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In the theory of [[Margo Schulter]], '''interseptimal''' is a category of intervals which occupy regions intermediate between two septimal ratios such as [[8/7]] and [[7/6]], or [[12/7]] and [[7/4]]. There are four interseptimal regions given below, with approximate cents ranges from Schulter's article [http://www.bestii.com/%7Emschulter/IntervalSpectrumRegions.txt Regions of the Interval Spectrum]:
In the theory of [[Margo Schulter]], an '''interseptimal interval''' is an [[interval]] that belongs in one of four [[interval region]]s which are intermediate between two septimal ratios such as [[8/7]] and [[7/6]], or [[12/7]] and [[7/4]]. There are four interseptimal regions given below, with approximate cents ranges from Schulter's essay [http://www.bestii.com/%7Emschulter/IntervalSpectrumRegions.txt ''Regions of the Interval Spectrum'']:


* Maj2–min3 – intermediate between [[8/7]] and [[7/6]] – 240¢–260¢
* Maj2–min3 – intermediate between [[8/7]] and [[7/6]] – 240–260{{c}}
* Maj3–4 – intermediate between [[9/7]] and [[21/16]] – 440¢–468¢
* Maj3–4 – intermediate between [[9/7]] and [[21/16]] – 440–468{{c}}
* 5–min6 – intermediate between [[32/21]] and [[14/9]] – 732¢–760¢
* 5–min6 – intermediate between [[32/21]] and [[14/9]] – 732–760{{c}}
* Maj6–min7 – intermediate between [[12/7]] and [[7/4]] – 940¢–960¢
* Maj6–min7 – intermediate between [[12/7]] and [[7/4]] – 940–960{{c}}


Interseptimal intervals are well-represented in [[24edo]] at 250¢, 450¢, 750¢ and 950¢. They also appear in [[19edo]] and [[29edo]]. As they fall in ambiguous zones between simpler categories, they are inevitably xenharmonic. In other words, interseptimals are the [[interordinal]] intervals with respect to the diatonic [[mos]] [[5L 2s]].
Additionally, there are also these 2 interseptimal regions near the unison and octave:
* 1–min2 – intermediate between [[64/63]] and [[28/27]] – 40–60{{c}}
* Maj7-8 – intermediate between [[27/14]] and [[63/32]] – 1140–1160{{c}}


A JI-agnostic synonym is '''interordinal'''; here, ''ordinal'' refers to the [[interval class]]es of the diatonic scale the interordinal intervals lie between, conventionally denoted with ordinal numbers.
Interseptimal intervals are well-represented in [[24edo]] at 250{{c}}, 450{{c}}, 750{{c}}, and 950{{c}}. They also appear in [[19edo]] and [[29edo]]. As they fall in ambiguous zones between both [[5L 2s|diatonic]] and [[chromatic]] categories, they are inevitably xenharmonic.


== Categorical and Notational Approaches ==
A JI-agnostic synonym is '''interordinal'''; here, ''ordinal'' refers to the [[interval class]]es of the diatonic scale the interordinal intervals lie between, conventionally denoted with ordinal numbers.
 
See [[Neutral and interordinal k-mossteps]] for a partial generalization of interseptimal categories to other mosses.
 
== Categorical and notational approaches ==
While interseptimals are interesting for falling right in between the typical western interval categories, this also makes them difficult to name and notate: do we classify a 250-cent interval as a second, a third, both, or neither?
While interseptimals are interesting for falling right in between the typical western interval categories, this also makes them difficult to name and notate: do we classify a 250-cent interval as a second, a third, both, or neither?


