Xenharmonic Wiki

Adaptive diatonic interval names: Difference between revisions

← Older edit
VisualWikitext
No edit summary
Overthink (talk | contribs)
autopatrolled
2,767 edits
a few links
 
(15 intermediate revisions by 4 users not shown)
Line 1: Line 1:
Adaptive diatonic interval names are Vector's attempt to characterize the labelling of certain EDOs' degrees of thirds in a manner inconsistent with conventional diatonic notation in a formal, systematic way.
{{Idiosyncratic terms}}
[[File:22edo ADIN.png|thumb|291x291px|22edo with adaptive diatonic interval names]]
Adaptive diatonic interval names are Vector's attempt to characterize the labelling of certain [[EDO]]s' degrees of thirds in a manner inconsistent with conventional [[diatonic]] notation in a formal, systematic way.


They replace [[Walker Brightness Notation]], which is now deprecated.


The fundamental premise of ADIN goes like this.
== Step 1: Identify central intervals ==
For an EDO (let's say, [[58edo]]), identify the "central intervals" for each category. These are the diatonic [[neutral (interval quality)|neutral intervals]], which may be between edosteps.


For an EDO (let's say, 58edo), identify the "central intervals" for each category. These are the diatonic neutral intervals, which may be between edosteps, with the exception of the fifth and fourth, which use their perfect varieties, and the tritone, which is always the semioctave.
== Step 2: Identify anchors ==
What [[interval quality|interval qualities]] will label is distances from these "central intervals". For each interval degree, follow the given 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 the smallest interval greater than or equal to 25{{c}} 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 85{{c}} above the neutral interval or b) the smallest interval greater than 75{{c}} 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 there are no unlabelled steps between submajor and supermajor, label whichever is closer to 50{{c}} above the neutral interval "major". If this is supermajor, label the next interval up "supermajor"; if this is submajor, label the next interval down "submajor" if its offset is still positive.
[[File:Qetqet.png|thumb|434x434px|Result after Step 2]]


* 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).
== Step 3: Fill in gaps ==
* 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 there are no unlabelled steps between submajor and supermajor, label whichever is closer to 50c above the neutral interval "major". If this is supermajor, label the next interval up "supermajor"; if this is submajor, label the next interval down "submajor" if its offset is still positive.
* There should now be unlabelled steps between submajor and supermajor. These correspond to different varieties of major. Follow the table:
* There should now be unlabelled steps between submajor and supermajor. These correspond to different varieties of major. Follow the table:


Line 20: Line 26:
|-
|-
|1
|1
|major
|pentamajor
|-
|-
|2
|2
Line 36: Line 42:
|6
|6
|magimajor, pentamajor, novamajor, trimajor, neomajor, shrubmajor
|magimajor, pentamajor, novamajor, trimajor, neomajor, shrubmajor
|-
|7
|magimajor, pentamajor, novamajor, dodecamajor, trimajor, neomajor, shrubmajor
|}
|}
 
This might lead to nonsensical assignments of "pentamajor" in edos like 17 (in fact, 17 is the only edo to have this property) - but do not worry! Those will be cleaned up in later steps.
* Find the closest otherwise unlabelled interval to 103c above the neutral interval which is higher than 75c; call it "ultramajor". (Note: you may be ending up with this interval being many semitones above the neutral, that's okay because extraneous names will be clipped off once we actually apply everything to interval classes.)
* Find the closest otherwise unlabelled interval to 103c above the neutral interval which is higher than 75{{c}}; call it "ultramajor". (Note: you may be ending up with this interval being many semitones above the neutral, that's okay because extraneous names will be clipped off once we actually apply everything to interval classes.) Label the step above ultramajor "tendo" if it is at most 1/6 of the size of the perfect fifth above neutral - otherwise, treat tendo as synonymous with the largest step that is at most 1/6 the size of the perfect fifth above neutral.
* There may now be one or more unlabelled steps between supermajor and ultramajor: this scheme currently supports one interval between these two points, which is labelled "sensamajor".
* There may now be one or more unlabelled steps between supermajor and ultramajor (ignoring tendo): this scheme currently supports one interval between these two points, which is labelled "sensamajor".
* There may also be one or more unlabelled steps below submajor. Follow the table:
* There may also be one or more unlabelled steps below submajor. Follow the table:[[File:Qyrewywe.png|thumb|432x432px|Result after Step 3]]


{| class="wikitable"
{| class="wikitable"
Line 52: Line 61:
|tendoneutral, supraneutral
|tendoneutral, supraneutral
|}
|}
The neutral interval, if present in the edo, is called neutral.
The neutral interval, if present in the edo, is simply called neutral.  
 
