Adaptive diatonic interval names: Difference between revisions
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[[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 EDOs' degrees of thirds in a manner inconsistent with conventional diatonic notation in a formal, systematic way. | 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. | ||
They replace Walker Brightness Notation, which is now deprecated. | |||
== Step 1: Identify central intervals == | |||
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, | 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, | ||
== Step 2: Identify anchors == | |||
What 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 each interval degree, follow the given procedure: | ||
* Find | * Find 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". | ||
** 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. | ** 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. | ||
[[File:Qetqet.png|thumb|434x434px|Result after Step 2]] | |||
== Step 3: Fill in gaps == | |||
* 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 25: | ||
|- | |- | ||
|1 | |1 | ||
| | |pentamajor | ||
|- | |- | ||
|2 | |2 | ||
| Line 37: | Line 42: | ||
|magimajor, pentamajor, novamajor, trimajor, neomajor, shrubmajor | |magimajor, pentamajor, novamajor, 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 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.) 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 57: | ||
|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. | ||
== 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). | ||
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". | == 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. | |||
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"). | 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). | |||
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. | |||
The semioctave may be explicitly labelled. | |||
== 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). | 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). | ||
== 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.]] | |||
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!]] | |||
| | |||
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| - | |||
! | |||
Revision as of 07:58, 10 October 2025
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.
They replace Walker Brightness Notation, which is now deprecated.
Step 1: Identify central intervals
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,
Step 2: Identify anchors
What interval qualities will label is distances from these "central intervals". For each interval degree, follow the given procedure:
- Find 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".
- 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.
Step 3: Fill in gaps
- There should now be unlabelled steps between submajor and supermajor. These correspond to different varieties of major. Follow the table:
| Number of steps | Qualities |
|---|---|
| 1 | pentamajor |
| 2 | pentamajor, neomajor |
| 3 | pentamajor, novamajor, neomajor |
| 4 | pentamajor, novamajor, trimajor, neomajor |
| 5 | pentamajor, novamajor, trimajor, neomajor, shrubmajor |
| 6 | magimajor, pentamajor, novamajor, 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.) 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 (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:
Result after Step 3
| Number of steps | Qualities |
|---|---|
| 1 | tendoneutral |
| 2 | tendoneutral, supraneutral |
The neutral interval, if present in the edo, is simply called neutral.
Step 4: Repeat for minor
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
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.
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").
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).
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.
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).
Composite
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.
