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Adaptive diatonic interval names: Difference between revisions

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What interval qualities will label is distances from these "central intervals". For the most complex case (intervals with major and minor quality), follow the following procedure.
What interval qualities will label is distances from these "central intervals". For the most complex case (intervals with major and minor quality), follow the following procedure.


* Find a) the smallest interval greater than 25c above the neutral interval. Take the interval BEFORE this and label it "submajor" (if its offset is still positive).
* Find a) the smallest interval greater than 25c above the neutral interval. Take the interval BEFORE this and label it "submajor" (if its offset is still positive, otherwise, interpret it as submajor but do not actually assign it that name).
* Find a) the closest interval to 85c above the neutral interval or b) the smallest interval greater than 75c above the neutral interval, whichever is higher. Label this interval "supermajor".  
* Find a) the closest interval to 85c above the neutral interval or b) the smallest interval greater than 75c above the neutral interval, whichever is higher. Label this interval "supermajor".  
** If supermajor and submajor coincide, label the interval "major" and then label the next interval up "supermajor".
** If supermajor and submajor coincide, label the interval "major" and then label the next interval up "supermajor".
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* If the step above ultramajor lies below 120 cents above neutral, label it tendo.
* If the step above ultramajor lies below 120 cents above neutral, label it tendo.
For the minor side of any interval class, swap "major" and "minor", "sub" and "super/supra" (super applies only to major and unqualified intervals, other types use supra), "ultra" and "infra", and "arto" and "tendo" (in all senses).


The perfect fourth is, as a rule, the same degree as the diatonic minor third (whatever that happens to be) - for example, if the diatonic minor third is subminor, the perfect fourth is also interpreted as subminor, and "perfect fourth" replaces "subfourth". 


As a result, in standard diatonic edos, "minor fourth" contracts to "fourth", and "major fifth" contracts to "fifth" For example, "subminor fourth" -> "subfourth". Instead of "minor fifth", there is "diminished fifth" (and instead of "major fourth", there is "augmented fourth").


For the minor side of any interval class, swap "major" and "minor", "sub" and "super/supra" (super applies only to major and unqualified intervals, other types use supra), "ultra" and "infra", and "arto" and "tendo" (in all senses).  
This behavior is reversed in antidiatonic edos, so that the perfect fourth is major and the perfect fifth is minor. No contraction is done in equiheptatonic edos (where the perfect fourth is neither major nor minor) or oneirotonic edos (where labels as thirds/sixths take priority over the assignment for the perfect fourth).  


For tritones, start at the semi-octave. If the semioctave exists in the scale, label it "neutral tritone", otherwise label the two closest intervals "artoneutral tritone" for the smaller and "tendoneutral tritone" for the larger. Label the sharpest tritone interval flatter than the neutral or artoneutral tritone "narrow tritone" (and label its complement "wide tritone"). If the interval closest to 565 cents has not already been labelled as a tritone, label it a "subtritone". If there are intervals between the subtritone and narrow tritone, label "grave tritone" if only 1 interval and "grave tritone" (for the smaller) and "small tritone" (for the larger) if there are two. Label the tritone interval one step smaller than the subtritone the infratritone. Similarly, do the same thing for fourths and fifths with the best 4/3 and 3/2 (and checking for n +- 35c rather than specifically 565c), but you won't need the artoneutral/tendoneutral case.  
Similar logic applies for the perfect fifth and the diatonic major third.  


For unisons, the term "diesis" is used in place of "major unison" and "counterdiesis" in place of "minor octave"; "comma" is "submajor unison" and "countercomma" is "supraminor octave".  
The semioctave may be explicitly labelled.  


Priority: "perfect fourth", "perfect fifth", "tritone", "unison", and "octave" have the same priority as a "novamajor third", "neomajor third", or "major third", whichever is the first one listed that exists in the labeling; priority decreases by one as you go further from the central interval (or perfect interval in the case of fourths and fifths), and increases by one as you go closer to it. Where the central interval is halfway between an edostep, the two closest intervals have a priority of -0.5. The interval name with the highest priority is used.
The unison/octave are treated as a diatonic major interval when going sharper than it, and a diatonic minor interval when going flatter than it. (Again, reversed when antidiatonic).  


If "narrow tritone", "subtritone", or "small tritone" (in decreasing order of precedence) do not appear in the list of highest-priority names, and "artoneutral tritone" exists, then relabel artoneutral to whichever is available.  
Priority decreases by one as you go further from the central interval and increases by one as you go closer to it. Where the central interval is halfway between an edostep, the two closest intervals have a priority of -0.5. The interval name with the highest priority is used.


If there is only one type of major or minor (not including submajor or supraminor), the qualifier on major or minor (i.e. penta- or magi-) is removed.
If there is only one type of major or minor (not including submajor or supraminor), the qualifier on major or minor (i.e. penta- or magi-) is removed.
Finally, if all this results in a distinction between "perfect fourth" and "fourth" (which happens in 22edo), its abbreviated quality is reinstated (usually minor). Same for the fifth.


