the 5/4 and 81/64 problem''': The 5/4 is named major third and this notation is split in half, while 81/64 is named high major third. 10/9 and 9/8 are both named major second. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]). </li></ul>Size-based systems are completely generalisable, but do not conserve interval arithmetic.

=== Ups and Downs ===
One final interval naming system, associated with the [[Ups and Downs Notation|Ups and Downs Notation]] system, belonging to microtonal theorist and musicians [[KiteGiedraitis|Kite Giedraitis]], like Sagittal is based on deviations from the diatonic scale. In this system however, deviations (from major, minor, perfect, augmented and diminished) are notated simply by the addition of up or down arrows: '^' or 'v', corresponding to raising or lowering of a single step of an edo. In some tunings ([[12edo]], [[19edo]] or [[31edo]] for example) 5/4 may be a M3, and in others a vM3 (downmajor 3rd) (e.g. [[15edo]], [[22edo]], 41edo, 72edo), or even an up-major 3rd (e.g. [[21edo]]). Ups and downs also includes neutrals, which lay exactly in-between major and minor intervals of the same degree, labelled '~' (mid). 'Up' and 'down' prefixes may be used before mid also, i.e. 'v~ 3). This system benefits from it's simplicity as well as it's conservation of interval arithmetic. It can be used for some MOS scales where one of the generators is a perfect fifth or a fraction of a perfect fifth, but not all of these (e.g. Diminished[8]), and not all MOS scales (if such scales are to be described, an additional pair of accidentals/qualifiers is used. Although the scales then are described, their intervals still are not given the same names in Ups and Downs' edo names). Another criticism of Kite's system that does not apply to the others is the fact that when an edo is doubled or multiplied by some simple fraction, and the best fifth is constant across the two edos, the same intervals may be be given different names.

[[Igliashon Jones]] is a supporter of this system, but for the relabeling of 'down' as 'sub' and 'up' as 'super' (or supra) and 'mid' as 'neutral', so that more common names are used, wherein 'super' infers a raise of 1 step of the edo, and 'sub' a lowering of one step. In this 'Extra-diatonic' system 'super' and 'sub' may be doubly applied, as in Ups and Downs, but they may not be applied before 'neutral' where in Ups and Downs they may be applied before 'mid'. The author's own extra-diatonic system is developed as a departure with caveat that 'S' and 's' prefixes are defined not as alterations by a single step of the edo, but by comma alterations as in Sagittal, in order that interval of MOS scales may be represented consistently across different tunings. Throughout the rest of the article the development is detailed, and the system defined.

== Premise: ==
Extra-diatonic names should be simple, generalisable, widely applicable, backwards compatible with standard diatonic notation and reflecting current informal practice as closely as possible. Extra-diatonic interval names are fifth based; extended from the familiar major, minor and perfect interval names so that diatonic interval arithmetic is conserved. ‘M’, ‘m’, and ‘P’ remain the short-hand for major, minor and perfect. ‘A’ and ‘d’ for Augmented and diminished may also be used in the familiar way. In cases where the chroma (the chromatic semitone, or augmented unison) is represented by multiple steps in the tuning the prefix ‘super’ raises major and perfect intervals by a single step while ‘sub’ lowers minor and perfect intervals, with short-hand ‘S’ and ‘s’. ‘S’ and ‘s’ may also be used to raise minor and lower major intervals respectively, reflecting occasion practice. In this case ‘S’ is short-hand for ‘supra’, and 's' for 'small'. They may also be used to raise or lower diminished and augmented intervals similarly. In this way this scheme is equivalent thus far to Ups and Downs notation, where ‘^’ or ‘up’ corresponds to ‘S’, ‘super’ or ‘supra’ and ‘v’ or ‘down’ to ‘sub’ or 'small' .

== Additions and examples: ==
''Neutrals'' and ''intermediates'' are also included, where neutrals occur between opposing sizes of a single generic interval the intermediates between each generic interval and the next.

Interval names for equal tunings are ranked in five tiers.

Interval names for equal tunings are ranked in eight tiers.
# Perfect
# Neutral
# Major, minor, A4 and d5 and, if the chroma is subtended by a single (positive) step of the edo, other augmented and diminished intervals
# Super, sub, supra and small prefixes to major, minor, perfect interval and to A4 and d5
# Intermediates
# Augmented and diminished intervals (for when the chroma is subtended by more than a single (positive) step of the edo)
# Super, sub, supra and small prefixes to augmented and diminished intervals
# Intervals augmented and diminished more than singularly
When more than one interval name corresponds to a specific interval, the names are privileged in order of the tiers. By this ordering, the first available name is the ‘primary’ for that interval, the second available ‘secondary’ and third 'tertiary'.

I initially intended for Intermediates to lay in the second tier with neutrals, but that led to primary intervals results that were not preferred by my colleagues over other options and I was persuaded to avoid using intermediates unless they are necessary.

=== Neutrals ===
N2, N3, N6 and N7, i.e. neutral 2nds, 3rds, 6ths and 7ths, falling exactly in-between the major and minor intervals of the same interval class, add native support for neutral-thirds temperament, where the N3 divides the P5 in exact halves and N2 divides the m3 is exact halves. In ups and downs neutrals indicated with '~' and said 'mid'.

Extending this familiar application to provide support for larger neutral scales, we add that neutrals occur also between P4 and A4; P5 and d5; P1 and A1; and P8 and d8.

Then [[Neutral7|Neutral[7]]] 3|3 can then be written:

P1 N2 N3 P4 P5 N6 N7 P8.

The same names give the primary interval names for [[7edo]], whose secondary intervals names are:

N1 m2/M2 m3/M3 N4 N5 m6/M6 m7/M7 N8.

The secondary interval names show that the chroma is equivalent to a unison in 7edo.

[[Neutral10|Neutral[10]]] 5|4 may then be written as

P1 N2 M2 N3 P4 N4 P5 N6 m7 N7 P8

[[Neutral17|Neutral[17]]] 8|8 may be written as

P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8,

which is equivalent to the primary interval names of [[17edo]].

=== Intermediates ===
To provide native support for [[The Archipelago|Barbados]] and Diminishes temperaments, intermediates are also added to the system. It should be noted immediately that intermediates are not as common to common microtonal interval naming as neutrals and though are a useful addition to this scheme, may be left out if desired. The appendix includes the MOS scales and edos from 'lists of all edos and MOS Scales', but without any intermediates.

‘2-3’ lies exactly half-way between M2 and m3 and divides the P4 in half. It may be read ‘second-third’ or ‘serd’. ‘6-7’, it’s octave-inverse lies exactly half-way between M6 and m7 and may be read ‘sixth-seventh’ or ‘sinth’.

‘1-2’ lies exactly half-way between P1 and m2, dividing the m2 in half. It may be read ‘unison-second’ or ‘unicond’. Its octave-inverse, ‘7-8’, lies exactly half-way between M7 and P8 and may be read ‘seventh-octave’ or ‘sevtave’.


‘3-4’ lies exactly half-way between M3 and P4, dividing the M6 in half. It may be read ‘third-fourth’ or ‘thourth’. It’s octave-inverse, ‘5-6’, lies exactly half-way between P5 and m6 and may be read ‘fifth-sixth’ or ‘fixth’. 

[[5edo]] can be spelled with the list of only these intermediates:

1-2 2-3 3-4 5-6 6-7 7-8.

The primary interval names for 5edo do not include them however, as they may be described by diatonic intervals:

P1 M2/m3 P4 P5 M6/m7 P8

Barbados[5] 2|2 may be described as

P1 2-3 P4 P5 6-7 P8,

and Barbados[9] 4|4 as

P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8.

The final intermediate, ‘4-5’ lies exactly between P4 and P5 and divides the octave in half. It may be read ‘fourth-fifth’ or ‘firth’. It is necessary for [[Diminished]] temperament, where the half-octave cannot be represented as any alteration of A4 or d5. Diminished temperament has a period of 1/4 of an octave, an approximation of 6/5. Therefore, the difference between four 6/5's and 2, 648/625, is tempered out.

Diminished[8] 1|0 (4) can be written as P1 sM2 Sm3 P4 4-5 Sm6 sM6 sM7 P8, and

[[Diminished12|Diminished[12]]] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 4-5 P5 Sm6 sM6 Sm7 sM7 P8,

== [[12edo]] and a problem ==
The primary interval names for 12edo are as we are familiar:

P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8,

but the interval labeled A4/d5 has the secondary name '4-5'.

We know that 12edo supports Diminished temperament, but we can't see that 12edo as equivalent to Diminished[12]: If we follow the rules from our premise, the notes of Diminished[12] in 12edo would give: P1 M2 m2 M3 m3 P4 A4/d5 P5 M6 m6 M7 m7 P8, with the majors and minor flipped. To be equivalent, sM must equal M and Sm must equal m.

=== [[10edo]], Pajara and a problem ===
10edo may be written as only neutrals and intermediates:

N1/1-2 N2 2-3 N3 3-4 N4/4-5/N5 5-6 N6 6-7 N7 7-8/N8

However it much more sense to write is using the primary interval name set:

P1 N2 M2/m3 N3 P4 N4/N5 P5 N6 M6/m7 N7 P8

It's secondary intervals as follows:

m2 Sm2/sM2 2-3 Sm3/sM3 M3 S4/s5 m6 Sm6/sM6 6-7 Sm7/sM7 M7

We can see that 10edo supports [[Neutral third scales]], given that we can make the interval names for Neutral[10] using the primary interval names for 10edo.

We know 10edo and 12edo both support Pajara temperament. Pajara[10] 2|2 (2) consists of:

P1 Sm2 M2 sM3 P4 sA4/Sd5 P5 Sm6 m7 sM7 P8,

and [[Pajara12|Pajara[12]]] 3|2 (2) of

P1 Sm2 M2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 m7 sM7 P8.

We can see Pajara[10] in 10edo, but in 12edo, wouldn’t sM3 be m3?

This is where my system diverges from Igliashon Jones’. We have to break our first rule here, or at least add some conditions to it.

== Comma association ==
To address this problem of consistency, we now state that when 81/80 is tempered out, M=sM and m=Sm, and when 64/63 is tempered out, M=SM and m=sm. In the case of sm and SM, ‘S’ and ‘s’ raise and lower by 64/63, and in the case of Sm and sM, ‘S’ and ‘s’ raise and lower by 81/80. In this way extra-diatonic interval names are equivalent to [http://forum.sagittal.org/viewforum.php?f=9 Sagispeak] interval names, where for sm and SM ‘S’ and ‘s’ are equivalent to ‘tai’ and ‘pao’ and for Sm and sM ‘S’ and ‘s’ are equivalent to ‘pai’ and ‘pao’.

It is important to note that given this change, 'S' and 's' may alter an interval by a different number of steps in an edo depending on which interval names they prefix. This may seem confusing, but it seems to reflect existing informal practice.

In 12edo, which represents the union between the two, where both 64/63 and 81/80 are tempered out, ‘S’ and ‘s’ do not raise or lower intervals at all. We can now easily see that 12edo supports Pajara, where simply removing all the ‘s’s and ‘S’s from Pajara[12] gives us our primary interval names of 12edo.

=== [[14edo]] and Injera ===
Like 10edo, 14edo may be written using all the neutrals and intermediates, but without any intervals described both as a neutral and as an intermediate:

N1 1-2 N2 2-3 N3 3-4 N4 4-5 N5 5-6 N6 6-7 N7 7-8 N8.

The primary interval names for 14edo are as follows:

P1 sm2 N2 SM2/sm3 N3 SM3/s4 P4 SA4/sd5 P5 S5/sm6 N6 SM6/sm7 N7 SM7 P1

We know that 12edo and 14edo support Injera, where Injera[12] 3|2 (2) may be labelled

P1 sm2 M2 sm3 SM3 P4 SA4/sd5 P5 sm6 M6 m7 SM7 P8,

and Injera[14] 3|3 (2) labelled

P1 sm2 M2 sm3 m3 SM3 P4 SA4/sd5 P5 sm6 M6 SM6 m7 SM7 P8.

We can see both in 14edo, and to get 12edo from Injera[12], as with Pajara, we remove all the ‘s’ and ‘S’ prefixes.

=== [[Blacksmith]] and further extension ===
10edo also supports Blacksmith temperament, and we may think to write Blacksmith[10] 1|0 (5) as:

P1 sM2 M2/m3 sM3 P4 s5 P5 sM6 M6/m7 sM7 P8

But we have now added mappings, but are yet to define the use of ‘S’ and ‘s’ for perfect intervals. In Blacksmith, the interval we might call ‘s5’ is 81/80 below P5, however, more commonly ‘s5’ is used to refer to [[16/11]], and S4 [[11/8]]. Since these intervals have above been labelled N4 and N5 above however, we do not need to worry about that, and can add that s5, a 'small 5th', is 81/80 below 3/2, and S4, a 'supra 4th' lies 81/80 above 4/3. where ‘s4’ has been typically been used to refer to [[21/16]], and ‘S5’ to [[32/21]], we add that s4 is lower than P4 by 64/63 and that S5 is higher than P5 by 64/63.

The primary intervals names of Blacksmith[15] 1|1 (5) then are as follows:

P1 Sm2 sM2 M2/m3 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 M6/m7 Sm7 sM7 P8,

these being identical to the the primary interval names of 15edo.

We may further add that ‘S’ (supra) and ‘s’ (small) may raise diminished, and lower augmented intervals by 81/80 as they do to minor and major respectively and that when ‘S’ (super) raised an augmented interval, or ‘s’ (sub) lowers it, the change is by 64/63.

In edos where 81/80 or 64/63 are represented by a single step, or when the apotome is represented by a single step, Ups and Downs with 'S', 's', and 'N' can name the intervals equivalently to this system. This can be seen in add edos considered thus far, as well as in those listed directly below. What this system adds is that it may also describe the intervals of MOS scales (as well as JI scales), such that these interval can be identically named in edos that approximate the scales they belong to. This is true for all MOS scales mentioned so far, as well as those listed below.

=== Further application in edos ===
The primary interval names are shown below for some larger edos.

[[22edo]]: P1 m2 Sm2 sM2 M2 m3 Sm3 sM3 M3 P4 S4 4-5 s5 P5 m6 Sm6 sM6 M6 m7 Sm7 sM7 M7 P8

[[24edo]]: P1 N1 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 N4 A4/d5 N5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 N8 P8

[[26edo]]: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 SA4/sd5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8

[[27edo]]: P1 m2 Sm2 N2 sM2 M2 m3 Sm3 N3 sM3 M3 P4 N4 d6 A3 N5 P5 m6 Sm6 N6 sM6 M6 m7 Sm7 N7 sM7 M7 P8

[[29edo]]: P1 sm2 m2 Sm2 sM2 M2 SM2/sm3 m3 Sm3 sM3 M3 SM3/s4 P4 S4 d5 A4 s5 P5 sm6 m6 Sm6 sM6 M6 SM6/sm7 m7 Sm7 sM7 M7 SM7 P8

[[31edo]]: P1 N1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 N4 A4 d5 N4 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 N8 P8

[[34edo]]: P1 1-2 m2 Sm2 N2 sM2 M2 2-3 m3 Sm3 N3 sM3 M3 3-4 P4 S4 N4 4-5 N5 s5 P5 5-6 m6 Sm6 N6 sM6 M6 6-7 m7 Sm7 N7 sM7 M7 7-8 P8

[[38edo]]: P1 S1 1-2 sm2 m2 N2 M2 SM2 2-3 sm3 m3 N3 M3 SM3 3-4 s4 P4 N4 A4 SA4/sd5 d5 N5 P5 S5 5-6 sm6 m6 N6 M6 SM6 6-7 sm7 m7 N7 M7 SM7 7-8 s8 P8

[[41edo]]: P1 N1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 s4 P4 S4 N4 d5 A4 N5 sM5 P5 S5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 N8 P8

[[43edo]]: P1 S1 1-2 A1/sm2 m2 dd3 AA1 M2 SM2/d3 2-3 A2/sm3 m3 dd4 AA2 M3 SM3/d4 3-4 A3/s4 P4 dd5 AA3 A4 d5 dd6 AA4 P5 S5/A6 5-6 A5/sm6 m6 dd7 AA5 M6 SM6/d7 6-7 A6/sm7 m7 dd8 AA6 M7 SM7/d8 7-8 s8 P8

[[46edo]]: P1 S1 sm2 m2 Sm2 A1 d3 sM2 M2 SM2 sm3 m3 Sm3 A2 d4 sM3 M3 SM3 s4 P4 S4 sd5 d5 sA4/Sd5 A4 SA4 s5 P5 S5 sm6 m6 Sm6 A4 d7 sM6 M6 SM6 sm7 m7 Sm7 A6 d8 sM7 M7 SM7 s8 P8

In 43edo we encounter the first time we have to use double Augmented and diminished intervals. 43edo marks the first instance in which Jones' alternative ups and downs interval names do not match those from this system. In his system, for example, AA2 would simply be sM3, but in this system since sM3 implies an approximation to 5/4 and the M3 already represents 5/4, and therefore is equivalent to sM3, we cannot do this. This tells us however that no simple ratio is approximated by the interval, and perhaps it is better understood as an AA2. Larger edos contain unlabeled intervals (without resorting to extended diatonic interval names). The association of 'super' and 'sub' with 64/63 and with 'supra' and 'small' with 81/80 may effect the assignment of primary interval names, but for all of these edos, as well as all those mentioned before, when 'S' and 's' are used, they still signify a raising or lowering by a single step of the edo, and thus appear equivalent to the Ups and Downs version. The comma associations add that, though use of enharmonic equivalences and secondary interval names may be necessary, intervals from MOS scales may be spelled in a consistent way across tuning to different edos.
=== Other rank-2 temperaments' MOS scales ===
On top of those discussed thus far, other temperaments generated by the P5 or a fraction of it are also supported to some extent, where their MOS scales may be represented, including [[Semaphore]], [[Augmented]], [[Porcupine]], [[Diminished]], [[Negri]], [[Tetracot]] and [[Slendric]].