Line 16: Line 22:
One option is to give each region a distinct name (analogous to using the word ''tritone'' rather than diminished fifth or augmented fourth). Possible names that could be used are:
One option is to give each region a distinct name (analogous to using the word ''tritone'' rather than diminished fifth or augmented fourth). Possible names that could be used are:
* 240¢–260¢ – '''semifourth''' – an interval of this size is around half the size of a perfect fourth.
* 240¢–260¢ – '''semifourth''' – an interval of this size is around half the size of a perfect fourth.
** The term '''chthonic''' (from ''khthon'', an ancient Greek word referring to spirits of the underworld) refers to the 240-260¢ region by [[Zhea Erose]].<ref>as per [[Primodal Archive]]</ref>
** The term '''chthonic''' (from ''khthon'', an ancient Greek word referring to spirits of the underworld) refers to the 240–260{{c}} region by [[Zhea Erose]].<ref group="note">As per [[Primodal Archive]].</ref>
* 440¢–468¢ – '''semisixth''' – an interval of this size is around half the size of a major sixth.
* 440¢–468¢ – '''semisixth''' – an interval of this size is around half the size of a major sixth.
** The term '''naiadic''' (from ''naiad'', a kind of ancient Greek water spirit) refers to the 440–464¢ region by [[Zhea Erose]], who uses it frequently.
** The term '''naiadic''' (from ''naiad'', a kind of ancient Greek water spirit) refers to the 440–464{{c}} region by [[Zhea Erose]], who uses it frequently.
* 732¢–760¢ – '''semitenth''' – an interval of this size is around half the size of a minor tenth (i. e., an octave plus a minor third). Another possible name is sesquifourth (since this is also about one and a half times the size of a perfect fourth).
* 732¢–760¢ – '''semitenth''' – an interval of this size is around half the size of a minor tenth (i. e., an octave plus a minor third). Another possible name is sesquifourth (since this is also about one and a half times the size of a perfect fourth).
** The term '''cocytic''' was proposed by [[Inthar]], by analogy with ''naiadic''.<ref>Flora Canou criticizes ''semisixth'' and ''semitenth'' as they fail to make clear whether the interval to be split is major or minor, and prefers ''naiadic'' and ''cocytic''.</ref>
** The term '''cocytic''' was proposed by [[Inthar]], by analogy with ''naiadic''.
* 940¢–960¢ – '''semitwelfth''' – an interval of this size is around half the size of a perfect twelfth (i.e. a compound perfect fifth, or tritave). All even [[edt]]s have a semitwelfth of approximately 951 cents, analogous to the 600 cent tritone shared by all even edos.
* 940¢–960¢ – '''semitwelfth''' – an interval of this size is around half the size of a perfect twelfth (i.e. a compound perfect fifth, or tritave). All even [[edt]]s have a semitwelfth of approximately 951{{c}}, analogous to the 600{{c}} tritone shared by all even edos.
** The term '''ouranic''' (by analogy with chthonic, and to match with the other terms) is proposed by [[User:Kaiveran|Kaiveran]].
** The term '''ouranic''' (by analogy with chthonic, and to match with the other terms) is proposed by [[User:Kaiveran|Kaiveran]].


This makes notating these intervals very easy as long as we have an agreed-upon symbol for "semi". By analogy with the "semi" names, the tritone could also be called a semioctave, although the term tritone is so well-established (and so well represented by an unsplit 3-limit) that there seems little reason to change it now. A key difference is that the tritone is intermediate between two septimal ratios separated by a jubilisma ([[50/49]]), whereas the other interseptimal ranges listed above are between two septimal ratios separated by a slendro diesis ([[49/48]]).
One might want to use a mixture of above terms. [[Flora Canou]] criticizes ''semisixth'' and ''semitenth'' as they fail to make clear whether the interval to be split is major or minor, and prefers ''naiadic'' and ''cocytic''. However, ''semifourth'' and ''semitwelfth'' are clear enough, so the Greek terms seems practically redundant.
 
The terminology makes notating these intervals very easy as long as we have an agreed-upon symbol for "semi". By analogy with the "semi" names, the tritone could also be called a semioctave, although the term tritone is so well-established (and so well represented by an unsplit 3-limit) that there seems little reason to change it now. A key difference is that the tritone is intermediate between two septimal ratios separated by a jubilisma ([[50/49]]), whereas the other interseptimal ranges listed above are between two septimal ratios separated by a slendro diesis ([[49/48]]).