The full notation of 68edo is as follows:
The full notation of 68edo is as follows:
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|-
|-
|1
|1
|comma
|superunison
| -4
| -5
|18
|18
|pentaminor third
|pentaminor third
Line 88: Line 93:
|-
|-
|2
|2
|diesis
|ultraunison / inframinor second
| -5
| -6
|19
|19
|supraminor third
|supraminor third
Line 140: Line 145:
| -1
| -1
|26
|26
|subfourth
|ultramajor third, infrafourth
| -5
| -6
|-
|-
|10
|10
Line 147: Line 152:
| -2
| -2
|27
|27
|narrow fourth
|subfourth
| -4
| -5
|-
|-
|11
|11
Line 155: Line 160:
|28
|28
|perfect fourth
|perfect fourth
| -3
| -4
|-
|-
|12
|12
Line 161: Line 166:
| -4
| -4
|29
|29
|wide fourth
|novafourth
| -4
| -3
|-
|-
|13
|13
Line 168: Line 173:
| -5
| -5
|30
|30
|superfourth
|pentafourth
| -5
| -2
|-
|-
|14
|14
Line 175: Line 180:
| -6
| -6
|31
|31
|ultrafourth/infratritone
|suprafourth
| -6
| -1
|-
|-
|15
|15
Line 182: Line 187:
| -5
| -5
|32
|32
|subtritone
|neutral fourth
| -5
| 0
|-
|-
|16
|16
Line 189: Line 194:
| -4
| -4
|33
|33
|narrow tritone
|subaugmented fourth
| -4
| -1
|-
|-
|17
|17
Line 196: Line 201:
| -3
| -3
|34
|34
|neutral tritone
|pentaaugmented fourth / pentadiminished fifth
| -3
| -2
|}
|}
The system also holds up in antidiatonic cases, such as 25edo as notated with its flat fifth:
The system also holds up in antidiatonic cases, such as 25edo as notated with its flat fifth:
Line 206: Line 211:
|-
|-
|1
|1
|diesis
|superunison
| -2
| -2
|-
|-
Line 238: Line 243:
|-
|-
|9
|9
|supermajor third
|diminished fourth
| -2
| -1
|-
|-
|10
|10
|narrow fourth
|neutral fourth
| -2
| 0
|-
|-
|11
|11
Line 250: Line 255:
|-
|-
|12
|12
|narrow tritone
|supermajor fourth
| -2
|}
And 23edo:
{| class="wikitable"
!
!
!
|-
|1
|superunison
| -1.5
|-
|2
|subminor second
| -1.5
|-
|3
|minor second
| -0.5
|-
|4
|major second
| -0.5
|-
|5
|supermajor second / subminor third
| -1.5
|-
|6
|minor third
| -0.5
|-
|7
|major third
| -0.5
|-
|8
|supermajor third / subdiminished fourth
| -1.5
|-
|9
|diminished fourth
| -0.5
|-
|10
|perfect fourth
| -0.5
|-
|11
|superfourth
| -1.5
| -1.5
|}
|}
Note that superfourth instead of suprafourth is used because the fourth is technically major.
In equiheptatonic cases, such as 28edo:
In equiheptatonic cases, such as 28edo:
{| class="wikitable"
{| class="wikitable"
Line 261: Line 318:
|1
|1
|diesis
|diesis
| -2
| -1
|-
|-
|2
|2
Line 296: Line 353:
|-
|-
|10
|10
|supermajor third
|supermajor third, subminor fourth
| -2
| -2
|-
|-
|11
|11
|narrow fourth
|minor fourth
| -2
| -1
|-
|-
|12
|12
|perfect fourth
|perfect fourth
| -1
| 0
|-
|-
|13
|13
|narrow tritone, wide fourth
|major fourth
| -2
| -1
|-
|-
|14
|14
|neutral tritone
|supermajor fourth, subminor fifth
| -1
| -2
|}
|}
And in equipentatonic cases, such as 15edo:
And in equipentatonic cases, such as 15edo:
Line 342: Line 399:
|-
|-
|6
|6
|perfect fourth
|supermajor third, perfect fourth
| -0.5
| -1.5
|-
|-
|7
|7
|narrow tritone
|minor fourth
| -0.5
| -0.5
|}
|}
It can even handle EDOs with only oneirotonic fifths, because the numbering from neutral intervals and the priority system balances out the negative limmas in these systems. Here's 18edo:
It can even handle EDOs with only oneirotonic fifths, because the numbering from neutral intervals and the priority system balances out the negative limmas in those systems - though fifths and fourths kinda swap places. Here's 18edo:
{| class="wikitable"
{| class="wikitable"
!
!
Line 380: Line 437:
|-
|-
|7
|7
|perfect fourth
|supermajor third / subminor fifth
|<nowiki>-0.5</nowiki>
| -1.5
|-
|-
|8
|8
|wide fourth, narrow tritone
|minor fifth
| -1.5
| -1.5
|-
|-
|9
|9
|neutral tritone
|major fifth / minor fourth
|<nowiki>-0.5</nowiki>
|<nowiki>-0.5</nowiki>
|}
|}
Line 420: Line 477:
|-
|-
|6
|6
|neutral tritone
|augmented fourth / diminished fifth
| -0.5
| -0.5
|}
|}
Line 431: Line 488:
|-
|-
|1
|1
|subminor second / diesis
|subminor second / superunison
|-
|-
|2
|2
Line 452: Line 509:
|-
|-
|8
|8
|supermajor third, narrow fourth
|supermajor third
|-
|-
|9
|9
Line 458: Line 515:
|-
|-
|10
|10
|wide fourth
|minor fourth
|-
|-
|11
|11
|neutral tritone
|major fourth
|}
|}
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