User:VectorGraphics/Walker brightness notation: Difference between revisions

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WBN is intended to be used with the assumption of just octaves.  
WBN is intended to be used with the assumption of just octaves.  


== Example ==
WBN uses increasing levels of detail to specify intervals more or less precisely. For example, 435c is a supermajor third, and "supermajor third", "major third", and "third" are all valid descriptions. WBN can be simply used to refer to the sizes of intervals.
As WBN (not WBM, sorry, I'm bad at acronyms) is not a rigid system, an example scale is probably the best way to explain how it works:
{| class="wikitable"
|+WBN Exemplary Scale
!Degree
![[Cents]]
!Name
|-
|0
|0
|
|-
|1
|53
|
|-
|2
|134
|
|-
|3
|156
|
|-
|4
|188
|
|-
|5
|206
|
|-
|6
|220
|
|-
|7
|248
|
|-
|8
|266
|
|-
|9
|300
|
|-
|10
|315
|
|-
|11
|366
|
|-
|12
|435
|
|-
|13
|542
|
|-
|14
|588
|
|-
|15
|611
|
|-
|16
|684
|
|-
|17
|688
|
|-
|18
|969
|
|-
|19
|992
|
|-
|20
|1200
|
|}


So first, we name each interval according to its general interval class.
However, it can also name the intervals in a scale as follows:


{| class="wikitable"
== Steps to name the intervals in a scale: ==
|+WBN Exemplary Scale
Note: Intervals at the boundary always go in the category closer to the octave or unison, so that all octave complements work.
!Degree
!Cents
!Name
|-
|0
|0
|unison
|-
|1
|53
|second
|-
|2
|134
|second
|-
|3
|156
|second
|-
|4
|188
|second
|-
|5
|206
|second
|-
|6
|220
|second
|-
|7
|248
|semifourth
|-
|8
|266
|third
|-
|9
|300
|third
|-
|10
|315
|third
|-
|11
|366
|third
|-
|12
|435
|third
|-
|13
|542
|fourth
|-
|14
|588
|tritone
|-
|15
|611
|tritone
|-
|16
|684
|fifth
|-
|17
|688
|fifth
|-
|18
|969
|seventh
|-
|19
|992
|seventh
|-
|20
|1200
|octave
|}


Note that only 0c and 1200c (or the closest match to 1200c for non-octave scales) are called the [[1/1|unison]] and [[octave]]. Also, here 248c has been given the name "semifourth", we'll talk more about that later.
=== 1. Give each interval in the scale its base label ===


Also, note that the scale lacks a sixth entirely. This is in alignment with the sizes of "sixths" - there is no interval between ~750 to ~950 cents that could reasonably be called a sixth. 969c gets close, but along with it already being in the seventh range (albeit the lower end of it), there are... [https://wiki.lspace.org/Pune harmonic reasons for calling it a seventh.]
* Label the unison and the octave at 0 and 1200 cents.
* Label any intervals between 650 and 750 cents "fifth".
* Label any intervals between 450 and 550 cents "fourth".
* Label any intervals between 550 and 650 cents "tritone".
* Label any intervals between 50 and 250 cents "second".
* Label any intervals between 250 and 450 cents "third".
* Label any intervals between 750 and 950 cents "sixth".
* Label any intervals between 950 and 1150 cents "seventh".
* Label any intervals between 1150 and 1200 cents (but not 1200 cents exactly) "suboctave".
* Label any intervals between 0 and 50 cents (but not 0 cents exactly) "comma".  


First of all, there are some special names to apply: [[7/4|harmonic seventh]] for scale degree 18, and [[diesis]] for scale degree 1. (If there were a scale degree of around 10-30 cents, it would be called a [[81/80|comma]].)  
=== 2. Resolve ambiguities (stage 1: major and minor) ===
{| class="wikitable"
!Degree
!Cents
!Name
|-
|0
|0
|unison
|-
|1
|53
|diesis
|-
|2
|134
|second
|}
{| class="wikitable"
!Degree
!Cents
!Name
|-
|17
|688
|fifth
|-
|18
|969
|[[7/4|harmonic seventh]]
|-
|19
|992
|seventh
|}


Note that "harmonic seventh" still requires other sevenths to be distinguished from it, unlike diesis and comma.
==== If there are multiple intervals labeled "third": ====


* Label any intervals between 250 and 340 cents "minor third".
* Label any intervals between 360 and 450 cents "major third".
* Label any intervals between 340 and 360 cents "neutral third".