=== Dual "semichromatic" names ===
=== Dual "semichromatic" names ===
Since interseptimal intervals are typically well represented by any [[EDO]] or [[pergen]] that divides its approximate 3/1 into 2''n'' parts, another option is to repurpose [[24edo#Quartertone Accidentals|quartertone accidentals]] to represent them, which is easy as long as we define our "half-sharps" or "half-flats" to be precisely half of a chromatic semitone. With this in mind, we get the following twinned identities for our interseptimals, with the simplest ones (assuming a half-fifth genchain) listed first:
Since interseptimal intervals are typically well represented by any [[EDO]] or [[pergen]] that divides its approximate 3/1 into 2''n'' parts, another option is to repurpose [[24edo#Quartertone Accidentals|quartertone accidentals]] to represent them, which is easy as long as we define our "half-sharps" or "half-flats" to be precisely half of a chromatic semitone. With this in mind, we get the following twinned identities for our interseptimals, with the simplest ones (assuming a half-fifth genchain) listed first:


* semifourth/chthonic = semi-augmented second (+11/2), semi-diminished third (-13/2)
* semifourth/chthonic = semi-augmented second (+11/2), semi-diminished third (−13/2)
* semisixth/naiadic = semi-diminished fourth (-9/2), semi-augmented third (+15/2)
* semisixth/naiadic = semi-diminished fourth (−9/2), semi-augmented third (+15/2)
* semitenth/cocytic = semi-augmented fifth (+9/2), semi-diminished sixth (-15/2)
* semitenth/cocytic = semi-augmented fifth (+9/2), semi-diminished sixth (−15/2)
* semitwelfth/ouranic = semi-diminished seventh (-11/2), semi-augmented sixth (+13/2)
* semitwelfth/ouranic = semi-diminished seventh (−11/2), semi-augmented sixth (+13/2)


While this does not give the interseptimals a single distinct ''notational'' name, it does reflect their ambiguity and flexibility with regards to the surrounding interval categories that many are so fond of. Furthermore, as both identities are exactly 12 notational fifths apart (i.e a direct analogue of the [[Pythagorean comma]]), composers can use a mechanism similar to the [[Color notation|"po and qu" of Color Notation]], or the plus and minus accidentals (+/-) proposed in [[Rational Comma Notation (RCN)|Rational Comma Notation]], to freely switch between the two identities.
While this does not give the interseptimals a single distinct ''notational'' name, it does reflect their ambiguity and flexibility with regards to the surrounding interval categories that many are so fond of. Furthermore, as both identities are exactly 12 notational fifths apart (i.e a direct analogue of the [[Pythagorean comma]]), composers can use a mechanism similar to the [[Color notation|"po and qu" of Color Notation]], or the plus and minus accidentals (+/−) proposed in [[Rational Comma Notation (RCN)|Rational Comma Notation]], to freely switch between the two identities.


Alternatively, one can use the ''ultra-'' prefix for sharpening by ~50¢ and ''infra-'' for flattening by ~50¢, analogous to ''super-'' and ''sub-'' for modifications by [[64/63]] (in a [[12edo]]-related context such as [[36edo]], 33¢).
Alternatively, one can use the ''ultra-'' prefix for sharpening by ~50¢ and ''infra-'' for flattening by ~50{{c}}, analogous to ''super-'' and ''sub-'' for modifications by ~30{{c}}.
* semifourth/chthonic = ultramajor second, inframinor third
* semifourth/chthonic = ultramajor second, inframinor third
* semisixth/naiadic = ultramajor third, infrafourth
* semisixth/naiadic = ultramajor third, infrafourth
* semitenth/cocytic = ultrafifth, inframinor sixth
* semitenth/cocytic = ultrafifth, inframinor sixth
* semitwelfth/ouranic = ultramajor sixth, inframinor seventh
* semitwelfth/ouranic = ultramajor sixth, inframinor seventh
''Ultra-'' and ''infra-'' also work for intervals that are very close to 11/8 or 16/11:
''Ultra-'' and ''infra-'' also work for intervals that are very close to 11/8 or 16/11:
* ~11/8 or ~550¢ = ultrafourth, infratritone, infrasemioctave
* ~11/8 or ~550{{c}} = ultrafourth, infratritone, infrasemioctave
* ~16/11 or ~650¢ = infrafifth, ultratritone, ultrasemioctave
* ~16/11 or ~650{{c}} = infrafifth, ultratritone, ultrasemioctave
 