Semaphore[5] 2|2: P1 SM2/sm3 P4 P5 SM6/sm7 P8

Semaphore[9] 4|4: P1 M2 SM2/sm3 SM3/s4 P4 P5 S5/sm6 SM6/sm7 m7 P8

Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8

Augmented[9] 1|1 (3): P1 Sm2 Sm3 sM3 P4 P5 Sm6 SM6 sM7 P8

Augmented[12] 2|1 (3): P1 Sm2 M2 Sm3 sM3 P4 sA4 P5 Sm6 Sm7 SM6 sM7 P8

Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8

Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8

Porcupine[8] 4|3: P1 sM2 Sm3 P4 s5 P5 sM6 Sm7 P8

Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8

Negri[9] 4|4: P1 Sm2 SM2/sm3 sM3 P4 P5 Sm6 SM6/sm7 sM7 P8

Negri[10] 4|4: P1 Sm2 SM2/sm3 sM3 P4 A4 P5 Sm6 SM6/sm7 sM7 P8

Tetracot[6] 3|2: P1 sM2 N3 S4 N6 Sm7 P8

Tetracot[7] 4|2: P1 sM2 N3 S4 P5 N6 Sm7 P8

Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 P8

[[Slendric5|Slendric[5]]] 2|3: P1 SM2 s4 S5 sm7 P8

[[Slendric6|Slendric[6]]] 3|2: P1 SM2 s4 P5 S5 sm7 P8

[[Slendric11|Slendric[11]]] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8

We may also write temperaments with a 9/8 but no 3/2. The most well known of these is [[Machine]]:

[[Machine5|Machine[5]]] 2|2: P1 M2 M3 m6 m7 P8

[[Machine6|Machine[6]]] 3|2: P1 M2 M3 A4 m6 m7 P8

[[Machine11|Machine[11]]] 5|5: P1 d3 M2 d4 M3 d5 A4 m6 A5 m7 A6 P8
=== Formal summary ===
The resultant system may be formally summarised as follows:

'''Definition 1a.''' M and m label the two sizes of 2nd, 3rd, 6th and 7th in the Pythagorean diatonic scale.

'''Definition 1b.''' The smaller 4th and larger 5th are labelled P.

'''Definition 1c.''' The single size of 1 and 8 is labelled P.

'''Definition 2.''' A chroma above M or P is A and below m or P is d.

'''Definition 3a.''' Within generic interval classes 2, 3, 6 and 7, half way between M and m is N.

'''Definition 3b'''. Within generic interval classes 1 and 4, half way between P and A is N.

'''Definition 3c.''' Within generic interval classes 5 and 8, half way between P and d is N.

'''Corollary:''' 7edo can be written P1 N2 N3 P4 P5 N6 N7 N8.

'''Definition 4.''' Half way between adjacent generic interval classes lie the intermediates.

'''Corollary:''' 5edo can be written 1-2 2-3 3-4 5-6 6-7 7-8.

'''Definition 5a.''' 'S' when applied before M, and 's' when applied before m, raise or lower by 64/63 respectively.

'''Definition 5b.''' 'S' when applied before m, and 's' when applied before M, raise or lower by 81/80 respectively.

'''Definition 5c.''' 'S' when applied to P5 (leading to S5) and 's' when applied to P4 (leading to s4) raise and lower by 64/63 respectively.

'''Definition 5d.''' 'S' when applied to P4 (leading to S4) and 's' when applied to P5 (leading to s5) raise and lower by 81/80 respectively.

'''Definition 5e.''' 'S' when applied to P1 and 's' when applied to P8 may raise and lower by 64/63, or by 81/80.

'''Definition 6a.''' When 'S' and 's' imply alterations of 64/63, they have long-form 'super' and 'sub' respectively.

'''Definition 6b.''' When 'S' and 's' imply alterations of 81/80, they have long-form 'supra' and 'small' respectively.

'''Definition 7.''' If alterations of 81/80 and of 64/63 need to be distinguished from one another in short-form, alterations of 81/80 can be written 'SR' and 'sl'.

== edos with extreme fifths ==
Thus far the best fifth of all edos described lies in the 'diatonic range', between 4 steps of 7edo and 3 steps of 5edo. The best fifths of some edos lies outside this range, in either directly. Whereas in edos where the best fifth lies between 7 steps of 12edo and 4 steps of 7edo, 81/80, the meantone comma is tempered out, wherein four fifths approximate the fifth harmonic (a sM3 raised two octave), in edos with fifths flatter than this, 135/128, a meantone chromatic semitone is instead tempered out, resulting in the four fifths instead approximate the Sm3, 6/5 (raised two ocaves). This system is called Mavila temperament. In the other direction, whereas the best fifth lies between 7 steps of 12edo and 3 steps of 5edo, 64/63, the septimal or Archytas comma is tempered out, wherein four fifths approximate a SM3, 9/7, raised two octaves, in edos who's best fifth is sharper than this, 9/7, as well as 5/4 are approximated by the perfect fourth, tempering out 16/15 and 28/27. This system is called Father temperament. The application of this system to edos of both of these fifth sizes is addressed below.

=== [[Mavila]] ===
In Mavila, the perfect 5th is flatter than in 7edo, so major intervals are below minor and augmented below major. In Mavila the major third approximates 6/5 and the minor third 5/4, tempering out [[135/128]]. This presents no problem to the scheme however, and the rules are applied just the same. The small major third, 81/80 below 6/5 or 81/64 comes to [[32/27]], the minor 3rd, and the sub minor 3rd remains 7/6.

Mavila[7] 3|3 can be written

P1 M2 m3 P4 P5 M6 m7 P8,

the same as the diatonic scale.

Mavila[9] 4|4 can be written

P1 M2 M3 m3 P4 P5 M6 m6 m7 P8,

which is also the primary interval name set for 9edo.

If we don't want to have major being below minor, we can hide it with some secondary interval names:

P1 M2 Sm3 sM3 P4 P5 Sm6 sM6 m7 P8, arriving at Augmented[9].

We say that our first Mavila[9] interval names are not ''well-ordered'', where for an interval name set to be well-ordered, for each degree major must be above minor. By extension we defined a well-ordered interval names set as one in which ... ≤ dd ≤ d ≤ m ≤ M ≤ A ≤ AA ≤ ... or ... ≤ dd ≤ d ≤ P ≤ A ≤ AA ≤ ..., and where s_ ≤ _ ≤ S_ (where '_' represents any of ... dd, d, m, (P), M, A, AA ...).

Using the fact that 64/63 is represented by a single step in 9edo, it is left as an exercise for the reader to prove that 9edo supports Negri temperament (by replacing some names with sub and super prefixed names to arrive at Negri[9] 4|4).

The primary interval names for Augmented[9] 1|1 (3) are well-ordered.

Mavila[16] 8|7 can be writtten

P1 d1 M2 m2 M3 m3 A4 P4 A5 P5 d5 M6 m6 M7 m7 A8 P8.

The enharmonic equivalences for 16edo can be generated by: M=Sm=sm or m=sM=SM

This leads us to our primary well-ordered interval name set for 16edo:

P1 S1 Sm2 sM2 Sm3 sM3 s4 P4 4-5 P5 S5 Sm6 sM6 Sm7 sM7 s8 P8,

wherein we can see that it supports Diminished temperament.

Mavila[23]11|11 can be written as:

P1 d1 A2 M2 m2 A3 M3 m3 d3 A4 P4 d4 A5 P5 d5 A6 M6 m6 d6 M7 m7 d7 A8 P8

And the primary well-ordered interval names for [[23edo]] are appear as:

P1 S1 1-2 M2 m2 2-3 M3 m3 3-4 A4 P4 d4 A5 P5 d5 5-6 M6 m6 6-7 M7 m7 7-8 d8 P8

=== [[Father]] ===
In the other direction, the fifths of Father temperament are sharper than those of 5edo, leading to minor second going backwards. In Father temperament, 5/4 and 4/3 are tempered to a unison, along with 9/7. As 64/63 is tempered out, alterations of 64/63 act as identity alterations.The M2, larger than m3 is also a Sm3. sM2, then, returns to m3.

The primary interval names for Father[5] 2|2,

P1 M2 P4 P5 m7 P8,

present no problems.

In the primary interval names for Father[8] 4|3, however:

P1 m3 M2 P4 M3 P5 m7 M6 P8,

which are the same as the primary intervals for [[8edo]], but with M3 rather than 4-5, we see our diatonic interval names begin to cross over. We will add to our definition of well-ordered interval names that no interval names from interval-class ''n''-1 may be subtended by a larger number of steps that any interval names from interval-class ''n.'' As above, we can may use some secondary interval names to address it, leading to

P1 sM2 M2 P4 S4 P5 m7 Sm7 P8,

as the primary well-ordered interval name set.

If we also re-write M2 and m7 as Sm3 and sM6, we get Porcupine[8] 4|3.

Using some secondary interval names to 'fix' the order leads us to

P1 sM2 M2 P4 S4 P5 m7 M6 P8

The primary interval names for Father[13] 6|6:

P1 M7 m3 M2 d5 P4 M3 m6 P5 A4 m7 M6 m2 P8,

look very unruly.

We will fix up the ordering again with secondary interval names:

P1 S1 sM2 M2 Sm3 P4 S4 s5 P5 sM6 m7 Sm7 s8 P8,.

The primary interval names for [[13edo]] are similar:

P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8.

Minimally fixing the order leads us to

P1 N2 sM2 M2 N3 P4 S4 s5 P5 N6 m7 Sm7 N7 P8,

which we have seen before as Tetracot[13] 6|6.

=== [[Miracle]], [[11edo]] and [[21edo]] ===
All the scales discusses to this point use either smalls and supras or subs and supers, so in the rare instance that we see a S1 or s8, we can infer whether or not it's small/supra or sub/super, probably without even thinking too much about it. Rarely do MOS scales in this scheme require alterations of both 81/80 and 64/63. One important temperament that includes such scales is Miracle. We do not encounter either S1 in the 10 and 11-note MOS:

[[Miracle 10|Miracle[10]]] 5|4: P1 Sm2 SM2 N3 s4 sA4 S5 N6 sm7 sM7 P8

Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8

We first see S1 and s8 in the 21-note MOS:

[[Blackjack|Miracle[21]]] 10|10: P1 S1 Sm2 N2 SM2 sm3 N3 sM3 s4 P4 Sd5 sA4 P5 S5 Sm6 N6 SM6 sm7 N7 sM7 s8 P8,

the largest MOS scale we have attempted yet to write. S1 in this scale is 64/63. We could maybe guess this from the presence of S5, but it is not obvious. If we need to make it clear when we are referring to small/supra, we can write their short-hand instead as 'sl' for 'small' and 'Sa' for supra. Miracle[21] would then be re-written:

P1 S1 Sam2 N2 SM2 sm3 N3 slM3 s4 P4 Sad5 slA4 P5 S5 Sam6 N6 SM6 sm7 N7 slM7 s8 P8, if complete clarity is needed, otherwise if long-form names are provided there is no need. It looks unwieldy to me and so I would avoid it, but it is there as a possibility, if the intervals of Mavila[21] need be written with only 4 letters allowed for each interval name.

Having encountered 11 and 21-note scales now, and haven't not described 11 and 21edo, I will add these here.

The fifth in 11edo is too flat even for it to be considered to support Mavila. Let's see what happens:

P1 M2 M3 m2 m3 P4 P5 M6 M7 m6 m7 P8.

Adding neutrals gives us our primary interval names for 11edo:

P1 M2 M3 N3 m3 P4 P5 M6 N6 m6 m7 P8.

This is clearly not Mavila, so we don't know what's tempered out, such that we might add our alterations to arrive at a well-ordered interval name set. Let's review the 11-note scales we have encountered above:

Slendric[11] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8

Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8

From these union of these scales we can see from P4=Sd5 that 135/128, the Mavila comma is tempered out. We apply our Mavila re-spellings to arrive at:

P1 Sm2 Sm3 N3 sM3 P4 P5 Sm6 N6 sM6 sM7 P8 as a well-ordered interval name set.

We should not expect our Machine[11] scale to be represented in this spelling of 11edo: A spelling of 11edo that shows that it supports Machine uses a different mapping, using the 9/8 from two 22edo P5s. We could spell 11edo as every other note of 22edo if we wish to see how it supports Machine.

21edo can be written as three 7edos as it's best fifth is that of 7edo. Since 81/80 is -1 steps in 21edo, we use 64/63 alterations:

P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8.

Using the available conversion sm = Sm (and therefor SM = sM), we can confirm that 21edo supports Miracle and Whitewood temperaments. This is left as an exercise for the the inspired reader.

=== [[6edo]] ===
Though 6edo is normally only ever considered as a subset of 12edo, given that we have encountered 6-note MOS I'll give it a red hot go.

Our P5 and P4 in 6edo is our half-octave, 4-5, so 9/8 is tempered out and our chromatic scale only covers 2edo: P1/M2/M3/A4 m2/m3/P4/4-5/P5/M6/M7 m6/m7/P8. If we want to write 6edo is a well-ordered way, we might choose:

P1 SM2 sm3 P4/4-5/P5 SM6 sm7 P8.

Writing s4 and S5 instead of sm3 and SM6 would give us Slendric[6] 3|2.

What of Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8?

This tells us that in 6edo 81/80 is mapped to -2 steps of 6edo. This is not a problem, as we can use alterations of 64/63, mapped to 1 step, though I don't see why anyone would want to think of 6edo in this way.

Similarly to Machine in 11edo, Machine in 6edo uses a different (much better) mapping of 9/8: That of 12edo. 6edo is much better spelled as a subset of 12edo, where we can see if supports Machine.

The primary interval names for the remaining trivial edos are trivially derived and are given along with all those described so far in the section below.

=== [[28edo]] ===
We encounter a new problem with 28edo. 28edo's best fifth is that of 7edo. It's primary intervals names are as such:

P1 sA1 1-2 Sm2 N2 sM2 2-3 Sm3 N3 sM3 3-4 S4 P4 sA4 4-5 Sd5 P5 s5 5-6 Sm6 N6 sM6 6-7 Sm7 N7 sM7 7-8 Sd8 P8

in 28edo 81/80 is represented by -1 steps. We en-devour to maintain as best we can in our primary interval names the original premise behind the notation - that the prefix 's' takes an interval down a single step of the edo and 'S' a single step up. In our primary interval names for 28edo we have S4 below P4 and d5 above P5. We can avoid this confusion however by using the secondary interval names for P4 and P5 in 28edo - N4 and N5. In our list of edos below this change is made.

== How to ==
Though we have derived our interval names through many examples thus far, the process has not been explained as yet such that the reader may immediately apply them. This will be addressed here, with a step-by-step derivation guide. This guide gives all possible labels to each interval.