So, we add qualifiers to distinguish intervals of the same general interval class.
==== If there are multiple intervals labeled "second": ====


{| class="wikitable"
* Label any intervals between 50 and 140 cents "minor second".
!Degree
* Label any intervals between 160 and 250 cents "major second".
!Cents
* Label any intervals between 140 and 160 cents "neutral second"
!Name
|-
|0
|0
|unison
|-
|1
|53
|diesis
|-
|2
|134
|minor second
|-
|3
|156
|neutral second
|-
|4
|188
|major second
|-
|5
|206
|major second
|-
|6
|220
|supermajor second
|-
|7
|248
|semifourth
|-
|8
|266
|subminor third
|-
|9
|300
|minor third
|-
|10
|315
|minor third
|-
|11
|366
|neutral third
|-
|12
|435
|major third
|-
|13
|542
|fourth
|-
|14
|588
|tritone
|-
|15
|611
|tritone
|-
|16
|684
|fifth
|-
|17
|688
|fifth
|-
|18
|969
|harmonic seventh
|-
|19
|992
|minor seventh
|-
|20
|1200
|octave
|}
Note that there are some intricacies with the way I've named these, starting off with the minor second, which is considerably on the sharper end of the "minor second" range - had there been another smaller minor second, this one would be called sup[er|ra]minor. Similarly with the major third, which is actually [[9/7|well into the supermajor range]] - however, there is no "normal" major third to distinguish it from, so "supermajor" is redundant. Onto the major seconds, you may notice there are two of them, along with the minor thirds. This will be resolved in the next step.


As for the semifourth, think of "semifourth" here as being a cover name for an interval with two names: 248c here serves as both an inframinor third and an ultramajor second, the same way a tritone can serve as both a diminished fifth and augmented fourth. (Side note - Vector has complaints about [[Nominal-accidental chain|the way "diminished" and "augmented" are traditionally used]]. But that is beside the point...for now.) If another interval, say 242c, were in the scale, that would be the ultramajor second and 248c would be the inframinor third.
==== If there are multiple intervals labeled "tritone": ====


Here, the "minor thirds" closer to [[6/5]] have been given higher priority - 266c is subminor. Conversely, there is no major third close to [[5/4]], which was mentioned prior. Similarly with the fourth, which is considerably sharp.
* Label any intervals between 550 and 600 cents "narrow tritone".
* If an interval is 600 cents exactly, label it "mid tritone".


Also, there is a minor seventh but no major seventh.
Do this for the octave complements of these interval ranges, changing the qualities and interval sizes respectively, for example 860 to 950 is a major sixth.  
{| class="wikitable"
!Degree
!Cents
!Name
|-
|0
|0
|unison
|-
|1
|53
|diesis
|-
|2
|134
|minor second
|-
|3
|156
|neutral second
|-
|4
|188
|minor tone
|-
|5
|206
|major tone
|-
|6
|220
|supermajor second
|-
|7
|248
|semifourth
|-
|8
|266
|subminor third
|-
|9
|300
|common minor third
|-
|10
|315
|just minor third
|-
|11
|366
|neutral third
|-
|12
|435
|major third
|-
|13
|542
|fourth
|-
|14
|588
|small tritone
|-
|15
|611
|large tritone
|-
|16
|684
|small fifth
|-
|17
|688
|large fifth
|-
|18
|969
|harmonic seventh
|-
|19
|992
|minor seventh
|-
|20
|1200
|octave
|}
Here, the smaller and larger intervals have been named... "small" and "large", following the convention for tritones. There are a couple exceptions, which it is optional to include, but this has been done here:


The major seconds have been named "minor tone" and "major tone" - a reference to nicetone and zarlino, and to take advantage of the unique name for the major second - a tone. Note that the 220c is still called supermajor.
=== 3. Resolve ambiguities (stage 2: supermajor and subminor) ===


The minor thirds have been named based on which tunings of the minor third they approximate, which is an idea I've taken from Lumatone's [[53edo]] naming scheme - "common" is used as a shorthand for [[12edo]]. Similarly, if ~195c were in the scale it could be called a mean tone.
==== If there are multiple intervals labeled "fourth": ====


You could also refer to the fifths by the scales they generate - [[mavila]] and [[diatonic]] - however, this will likely not be very helpful as neither of these fifths are actually being used to generate scales.
* Label any intervals between 450 and 480 cents "subfourth".
* Label any intervals between 520 and 550 cents "superfourth".
* Continue to call any intervals between 480 and 520 cents "fourth".