* If the step above ultramajor lies below 120 cents above neutral, label it tendo.
 
 


== Step 4: Repeat for minor ==
[[File:With minor.png|thumb|429x429px|Result after Step 4]]
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).  
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).  


== Step 5: Adjust fourths and fifths ==
[[File:4th 1.png|thumb|430x430px|4ths labelled like any standard interval.]]
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 "subminor fourth". Similar logic applies for the perfect fifth and the diatonic major third. 


Fourths/fifths and tritones utilize the exact same interval qualities, except that the "neutral" is removed from the neutral qualities. Note that the terms "penta-" and "tri-" now no longer directly correspond to 5- or 3-limit JI intervals in the case of intervals in this region.
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").
[[File:4ths er.png|thumb|428x428px|Result after Step 5]]
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 fifths and fourths switch places).  


Note that here, "infra" is preferred in lieu of "supraminor", and likewise "ultra" in place of "submajor" (unless "sub-" and "super-" are not already taken, in which case, those are used).  
The semioctave may be explicitly labelled if it is present, or if two tritones are present "narrow tritone" and "wide tritone" - but these are not standard degree-based interval names.  


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".  
== Unisons and octaves ==
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).  


Priority: "fourth", "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, 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.
== Composite ==
 
[[File:Compie Compie.png|thumb|417x417px|Composite interval name lists take only the interval name with the highest priority out of the options from each individual list.]]
An unqualified fourth or fifth is labelled "perfect"; an unqualified tritone is labelled "neutral" (importantly, subtritone and supratritone remain unchanged).
In order to composite our independent naming schemes for different degrees into a complete interval name list (and trim off nonsensical names near the outer edges), we'll use a system of priority. Priority starts at 0 for any neutral interval and decreases by 0.5 for every half of an edostep you travel away from the neutral (meaning that if the neutral isn't in the edo, the two closest intervals have a priority of -0.5) or more simply 1 for every edostep. The interval name with the highest priority is chosen as the canonical interval name.


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.
[[File:Composite 2.png|thumb|420x420px|See? Nonsensical penta- names are all gone!]]