For ''n-''edo:
# Label P1 = 1, P8 = ''n''
# Find the best approximations of 3/2, 5/4 and 7/4, which are to be labelled P5, sM3 and sm7 (This is equivalent to finding the [[7-limit]] [[patent val]]).
#* P5 = round(ln(3/2)/ln(''n'')) steps
#* sM3 = round(ln(5/4)/ln(''n'')) steps
#* sm7 = round(ln(7/4)/ln(''n'')) steps
# Using the best fifth, label the diatonic intervals. i.e.
#* M2 = (2*P5) mod ''n''
#* M6 = (3*P5) mod ''n''
#* M3 = (4*P5) mod ''n''
#* M7 = (5*P5) mod ''n''
#* (P4, m7, m3, m6, m2) = ''n'' - (P4, M2, M6, M3, M7)
# Determine the size of the apotome: A = M-m (for any of degree 2, 3, 6 or 7).
# The initial chromatic intervals may be labelled:
#* A1 = A
#* d8 = P8 - A
#* A4 = P4 + A
#* d5 = P5 - A
# If A is even, then intervals half way between M and m within a degree may be labelled N.
# Determine the size of meantone and septimal comma alterations (steps 7-12 may be skipped if A = 1)
#* meantone comma = M3 - sM3. If equal to 0 then 81/80 is tempered out, the edo is said to support Meantone temperament and alterations of 81/80 are not needed.
#* septimal comma = m7 - sm7. If equal to 0 then 64/63 is tempered out, the edo is said to support Superpyth temperament, and alterations of 64/63 are not needed.
# If sub/super < 0, alterations of 64/63 should not be applied.
# If small/supra < 0, alterations of 81/80 my be applied only if A = 0, wherein all diatonic intervals are given the label 'N'.
# Acknowledging points 7, 8 and 9, label 's' and 'S' alterations to diatonic intervals, where:
#* sM (small major) = M - meantone comma
#* Sm (supra minor) = m + meantone comma
#* SM (super major) = M + septimal comma
#* sm (sub minor) = m - septimal comma
# Acknowledging points 7, 8 and 9, label 's' and 'S' alterations to P4 and P5 where:
#* S4 (supra 4th) = P4 + meantone comma
#* s5 (small 5th) = P5 - meantone comma
#* s4 (sub 4th) = P4 - septimal comma
#* S5 (super 5th) = P5 + septimal comma
# If meantone comma alterations are used, S1 (supra unison) may be labelled at P1 + meantone comma and d8 (small octave) at P8 - meantone comma. If septimal comma alterations are used, S1 (super unison) may be labelled at P1 + septimal comma and s8 (sub octave) may be labelled at P8 - septimal comma. If both are needed and not equal, and after steps 13-16 are completed as desired the interval corresponding to a super or supra unison (equivalently, sub or small octave) is yet otherwise unlabeled then to ascertain whether S1 is a supra or super unison (or equivalently s8 a small or sub octave), for all degrees, the short hand for 'small' is to be 'sl' rather than 's', and for 'supra' is to be 'SR' rather than 'S'.
# Remaining augmented (M + A) and diminished (m - A) intervals may be labelled 'A' and 'd'.
# If desired, augmentation and diminution may be iterated through the continued adding or subtracting of A, labelled by additional 'A' or 'd' prefixes respectively.
# 'S' and 's' may be further applied as if 'A' were 'M' or 'd' were 'm'.
# If desired and if ''n'' is even, the interval subtended by ''n''/2 steps may be labelled 4-5.
# If desired and if m2 is subtended by an even number of steps, the remaining intermediates may be labelled: (1-2, 2-3, 3-4, 5-6, 6-7, 7-8) = (P1, M2, M3, P5, M6, M7) + m2/2.
#

== Lists of edos and MOS scales ==

=== Primary well-ordered (unless otherwise noted) interval names for edos ===
2edo: P1 P4/P5 P8

3edo: P1 P4 P5 P8

4edo: P1 SM2/sm3 P4/P5 SM6/sm7 P8

5edo: P1/m2 M2/m3 M3/P4 P5/m6 M6/m7 M7/P8

6edo: P1 SM2 sm3/s4 P4/P5 S5/sM6 sm7 P8

7edo: P1 N2 N3 P4 P5 N6 N7 P8

8edo: P1 sM2 M2 P4 S4/s5 P5 m7 Sm7 P8

9edo: P1 M2 M3 m3 P4 P5 M6 m6 m7 P8

10edo: P1/m2 N2 M2/m3 N3 M3/P4 S4/s5 P5/m6 N6 M6/m7 N7 M7/P8

11edo: P1 Sm2 Sm3 N3 sM3 P4 P5 Sm6 N6 sM6 sM7 P8

12edo: P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8

13edo: P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8

14edo: P1 S1/sm2 N2 SM2/sm3 N3 SM3/s4 P4 SA4/sd5 P5 S5/sm6 N6 SM6/sm7 N7 SM7/s8 P8

15edo: P1/m2 S1/Sm2 sM2 M2/m3 Sm3 sM3 M3/P4 S4 s5 P5/m6 Sm6 sM6 M6/m7 Sm7 sM7/s8 M7/P8

16edo: P1 1-2 Sm2 sM2 Sm3 sM3 3-4 P4 4-5 P5 5-6 Sm6 sM6 Sm7 sM7 7-8 P8

17edo: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8

19edo: P1 S1/sm2 m2 M2 SM2/sm3 m3 M3 SM3/s4 P4 A4 d5 P5 S5/sm6 m6 M6 SM6/sm7 m7 M7/s8 P8

21edo: P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8

22edo: P1 m2 Sm2 sM2 M2 m3 Sm3 sM3 M3 P4 S4 4-5 s5 P5 m6 Sm6 sM6 M6 m7 Sm7 sM7 M7 P8

23edo: P1 S1 1-2 Sm2 sM2 2-3 Sm3 sM3 3-4 A4 P4 S4 s5 P5 d5 5-6 Sm6 sM6 6-7 Sm7 sM7 7-8 s8 P8 (can't quite get it well-ordered)

24edo: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 N4 A4/d5 N5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8

26edo: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 4-5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8

27edo: P1 m2 Sm2 N2 sM2 M2 m3 Sm3 N3 sM3 M3 P4 N4 d6 A3 N5 P5 m6 Sm6 N6 sM6 M6 m7 Sm7 N7 sM7 M7 P8

28edo: P1 sA1 1-2 Sm2 N2 sM2 2-3 Sm3 N3 sM3 3-4 S4 N4 sA4 4-5 Sd5 N5 s5 5-6 Sm6 N6 sM6 6-7 Sm7 N7 sM7 7-8 Sd8 P8

29edo: P1 S1 m2 Sm2 sM2 M2 SM2/sm3 m3 Sm3 sM3 M3 SM3/s4 P4 S4 d5 A4 s5 P5 S5/sm6 m6 Sm6 sM6 M6 SM6/sm7 m7 Sm7 sM7 M7 s8 P8

31edo: P1 N1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 N4 A4 d5 N4 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 N8 P8

34edo: P1 S1 m2 Sm2 N2 sM2 M2 2-3 m3 Sm3 N3 sM3 M3 3-4 P4 S4 N4 4-5 N5 s5 P5 5-6 m6 Sm6 N6 sM6 M6 6-7 m7 Sm7 N7 sM7 M7 s8 P8

38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 N4 A4 SA4/sd5 d5 N5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8

41edo: P1 N1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 s4 P4 S4 N4 sA4 Sd5 N5 sM5 P5 S5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 N8 P8

43edo: P1 S1 1-2 A1/sm2 m2 dd3 AA1 M2 SM2/d3 2-3 A2/sm3 m3 dd4 AA2 M3 SM3/d4 3-4 A3/s4 P4 dd5 AA3 A4 d5 dd6 AA4 P5 S5/A6 5-6 A5/sm6 m6 dd7 AA5 M6 SM6/d7 6-7 A6/sm7 m7 dd8 AA6 M7 SM7/d8 7-8 s8 P8 (not possible to find a well-ordered interval name set)

46edo: P1 S1 sm2 m2 Sm2 A1 d3 sM2 M2 SM2 sm3 m3 Sm3 A2 d4 sM3 M3 SM3 s4 P4 S4 sd5 d5 4-5 A4 SA4 s5 P5 S5 sm6 m6 Sm6 A4 d7 sM6 M6 SM6 sm7 m7 Sm7 A6 d8 sM7 M7 SM7 s8 P8 (not possible to find a well-ordered interval name set)

=== Interval names for MOS scales ===
Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8

Augmented[9] 1|1 (3): P1 Sm2 Sm3 sM3 P4 P5 Sm6 SM6 sM7 P8

Augmented[12] 2|1 (3): P1 Sm2 M2 Sm3 sM3 P4 sA4 P5 Sm6 Sm7 SM6 sM7 P8

Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8

Barbados[5] 2|2: P1 2-3 P4 P5 6-7 P8

Barbados[9] 4|4: P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8

Blackwood[10] 1|0 (5): P1 sM2 M2/m3 sM3 P4 s5 P5 sM6 M6/m7 sM7/s8 P8

Blackwood[15] 1|1 (5): P1 S1/Sm2 sM2 M2/m3 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 M6/m7 Sm7 sM7/s8 P8

Diminished[8] 1|0 (4): P1 sM2 Sm3 P4 sA4/Sd5 Sm6 sM6 sM7 P8

Diminished[12] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 Sm7 sM7 P8

Injera[12] 3|2 (2): P1 sm2 M2 sm3 SM3 P4 SA4/sd5 P5 sm6 M6 m7 SM7 P8

Injera[14] 3|3 (2): P1 sm2 M2 sm3 m3 SM3 P4 SA4/sd5 P5 sm6 M6 SM6 m7 SM7 P8

Machine[5] 2|2: P1 M2 M3 m6 m7 P8

Machine[6] 3|2: P1 M2 M3 A4 m6 m7 P8

Machine[11] 5|5: P1 d3 M2 d4 M3 d5 A4 m6 A5 m7 A6 P8

Mavila[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8

Mavila[9] 4|4: P1 M2 M3 m3 P4 P5 M6 m6 m7 P8

Mavila[16] 8|7: P1 d1 M2 m2 M3 m3 A4 P4 A5 P5 d5 M6 m6 M7 m7 A8 P8

Meantone[5] 2|2: P1 M2 P4 P5 m7 P8

Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8

Meantone[12] 6|5: P1 m2 M2 m3 M3 P4 A4 P5 m6 M6 m7 M7 P8

Meantone[19] 9|9: P1 A1 m2 M2 A2 m3 M3 d4 P4 A4 d5 P5 A5 m6 M6 d7 m7 M7 d8 P8

Miracle[10] 5|4: P1 Sm2 SM2 N3 s4 sA4 S5 N6 sm7 sM7 P8

Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8

Negri[9] 4|4: P1 Sm2 SM2/sm3 sM3 P4 P5 Sm6 SM6/sm7 sM7 P8

Negri[10] 4|4: P1 Sm2 SM2/sm3 sM3 P4 A4 P5 Sm6 SM6/sm7 sM7 P8

Neutral[7] 3|3: P1 N2 N3 P4 P5 N6 N7 P8

Neutral[10] 5|4: P1 N2 M2 N3 P4 N4 P5 N6 m7 N7 P8

Neutral[17] 8|8: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8

Pajara[10] 2|2 (2): P1 Sm2 M2 sM3 P4 sA4/Sd5 P5 Sm6 m7 sM7 P8

Pajara[12] 3|2 (2): P1 Sm2 M2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 m7 sM7 P8

Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8

Porcupine[8] 4|3: P1 sM2 Sm3 P4 s5 P5 sM6 Sm7 P8

Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8

Semaphore[5] 2|2: P1 SM2/sm3 P4 P5 SM6/sm7 P8

Semaphore[9] 4|4: P1 M2 SM2/sm3 SM3/s4 P4 P5 S5/sm6 SM6/sm7 m7 P8

Slendric[5] 2|3: P1 SM2 s4 S5 sm7 P8

Slendric[6] 3|2: P1 SM2 s4 P5 S5 sm7 P8

Slendric[11] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8

Superpyth[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8

Superpyth[12] 6|5: P1 m2 M2 m3 M3 P4 A4 P5 m6 M6 m7 M7 P8

Superpyth[17] 8|8: P1 m2 A1 M2 m3 d4 M3 P4 d5 A4 P5 m6 A5 M6 m7 d8 M7 P8

Tetracot[6] 3|2: P1 sM2 N3 S4 N6 Sm7 P8

Tetracot[7] 4|2: P1 sM2 N3 S4 P5 N6 Sm7 P8

Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 P8

== Conclusion ==
Using only 'S' and 's' as qualifiers to M, m, P, A and d, along with N and intermediate degrees, a system is developed wherein for most edos below 50 intervals can be systematically named such that for primary interval names, 'S' and 's' raise and lower, respectively, by 1 step of an edo , whilst the intervals of many MOS scales may be consistently named across different tunings, taking the best of both Igliashon Jones' extra diatonic interval names and Sagispeak. One does not need to understand the comma associations to make use of the interval names. While the intervals of some MOS scales may hold consistent names in edos in an Ups and Downs based scheme, there are many common scales that do not in such a system, that do in this one, such as scales of Diminished and Augmented temperament. What's more, the use of neutrals and intermediates leads to quicker recognition of MOS scales that may be supported in edos.

User:Lhearne/Extra-Diatonic Intervals

2018-10-15T06:33:22Z

PiotrGrochowski:

Today a small under of competing interval naming schemes exist for the description of microtonal music. More common than any particular defined standard are certain tendencies for microtonal interval naming, or names for specific intervals. While risking the creation of simply another competing standard, an effort is made to develop a scheme that is able to take the best aspects of the existing standards and apply them in a formal interval naming system built on common undefined practice. Such a system is developed, where in addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', only the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees. Using this system all intervals in three fifths of all [[Equal division of the octave|edo]]<nowiki/>s up to 50 can be named such that 'S' and 's' correspond to a displacement of an interval up or down a single interval of the edo, respectively. Many commonly used [[MOS scale|MOS scales]] may also be described using this scheme such that these scales' interval names are consistent expression in any tuning that supports them. The resultant scheme can also be easily mapped to any of the current naming standards, and may even facilitate translation between. The resulting scheme should improve pedagogy and communication in microtonal music.

== Background ==

=== The origin of diatonic interval names ===
[[File:Mesopotamian interval names table.jpg|thumb|500x500px|Mesopotamian interval names, from http://www.historyofmusictheory.com/?page_id=130, accessed October 7, 2018.]]
Music theory describing the use of heptatonic-diatonic scales, including interval names, has been traced back as far as 2000BC, deciphered from a Sumerian cuneiform tablet from Nippur by Kilmer (1986). From Kummel (1970) we know that 'the names given to the seven tunings/scales were derived from the specific intervals on which the tuning procedure started' (Kilmer, 1986). This formed the basis of their musical notation ([http://www.jstor.org/stable/985853. Kilmer, 2016]). The table to the right following table displays the Ancient Mesopotamian interval names accompanied by their modern names.

Kilmer also writes that 'the ancient Mesopotamian musicians/“musicologists” knew what we call today the Pythagorean series of fifths, and that the series could be accomplished within a single octave by means of “inversion.” '. The Mesopotamian's music and theory was passed down through the Babylonians and the Assyrians to the Ancient Greeks, as well as their mathematics, particularly concerning musical and acoustical sound [[ratios]] (Ibid, [http://math-cs.aut.ac.ir/~shamsi/HoM/Hodgkin%20-%20A%20History%20of%20Mathematics%20From%20Mesopotamia%20to%20Modernity.pdf Hodgekin, 2005]).

Such mathematical and musical ideas are attributed to Pythagoras, who undoubtedly made them popular., although many scholars suggest he may have learned these ideas from his Babylonian and Egyptian mentors. None the less, Pythagoras' idea that that by dividing the length of a string into ratios of halves, thirds, quarters and fifths created the musical intervals of an octave, a perfect fifth, an octave again, and a major third form the basis of Ancient Greek music theory (http://www.historyofmusictheory.com/?page_id=20). His tuning of the diatonic scale by only octaves and perfect fifths ([[Pythagorean tuning]]) is influential through to today.

=== Ancient Greek interval names ===
Intervals in Ancient Greek music were written either as string length ratios, after Pythagoras, or as positions in a [[tetrachord]].