== What about accidentals? ==
==== If there are multiple intervals labeled "major third": ====
This system does not specify accidentals. A recommendation is to use "sharp" and "flat" to cover the distances between some kind of minor third and some kind of major third (if not thirds, then seconds, sixths, or sevenths), preferably those close to either the common (300c and 400c) or just (315c and 386c) thirds, but the two are usually defined in terms of the circle of fifths (as in ups-and-downs notation and in Pythagorean tuning) and that can be used as well, where such systems are applicable.  
 
* Label any intervals between 425 and 450 cents "supermajor third".
* Continue to call any intervals between 360 and 425 cents "major third".
 
==== If there are multiple intervals labeled "minor third": ====
 
* Label any intervals between 250 and 280 cents "subminor third".
* Continue to call any intervals between 280 and 340 cents "minor third".
 
==== If there are multiple intervals labeled "major second": ====
 
* Label any intervals between 220 and 250 cents "supermajor second".
* Continue to call any intervals between 160 and 220 cents "major second".
 
==== If there are multiple intervals labeled "minor second": ====
 
* Label any intervals between 50 and 80 cents "subminor second".
* Continue to call any intervals between 80 and 140 cents "minor second".
 
==== If there are multiple intervals labeled "comma": ====
 
* Label any intervals between 30 and 50 cents "diesis".
* Continue to call any intervals smaller than 30 cents "comma".
 
==== If there are multiple intervals labeled "narrow tritone": ====
 
* Label any intervals between 550 and 575 cents "subtritone".
* Continue to call any intervals between 575 and 600 cents "narrow tritone".
 
Do this for the octave complements of these interval ranges. Note that 1150-1170c becomes an infraoctave, due to "diesis" not having a well-defined octave complement, and that the octave complement of a narrow tritone is a wide tritone.
 
=== 4. Resolve ambiguities (stage 3: inframinor and ultramajor) ===
 
==== If one of these steps resolves ambiguity, perform it: ====
 
* '''Seconds:'''
** Label any intervals between 50 and 65 cents "inframinor second".
** Label any intervals between 125 and 140 cents "superminor second".
** Label any intervals between 160 and 175 cents "submajor second".
** Label any intervals between 235 and 250 cents "ultramajor second".
* '''Thirds:'''
** Label any intervals between 250 and 260 cents "inframinor third".
** Label any intervals between 325 and 340 cents "superminor third".
** Label any intervals between 360 and 375 cents "submajor third".
** Label any intervals between 440 and 450 cents "ultramajor third".
* '''Fourths:'''
** Label any intervals between 450 and 460 cents "infrafourth".
** Label any intervals between 535 and 550 cents "ultrafourth".
* '''Tritones:'''
** Label any intervals between 550 and 560 cents "infratritone".
 
Do this for the octave complements of these interval ranges. Remember to do the octave complements for the tritones as well.
 
=== 5. Resolve ambiguities (stage 4: arto and tendo) ===
 
==== If one of these steps resolves ambiguity, perform it: ====
 
* '''Seconds:'''
** Label any intervals between 80 and 95 cents "artominor second".
** Label any intervals between 110 and 125 cents "tendominor second".
** Label any intervals between 175 and 190 cents "artomajor second".
** Label any intervals between 205 and 220 cents "tendomajor second".
* '''Thirds:'''
** Label any intervals between 280 and 295 cents "artominor third".
** Label any intervals between 310 and 325 cents "tendominor third".
** Label any intervals between 375 and 390 cents "artomajor third".
** Label any intervals between 405 and 425 cents "tendomajor third".
* '''Fourths:'''
** Label any intervals between 480 and 495 cents "artofourth".
** Label any intervals between 510 and 520 cents "tendofourth".
* '''Tritones:'''
** Label any intervals between 575 and 585 cents "artotritone".
 