=== "Inter" names ===
=== "Inter" names ===
Both the "semi-nth" names and the Greek-derived names above are less intuitive than they could be and require some amount of memorization. For this reason, Inthar has proposed the following terms that explicitly name the diatonic interval categories that the interseptimals fall between:
Both the "semi-nth" names and the Greek-derived names above are less intuitive than they could be and require some amount of memorization. For this reason, Inthar has proposed the following terms that explicitly name the diatonic interval categories that the interseptimals fall between:
Line 50: Line 59:
* semitenth/cocytic = fifth-inter-sixth (5×6)
* semitenth/cocytic = fifth-inter-sixth (5×6)
* semitwelfth/ouranic = sixth-inter-seventh (6×7)
* semitwelfth/ouranic = sixth-inter-seventh (6×7)
These names and this notation are inspired by analogous names for [[interordinal]]s in other mosses, but they refer to interval regions rather than exact intervals implied by a concrete mos tuning.
 
=== "Plus" names ===
To combine intuitiveness with conciseness, Kite Giedraitis has proposed using "plus" to indicate interordinals.
 
* semifourth = plus-second (+2nd or +2)
* semisixth = plus-third (+3rd or +3)
* semitenth = plus-fifth (+5th or +5)
* semitwelfth = plus-sixth (+6th or +6)
 
See [[User:TallKite/Midpoints]] (work in progress).
 
=== Decimal ordinal names ===
CompactStar has proposed names using decimal ordinals to indicate how these fall between diatonic categories:
* semifourth/chthonic = 2.5th
* semisixth/naiadic = 3.5th
* semitenth/cocytic = 5.5th
* semitwelfth/ouranic = 6.5th


=== Within a pentatonic framework ===
=== Within a pentatonic framework ===
A pentatonic framework, as elucidated in Kite Giedraitis's [http://www.tallkite.com/AlternativeTunings.html Alternative Tuning guide], is far more amenable to interseptimal intervals than the traditional Western heptatonic framework.
A pentatonic framework, as elucidated in Kite Giedraitis's [http://www.tallkite.com/AlternativeTunings.html Alternative Tuning guide], is far more amenable to interseptimal intervals than the traditional Western heptatonic framework. Such a framework is also discussed on the page [[Pentatonic Functional Just System]].
 