[[2/1]], the [[octave]], was named ''diapason'' meaning ''<nowiki/>'''through all [strings]'

[[3/2]], the [[perfect fifth]] was labelled ''diapente,'' meaning 'through 5 [strings]'

[[4/3]], the [[perfect fourth]], was labelled ''diatessaron'', meaning 'through 4 [strings]'

''dieses,'' 'sending through', refers to any interval smaller than about 1/3 of a perfect fourth

''tonos'' referred both to the interval of a whole tone, and something more akin to [[mode]] or key in the modern sense ([http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf Chalmers, 1993])

''ditone'' referred to the interval made by stacking two [[9/8]] whole tones, resulting in [[81/64]], the Pythagorean major third. ([[Joe Monzo|Monzo]], http://www.tonalsoft.com)

[[256/243]] - the ''limma'', which is the ratio between left over after subtracting two 9/8 tones (together making a ditone) a perfect fourth, the ''diatonic semitone'' of the Pythagorean diatonic scale

[[2187/2048]] - the ''apotome'', which is the ratio between the tone and the limma, the ''chromatic semitone'' of the Pythagorean diatonic scale

The Ancient greek term ''diatonon'', meaning 'through tones', refers to the genus with two whole tones and a semitone, or any genus in which no interval is greater than one half of the fourth (Chalmers, 1993). The Pythaogrean diatonic scale is the scale that may be built from one two Pythagorean tetrachords, and the left over interval of 9/8.

=== Zarlino and Meantone ===
[[File:Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1.png|thumb|566x573px|''Le institutioni harmoniche,'' Zarlino, 1558, Cap. 15: Della proprietà del numero Senario & della sue parti; & come in esse si ritroua ogni consonanze musicale, figura 1, pg. 25.]]
Intervals were referred to by the Ancient Greek names through the the 18th century, as Latin names. By the Renaissance it had been discovered that a Pythagorean diminished fourth sounded sweet, and approximated the string length ratio 5/4. This just tuning for the major third was sought after, along with the complementary 6/5 tuning for the minor third, and octave complements to both - 8/5 for the minor sixth and 5/3 for the major sixth. Influential Italian music theorist and composer Gioseffo Zarlino put forth that choirs tuned the diatonic scale to the tuning built from this tetrachord, the ''intense diatonic scale'', also known as the ''syntonic or syntonus diatonic scale'' or the ''Ptolemaic sequence'':

1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

Included in this scale, however, were ''wolf intervals:'' imperfect consonances that occurred as tunings of the same interval as perfect consonances. For example, between 1/1 and 3/2, 4/3 and 1/1, 5/3 and 5/4; and 5/4 and 15/8 occurs the perfect fifth, 3/2, whereas between 9/8 and 5/3 occurs the wolf fifth, 40/27, flat of 3/2 by 81/80. This was also the interval by which four 3/2 fifths missed 5/1 (the interval two octaves above 5/4). It was named the ''syntonic comma'' after Ptolemy's ''syntonus'' or ''intense diatonic tetrachord'' which consists of the intervals 9/8, 10/9 and 16/15, where 9/8 and 10/9 differ by this interval. By making the syntonic comma a unison the wolf fifth could be made a perfect fifth. It was discovered that this could be achieved by flattening (tempering) the perfect fifth by some fraction of this comma such that four of these fifths less two octaves gave an approximation of 5/4. Where two fifths less an octave give 9/8, the next two add another 10/9 to result in the 5/4. 9/8 and 10/9 were referred to as the ''major tone'' (''tunono maggiore'') and ''minor tone'' (''tunono minore''), respectively, and where this tuning led to them being equated, it was referred to as Meantone temperament, which is said to 'temper out' the syntonic comma. Zarlino advocated the flattening of the fifth by 2/7 of a comma, leading to 2/7-comma Meantone, but also described 1/3-comma and 1/4-comma Meantone as usable (Zarlino, 1558).

The diagram on the right, from Zarlino's 1558 treatise ''Le istitutioni harmoniche'' associates many intervals with their tuning as perfect consonances. The perfect tuning for the ditone was considered then to be 5/4, rather than 81/64. The interval for which 6/5 is considered a perfect tuning was referred to as a ''semiditone'' (labelled also in ''Le istitutioni harmoniche'' by as ''Trihemituono)''. This may seem odd to us now, but in Latin 'semi' referred not to 'half', but to 'smaller', so 'semiditone' translated to something like 'smaller ditone'. Additionally 'semitone' referred to the interval smaller than the 'tone'. Like the tone, this interval possessed two alternative perfect tunings: 16/15, the difference between 15/8 and 2/1, or 5/4 and 4/3, and 25/24, the difference between 6/5 and 5/4. 16/15 was referred to as the ''major semitone'' (''semituono maggiore'') and 25/24 as the ''minor semitone (semituono maggiore'').

In addition to the Latin interval names, derived from the Ancient Greek interval names, we see on the diagram a single interval name in Italian: ''Essachordo maggiore'', referring to the ratio 5/3, which we are tempted to translate to 'major sixth'. Chapter 16, ''Quel che sia Consonanze semplice, e Composta; & che nel Senario si ritouano le sorme di tutte le somplici consonanze; & onde habbia origine l'Essachordo minore'', puts forward that the ''Essachordo minore,'' or perhaps 'minor sixth' be tuned to 8/5.

In the 1691 ''Lettre de Monsieur Huygens à l'Auteur [Henri Basnage de Beauval] touchant le Cycle Harmonique,'' theorist Christiaan Huygens gave names and ratios to common intervals and mapped them to 31-tET, which very closely approximates 1/4-comma Meantone. Translated from French, 3/2 was labelled a Fifth, 4/3 a Fourth, 5/4 a major Third, 6/5 and minor Third, 5/3 a major Sixth and 8/5 a minor Sixth. Here we really begin to see today's interval names.

=== English interval names in the Baroque ===
[[File:Harmonics, or The Philosophy of Musical Sounds, Section 2 figure 3.png|thumb|517x548px|''Harmonics, or The Philosophy of Musical Sounds'', Edition 2, Smith, 1759, Section 2: On the Names and Notation of consonance and their intervals, Fig. 2 & 3 , pg. 10|547.986x547.986px]]
After English superseded Latin as the the main language of scholarship, the Latin interval names were rejected and the convention we saw in Zarlino's Italian for naming the smaller of a pair of sizes of an interval 'minor' and the larger 'major' was further applied.

Music theorist and mathematician Robert Smith provides the diagram and table on the right in his 1749 ''Harmonics, or, The Philosophy of Musical Sounds,'' with the description: <blockquote>'Fig. 2. If a musical string ''CO'' and it's parts ''DO'', ''EO'', ''FO'', ''GO'', ''AO'', ''BO'', ''cO'', be in proportion to one another as the numbers 1, 8/9, 4/5, 3/4, 2/3, 3/5, 8/15, 1/2, their vibrations will exhibit the system of 8 sounds which musicians donate by the letters ''C'', ''D'', ''E'', ''F'', ''G'', ''A'', ''B'', ''c''.</blockquote><blockquote>Fig. 3. And supposing those strings to be ranged like ordinates to a right line ''Cc'', and their distances ''CD'', ''DE'', ''EF'', ''FG'', ''GA'', ''AB'', ''BC'', not to be the differences of their lengths, as in fig. 2. but to be the magnitudes proportional to the intervals of their sounds, the received Names of these intervals are shewn in the following Table; and are taken from the numbers of the strings or sounds in each interval inclusively; as a Second, Third, Fourth, Fifth, &c, with the epithet of ''major'' or ''minor'', according as the name or number belongs to a greater of smaller total interval; the difference of which results chiefly from the different magnitudes of the major and minor second, called the Tone and Hemitone.'</blockquote>We note that Smith uses the tuning of the diatonic scale that Zarlino put forward: the Ptolemaic Sequence.

We may be surprised to see 4:3 here labelled as a minor Fourth, and 3/2 as a major Fifth, but it is obvious that this naming is more consistent than today's. Smith adds that <blockquote>'Any one of the ratios in the third column of the foregoing Table, except 80 to 81, or any one of them compounded once of oftener with the ratio 2 to 1 or 1 to 2, is called a Perfect ratio when reduced to it's least terms. And when the times of the single vibrations of any two sounds have a perfect ratio, the consonance and it's interval too after called Perfect; and is called Imperfect or Tempered when that perfect ratio and interval is a little increased or decreased.'</blockquote><blockquote>'Any small increment of decrement of a perfect interval is called respectively the Sharp or Flat Temperament of the imperfect consonance, and is measured most conveniently by the proportion it bears to the comma'</blockquote>Therefore in this system 3/2 is the ''Perfect major Fifth'' and 5/4 the ''Perfect major Third''. 81/64 might be labelled a ''comma sharp major Third'', 32/27 a ''comma flat minor Third'', and the 1/4-comma Meantone fifth a ''1/4-comma flat major Fifth''. The interval naming scheme Smith describes may be immediately applied to 5-limit microtonal systems. There is an inconsistency, however, where it seems that 9/8 should be called a ''Perfect major Second,'' but that, while 9/5 be named a ''comma sharp minor Seventh'', it's inverse, 10/9, is a ''Perfect minor Tone.''

In ''An Elementary Treatise on Musical Intervals and Temperament,'' published in 1876'','' R. H. M Bosanquet refers to 5/4 as the ''perfect third'', and 81/64 as the P''ythagorean third''. Bosanquet also labels other intervals of the Pythagorean diatonic scale similarly, i.e. 256/243, the limma, is labelled the ''Pythagorean semitone'', and 27/16 the ''Pythagorean sixth''. 81/80 is labelled the ''ordinary comma'', or simple the ''comma,'' and the Pythagorean comma is defined as the difference between twelve fifths and seven octaves. The apotome of 2187/2048 is referred to as Apatomè Pythagoria. The following relationships are then described: (<span style="color:#FF0000">the image is too inappropriate to show, see pg 28 for rules regarding posting images to the Internet<span>)

[[File:Helmholtz consonances table.png|thumb|Table describing the influence of the different consonances on one another, up to the 9th partial, from ''On the Sensations of Tone as a Phycological Basis for Theory of Music'', Helmholtz, 1863, Translation by Ellis, 1875, pg. 187]]

=== Helmholtz and Ellis ===
Through the investigations of Galileo (1638), Newton, Euler (1729), and Bernouilli (1771), theorist Hermann von Helmholtz was aware that ratios governing the lengths of strings existed also for the vibrations of the tones they produced. His investigation of the harmonic series associated with these ratios of vibration led him to the consideration of ratios above the 5-limit. In his seminal ''On the Sensations of Tone as a Physiological Basis for the Theory of Music'', published in German in 1863 and translated into English in 1875 by Alexander Ellis, he listed just intervals as show in the table to the right. It is interesting to note that 8:9 is labelled a 'Second' rather than a 'major Second'. The minor Seventh is shown as 5:9 rather than as 9:16 seemingly because of the 9 partial limit imposed on the table. It is also worth noting that 5:7 is labelled a subminor Fifth. 'Super', indicated in notation with a '+', raises an interval by 35:36, the septimal quarter tone, and 'sub', indicated with a '-' lowers by the same interval with the exception of the Supersecond 7:8, which lies 63:64, the septimal comma above the Second. The subminor fifth is not included in this as no minor Fifth is shown. If we assume that 'sub' lowers an interval 35:36, then the minor Fifth would be 18:25, 80:81 above Smith's 45:64 minor Fifth, however in table 2 below, Ellis labels 18:25 a ''superfluous Fourth'', and it's inverse, 25:32, an ''acute diminished Fifth'', whilst 64:45 is labelled a ''diminished Fifth'' and its inverse 32:45 a ''false Fourth or Tritone.'' If we label 9:10 as a 'major Second', and 7:8 as a 'supermajor Second' then they differ by 35:36, the major Second is the inverse of the minor Seventh, and the supermajor Second is the octave inverse of the subminor Seventh. Perhaps 'major' has been left off name of the major Fifth, and minor off the name of the minor Fourth since the time of Smith. We can add to this table the remaining octave inversions as well as the super Fourth and sub Fifth.
{| class="wikitable"
|+Table 1. Additional Intervals
!Intervals
!Notation
!Ratio
!Cents in the interval
|-
|Major Second
|D
|9:10
|182
|-
|Sub Fourth
|F-
|16:21
|471
|-
|Supermajor Fourth
|F#+
|7:10
|617
|-
|Super Fifth
|G+
|21:32
|729
|-
|Subminor Sixth
|A♭-
|9:14
|765
|-
|Supermajor Sixth
|A+
|7:12
|933
|}
Helmholtz defined the ''perfect consonances'' as the Octave, Twelfth and Double Octave as well as Fourth and Fifth. The major Sixth and major Third are next called ''medial consonances'', considered to in the era of Pythagorean tuning to be ''imperfect consonances'', which Helmholtz defined instead to be the minor Third and the minor Sixth.

Regarding the tuning of the intervals however, those corresponding to simple ratios of vibration are, as in Smith, referred to as perfect, however hey are also described as 'justly-intoned', or by Ellis as 'just'. The perfect tuning for the semitone is listed as 16/15, or 182c. The perfect tunings are compared to the Pythagorean tunings, where the Pythagorean tuning of the major Third and sixth are described as 81/80 above the perfect tunings, and of the minor Third, minor Sixth and semitone to be 81/80 below the perfect tunings. Helmholtz notes that the Pythagorean tunings are closer to the equal tempered tunings than the perfect tunings. Helmholtz also describes the Pythagorean Tritone as of 612c.

Ellis includes an [[Ellis' interval table|additional table]] providing names for many different just and tempered intervals, perfect and imperfect. The interval names do not appear to follow any sort of consistent naming system, rather intervals seem to be named case-by-case. They also largely do not correspond to the interval names used by Helmholtz.

In ''Notes on the Observations of Musical Beats'', Proc. R. Soc. Lond. 1880, Ellis named many just intervals of the 7-limit (including 3 and 5-limit intervals):<blockquote>‘Fifth 3:2, Fourth 4:3, Major Third 5:4, Minor Third 6:5, Major Sixth 5:3, Sub-Fifth 7:5, Super-Fourth 10:7, Super-major Third 9:7, Sub-minor Sixth 14:9, Sub-minor Third 7:6, Super-major Sixth 12:7, Sub-minor or Harmonic Seventh 7:4, Super-major Second 8:7, Major Tone 9:8, Minor Tone 10:9, Small Major Seventh 9:5 and Diatomic (sic.) Semitone 16:15’</blockquote>He later lists ‘The Major Sevenths 16:9 and 15:8’. The labeling of 16:9 as a Major Seventh and 9:5 as a Small Major Seventh is interesting and at odds with Smith's interval names. Given that 9:5 is larger than 16:9, and no Minor Seventh is mentioned, we can assume 16:9 was mislabeled as a Major Seventh and was understood to be a Minor Seventh, as referred to by Smith.

There is an inconsistency associated with the labeling of 9:5 as a Small Major Seventh also, as it lies a 3:2 Fifth above the 6:5 Minor Third, and we know a fifth and a minor third when added together to give a minor, rather than major seventh.

Ellis here uses the name for 8:7 we suggested above, Super-major Second, and includes our suggested Sub-minor Sixth and Super-major Sixth, however rather than Subminor Fifth and Supermajor Fourth, 7:5 and 10:7 are labelled Sub-Fifth and Super-Fourth, where in this instance sub and super are seen to raise and lower by 21:20 instead of by 36:35.

Ellis, in a footnote to his translation of Helmholtz,'s treatise also provides names for a single 11-limit interval. The interval 22:27, of 355c, introduced by Zalzal, says Ellis was termed a ''neutral Third'' by Herr J. P. N. Land originally in ''Over de Toonladders der Arabische Musiek'' (On the Scales of Arabic Music) in 1880. An interval a fourth higher than this is mentioned, but a ratio is not given, and it is not named. We can ourselves however find it's ratio as 11:18, and guess it's name to be a ''neutral Sixth'', given that it lies a perfect Fourth above the neutral Third. Following a similar process as in our completion of Helmholtz table above, and assuming that the octave inverse of a neutral Third should be a neutral Sixth we may introduce the following 11-limit intervals that see common use among music theorists and microtonal musicians through to today:
{| class="wikitable"
|+Table 3. 11-limit intervals
!Intervals
!Notation
!Ratio
!Cents in the interval
|-
|Neutral Second
|D♭^
|81:88
|143
|-
|Neutral Second
|Dv
|11:12
|151
|-
|Neutral Third
|E♭^
|9:11
|347
|-
|Neutral Third
|Ev
|22:27
|355
|-
|Neutral Sixth
|A♭^
|27:44
|845
|-
|Neutral Sixth
|Av
|11:18
|853
|-
|Neutral Seventh
|B♭^
|6:11
|1049
|-
|Neutral Seventh
|Bv
|44:81
|1057
|}
Each interval name has two sizes that differ by the comma 242:243. The notation included in the table is from HEWM notation, developed as an extension to the Helmholtz-Ellis use of '+' and '-' by Joe Monzo (http://www.tonalsoft.com/enc/h/hewm.aspx<nowiki/>).'^' indicates raising 'v' a lowered of 33/32. In HEWM notation '+' and '-' are refined to mean raising and lowering of 81/80 respectively and '>' and '<' are added instead to indicate raising and lowering of 64/63. Letter names correspond instead of the the Ptolemaic sequence, as in Smith's and Helmholtz' descriptions, but to a Pythagorean tuning of the diatonic scale, where '#' and '♭' and respectively raise and lower the apotome, 2187/2048. HEWM notation is not accompanied by an interval naming system.