Do this for the octave complements of these interval ranges. Remember to do the octave complements for the tritones as well. Though there are some notes to keep in mind:
 
* If a pair of intervals ends up distinguished by ''arto'' vs. no stage 4 prefix, i.e. artomajor third and major third, the other interval can be labeled tendo, regardless of its actual size. The same applies in reverse, for if a pair of intervals is distinguished by ''tendo'' vs. no stage 4 prefix.
* If there is only one interval simply labelled ''fourth'', call it a "perfect fourth".
** Same with fifths.
 
=== 6. Resolve ambiguities (stage 5) ===
By this point, except for commas, the octave has been subdivided into 10-20c interval regions, suitable for notating edos as large as 71edo without any problems. Where intervals need to be named more precisely than this, however, one may resort to just applying other descriptors like "small", "medium", "large", "grave", "acute", etc, depending on the needs of the scale.




== Examples applied to actual scales ==




12edo is the same as normal:
{| class="wikitable"
!Degree
!Cents
!Name
|-
|0
|0
|unison
|-
|1
|100
|minor second
|-
|2
|200
|major second
|-
|3
|300
|minor third
|-
|4
|400
|major third
|-
|5
|500
|perfect fourth
|-
|6
|600
|tritone
|-
|7
|700
|perfect fifth
|-
|8
|800
|minor sixth
|-
|9
|900
|major sixth
|-
|10
|1000
|minor seventh
|-
|11
|1100
|major seventh
|-
|12
|1200
|octave
|}
So is 17edo, due to its lack of common major and minor thirds:
{| class="wikitable"
!Degree
!Cents
!Name
|-
|0
|0.00
|unison
|-
|1
|70.59
|minor second
|-
|2
|141.18
|neutral second
|-
|3
|211.76
|major second
|-
|4
|282.35
|minor third
|-
|5
|352.94
|neutral third
|-
|6
|423.53
|major third
|-
|7
|494.12
|fourth
|-
|8
|564.71
|small tritone
|-
|9
|635.29
|large tritone
|-
|10
|705.88
|fifth
|-
|11
|776.47
|minor sixth
|-
|12
|847.06
|neutral sixth
|-
|13
|917.65
|major sixth
|-
|14
|988.24
|harmonic seventh
|-
|15
|1058.82
|neutral seventh
|-
|16
|1129.41
|major seventh
|-
|17
|1200.00
|octave
|}
However, 27edo is noticeably different from ups-and-downs naming.
{| class="wikitable"
!Degree
!Cents
!Name
!Name (Ups and downs)
|-
|0
|0.00
|unison
|perfect unison
|-
|1
|44.44
|diesis
|'''minor 2nd'''
|-
|2
|88.89
|'''minor second'''
|upminor 2nd
|-
|3
|133.33
|neutral second
|mid 2nd
|-
|4
|177.78
|small major second
|downmajor 2nd
|-
|5
|222.22
|large major second
|major 2nd
|-
|6
|266.67
|subminor third
|'''minor 3rd'''
|-
|7
|311.11
|'''minor third'''
|upminor 3rd
|-
|8
|355.56
|neutral third
|mid 3rd
|-
|9
|400.00
|'''major third'''
|downmajor 3rd
|-
|10
|444.44
|supermajor third
|'''major 3rd'''
|-
|11
|488.89
|fourth
|perfect 4th
|-
|12
|533.33
|superfourth
|up 4th
|-
|13
|577.78
|small tritone
|mid 4th
|-
|14
|622.22
|large tritone
|mid 5th
|-
|15
|666.67
|subfifth
|down 5th
|-
|16
|711.11
|fifth
|perfect 5th
|-
|17
|755.56
|subminor sixth
|'''minor 6th'''
|-
|18
|800.00
|'''minor sixth'''
|upminor 6th
|-
|19
|844.44
|neutral sixth
|mid 6th
|-
|20
|888.89
|'''major sixth'''
|downmajor 6th
|-
|21
|933.33
|supermajor sixth
|'''major 6th'''
|-
|22
|977.78
|harmonic seventh
|minor 7th
|-
|23
|1022.22
|large minor seventh
|upminor 7th
|-
|24
|1066.67
|neutral seventh
|mid 7th
|-
|25
|1111.11
|'''major seventh'''
|downmajor 7th
|-
|26
|1155.56
|supermajor seventh
|'''major 7th'''
|-
|27
|1200.00
|octave
|8ve
|}
The notation works in tunings without a diatonic scale:
{| class="wikitable"
!Degree
!Cents
!Name
|-
|0
|0
|unison
|-
|1
|75
|minor second
|-
|2
|150
|neutral second
|-
|3
|225
|major second
|-
|4
|300
|minor third
|-
|5
|375
|major third
|-
|6
|450
|semisixth
|-
|7
|525
|fourth
|-
|8
|600
|tritone
|-
|9
|675
|fifth
|-
|10
|750
|semitenth
|-
|11
|825
|minor sixth
|-
|12
|900
|major sixth
|-
|13
|975
|harmonic seventh
|-
|14
|1050
|neutral seventh
|-
|15
|1125
|major seventh
|-
|16
|1200
|octave
|}
And in tunings where the diatonic scale is degenerate:
{| class="wikitable"
!Degree
!Cents
!Name
|-
|0
|0
|unison
|-
|1
|80
|minor second
|-
|2
|160
|neutral second
|-
|3
|240
|major second
|-
|4
|320
|minor third
|-
|5
|400
|major third
|-
|6
|480
|fourth
|-
|7
|560
|small tritone
|-
|8
|640
|large tritone
|-
|9
|720
|fifth
|-
|10
|800
|minor sixth
|-
|11
|880
|major sixth
|-
|12
|960
|harmonic seventh
|-
|13
|1040
|neutral seventh
|-
|14
|1120
|major seventh
|-
|15
|1200
|octave
|}
Here is an example for 13edo:
{| class="wikitable"
!Degree
!Cents
!Name
|-
|0
|0.00
|unison
|-
|1
|92.31
|minor second
|-
|2
|184.62
|major second
|-
|3
|276.92
|minor third
|-
|4
|369.23
|major third
|-
|5
|461.54
|fourth
|-
|6
|553.85
|small tritone
|-
|7
|646.15
|large tritone
|-
|8
|738.46
|fifth
|-
|9
|830.77
|minor sixth
|-
|10
|923.08
|major sixth
|-
|11
|1015.38
|minor seventh
|-
|12
|1107.69
|major seventh
|-
|13
|1200.00
|octave
|}
This major third is extremely flat, however it still falls into the submajor range, so it is still reasonable to call it major.