{| class="wikitable"
{| class="wikitable"
|+The pentatonic framework
|+ style="font-size: 105%;" | The pentatonic framework
! colspan="2" |names
|-
!quality
! colspan="2" | Names
!boundaries
! Quality
! colspan="2" |heptatonic equivalent
! Boundaries
! colspan="2" | Heptatonic equivalent
|-
|-
| rowspan="3" |1sn
| rowspan="3" | 1sn
| rowspan="3" |unison
| rowspan="3" | Unison
|perfect
| Perfect
|1/1 to 64/63
| 1/1 to 64/63
|perfect
| Perfect
|1sn
| 1sn
|-
|-
|half-augmented
| Half-augmented
|(interseptimal)
| (Interseptimal)
! colspan="2" |
! colspan="2" |  
|-
|-
|augmented
| Augmented
|28/27 to 16/15
| 28/27 to 16/15
|minor
| Minor
| rowspan="3" |2nd
| rowspan="3" | 2nd
|-
|-
! colspan="3" |
! colspan="3" |  
|(interpental)
| (Interpental)
|neutral
| Neutral
|-
|-
| rowspan="3" |penta-2nd
| rowspan="3" | Penta-2nd
| rowspan="3" |subthird
| rowspan="3" | Subthird
|minor
| Minor
|10/9 to 8/7
| 10/9 to 8/7
|major
| Major
|-
|-
|neutral
| Neutral
|(interseptimal)
| (Interseptimal)
! colspan="2" |
! colspan="2" |  
|-
|-
|major
| Major
|7/6 to 6/5
| 7/6 to 6/5
|minor
| Minor
| rowspan="3" |3rd
| rowspan="3" | 3rd
|-
|-
! colspan="3" |
! colspan="3" |  
|(interpental)
| (Interpental)
|neutral
| Neutral
|-
|-
| rowspan="5" |penta-3rd
| rowspan="5" | Penta-3rd
| rowspan="5" |fourthoid
| rowspan="5" | Fourthoid
|diminished
| Diminished
|5/4 to 9/7
| 5/4 to 9/7
|major
| Major
|-
|-
|half-diminished
| Half-diminished
|(interseptimal)
| (Interseptimal)
! colspan="2" |
! colspan="2" |  
|-
|-
|perfect
| Perfect
|21/16 to 27/20
| 21/16 to 27/20
|perfect
| perfect
| rowspan="3" |4th
| rowspan="3" | 4th
|-
|-
|half-augmented
| Half-augmented
|(interpental)
| (Interpental)
|half-augmented
| Half-augmented
|-
|-
|augmented
| Augmented
| rowspan="2" |7/5 to 10/7
| rowspan="2" | 7/5 to 10/7
|augmented
| Augmented
|-
|-
| rowspan="5" |penta-4th
| rowspan="5" | Penta-4th
| rowspan="5" |fifthoid
| rowspan="5" | Fifthoid
|diminished
| Diminished
|diminished
| Diminished
| rowspan="3" |5th
| rowspan="3" | 5th
|-
|-
|half-diminished
| Half-diminished
|(interpental)
| (Interpental)
|half-diminished
| Half-diminished
|-
|-
|perfect
| Perfect
|40/27 to 32/21
| 40/27 to 32/21
|perfect
| Perfect
|-
|-
|half-augmented
| Half-augmented
|(interseptimal)
| (Interseptimal)
! colspan="2" |
! colspan="2" |  
|-
|-
|augmented
| Augmented
|14/9 to 8/5
| 14/9 to 8/5
|minor
| Minor
| rowspan="3" |6th
| rowspan="3" | 6th
|-
|-
! colspan="3" |
! colspan="3" |  
|(interpental)
| (Interpental)
|neutral
| Neutral
|-
|-
| rowspan="3" |penta-5th
| rowspan="3" | Penta-5th
| rowspan="3" |subseventh
| rowspan="3" | Subseventh
|minor
| Minor
|5/3 to 12/7
| 5/3 to 12/7
|major
| Major
|-
|-
|neutral
| Neutral
|(interseptimal)
| (Interseptimal)
! colspan="2" |
! colspan="2" |  
|-
|-
|major
| Major
|7/4 to 9/5
| 7/4 to 9/5
|minor
| Minor
| rowspan="3" |7th
| rowspan="3" | 7th
|-
|-
! colspan="3" |
! colspan="3" |  
|(interpental)
| (Interpental)
|neutral
| Neutral
|-
|-
| rowspan="3" |hexave
| rowspan="3" | Hexave
| rowspan="3" |octoid
| rowspan="3" | Octoid
|diminished
| Diminished
|15/8 to 27/14
| 15/8 to 27/14
|major
| Major
|-
|-
|half-diminished
| Half-diminished
|(interseptimal)
| (Interseptimal)
! colspan="2" |
! colspan="2" |  
|-
|-
|perfect
| Perfect
|63/32 to 2/1
| 63/32 to 2/1
|perfect
| Perfect
|8ve
| 8ve
|}
|}
Note the two additional interseptimal regions. The boundary ratios are mostly either 81/80 or 64/63 away from a 3-limit interval. The exceptions are 7/5 and 10/7, which are only a [[5120/5103|Saruyo]] comma away from the 3-limit diminished 5th and augmented 4th respectively.  
Note the two additional interseptimal regions. The boundary ratios are mostly either 81/80 or 64/63 away from a 3-limit interval. The exceptions are 7/5 and 10/7, which are only a [[5120/5103|Saruyo]] comma away from the 3-limit diminished 5th and augmented 4th respectively.  