=== Common interval names today ===
These interval names are used by theorists and microtonal musicians today, though 7:5 and 10:7 are given many different names, today also considered to be an augmented fourth and diminished fifth, lesser septimal tritone and greater septimal tritone, or simply as tritones. The fourth and fifth are today called perfect fourth and perfect fifth, and Smith's major Fourth and minor Fifth referred to as augmented fourth and diminished fifth respectively. As can be seen in Tchaikovsky's ''A Guide to the Practical Study of Harmony,'' by the beginning of end of the 19th century the familiar conventions for the naming of intervals were set, wherein
* Seconds, thirds, sixths and sevenths appear in the diatonic in two sizes, the larger labelled 'major' and the smaller, 'minor'.
* Major, when raised by a semitone, becomes 'augmented', and minor, lowered by a semitone, 'diminished'.
* The smaller of the two sizes of fourth and the larger of the two sizes of fifth are labelled 'perfect', along with the unison and octave.
* A perfect interval, when raised a semitone is labelled 'augmented', and when lowered a semitone, 'diminished'.

=== Fokker/Keenan Extended-diatonic interval-names ===

Considering the 11-limit otonal chord 4:5:6:7:9:11 a chain of thirds, in addition to the familiar major, minor, subminor, supermajor and neutral thirds, Dave Kennan labelled 5:7 a sub-diminished fifth and 7:11 an augmented fifth. 7:10, the inversion of 5:7, is labelled a diminished. 5:7, therefore, is also an augmented fourth. In terms of sevenths, 4:7 is subminor, 5:9 is minor and 11:6 is neutral.

Super and sub were further generalised by some theorists and musicians such that an interval a bit smaller than a major is referred to as a ''subminor third'', and an interval a bit larger than a minor third as a ''supraminor third''.

From this, Keenan defines a consistent interval naming system, meaning one which obeys diatonic interval arithmetic (In each column, the parenthesised prefix is the one that is implied when there is no prefix). When adding intervals the indexes are added together to give the index of the resulting interval. Keenan also adds corrections for each interval class to the indexes in order to account for inconsistencies that occur within diatonic interval arithmetic when concerning intervals greater than an octave, so that his system, unlike regular diatonic interval names, may be completely consistent.

As can be seen above, sub, super, augmented and diminished have also carried inconsistent meaning historically, where in Keenan's system they always alter intervals by the same amount.
{| class="wikitable"
|+
!Index
!Prefix for unisons, fourths, fifths, octaves
!Prefix for seconds, thirds, sixths, sevenths, ninths
|-
| -4
|double diminished
|subdiminished
|-
| -3
|subdiminished
|diminished
|-
| -2
|diminished
|subminor
|-
| -1
|sub
|minor
|-
|0
|(perfect)
|neutral
|-
|1
|super
|(major)
|-
|2
|augmented
|supermajor
|-
|3
|superaugmented
|augmented
|-
|4
|double augmented
|superaugmented
|}
The index values correspond most directly to degrees of 31edo, whose interval names by this method are given in the following table:
{| class="wikitable"
|+Extended-diatonic interval names in 31-tET
!31-tET degree
!Ratios
!Names
!
|-
|0
|1:1
|
|unison
|-
|1
|48:49 44:45 36:35 33:32
|(diminished second)
|super unison
|-
|2
|27:28 24:25 20:21
|subminor second
|(augmented unison)
|-
|3
|15:16 14:15
|minor second
|
|-
|4
|11:12 10:11
|neutral second
|
|-
|5
|9:10 8:9
|major second
|
|-
|6
|7:8
|supermajor second
|(diminished third)
|-
|7
|6:7
|(augmented second)
|subminor third
|-
|8
|5:6
|
|minor third
|-
|9
|9:11
|
|neutral third
|-
|10
|4:5
|(subdiminished fourth)
|major third
|-
|11
|11:14 7:9
|(diminished fourth)
|supermajor third
|-
|12
|16:21
|subfourth
|(augmented third)
|-
|13
|3:4
|perfect fourth
|
|-
|14
|8:11
|super fourth
|
|-
|15
|5:7
|augmented fourth
|(subdiminished fifth)
|-
|16
|7:10
|(superaugmented fourth)
|diminished fifth
|-
|17
|11:16
|
|sub fifth
|-
|18
|2:3
|
|perfect fifth
|-
|19
|21:32
|(diminished sixth)
|super fifth
|-
|20
|9:14 7:11
|subminor sixth
|(augmented fifth)
|-
|21
|5:8
|minor sixth
|(superaugmented fifth)
|-
|22
|11:18
|neutral sixth
|
|-
|23
|3:5
|major sixth
|
|-
|24
|7:12
|supermajor sixth
|(diminished seventh)
|-
|25
|4:7
|(augmented sixth)
|subminor seventh
|-
|26
|9:16 5:9
|
|minor seventh
|-
|27
|11:20 6:11
|
|neutral seventh
|-
|28
|8:15
|
|major seventh
|-
|29
|14:27
|(diminished octave)
|supermajor seventh
|-
|30
|18:35
|sub octave
|(augmented seventh)
|-
|31
|1:2
|octave
|
|}
The interval names shown in brackets could be said to be 'secondary', the others, 'primary'.

After releasing his system Keenan was informed that is was identical to the extended-diatonic interval-naming scheme of Adriaan Fokker but for the acknowledgment of more 11-limit ratios.This system depends on the tempering out of 81/80, where the diatonic major third, from four stacked fifths, approximates the just major third, 5/4. It also depends on the existence of neutral intervals, i.e., that the perfect fifth or equivalently, the chromatic semitone, subtends an even number of degrees of the ET. To simply to our familiar naming scheme for 12-tET, we observe that it applies to 24edo equally as directly as in 31-tET, where the prefixes correspond to degrees of the edo. Exactly the same is also true for 38-tET, twice 19-tET, a meantone which very closely approximates 1/3-comma meantone. Meantone temperament wherein the fifth is divided into two equally sized neutral thirds is referred to as neutral temperament. Whereas meantone temperament is generated by the fifth, iin neutral temperament the generator is half this interval, the neutral third. Where it was seen above that there are two neutral thirds, 9:11 and 22:27 that differ by 243/242, neutral temperament is at its most simple the temperament defined by this equivalence: the tempering out of 243/242, as meantone is defined by the tempering out of 81/80. The temperament that tempers out both 81/80 and 243/242 is called Mohajira, upon which Keenan's scheme can be said to be based.

The primary interval names resulting in this system's application to 24-tET and 38-tET is now show, along with 31-tET again for easy comparison, where 'M', 'm', 'P', 'N', 'A', 'd', 'S' and 's' are shorthand for major, minor, perfect, neutral, augmented, diminished, super and sub, respectively:

24edo: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 S4 A4/d5 s5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8

31edo: P1 S1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 S4 A4 d5 s5 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 s8 P8

38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 S4 A4 SA4/sd5 d5 s5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8

to Meantone tunings that are not Mohajira tunings, the regular diatonic interval names can be applied, but with the addition of double augmented and double diminished:

19edo: P1 A1 m2 M2 A2/d3 m3 M3 d4 P4 A4 d5 P5 A5 m6 M6 A6/d7 m7 d8 P8

26edo: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 AA4/AA5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8

Keenan adds further that if it is desired to distinguish between ratios that are in 31-tET approximated by the same number of steps, an addition prefix be added to describe the prime limit of the approximated interval. For 3-limit intervals, the obvious choice is 'Pythagorean', for 5-limit Keenan chooses 'classic', for 7, 'septimal, 11, 'undecimal' and 13, 'tridecimal'. When the highest prime is the same, Keenan suggests adding 'small' and 'large' as final prefixes for this purpose.

In non-Meantone tunings, the two definitions of major third - 4:5 and 64:81, the just (or classic) and Pythagorean major thirds no-longer correspond. If intervals are to receive unique names then to one or both of these major thirds must be added a prefix. Keenan has been involved with the development of both types of systems. Only when the major is defined by it's mapping as fourth fifths, i.e. 81/64, can conserve interval arithmetic.

=== [[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak] ===
One system in which 81/64 is the major third, sagispeak, was developed by [[Dave Keenan]] and others as an interval naming system that maps 1-1 with the Sagittal microtonal music notation system. Sagittal notation was developed as a generalised diatonic-based notation system applicable equally to [[just intonation]], [[Equal Temperaments|equal tunings]] and rank-''n'' [[temperaments]]. Dozens of different accidentals can be used on a regular diatonic [[Staff notation|staff]] to notate up to extremely fine divisions, however in most cases only a handful are needed. In sagispeak, each accidental is presented by a prefix, made up of a single letter, in most cases, followed by either 'ai' if the accidental raises a note, or 'ao' if it lowers a note. As in HEWM notation, Pythagorean intonation is assumed as a basis. Then the prefixes depart from Pythaogrean intonation, altering by commas and introducing other primes. In place of the prefixes 'sub' and 'super', generally signifying an alteration of 36/35 from 5-limit intervals or 64/63 for 3-limit, Sagittal features an accidental of [[64/63]], which may be used to take a Pythagorean major interval to a supermajor, minor to subminor, or perfect to super or sub. The prefix 'tao' indicates a decrease of 64/63 and and the prefix 'tai' an increase. Whereas in previous interval naming schemes 'major' and 'minor' were synonymous with the 5-limit tunings, in sagispeak they map instead to Pythagorean. A prefix is needed then to take a Pythagorean intoned interval to a 5-limit tuning. Where 5/4 is 81/80 below the the Pythagorean third, the prefixes 'pai' and 'pao' (where 'p' is for 'pental', as in, involving prime 5), which raise or lower a note by [[81/80]]<nowiki/>respectively. Similarly, 'vai' and 'vao', which raise or lower a note by [[33/32]] respectively, leading to ratios of 11.

Because it is built off of the diatonic scale, sagispeak conserves diatonic interval arithmetic, i.e. familiar relations in the diatonic scale, i.e. M2 + m3 = P4. As in Fokker/Keenan Extended-diatonic Interval-names, diatonic interval arithmetic is also extended, where, for example, tai-major 2 + tao-minor 3 = P4 (8/7 + 7/6 = 4/3), where opposite alterations cancel each other out, and diatonic interval arithmetic is conserved, a very useful property for a microtonal interval naming system to possess. Another helpful property of sagispeak is its generalised applicability to edos, just intonation and other tunings, where the same intervals maintain their spelling across different tunings. Despite these benefits however, many see Sagittal and Sagispeak as overly complex (even though the entire extended system need hardly ever be applied), and requiring too many new terms to be learnt.

We have seen that there are two competing definitions of a major third, the ratio '5/4' or the interval built from fourth stacked fifths, that may or may not correspond to 5/4. In meantone systems, those we are used to, they correspond, but in most edos they do not. Interval naming systems wherein the major third is defined as an approximation to 5/4 rather than as four fifths minus two octaves may benefit from a familiar name for 5/4, but they are unable to conserve diatonic interval arithmetic.

For comparison, 31edo is shown below in sagispeak:

31edo: P1 tai-1/vai-1 tao-m2 m2 vai-2m/vao-M2 M2 tai-M2 tao-m3 m3 vai-m3/vao-M3 M3 tai-M3 tao-4 P4 vai-4 A4 d5 vao-5 P5 tai-5 tao-m6 m6 vai-m6/vao-M6 M6 tai-M6 tao-m7 m7 vai-m7.vao-M7 M7 tai-M7 vao-8/tao-8 P8

=== Dave Keenan's most recent system ===
In 2016 Dave Keenan proposed an alternative generalised [http://dkeenan.com/Music/EdoIntervalNames.pdf microtonal interval naming system for edos]. In what might be understood as a generalisation of his extended-diatonic interval-naming system described above onto any equal tuning. Employing as prefixes the familiar 'sub', 'super', and 'neutral'. His scheme is based on the diatonic scale, however the diatonic interval names are not defined by their position in a cycle of fifths like is Sagispeak. In Keenan's system the ET's best 3/2 is first labelled P5, and the fourth P4. The interval half-way between the tonic and fifth is labelled the neutral third, or 'N3', and halfway between the fourth and the octave N6. Then the interval a perfect fifth larger than N3 is labelled N7, and the interval a fifth smaller than N6 labelled N2. The neutral intervals then lie either at a step of the ET, or between two steps. After this the remaining interval names are decided based on the distance they lie in pitch from the 7 labelled intervals, which make up the ''Neutral scale'', P1 N2 N3 P4 P5 N6 N7, which, like the diatonic, is an MOS scale, which may be labelled [[Neutral7|Neutral[7]]] 3|3 using [[Modal UDP Notation|Modal UPD notation]]. To name an interval in an edo, the number of steps of [[72edo]] that most closely approximate the size of the interval difference from a note of the neutral scale is first found. Then the prefix corresponding to that number of steps of 72edo is applied to the interval name. The following chart details this process (can't load the chart :( ). An interval just smaller than a major third in Keenan's system is labelled a ''narrow major third'', and an interval just wider than a [[6/5]] minor third a ''wide minor third'', however he notes that 'narrow' and 'wide' are only necessary in edos greater than 31. Note that the interval just wide of a minor third is labelled a 'supraminor third' rather than a 'superminor third'. This reflect recent tendencies among microtonal musicians and theorists.

Keenan's system is an elegant way to keep the 'major 3rd' label for 5/4, where labels depend on the size of the best fifth, however it suffers from it's applicability only to edos, and that it does not conserve interval arithmetic. Another potentially undesirable result of the system is that the major second approximates 10/9, and a ''wide major second'' 9/8, where as 9/8 is almost always considered a major second, and [[10/9]] often a narrow or small major second. One such system that considers 10/9 a narrow major second is that of Aaron Hunt.

=== Size-based systems ===
Microtonal theorist [[Aaron Andrew Hunt]] devised [http://musictheory.zentral.zone/huntsystem4.html the Hunt system], which includes interval name assignments for JI (just intonation) and edos based on [[41edo]]. Compared to Keenan's 72 interval names, Aaron's system includes 41. His system is based directly on 41edo, and unlike Keenan's system, interval are given the name of the closest step of 41edo, and no account is taken of the size of the edos fifth. In 41edo, Major, minor, augmented and diminished intervals are those obtained through the approximately Pythagorean cycle of fifths. Intervals one step of 41edo above these are given the prefix 'small', one step larger are given the prefix 'large', two steps smaller the prefix 'narrow' and two larger the prefix 'wide'. As a result, 5/4 is labelled a 'small major 3rd', or SM3 (not to be confused with a super major third, a label that does not exist in this system).

Neo-medieval musicians and early music historian and theorist [[Margo Schulter]] described her own [http://www.bestii.com/~mschulter/IntervalSpectrumRegions.txt interval naming scheme] built on approximations to JI intervals. Each interval names corresponds to an approximate size, and no particular edo is referenced. In her scheme middle major thirds range in size from 400-423 cents, and small major thirds from 372-400c. 5/4 is labelled a small major third, 81/64 a middle major third and 9/7 a large major third. Margo's scheme includes small, middle and large varieties of major, minor and neutral 2nds, 3rds, 6ths, 7ths; perfect fourth and fifths; and tritones, as well as a sub fifth and super fourth a dieses and comma and an octave less dieses and comma and ''interseptimals'', which correspond to intermediates, her name referencing the fact that they may each approximate two ratios of 7.

In Hunt's system when used in 41edo or JI diatonic interval arithmetic is conserved, but in other tunings it may not be, and Margo's system may not conserve diatonic interval arithmetic either. Both systems may be applied to arbitrary tunings, but the same intervals (defined, perhaps by a MOS scale) may not be given the same interval names across different tunings.