Here is a table of interval names for 72edo up to the tritone:
{| class="wikitable center-all right-2 left-3"
!Degree
!Cents
!Name
|-
|0
|0.000
|unison
|-
|1
|16.667
|comma
|-
|2
|33.333
|small diesis
|-
|3
| 50.000
|large diesis
|-
|4
|66.667
|subminor second
|-
|5
|83.333
|small semitone
|-
|6
|100.000
|medium semitone
|-
|7
|116.667
|large semitone
|-
|8
|133.333
|<nowiki>sup[er|ra]minor second</nowiki>
|-
|9
|150.000
|neutral second
|-
|10
|166.667
|submajor second
|-
|11
|183.333
|ptolemaic tone
|-
|12
|200.000
|pythagorean tone
|-
|13
|216.667
|large tone
|-
|14
|233.333
|supermajor second
|-
|15
|250.000
|inframinor third
|-
|16
| 266.667
|small subminor third
|-
|17
|283.333
|large subminor third
|-
|18
|300.000
|common minor third
|-
|19
|316.667
|just minor third
|-
|20
|333.333
|<nowiki>sup[er|ra]minor third</nowiki>
|-
|21
|350.000
|neutral third
|-
| 22
| 366.667
|submajor third
|-
|23
| 383.333
|just major third
|-
|24
|400.000
|common major third
|-
|25
|416.667
|small supermajor third
|-
|26
|433.333
|large supermajor third
|-
|27
| 450.000
|ultramajor third
|-
|28
| 466.667
|subfourth
|-
| 29
|483.333
|small fourth
|-
|30
|500.000
|perfect fourth
|-
| 31
|516.667
|small superfourth
|-
|32
|533.333
|large superfourth
|-
|33
|550.000
|ultrafourth
|-
|34
|566.667
|subtritone
|-
|35
|583.333
|small tritone
|-
|36
|600.000
|medium tritone
|}
Note that "ultrafourth" has been used to align the use of "superfourth" with other "super-" intervals. Alternatively, one could use "acute" in this case and reserve "superfourth" for the intervals around 550c.
The system fits rather nicely into 53edo:
{| class="wikitable center-all right-2 left-3 left-5"
!Degree
!Cents
!Name
|-
|0
|0.00
|unison
|-
|1
|22.64
|comma
|-
|2
|45.28
|diesis
|-
|3
|67.92
|subminor second
|-
|4
|90.57
|small minor second
|-
|5
|113.21
|large minor second
|-
|6
|135.85
|<nowiki>sup[er|ra]minor second</nowiki>
|-
|7
|158.49
|submajor second
|-
|8
|181.13
|ptolemaic tone
|-
|9
|203.77
|pythagorean tone
|-
|10
|226.42
|supermajor second
|-
| 11
|249.06
|inframinor third
|-
|12
| 271.70
|subminor third
|-
| 13
|294.34
|pythagorean minor third
|-
|14
| 316.98
|just minor third
|-
|15
|339.62
|<nowiki>sup[er|ra]minor third</nowiki>
|-
|16
|362.26
|submajor third
|-
|17
|384.91
|just major third
|-
|18
| 407.55
|pythagorean major third
|-
|19
|430.19
|supermajor third
|-
|20
| 452.83
|ultramajor third
|-
|21
|475.47
|subfourth
|-
|22
| 498.11
|fourth
|-
|23
|520.75
|superfourth
|-
|24
|543.40
|ultrafourth
|-
|25
|566.04
|subtritone
|-
|26
|588.68
|small tritone
|}
[[Category:Interval naming]]
[[Category:Interval naming]]