Interseptimal intervals are now easily named. However there are now hard-to-name "interpental" intervals which would be neutral intervals in the heptatonic framework, containing such ratios as 12/11, 11/9, etc. This is because interseptimal intervals are the neutral intervals with respect to the parent [[mos]] [[2L 3s]] of the diatonic mos [[5L 2s]]. See [[Neutral and interordinal k-mossteps]] for a partial generalization of this behavior to other mosses.
Interseptimal intervals are now easily named. However there are now hard-to-name "interpental" intervals which would be neutral intervals in the heptatonic framework, containing such ratios as 12/11, 11/9, etc. This is because interseptimal intervals are the neutral intervals with respect to the parent [[mos]] [[2L&nbsp;3s]] of the diatonic mos [[5L&nbsp;2s]].


So composing in a pentatonic framework may allow interseptimal intervals to play much more pivotal roles than usual.
Thus composing in a pentatonic framework may allow interseptimal intervals to play much more pivotal roles than usual.


== Examples ==
== Examples ==
Some interseptimal intervals in all four ranges, both just and tempered, are listed below.
Some interseptimal intervals in all four ranges, both just and tempered, are listed below.


=== Maj2–min3 – 240-260¢ ===
=== Maj2–min3 (semifourth/chthonic) ===
 
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
! Interval
! Interval
! Cents Value
! Size<br />(cents)
! Prime Limit (if applicable)
! Prime limit<br />(if applicable)
|-
|-
| [[147/128]]
| [[147/128]]
Line 206: Line 234:
| 1\[[5edo|5]]
| 1\[[5edo|5]]
| 240.000
| 240.000
| -
| —
|-
|-
| 54/47
| 54/47
Line 234: Line 262:
| 6\[[29edo|29]]
| 6\[[29edo|29]]
| 248.276
| 248.276
| -
| —
|-
|-
| 5\[[24edo|24]]
| 5\[[24edo|24]]
| 250.000
| 250.000
| -
| —
|-
|-
| [[52/45]]
| [[52/45]]
Line 254: Line 282:
| 4\[[19edo|19]]
| 4\[[19edo|19]]
| 252.632
| 252.632
| -
| —
|-
|-
| [[22/19]]
| [[22/19]]
Line 266: Line 294:
| 3\[[14edo|14]]
| 3\[[14edo|14]]
| 257.143
| 257.143
| -
| —
|-
|-
| 297/256
| 297/256
Line 278: Line 306:
| 5\[[23edo|23]]
| 5\[[23edo|23]]
| 260.870
| 260.870
| -
| —
|}
|}