<ul><li>[[User:PiotrGrochowski/Extra-Diatonic Intervals]] gives each 43edo interval a name, then maps each desired interval to a 43edo interval. [[User:PiotrGrochowski/Extra-Diatonic Intervals — 50edo]] does this with 50edo. '''It has a very amazingly excellent solution to the 5/4 and 81/64 problem''': The 5/4 is named major third and this notation is split in half, while 81/64 is named high major third. 10/9 and 9/8 are both named major second. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]). </li></ul>Size-based systems are completely generalisable, but do not conserve interval arithmetic.

=== Ups and Downs ===
One final interval naming system, associated with the [[Ups and Downs Notation|Ups and Downs Notation]] system, belonging to microtonal theorist and musicians [[KiteGiedraitis|Kite Giedraitis]], like Sagittal is based on deviations from the diatonic scale. In this system however, deviations (from major, minor, perfect, augmented and diminished) are notated simply by the addition of up or down arrows: '^' or 'v', corresponding to raising or lowering of a single step of an edo. In some tunings ([[12edo]], [[19edo]] or [[31edo]] for example) 5/4 may be a M3, and in others a vM3 (downmajor 3rd) (e.g. [[15edo]], [[22edo]], 41edo, 72edo), or even an up-major 3rd (e.g. [[21edo]]). Ups and downs also includes neutrals, which lay exactly in-between major and minor intervals of the same degree, labelled '~' (mid). 'Up' and 'down' prefixes may be used before mid also, i.e. 'v~ 3). This system benefits from it's simplicity as well as it's conservation of interval arithmetic. It can be used for some MOS scales where one of the generators is a perfect fifth or a fraction of a perfect fifth, but not all of these (e.g. Diminished[8]), and not all MOS scales (if such scales are to be described, an additional pair of accidentals/qualifiers is used. Although the scales then are described, their intervals still are not given the same names in Ups and Downs' edo names). Another criticism of Kite's system that does not apply to the others is the fact that when an edo is doubled or multiplied by some simple fraction, and the best fifth is constant across the two edos, the same intervals may be be given different names.

[[Igliashon Jones]] is a supporter of this system, but for the relabeling of 'down' as 'sub' and 'up' as 'super' (or supra) and 'mid' as 'neutral', so that more common names are used, wherein 'super' infers a raise of 1 step of the edo, and 'sub' a lowering of one step. In this 'Extra-diatonic' system 'super' and 'sub' may be doubly applied, as in Ups and Downs, but they may not be applied before 'neutral' where in Ups and Downs they may be applied before 'mid'. The author's own extra-diatonic system is developed as a departure with caveat that 'S' and 's' prefixes are defined not as alterations by a single step of the edo, but by comma alterations as in Sagittal, in order that interval of MOS scales may be represented consistently across different tunings. Throughout the rest of the article the development is detailed, and the system defined.

== Premise: ==
Extra-diatonic names should be simple, generalisable, widely applicable, backwards compatible with standard diatonic notation and reflecting current informal practice as closely as possible. Extra-diatonic interval names are fifth based; extended from the familiar major, minor and perfect interval names so that diatonic interval arithmetic is conserved. ‘M’, ‘m’, and ‘P’ remain the short-hand for major, minor and perfect. ‘A’ and ‘d’ for Augmented and diminished may also be used in the familiar way. In cases where the chroma (the chromatic semitone, or augmented unison) is represented by multiple steps in the tuning the prefix ‘super’ raises major and perfect intervals by a single step while ‘sub’ lowers minor and perfect intervals, with short-hand ‘S’ and ‘s’. ‘S’ and ‘s’ may also be used to raise minor and lower major intervals respectively, reflecting occasion practice. In this case ‘S’ is short-hand for ‘supra’, and 's' for 'small'. They may also be used to raise or lower diminished and augmented intervals similarly. In this way this scheme is equivalent thus far to Ups and Downs notation, where ‘^’ or ‘up’ corresponds to ‘S’, ‘super’ or ‘supra’ and ‘v’ or ‘down’ to ‘sub’ or 'small' .

== Additions and examples: ==
''Neutrals'' and ''intermediates'' are also included, where neutrals occur between opposing sizes of a single generic interval the intermediates between each generic interval and the next.

Interval names for equal tunings are ranked in five tiers.

Interval names for equal tunings are ranked in eight tiers.
# Perfect
# Neutral
# Major, minor, A4 and d5 and, if the chroma is subtended by a single (positive) step of the edo, other augmented and diminished intervals
# Super, sub, supra and small prefixes to major, minor, perfect interval and to A4 and d5
# Intermediates
# Augmented and diminished intervals (for when the chroma is subtended by more than a single (positive) step of the edo)
# Super, sub, supra and small prefixes to augmented and diminished intervals
# Intervals augmented and diminished more than singularly
When more than one interval name corresponds to a specific interval, the names are privileged in order of the tiers. By this ordering, the first available name is the ‘primary’ for that interval, the second available ‘secondary’ and third 'tertiary'.

I initially intended for Intermediates to lay in the second tier with neutrals, but that led to primary intervals results that were not preferred by my colleagues over other options and I was persuaded to avoid using intermediates unless they are necessary.

=== Neutrals ===
N2, N3, N6 and N7, i.e. neutral 2nds, 3rds, 6ths and 7ths, falling exactly in-between the major and minor intervals of the same interval class, add native support for neutral-thirds temperament, where the N3 divides the P5 in exact halves and N2 divides the m3 is exact halves. In ups and downs neutrals indicated with '~' and said 'mid'.

Extending this familiar application to provide support for larger neutral scales, we add that neutrals occur also between P4 and A4; P5 and d5; P1 and A1; and P8 and d8.

Then [[Neutral7|Neutral[7]]] 3|3 can then be written:

P1 N2 N3 P4 P5 N6 N7 P8.

The same names give the primary interval names for [[7edo]], whose secondary intervals names are:

N1 m2/M2 m3/M3 N4 N5 m6/M6 m7/M7 N8.

The secondary interval names show that the chroma is equivalent to a unison in 7edo.

[[Neutral10|Neutral[10]]] 5|4 may then be written as

P1 N2 M2 N3 P4 N4 P5 N6 m7 N7 P8

[[Neutral17|Neutral[17]]] 8|8 may be written as

P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8,

which is equivalent to the primary interval names of [[17edo]].

=== Intermediates ===
To provide native support for [[The Archipelago|Barbados]] and Diminishes temperaments, intermediates are also added to the system. It should be noted immediately that intermediates are not as common to common microtonal interval naming as neutrals and though are a useful addition to this scheme, may be left out if desired. The appendix includes the MOS scales and edos from 'lists of all edos and MOS Scales', but without any intermediates.

‘2-3’ lies exactly half-way between M2 and m3 and divides the P4 in half. It may be read ‘second-third’ or ‘serd’. ‘6-7’, it’s octave-inverse lies exactly half-way between M6 and m7 and may be read ‘sixth-seventh’ or ‘sinth’.

‘1-2’ lies exactly half-way between P1 and m2, dividing the m2 in half. It may be read ‘unison-second’ or ‘unicond’. Its octave-inverse, ‘7-8’, lies exactly half-way between M7 and P8 and may be read ‘seventh-octave’ or ‘sevtave’.


‘3-4’ lies exactly half-way between M3 and P4, dividing the M6 in half. It may be read ‘third-fourth’ or ‘thourth’. It’s octave-inverse, ‘5-6’, lies exactly half-way between P5 and m6 and may be read ‘fifth-sixth’ or ‘fixth’. 

[[5edo]] can be spelled with the list of only these intermediates:

1-2 2-3 3-4 5-6 6-7 7-8.

The primary interval names for 5edo do not include them however, as they may be described by diatonic intervals:

P1 M2/m3 P4 P5 M6/m7 P8

Barbados[5] 2|2 may be described as

P1 2-3 P4 P5 6-7 P8,

and Barbados[9] 4|4 as

P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8.

The final intermediate, ‘4-5’ lies exactly between P4 and P5 and divides the octave in half. It may be read ‘fourth-fifth’ or ‘firth’. It is necessary for [[Diminished]] temperament, where the half-octave cannot be represented as any alteration of A4 or d5. Diminished temperament has a period of 1/4 of an octave, an approximation of 6/5. Therefore, the difference between four 6/5's and 2, 648/625, is tempered out.

Diminished[8] 1|0 (4) can be written as P1 sM2 Sm3 P4 4-5 Sm6 sM6 sM7 P8, and

[[Diminished12|Diminished[12]]] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 4-5 P5 Sm6 sM6 Sm7 sM7 P8,

== [[12edo]] and a problem ==
The primary interval names for 12edo are as we are familiar:

P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8,

but the interval labeled A4/d5 has the secondary name '4-5'.

We know that 12edo supports Diminished temperament, but we can't see that 12edo as equivalent to Diminished[12]: If we follow the rules from our premise, the notes of Diminished[12] in 12edo would give: P1 M2 m2 M3 m3 P4 A4/d5 P5 M6 m6 M7 m7 P8, with the majors and minor flipped. To be equivalent, sM must equal M and Sm must equal m.

=== [[10edo]], Pajara and a problem ===
10edo may be written as only neutrals and intermediates:

N1/1-2 N2 2-3 N3 3-4 N4/4-5/N5 5-6 N6 6-7 N7 7-8/N8

However it much more sense to write is using the primary interval name set:

P1 N2 M2/m3 N3 P4 N4/N5 P5 N6 M6/m7 N7 P8

It's secondary intervals as follows:

m2 Sm2/sM2 2-3 Sm3/sM3 M3 S4/s5 m6 Sm6/sM6 6-7 Sm7/sM7 M7

We can see that 10edo supports [[Neutral third scales]], given that we can make the interval names for Neutral[10] using the primary interval names for 10edo.

We know 10edo and 12edo both support Pajara temperament. Pajara[10] 2|2 (2) consists of:

P1 Sm2 M2 sM3 P4 sA4/Sd5 P5 Sm6 m7 sM7 P8,

and [[Pajara12|Pajara[12]]] 3|2 (2) of

P1 Sm2 M2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 m7 sM7 P8.

We can see Pajara[10] in 10edo, but in 12edo, wouldn’t sM3 be m3?

This is where my system diverges from Igliashon Jones’. We have to break our first rule here, or at least add some conditions to it.

== Comma association ==
To address this problem of consistency, we now state that when 81/80 is tempered out, M=sM and m=Sm, and when 64/63 is tempered out, M=SM and m=sm. In the case of sm and SM, ‘S’ and ‘s’ raise and lower by 64/63, and in the case of Sm and sM, ‘S’ and ‘s’ raise and lower by 81/80. In this way extra-diatonic interval names are equivalent to [http://forum.sagittal.org/viewforum.php?f=9 Sagispeak] interval names, where for sm and SM ‘S’ and ‘s’ are equivalent to ‘tai’ and ‘pao’ and for Sm and sM ‘S’ and ‘s’ are equivalent to ‘pai’ and ‘pao’.

It is important to note that given this change, 'S' and 's' may alter an interval by a different number of steps in an edo depending on which interval names they prefix. This may seem confusing, but it seems to reflect existing informal practice.

In 12edo, which represents the union between the two, where both 64/63 and 81/80 are tempered out, ‘S’ and ‘s’ do not raise or lower intervals at all. We can now easily see that 12edo supports Pajara, where simply removing all the ‘s’s and ‘S’s from Pajara[12] gives us our primary interval names of 12edo.

=== [[14edo]] and Injera ===
Like 10edo, 14edo may be written using all the neutrals and intermediates, but without any intervals described both as a neutral and as an intermediate:

N1 1-2 N2 2-3 N3 3-4 N4 4-5 N5 5-6 N6 6-7 N7 7-8 N8.

The primary interval names for 14edo are as follows:

P1 sm2 N2 SM2/sm3 N3 SM3/s4 P4 SA4/sd5 P5 S5/sm6 N6 SM6/sm7 N7 SM7 P1

We know that 12edo and 14edo support Injera, where Injera[12] 3|2 (2) may be labelled

P1 sm2 M2 sm3 SM3 P4 SA4/sd5 P5 sm6 M6 m7 SM7 P8,

and Injera[14] 3|3 (2) labelled

P1 sm2 M2 sm3 m3 SM3 P4 SA4/sd5 P5 sm6 M6 SM6 m7 SM7 P8.

We can see both in 14edo, and to get 12edo from Injera[12], as with Pajara, we remove all the ‘s’ and ‘S’ prefixes.

=== [[Blacksmith]] and further extension ===
10edo also supports Blacksmith temperament, and we may think to write Blacksmith[10] 1|0 (5) as:

P1 sM2 M2/m3 sM3 P4 s5 P5 sM6 M6/m7 sM7 P8

But we have now added mappings, but are yet to define the use of ‘S’ and ‘s’ for perfect intervals. In Blacksmith, the interval we might call ‘s5’ is 81/80 below P5, however, more commonly ‘s5’ is used to refer to [[16/11]], and S4 [[11/8]]. Since these intervals have above been labelled N4 and N5 above however, we do not need to worry about that, and can add that s5, a 'small 5th', is 81/80 below 3/2, and S4, a 'supra 4th' lies 81/80 above 4/3. where ‘s4’ has been typically been used to refer to [[21/16]], and ‘S5’ to [[32/21]], we add that s4 is lower than P4 by 64/63 and that S5 is higher than P5 by 64/63.

The primary intervals names of Blacksmith[15] 1|1 (5) then are as follows:

P1 Sm2 sM2 M2/m3 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 M6/m7 Sm7 sM7 P8,

these being identical to the the primary interval names of 15edo.

We may further add that ‘S’ (supra) and ‘s’ (small) may raise diminished, and lower augmented intervals by 81/80 as they do to minor and major respectively and that when ‘S’ (super) raised an augmented interval, or ‘s’ (sub) lowers it, the change is by 64/63.

In edos where 81/80 or 64/63 are represented by a single step, or when the apotome is represented by a single step, Ups and Downs with 'S', 's', and 'N' can name the intervals equivalently to this system. This can be seen in add edos considered thus far, as well as in those listed directly below. What this system adds is that it may also describe the intervals of MOS scales (as well as JI scales), such that these interval can be identically named in edos that approximate the scales they belong to. This is true for all MOS scales mentioned so far, as well as those listed below.

=== Further application in edos ===
The primary interval names are shown below for some larger edos.

[[22edo]]: P1 m2 Sm2 sM2 M2 m3 Sm3 sM3 M3 P4 S4 4-5 s5 P5 m6 Sm6 sM6 M6 m7 Sm7 sM7 M7 P8

[[24edo]]: P1 N1 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 N4 A4/d5 N5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 N8 P8

[[26edo]]: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 SA4/sd5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8

[[27edo]]: P1 m2 Sm2 N2 sM2 M2 m3 Sm3 N3 sM3 M3 P4 N4 d6 A3 N5 P5 m6 Sm6 N6 sM6 M6 m7 Sm7 N7 sM7 M7 P8

[[29edo]]: P1 sm2 m2 Sm2 sM2 M2 SM2/sm3 m3 Sm3 sM3 M3 SM3/s4 P4 S4 d5 A4 s5 P5 sm6 m6 Sm6 sM6 M6 SM6/sm7 m7 Sm7 sM7 M7 SM7 P8

[[31edo]]: P1 N1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 N4 A4 d5 N4 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 N8 P8

[[34edo]]: P1 1-2 m2 Sm2 N2 sM2 M2 2-3 m3 Sm3 N3 sM3 M3 3-4 P4 S4 N4 4-5 N5 s5 P5 5-6 m6 Sm6 N6 sM6 M6 6-7 m7 Sm7 N7 sM7 M7 7-8 P8

[[38edo]]: P1 S1 1-2 sm2 m2 N2 M2 SM2 2-3 sm3 m3 N3 M3 SM3 3-4 s4 P4 N4 A4 SA4/sd5 d5 N5 P5 S5 5-6 sm6 m6 N6 M6 SM6 6-7 sm7 m7 N7 M7 SM7 7-8 s8 P8

[[41edo]]: P1 N1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 s4 P4 S4 N4 d5 A4 N5 sM5 P5 S5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 N8 P8

[[43edo]]: P1 S1 1-2 A1/sm2 m2 dd3 AA1 M2 SM2/d3 2-3 A2/sm3 m3 dd4 AA2 M3 SM3/d4 3-4 A3/s4 P4 dd5 AA3 A4 d5 dd6 AA4 P5 S5/A6 5-6 A5/sm6 m6 dd7 AA5 M6 SM6/d7 6-7 A6/sm7 m7 dd8 AA6 M7 SM7/d8 7-8 s8 P8

[[46edo]]: P1 S1 sm2 m2 Sm2 A1 d3 sM2 M2 SM2 sm3 m3 Sm3 A2 d4 sM3 M3 SM3 s4 P4 S4 sd5 d5 sA4/Sd5 A4 SA4 s5 P5 S5 sm6 m6 Sm6 A4 d7 sM6 M6 SM6 sm7 m7 Sm7 A6 d8 sM7 M7 SM7 s8 P8

In 43edo we encounter the first time we have to use double Augmented and diminished intervals. 43edo marks the first instance in which Jones' alternative ups and downs interval names do not match those from this system. In his system, for example, AA2 would simply be sM3, but in this system since sM3 implies an approximation to 5/4 and the M3 already represents 5/4, and therefore is equivalent to sM3, we cannot do this. This tells us however that no simple ratio is approximated by the interval, and perhaps it is better understood as an AA2. Larger edos contain unlabeled intervals (without resorting to extended diatonic interval names). The association of 'super' and 'sub' with 64/63 and with 'supra' and 'small' with 81/80 may effect the assignment of primary interval names, but for all of these edos, as well as all those mentioned before, when 'S' and 's' are used, they still signify a raising or lowering by a single step of the edo, and thus appear equivalent to the Ups and Downs version. The comma associations add that, though use of enharmonic equivalences and secondary interval names may be necessary, intervals from MOS scales may be spelled in a consistent way across tuning to different edos.
=== Other rank-2 temperaments' MOS scales ===
On top of those discussed thus far, other temperaments generated by the P5 or a fraction of it are also supported to some extent, where their MOS scales may be represented, including [[Semaphore]], [[Augmented]], [[Porcupine]], [[Diminished]], [[Negri]], [[Tetracot]] and [[Slendric]].