Revision as of 04:08, 14 July 2024

Walker brightness notation is a way of naming intervals "invented" by Jay Walker / VectorGraphics. It names intervals purely based on their sizes, ignoring just intonation and MOS scales almost entirely (though it takes names from them, specifically diatonic, for the sake of recognizability).

WBN is intended to be used with the assumption of just octaves.

WBN uses increasing levels of detail to specify intervals more or less precisely. For example, 435c is a supermajor third, and "supermajor third", "major third", and "third" are all valid descriptions. WBN can be simply used to refer to the sizes of intervals.

However, it can also name the intervals in a scale as follows:

Steps to name the intervals in a scale:

Note: Intervals at the boundary always go in the category closer to the octave or unison, so that all octave complements work.

1. Give each interval in the scale its base label

  • Label the unison and the octave at 0 and 1200 cents.
  • Label any intervals between 650 and 750 cents "fifth".
  • Label any intervals between 450 and 550 cents "fourth".
  • Label any intervals between 550 and 650 cents "tritone".
  • Label any intervals between 50 and 250 cents "second".
  • Label any intervals between 250 and 450 cents "third".
  • Label any intervals between 750 and 950 cents "sixth".
  • Label any intervals between 950 and 1150 cents "seventh".
  • Label any intervals between 1150 and 1200 cents (but not 1200 cents exactly) "suboctave".
  • Label any intervals between 0 and 50 cents (but not 0 cents exactly) "comma".

2. Resolve ambiguities (stage 1: major and minor)

If there are multiple intervals labeled "third":

  • Label any intervals between 250 and 340 cents "minor third".
  • Label any intervals between 360 and 450 cents "major third".
  • Label any intervals between 340 and 360 cents "neutral third".

If there are multiple intervals labeled "second":

  • Label any intervals between 50 and 140 cents "minor second".
  • Label any intervals between 160 and 250 cents "major second".
  • Label any intervals between 140 and 160 cents "neutral second"

If there are multiple intervals labeled "tritone":

  • Label any intervals between 550 and 600 cents "narrow tritone".
  • If an interval is 600 cents exactly, label it "mid tritone".

Do this for the octave complements of these interval ranges, changing the qualities and interval sizes respectively, for example 860 to 950 is a major sixth.