The full notation of 68edo is as follows:
== Functioning for various edos ==
{| class="wikitable"
{| class="wikitable"
|+
|+
!
!EDO
!
!Notes
!
!
!
!
|-
|-
|1
|1
|comma
|Functional: unison and octave.
| -4
|18
|pentaminor third
| -2
|-
|-
|2
|2
|diesis
|Mostly functional: unison, octave, and tritone, which is read as a perfect fifth and perfect fourth simultaneously.
| -5
|19
|supraminor third
| -1
|-
|-
|3
|3
|subminor second
|Oneirotonic; switches places of fifth and fourth and no perfect intervals. Neutral fifth is 400c; neutral fourth is 800c.
| -5
|20
|neutral third
|0
|-
|-
|4
|4
|neominor second
|Extreme case: 300c and 900c intervals are neutral fourth and fifth respectively. Alternatively, neutral third and sixth respectively.
| -4
|21
|submajor third
| -1
|-
|-
|5
|5
|novaminor second
|Functional as in standard notation. 4ths and fifths have exact same priority.
| -3
|22
|pentamajor third
| -2
|-
|-
|6
|6
|pentaminor second
|Oneirotonic; 600c interval is "supermajor" fifth.
| -2
|23
|novamajor third
| -3
|-
|-
|7
|7
|supraminor second
|Functional as in standard notation.
| -1
|24
|neomajor third
| -4
|-
|-
|8
|8
|neutral second
|Oneirotonic; switches places of fifth and fourth and no perfect intervals.
|0
|25
|supermajor third
| -5
|-
|-
|9
|9
|submajor second
|Functional. Major second and minor third coincide.
| -1
|26
|minor fourth
| -5
|-
|-
|10
|10
|pentamajor second
|Functional. 5edo + neutrals.
| -2
|27
|subfourth
| -4
|-
|-
|11
|11
|novamajor second
|Functional. Most intervals are neutral.
| -3
|28
|perfect fourth
| -3
|-
|-
|12
|12
|neomajor second
|Functional as in standard notation.
| -4
|29
|superfourth
| -4
|-
|-
|13
|13
|supermajor second
|Oneirotonic; switches places of fifth and fourth and no perfect intervals.
| -5
|30
|pentamajor fourth
| -5
|-
|-
|14
|14
|ultramajor second, inframinor third
|Functional. 7edo + major/minor intervals which are also perfect interordinals.
| -6
|31
|novaminor tritone, novamajor fourth
| -6
|-
|-
|15
|15
|subminor third
|Functional.
| -5
|32
|pentaminor tritone
| -5
|-
|-
|16
|16
|neominor third
|Functional.
| -4
|33
|subtritone
| -4
|-
|-
|17
|17
|novaminor third
|Functional. Contains the nonsensical pentamajor that's resolved by the compositing process.
| -3
|34
|tritone
| -3
|}
The system also holds up in antidiatonic cases, such as 25edo as notated with its flat fifth:
{| class="wikitable"
!
!
!
|-
|-
|1
|18
|diesis
|Oneirotonic; switches places of fifth and fourth and no perfect intervals.
| -2
|-
|2
|subminor second
| -2
|-
|3
|minor second
| -1
|-
|4
|neutral second
|0
|-
|5
|major second
| -1
|-
|6
|minor third
| -1
|-
|7
|neutral third
|0
|-
|8
|major third
| -1
|-
|9
|supermajor third
| -2
|-
|10
|subfourth
| -2
|-
|11
|perfect fourth
| -1
|-
|-
|12
|19+
|subtritone
|Functional.
| -1.5
|}
In equiheptatonic cases, such as 28edo:
{| class="wikitable"
!
!
!
|-
|1
|diesis
| -2
|-
|2
|subminor second
| -2
|-
|3
|minor second
| -1
|-
|4
|neutral second
|0
|-
|5
|major second
| -1
|-
|6
|subminor third, supermajor second
| -2
|-
|7
|minor third
| -1
|-
|8
|neutral third
|0
|-
|9
|major third
| -1
|-
|10
|supermajor third
| -2
|-
|11
|minor fourth
| -2
|-
|12
|perfect fourth
| -1
|-
|13
|minor tritone, major fourth
| -2
|-
|14
|neutral tritone
| -1
|}
And in equipentatonic cases, such as 15edo:
{| class="wikitable"
!
!
!
|-
|1
|minor second
| -0.5
|-
|2
|major second
| -0.5
|-
|3
|supermajor second, subminor third
| -1.5
|-
|4
|minor third
| -0.5
|-
|5
|major third
| -0.5
|-
|6
|perfect fourth
| -0.5
|-
|7
|minor tritone
| -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:
{| class="wikitable"
!
!
!
|-
|1
|minor second
|<nowiki>-0.5</nowiki>
|-
|2
|major second
|<nowiki>-0.5</nowiki>
|-
|3
|supermajor second
| -1.5
|-
|4
|subminor third
| -1.5
|-
|5
|minor third
| -0.5
|-
|6
|major third
|<nowiki>-0.5</nowiki>
|-
|7
|perfect fourth
|<nowiki>-0.5</nowiki>
|-
|8
|major fourth, minor tritone
| -1.5
|-
|9
|neutral tritone
|<nowiki>-0.5</nowiki>
|}
|}
The first edo that the current set of names is insufficient to cover is 152edo. If an extra name is added for an interval between sensamajor and ultramajor, then the first failure is at 156, and then at 159.


== Oneirotonic extension ==
[[File:Oneieh.png|thumb|18edo with oneirotonic vs. diatonic schemes compared. Note the lack of unison, octave, or perfect fourth/fifth in diatonic, and the tritone intervals in oneirotonic.]]
One possible extension to handle edos with an oneirotonic fifth (3, 6, 8, 13, and 18) is to use the neutralized oneirotonic scale as the basis instead of the neutralized diatonic scale. In this case, things function exactly the same as before, but based on oneirotonic instead of diatonic. The additional degree may be called a median or tritone - special tritone labels are not used as the semioctave is always the neutral tritone.


And just as a sanity check, here's 12edo:
[[Category:Interval naming]]
{| class="wikitable"
!
!
!
|-
|1
|minor second
| -0.5
|-
|2
|major second
| -0.5
|-
|3
|minor third
| -0.5
|-
|4
|major third
| -0.5
|-
|5
|perfect fourth
| -0.5
|-
|6
|neutral tritone
| -0.5
|}
Retrieved from "https://en.xen.wiki/w/Adaptive_diatonic_interval_names"