Semaphore[5] 2|2: P1 SM2/sm3 P4 P5 SM6/sm7 P8

Semaphore[9] 4|4: P1 M2 SM2/sm3 SM3/s4 P4 P5 S5/sm6 SM6/sm7 m7 P8

Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8

Augmented[9] 1|1 (3): P1 Sm2 Sm3 sM3 P4 P5 Sm6 SM6 sM7 P8

Augmented[12] 2|1 (3): P1 Sm2 M2 Sm3 sM3 P4 sA4 P5 Sm6 Sm7 SM6 sM7 P8

Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8

Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8

Porcupine[8] 4|3: P1 sM2 Sm3 P4 s5 P5 sM6 Sm7 P8

Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8

Negri[9] 4|4: P1 Sm2 SM2/sm3 sM3 P4 P5 Sm6 SM6/sm7 sM7 P8

Negri[10] 4|4: P1 Sm2 SM2/sm3 sM3 P4 A4 P5 Sm6 SM6/sm7 sM7 P8

Tetracot[6] 3|2: P1 sM2 N3 S4 N6 Sm7 P8

Tetracot[7] 4|2: P1 sM2 N3 S4 P5 N6 Sm7 P8

Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 P8

[[Slendric5|Slendric[5]]] 2|3: P1 SM2 s4 S5 sm7 P8

[[Slendric6|Slendric[6]]] 3|2: P1 SM2 s4 P5 S5 sm7 P8

[[Slendric11|Slendric[11]]] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8

We may also write temperaments with a 9/8 but no 3/2. The most well known of these is [[Machine]]:

[[Machine5|Machine[5]]] 2|2: P1 M2 M3 m6 m7 P8

[[Machine6|Machine[6]]] 3|2: P1 M2 M3 A4 m6 m7 P8

[[Machine11|Machine[11]]] 5|5: P1 d3 M2 d4 M3 d5 A4 m6 A5 m7 A6 P8
=== Formal summary ===
The resultant system may be formally summarised as follows:

'''Definition 1a.''' M and m label the two sizes of 2nd, 3rd, 6th and 7th in the Pythagorean diatonic scale.

'''Definition 1b.''' The smaller 4th and larger 5th are labelled P.

'''Definition 1c.''' The single size of 1 and 8 is labelled P.

'''Definition 2.''' A chroma above M or P is A and below m or P is d.

'''Definition 3a.''' Within generic interval classes 2, 3, 6 and 7, half way between M and m is N.

'''Definition 3b'''. Within generic interval classes 1 and 4, half way between P and A is N.

'''Definition 3c.''' Within generic interval classes 5 and 8, half way between P and d is N.

'''Corollary:''' 7edo can be written P1 N2 N3 P4 P5 N6 N7 N8.

'''Definition 4.''' Half way between adjacent generic interval classes lie the intermediates.

'''Corollary:''' 5edo can be written 1-2 2-3 3-4 5-6 6-7 7-8.

'''Definition 5a.''' 'S' when applied before M, and 's' when applied before m, raise or lower by 64/63 respectively.

'''Definition 5b.''' 'S' when applied before m, and 's' when applied before M, raise or lower by 81/80 respectively.

'''Definition 5c.''' 'S' when applied to P5 (leading to S5) and 's' when applied to P4 (leading to s4) raise and lower by 64/63 respectively.

'''Definition 5d.''' 'S' when applied to P4 (leading to S4) and 's' when applied to P5 (leading to s5) raise and lower by 81/80 respectively.

'''Definition 5e.''' 'S' when applied to P1 and 's' when applied to P8 may raise and lower by 64/63, or by 81/80.

'''Definition 6a.''' When 'S' and 's' imply alterations of 64/63, they have long-form 'super' and 'sub' respectively.

'''Definition 6b.''' When 'S' and 's' imply alterations of 81/80, they have long-form 'supra' and 'small' respectively.

'''Definition 7.''' If alterations of 81/80 and of 64/63 need to be distinguished from one another in short-form, alterations of 81/80 can be written 'SR' and 'sl'.

== edos with extreme fifths ==
Thus far the best fifth of all edos described lies in the 'diatonic range', between 4 steps of 7edo and 3 steps of 5edo. The best fifths of some edos lies outside this range, in either directly. Whereas in edos where the best fifth lies between 7 steps of 12edo and 4 steps of 7edo, 81/80, the meantone comma is tempered out, wherein four fifths approximate the fifth harmonic (a sM3 raised two octave), in edos with fifths flatter than this, 135/128, a meantone chromatic semitone is instead tempered out, resulting in the four fifths instead approximate the Sm3, 6/5 (raised two ocaves). This system is called Mavila temperament. In the other direction, whereas the best fifth lies between 7 steps of 12edo and 3 steps of 5edo, 64/63, the septimal or Archytas comma is tempered out, wherein four fifths approximate a SM3, 9/7, raised two octaves, in edos who's best fifth is sharper than this, 9/7, as well as 5/4 are approximated by the perfect fourth, tempering out 16/15 and 28/27. This system is called Father temperament. The application of this system to edos of both of these fifth sizes is addressed below.

=== [[Mavila]] ===
In Mavila, the perfect 5th is flatter than in 7edo, so major intervals are below minor and augmented below major. In Mavila the major third approximates 6/5 and the minor third 5/4, tempering out [[135/128]]. This presents no problem to the scheme however, and the rules are applied just the same. The small major third, 81/80 below 6/5 or 81/64 comes to [[32/27]], the minor 3rd, and the sub minor 3rd remains 7/6.

Mavila[7] 3|3 can be written

P1 M2 m3 P4 P5 M6 m7 P8,

the same as the diatonic scale.

Mavila[9] 4|4 can be written

P1 M2 M3 m3 P4 P5 M6 m6 m7 P8,

which is also the primary interval name set for 9edo.

If we don't want to have major being below minor, we can hide it with some secondary interval names:

P1 M2 Sm3 sM3 P4 P5 Sm6 sM6 m7 P8, arriving at Augmented[9].

We say that our first Mavila[9] interval names are not ''well-ordered'', where for an interval name set to be well-ordered, for each degree major must be above minor. By extension we defined a well-ordered interval names set as one in which ... ≤ dd ≤ d ≤ m ≤ M ≤ A ≤ AA ≤ ... or ... ≤ dd ≤ d ≤ P ≤ A ≤ AA ≤ ..., and where s_ ≤ _ ≤ S_ (where '_' represents any of ... dd, d, m, (P), M, A, AA ...).

Using the fact that 64/63 is represented by a single step in 9edo, it is left as an exercise for the reader to prove that 9edo supports Negri temperament (by replacing some names with sub and super prefixed names to arrive at Negri[9] 4|4).

The primary interval names for Augmented[9] 1|1 (3) are well-ordered.

Mavila[16] 8|7 can be writtten

P1 d1 M2 m2 M3 m3 A4 P4 A5 P5 d5 M6 m6 M7 m7 A8 P8.

The enharmonic equivalences for 16edo can be generated by: M=Sm=sm or m=sM=SM

This leads us to our primary well-ordered interval name set for 16edo:

P1 S1 Sm2 sM2 Sm3 sM3 s4 P4 4-5 P5 S5 Sm6 sM6 Sm7 sM7 s8 P8,

wherein we can see that it supports Diminished temperament.

Mavila[23]11|11 can be written as:

P1 d1 A2 M2 m2 A3 M3 m3 d3 A4 P4 d4 A5 P5 d5 A6 M6 m6 d6 M7 m7 d7 A8 P8

And the primary well-ordered interval names for [[23edo]] are appear as:

P1 S1 1-2 M2 m2 2-3 M3 m3 3-4 A4 P4 d4 A5 P5 d5 5-6 M6 m6 6-7 M7 m7 7-8 d8 P8

=== [[Father]] ===
In the other direction, the fifths of Father temperament are sharper than those of 5edo, leading to minor second going backwards. In Father temperament, 5/4 and 4/3 are tempered to a unison, along with 9/7. As 64/63 is tempered out, alterations of 64/63 act as identity alterations.The M2, larger than m3 is also a Sm3. sM2, then, returns to m3.

The primary interval names for Father[5] 2|2,

P1 M2 P4 P5 m7 P8,

present no problems.

In the primary interval names for Father[8] 4|3, however:

P1 m3 M2 P4 M3 P5 m7 M6 P8,

which are the same as the primary intervals for [[8edo]], but with M3 rather than 4-5, we see our diatonic interval names begin to cross over. We will add to our definition of well-ordered interval names that no interval names from interval-class ''n''-1 may be subtended by a larger number of steps that any interval names from interval-class ''n.'' As above, we can may use some secondary interval names to address it, leading to

P1 sM2 M2 P4 S4 P5 m7 Sm7 P8,

as the primary well-ordered interval name set.

If we also re-write M2 and m7 as Sm3 and sM6, we get Porcupine[8] 4|3.

Using some secondary interval names to 'fix' the order leads us to

P1 sM2 M2 P4 S4 P5 m7 M6 P8

The primary interval names for Father[13] 6|6:

P1 M7 m3 M2 d5 P4 M3 m6 P5 A4 m7 M6 m2 P8,

look very unruly.

We will fix up the ordering again with secondary interval names:

P1 S1 sM2 M2 Sm3 P4 S4 s5 P5 sM6 m7 Sm7 s8 P8,.

The primary interval names for [[13edo]] are similar:

P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8.

Minimally fixing the order leads us to

P1 N2 sM2 M2 N3 P4 S4 s5 P5 N6 m7 Sm7 N7 P8,

which we have seen before as Tetracot[13] 6|6.

=== [[Miracle]], [[11edo]] and [[21edo]] ===
All the scales discusses to this point use either smalls and supras or subs and supers, so in the rare instance that we see a S1 or s8, we can infer whether or not it's small/supra or sub/super, probably without even thinking too much about it. Rarely do MOS scales in this scheme require alterations of both 81/80 and 64/63. One important temperament that includes such scales is Miracle. We do not encounter either S1 in the 10 and 11-note MOS:

[[Miracle 10|Miracle[10]]] 5|4: P1 Sm2 SM2 N3 s4 sA4 S5 N6 sm7 sM7 P8

Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8

We first see S1 and s8 in the 21-note MOS:

[[Blackjack|Miracle[21]]] 10|10: P1 S1 Sm2 N2 SM2 sm3 N3 sM3 s4 P4 Sd5 sA4 P5 S5 Sm6 N6 SM6 sm7 N7 sM7 s8 P8,

the largest MOS scale we have attempted yet to write. S1 in this scale is 64/63. We could maybe guess this from the presence of S5, but it is not obvious. If we need to make it clear when we are referring to small/supra, we can write their short-hand instead as 'sl' for 'small' and 'Sa' for supra. Miracle[21] would then be re-written:

P1 S1 Sam2 N2 SM2 sm3 N3 slM3 s4 P4 Sad5 slA4 P5 S5 Sam6 N6 SM6 sm7 N7 slM7 s8 P8, if complete clarity is needed, otherwise if long-form names are provided there is no need. It looks unwieldy to me and so I would avoid it, but it is there as a possibility, if the intervals of Mavila[21] need be written with only 4 letters allowed for each interval name.

Having encountered 11 and 21-note scales now, and haven't not described 11 and 21edo, I will add these here.

The fifth in 11edo is too flat even for it to be considered to support Mavila. Let's see what happens:

P1 M2 M3 m2 m3 P4 P5 M6 M7 m6 m7 P8.

Adding neutrals gives us our primary interval names for 11edo:

P1 M2 M3 N3 m3 P4 P5 M6 N6 m6 m7 P8.

This is clearly not Mavila, so we don't know what's tempered out, such that we might add our alterations to arrive at a well-ordered interval name set. Let's review the 11-note scales we have encountered above:

Slendric[11] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8

Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8

From these union of these scales we can see from P4=Sd5 that 135/128, the Mavila comma is tempered out. We apply our Mavila re-spellings to arrive at:

P1 Sm2 Sm3 N3 sM3 P4 P5 Sm6 N6 sM6 sM7 P8 as a well-ordered interval name set.

We should not expect our Machine[11] scale to be represented in this spelling of 11edo: A spelling of 11edo that shows that it supports Machine uses a different mapping, using the 9/8 from two 22edo P5s. We could spell 11edo as every other note of 22edo if we wish to see how it supports Machine.

21edo can be written as three 7edos as it's best fifth is that of 7edo. Since 81/80 is -1 steps in 21edo, we use 64/63 alterations:

P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8.

Using the available conversion sm = Sm (and therefor SM = sM), we can confirm that 21edo supports Miracle and Whitewood temperaments. This is left as an exercise for the the inspired reader.

=== [[6edo]] ===
Though 6edo is normally only ever considered as a subset of 12edo, given that we have encountered 6-note MOS I'll give it a red hot go.

Our P5 and P4 in 6edo is our half-octave, 4-5, so 9/8 is tempered out and our chromatic scale only covers 2edo: P1/M2/M3/A4 m2/m3/P4/4-5/P5/M6/M7 m6/m7/P8. If we want to write 6edo is a well-ordered way, we might choose:

P1 SM2 sm3 P4/4-5/P5 SM6 sm7 P8.

Writing s4 and S5 instead of sm3 and SM6 would give us Slendric[6] 3|2.

What of Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8?

This tells us that in 6edo 81/80 is mapped to -2 steps of 6edo. This is not a problem, as we can use alterations of 64/63, mapped to 1 step, though I don't see why anyone would want to think of 6edo in this way.

Similarly to Machine in 11edo, Machine in 6edo uses a different (much better) mapping of 9/8: That of 12edo. 6edo is much better spelled as a subset of 12edo, where we can see if supports Machine.

The primary interval names for the remaining trivial edos are trivially derived and are given along with all those described so far in the section below.

=== [[28edo]] ===
We encounter a new problem with 28edo. 28edo's best fifth is that of 7edo. It's primary intervals names are as such:

P1 sA1 1-2 Sm2 N2 sM2 2-3 Sm3 N3 sM3 3-4 S4 P4 sA4 4-5 Sd5 P5 s5 5-6 Sm6 N6 sM6 6-7 Sm7 N7 sM7 7-8 Sd8 P8

in 28edo 81/80 is represented by -1 steps. We en-devour to maintain as best we can in our primary interval names the original premise behind the notation - that the prefix 's' takes an interval down a single step of the edo and 'S' a single step up. In our primary interval names for 28edo we have S4 below P4 and d5 above P5. We can avoid this confusion however by using the secondary interval names for P4 and P5 in 28edo - N4 and N5. In our list of edos below this change is made.

== How to ==
Though we have derived our interval names through many examples thus far, the process has not been explained as yet such that the reader may immediately apply them. This will be addressed here, with a step-by-step derivation guide. This guide gives all possible labels to each interval.