3. Resolve ambiguities (stage 2: supermajor and subminor)

If there are multiple intervals labeled "fourth":

  • Label any intervals between 450 and 480 cents "subfourth".
  • Label any intervals between 520 and 550 cents "superfourth".
  • Continue to call any intervals between 480 and 520 cents "fourth".

If there are multiple intervals labeled "major third":

  • Label any intervals between 425 and 450 cents "supermajor third".
  • Continue to call any intervals between 360 and 425 cents "major third".

If there are multiple intervals labeled "minor third":

  • Label any intervals between 250 and 280 cents "subminor third".
  • Continue to call any intervals between 280 and 340 cents "minor third".

If there are multiple intervals labeled "major second":

  • Label any intervals between 220 and 250 cents "supermajor second".
  • Continue to call any intervals between 160 and 220 cents "major second".

If there are multiple intervals labeled "minor second":

  • Label any intervals between 50 and 80 cents "subminor second".
  • Continue to call any intervals between 80 and 140 cents "minor second".

If there are multiple intervals labeled "comma":

  • Label any intervals between 30 and 50 cents "diesis".
  • Continue to call any intervals smaller than 30 cents "comma".

If there are multiple intervals labeled "narrow tritone":

  • Label any intervals between 550 and 575 cents "subtritone".
  • Continue to call any intervals between 575 and 600 cents "narrow tritone".

Do this for the octave complements of these interval ranges. Note that 1150-1170c becomes an infraoctave, due to "diesis" not having a well-defined octave complement, and that the octave complement of a narrow tritone is a wide tritone.

4. Resolve ambiguities (stage 3: inframinor and ultramajor)

If one of these steps resolves ambiguity, perform it:

  • Seconds:
    • Label any intervals between 50 and 65 cents "inframinor second".
    • Label any intervals between 125 and 140 cents "superminor second".
    • Label any intervals between 160 and 175 cents "submajor second".
    • Label any intervals between 235 and 250 cents "ultramajor second".
  • Thirds:
    • Label any intervals between 250 and 260 cents "inframinor third".
    • Label any intervals between 325 and 340 cents "superminor third".
    • Label any intervals between 360 and 375 cents "submajor third".
    • Label any intervals between 440 and 450 cents "ultramajor third".
  • Fourths:
    • Label any intervals between 450 and 460 cents "infrafourth".
    • Label any intervals between 535 and 550 cents "ultrafourth".
  • Tritones:
    • Label any intervals between 550 and 560 cents "infratritone".

Do this for the octave complements of these interval ranges. Remember to do the octave complements for the tritones as well.

5. Resolve ambiguities (stage 4: arto and tendo)

If one of these steps resolves ambiguity, perform it:

  • Seconds:
    • Label any intervals between 80 and 95 cents "artominor second".
    • Label any intervals between 110 and 125 cents "tendominor second".
    • Label any intervals between 175 and 190 cents "artomajor second".
    • Label any intervals between 205 and 220 cents "tendomajor second".
  • Thirds:
    • Label any intervals between 280 and 295 cents "artominor third".
    • Label any intervals between 310 and 325 cents "tendominor third".
    • Label any intervals between 375 and 390 cents "artomajor third".
    • Label any intervals between 405 and 425 cents "tendomajor third".
  • Fourths:
    • Label any intervals between 480 and 495 cents "artofourth".
    • Label any intervals between 510 and 520 cents "tendofourth".
  • Tritones:
    • Label any intervals between 575 and 585 cents "artotritone".

Do this for the octave complements of these interval ranges. Remember to do the octave complements for the tritones as well. Though there are some notes to keep in mind:

  • If a pair of intervals ends up distinguished by arto vs. no stage 4 prefix, i.e. artomajor third and major third, the other interval can be labeled tendo, regardless of its actual size. The same applies in reverse, for if a pair of intervals is distinguished by tendo vs. no stage 4 prefix.
  • If there is only one interval simply labelled fourth, call it a "perfect fourth".
    • Same with fifths.

6. Resolve ambiguities (stage 5)

By this point, except for commas, the octave has been subdivided into 10-20c interval regions, suitable for notating edos as large as 71edo without any problems. Where intervals need to be named more precisely than this, however, one may resort to just applying other descriptors like "small", "medium", "large", "grave", "acute", etc, depending on the needs of the scale.

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