For ''n-''edo:
# Label P1 = 1, P8 = ''n''
# Find the best approximations of 3/2, 5/4 and 7/4, which are to be labelled P5, sM3 and sm7 (This is equivalent to finding the [[7-limit]] [[patent val]]).
#* P5 = round(ln(3/2)/ln(''n'')) steps
#* sM3 = round(ln(5/4)/ln(''n'')) steps
#* sm7 = round(ln(7/4)/ln(''n'')) steps
# Using the best fifth, label the diatonic intervals. i.e.
#* M2 = (2*P5) mod ''n''
#* M6 = (3*P5) mod ''n''
#* M3 = (4*P5) mod ''n''
#* M7 = (5*P5) mod ''n''
#* (P4, m7, m3, m6, m2) = ''n'' - (P4, M2, M6, M3, M7)
# Determine the size of the apotome: A = M-m (for any of degree 2, 3, 6 or 7).
# The initial chromatic intervals may be labelled:
#* A1 = A
#* d8 = P8 - A
#* A4 = P4 + A
#* d5 = P5 - A
# If A is even, then intervals half way between M and m within a degree may be labelled N.
# Determine the size of meantone and septimal comma alterations (steps 7-12 may be skipped if A = 1)
#* meantone comma = M3 - sM3. If equal to 0 then 81/80 is tempered out, the edo is said to support Meantone temperament and alterations of 81/80 are not needed.
#* septimal comma = m7 - sm7. If equal to 0 then 64/63 is tempered out, the edo is said to support Superpyth temperament, and alterations of 64/63 are not needed.
# If sub/super < 0, alterations of 64/63 should not be applied.
# If small/supra < 0, alterations of 81/80 my be applied only if A = 0, wherein all diatonic intervals are given the label 'N'.
# Acknowledging points 7, 8 and 9, label 's' and 'S' alterations to diatonic intervals, where:
#* sM (small major) = M - meantone comma
#* Sm (supra minor) = m + meantone comma
#* SM (super major) = M + septimal comma
#* sm (sub minor) = m - septimal comma
# Acknowledging points 7, 8 and 9, label 's' and 'S' alterations to P4 and P5 where:
#* S4 (supra 4th) = P4 + meantone comma
#* s5 (small 5th) = P5 - meantone comma
#* s4 (sub 4th) = P4 - septimal comma
#* S5 (super 5th) = P5 + septimal comma
# If meantone comma alterations are used, S1 (supra unison) may be labelled at P1 + meantone comma and d8 (small octave) at P8 - meantone comma. If septimal comma alterations are used, S1 (super unison) may be labelled at P1 + septimal comma and s8 (sub octave) may be labelled at P8 - septimal comma. If both are needed and not equal, and after steps 13-16 are completed as desired the interval corresponding to a super or supra unison (equivalently, sub or small octave) is yet otherwise unlabeled then to ascertain whether S1 is a supra or super unison (or equivalently s8 a small or sub octave), for all degrees, the short hand for 'small' is to be 'sl' rather than 's', and for 'supra' is to be 'SR' rather than 'S'.
# Remaining augmented (M + A) and diminished (m - A) intervals may be labelled 'A' and 'd'.
# If desired, augmentation and diminution may be iterated through the continued adding or subtracting of A, labelled by additional 'A' or 'd' prefixes respectively.
# 'S' and 's' may be further applied as if 'A' were 'M' or 'd' were 'm'.
# If desired and if ''n'' is even, the interval subtended by ''n''/2 steps may be labelled 4-5.
# If desired and if m2 is subtended by an even number of steps, the remaining intermediates may be labelled: (1-2, 2-3, 3-4, 5-6, 6-7, 7-8) = (P1, M2, M3, P5, M6, M7) + m2/2.
#

== Lists of edos and MOS scales ==

=== Primary well-ordered (unless otherwise noted) interval names for edos ===
2edo: P1 P4/P5 P8

3edo: P1 P4 P5 P8

4edo: P1 SM2/sm3 P4/P5 SM6/sm7 P8

5edo: P1/m2 M2/m3 M3/P4 P5/m6 M6/m7 M7/P8

6edo: P1 SM2 sm3/s4 P4/P5 S5/sM6 sm7 P8

7edo: P1 N2 N3 P4 P5 N6 N7 P8

8edo: P1 sM2 M2 P4 S4/s5 P5 m7 Sm7 P8

9edo: P1 M2 M3 m3 P4 P5 M6 m6 m7 P8

10edo: P1/m2 N2 M2/m3 N3 M3/P4 S4/s5 P5/m6 N6 M6/m7 N7 M7/P8

11edo: P1 Sm2 Sm3 N3 sM3 P4 P5 Sm6 N6 sM6 sM7 P8

12edo: P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8

13edo: P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8

14edo: P1 S1/sm2 N2 SM2/sm3 N3 SM3/s4 P4 SA4/sd5 P5 S5/sm6 N6 SM6/sm7 N7 SM7/s8 P8

15edo: P1/m2 S1/Sm2 sM2 M2/m3 Sm3 sM3 M3/P4 S4 s5 P5/m6 Sm6 sM6 M6/m7 Sm7 sM7/s8 M7/P8

16edo: P1 1-2 Sm2 sM2 Sm3 sM3 3-4 P4 4-5 P5 5-6 Sm6 sM6 Sm7 sM7 7-8 P8

17edo: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8

19edo: P1 S1/sm2 m2 M2 SM2/sm3 m3 M3 SM3/s4 P4 A4 d5 P5 S5/sm6 m6 M6 SM6/sm7 m7 M7/s8 P8

21edo: P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8

22edo: P1 m2 Sm2 sM2 M2 m3 Sm3 sM3 M3 P4 S4 4-5 s5 P5 m6 Sm6 sM6 M6 m7 Sm7 sM7 M7 P8

23edo: P1 S1 1-2 Sm2 sM2 2-3 Sm3 sM3 3-4 A4 P4 S4 s5 P5 d5 5-6 Sm6 sM6 6-7 Sm7 sM7 7-8 s8 P8 (can't quite get it well-ordered)

24edo: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 N4 A4/d5 N5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8

26edo: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 4-5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8

27edo: P1 m2 Sm2 N2 sM2 M2 m3 Sm3 N3 sM3 M3 P4 N4 d6 A3 N5 P5 m6 Sm6 N6 sM6 M6 m7 Sm7 N7 sM7 M7 P8

28edo: P1 sA1 1-2 Sm2 N2 sM2 2-3 Sm3 N3 sM3 3-4 S4 N4 sA4 4-5 Sd5 N5 s5 5-6 Sm6 N6 sM6 6-7 Sm7 N7 sM7 7-8 Sd8 P8

29edo: P1 S1 m2 Sm2 sM2 M2 SM2/sm3 m3 Sm3 sM3 M3 SM3/s4 P4 S4 d5 A4 s5 P5 S5/sm6 m6 Sm6 sM6 M6 SM6/sm7 m7 Sm7 sM7 M7 s8 P8

31edo: P1 N1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 N4 A4 d5 N4 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 N8 P8

34edo: P1 S1 m2 Sm2 N2 sM2 M2 2-3 m3 Sm3 N3 sM3 M3 3-4 P4 S4 N4 4-5 N5 s5 P5 5-6 m6 Sm6 N6 sM6 M6 6-7 m7 Sm7 N7 sM7 M7 s8 P8

38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 N4 A4 SA4/sd5 d5 N5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8

41edo: P1 N1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 s4 P4 S4 N4 sA4 Sd5 N5 sM5 P5 S5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 N8 P8

43edo: P1 S1 1-2 A1/sm2 m2 dd3 AA1 M2 SM2/d3 2-3 A2/sm3 m3 dd4 AA2 M3 SM3/d4 3-4 A3/s4 P4 dd5 AA3 A4 d5 dd6 AA4 P5 S5/A6 5-6 A5/sm6 m6 dd7 AA5 M6 SM6/d7 6-7 A6/sm7 m7 dd8 AA6 M7 SM7/d8 7-8 s8 P8 (not possible to find a well-ordered interval name set)

46edo: P1 S1 sm2 m2 Sm2 A1 d3 sM2 M2 SM2 sm3 m3 Sm3 A2 d4 sM3 M3 SM3 s4 P4 S4 sd5 d5 4-5 A4 SA4 s5 P5 S5 sm6 m6 Sm6 A4 d7 sM6 M6 SM6 sm7 m7 Sm7 A6 d8 sM7 M7 SM7 s8 P8 (not possible to find a well-ordered interval name set)

=== Interval names for MOS scales ===
Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8

Augmented[9] 1|1 (3): P1 Sm2 Sm3 sM3 P4 P5 Sm6 SM6 sM7 P8

Augmented[12] 2|1 (3): P1 Sm2 M2 Sm3 sM3 P4 sA4 P5 Sm6 Sm7 SM6 sM7 P8

Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8

Barbados[5] 2|2: P1 2-3 P4 P5 6-7 P8

Barbados[9] 4|4: P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8

Blackwood[10] 1|0 (5): P1 sM2 M2/m3 sM3 P4 s5 P5 sM6 M6/m7 sM7/s8 P8

Blackwood[15] 1|1 (5): P1 S1/Sm2 sM2 M2/m3 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 M6/m7 Sm7 sM7/s8 P8

Diminished[8] 1|0 (4): P1 sM2 Sm3 P4 sA4/Sd5 Sm6 sM6 sM7 P8

Diminished[12] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 Sm7 sM7 P8

Injera[12] 3|2 (2): P1 sm2 M2 sm3 SM3 P4 SA4/sd5 P5 sm6 M6 m7 SM7 P8

Injera[14] 3|3 (2): P1 sm2 M2 sm3 m3 SM3 P4 SA4/sd5 P5 sm6 M6 SM6 m7 SM7 P8

Machine[5] 2|2: P1 M2 M3 m6 m7 P8

Machine[6] 3|2: P1 M2 M3 A4 m6 m7 P8

Machine[11] 5|5: P1 d3 M2 d4 M3 d5 A4 m6 A5 m7 A6 P8

Mavila[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8

Mavila[9] 4|4: P1 M2 M3 m3 P4 P5 M6 m6 m7 P8

Mavila[16] 8|7: P1 d1 M2 m2 M3 m3 A4 P4 A5 P5 d5 M6 m6 M7 m7 A8 P8

Meantone[5] 2|2: P1 M2 P4 P5 m7 P8

Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8

Meantone[12] 6|5: P1 m2 M2 m3 M3 P4 A4 P5 m6 M6 m7 M7 P8

Meantone[19] 9|9: P1 A1 m2 M2 A2 m3 M3 d4 P4 A4 d5 P5 A5 m6 M6 d7 m7 M7 d8 P8

Miracle[10] 5|4: P1 Sm2 SM2 N3 s4 sA4 S5 N6 sm7 sM7 P8

Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8

Negri[9] 4|4: P1 Sm2 SM2/sm3 sM3 P4 P5 Sm6 SM6/sm7 sM7 P8

Negri[10] 4|4: P1 Sm2 SM2/sm3 sM3 P4 A4 P5 Sm6 SM6/sm7 sM7 P8

Neutral[7] 3|3: P1 N2 N3 P4 P5 N6 N7 P8

Neutral[10] 5|4: P1 N2 M2 N3 P4 N4 P5 N6 m7 N7 P8

Neutral[17] 8|8: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8

Pajara[10] 2|2 (2): P1 Sm2 M2 sM3 P4 sA4/Sd5 P5 Sm6 m7 sM7 P8

Pajara[12] 3|2 (2): P1 Sm2 M2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 m7 sM7 P8

Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8

Porcupine[8] 4|3: P1 sM2 Sm3 P4 s5 P5 sM6 Sm7 P8

Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8

Semaphore[5] 2|2: P1 SM2/sm3 P4 P5 SM6/sm7 P8

Semaphore[9] 4|4: P1 M2 SM2/sm3 SM3/s4 P4 P5 S5/sm6 SM6/sm7 m7 P8

Slendric[5] 2|3: P1 SM2 s4 S5 sm7 P8

Slendric[6] 3|2: P1 SM2 s4 P5 S5 sm7 P8

Slendric[11] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8

Superpyth[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8

Superpyth[12] 6|5: P1 m2 M2 m3 M3 P4 A4 P5 m6 M6 m7 M7 P8

Superpyth[17] 8|8: P1 m2 A1 M2 m3 d4 M3 P4 d5 A4 P5 m6 A5 M6 m7 d8 M7 P8

Tetracot[6] 3|2: P1 sM2 N3 S4 N6 Sm7 P8

Tetracot[7] 4|2: P1 sM2 N3 S4 P5 N6 Sm7 P8

Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 P8

== Conclusion ==
Using only 'S' and 's' as qualifiers to M, m, P, A and d, along with N and intermediate degrees, a system is developed wherein for most edos below 50 intervals can be systematically named such that for primary interval names, 'S' and 's' raise and lower, respectively, by 1 step of an edo , whilst the intervals of many MOS scales may be consistently named across different tunings, taking the best of both Igliashon Jones' extra diatonic interval names and Sagispeak. One does not need to understand the comma associations to make use of the interval names. While the intervals of some MOS scales may hold consistent names in edos in an Ups and Downs based scheme, there are many common scales that do not in such a system, that do in this one, such as scales of Diminished and Augmented temperament. What's more, the use of neutrals and intermediates leads to quicker recognition of MOS scales that may be supported in edos.

Help talk:Editing

2018-10-15T06:28:42Z

PiotrGrochowski: /* Removed links to it */ new section

== Soft redirects and smart redirects ==

Hi Piotr - I like the soft and hard redirects. What is a "smart" redirect though? In your article you describe them as the same, but you have created separate "Soft redirects" and "Smart redirects" categories as though they are different. Are they the same thing or two different things? Thanks for helping to organize all this! [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 15:22, 20 September 2018 (UTC) [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 15:22, 20 September 2018 (UTC)
: I did not describe them, it came from http://xenharmonic.wikispaces.com/wiki+help. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 15:33, 20 September 2018 (UTC)
:: OK. In case it's helpful to anyone, here is a link to MediaWiki's redirects - https://www.mediawiki.org/wiki/Help:Redirects. I think the thing you are calling "regular" and "hard" redirects might be the same thing (use of <code>#REDIRECT</code>), but I don't want to change the page now since you are editing. [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 15:43, 20 September 2018 (UTC)
:::Regular redirect is <nowiki>#REDIRECT [[Page]]</nowiki>, while hard redirect is <nowiki>#REDIRECT [[:en:Page]]</nowiki>. We are including hard redirects in [[Wiki help#Redirects]] for maximum compatibility. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 15:46, 20 September 2018 (UTC)
:::: Hello Piotr - from MediaWiki's standpoint, both are the same type of redirect. The "en" prefix is just for inter-language compatibility: you can also create links to <nowiki>[[:de:Wilkommen]]</nowiki>, for example: [[:de:Wilkommen]]. Since we are already on "en", the links <nowiki>[[Home]]</nowiki> and <nowiki>[[:en:Home]]</nowiki> are the same. Likewise with redirects, putting ":en" adds no difference. Try it! You will see they are the same - both "hard" redirects. If you put "en" there, it's what MediaWiki calls an "Interwiki redirect." [[User:Battaglia01|Mike Battaglia]] ([[User talk:Battaglia01|talk]]) 18:03, 21 September 2018 (UTC)
::::: It does have a difference. If not for :en:, [[Crash]] wouldn't be [[crash]]. The page has a description of hard redirect that is not true for a regular redirect. While it is not intentional MediaWiki behavior, it's required for this migration. [[2 1]] and [[531441 524288]] have an excellent use of the hard redirect. [[User:PiotrGrochowski|PiotrGrochowski]] ([[User talk:PiotrGrochowski|talk]]) 18:08, 21 September 2018 (UTC)

== Removed links to it ==

This page is protected in a state where it breaks the [[Conventions]]. Therefore it is not a proper {{SITENAME}} page and all links to it are removed. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]) 06:28, 15 October 2018 (UTC)

Hard redirect

2018-10-15T06:26:40Z

PiotrGrochowski: Removed redirect to en:Help:Wiki help#Redirects

Error

Redirect

2018-10-15T06:25:54Z

PiotrGrochowski: Removed redirect to Help:Wiki help#Redirects

Error

Soft redirect

2018-10-15T06:25:41Z

PiotrGrochowski: Replaced content with "Error"

Error

Wiki help

2018-10-15T06:25:20Z

PiotrGrochowski: Changed redirect target from Help:Wiki help to Help:Help

#REDIRECT [[Help:Help]]