=== Maj3–4 – 440-468¢ ===
=== Maj3–4 (semisixth/naiadic) ===
 
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
! Interval
! Interval
! Cents Value
! Size<br />(cents)
! Prime Limit (if applicable)
! Prime limit<br />(if applicable)
|-
|-
| 5\[[88cET]] or 11\[[30edo|30]]
| 5\[[88cET]] or 11\[[30edo|30]]
| 440.000
| 440.000
| -
| —
|-
|-
| [[40/31]]
| [[40/31]]
Line 298: Line 326:
| 7\[[19edo|19]]
| 7\[[19edo|19]]
| 442.015
| 442.015
| -
| —
|-
|-
| [[31/24]]
| [[31/24]]
Line 306: Line 334:
| 10\[[27edo|27]]
| 10\[[27edo|27]]
| 444.444
| 444.444
| -
| —
|-
|-
| [[22/17]]
| [[22/17]]
Line 318: Line 346:
| 3\[[8edo|8]]
| 3\[[8edo|8]]
| 450.000
| 450.000
| -
| —
|-
|-
| 48/37
| 48/37
Line 330: Line 358:
| 11\[[29edo|29]]
| 11\[[29edo|29]]
| 455.172
| 455.172
| -
| —
|-
|-
| [[125/96]]
| [[125/96]]
Line 338: Line 366:
| 8\[[21edo|21]]
| 8\[[21edo|21]]
| 457.143
| 457.143
| -
| —
|-
|-
| 56/43
| 56/43
Line 354: Line 382:
| 5\[[13edo|13]]
| 5\[[13edo|13]]
| 461.538
| 461.538
| -
| —
|-
|-
| 47/36
| 47/36
Line 374: Line 402:
| 12\[[31edo|31]]
| 12\[[31edo|31]]
| 464.516
| 464.516
| -
| —
|-
|-
| 7\[[18edo|18]]
| 7\[[18edo|18]]
| 466.667
| 466.667
| -
| —
|-
|-
| [[38/29]]
| [[38/29]]
Line 385: Line 413:
|}
|}


=== 5–min6 – 732-760¢ ===
=== 5–min6 (semitenth/cocytic) ===
 
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
! Interval
! Interval
! Cents Value
! Size<br />(cents)
! Prime Limit (if applicable)
! Prime limit<br />(if applicable)
|-
|-
| 5\[[Bohlen-Pierce]]
| 5\[[13edt]]
| 731.521
| 731.521
| -
| —
|-
|-
| [[29/19]]
| [[29/19]]
Line 402: Line 430:
| 11\[[18edo|18]]
| 11\[[18edo|18]]
| 733.333
| 733.333
| -
| —
|-
|-
| 19\[[31edo|31]]
| 19\[[31edo|31]]
| 735.484
| 735.484
| -
| —
|-
|-
| [[26/17]]
| [[26/17]]
Line 438: Line 466:
| 13\[[21edo|21]]
| 13\[[21edo|21]]
| 742.857
| 742.857
| -
| —
|-
|-
| [[182/125]]
| [[182/125]]
Line 446: Line 474:
| 18\[[29edo|29]]
| 18\[[29edo|29]]
| 744.828
| 744.828
| -
| —
|-
|-
| [[20/13]]
| [[20/13]]
Line 458: Line 486:
| 5\[[8edo|8]]
| 5\[[8edo|8]]
| 750.000
| 750.000
| -
| —
|-
|-
| [[54/35]]
| [[54/35]]
Line 470: Line 498:
| 17\[[27edo|27]]
| 17\[[27edo|27]]
| 755.556
| 755.556
| -
| —
|-
|-
| [[48/31]]
| [[48/31]]
Line 478: Line 506:
| 12\[[19edo|19]]
| 12\[[19edo|19]]
| 757.895
| 757.895
| -
| —
|-
|-
| [[31/20]]
| [[31/20]]
Line 486: Line 514:
| 19\[[30edo|30]]
| 19\[[30edo|30]]
| 760.000
| 760.000
| -
| —
|}
|}


=== Maj6–min7 – 940-960¢ ===
=== Maj6–min7 (semitwelfth/ouranic) ===
 
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
! Interval
! Interval
! Cents Value
! Size<br />(cents)
! Prime Limit (if applicable)
! Prime limit<br />(if applicable)
|-
|-
| 18\[[23edo|23]]
| 18\[[23edo|23]]
| 939.130
| 939.130
| -
| —
|-
|-
| [[31/18]]
| [[31/18]]
Line 510: Line 538:
| 11\[[14edo|14]]
| 11\[[14edo|14]]
| 942.857
| 942.857
| -
| —
|-
|-
| [[50/29]]
| [[50/29]]
Line 526: Line 554:
| 15\[[19edo|19]]
| 15\[[19edo|19]]
| 947.368
| 947.368
| -
| —
|-
|-
| 64/37
| 64/37
Line 538: Line 566:
| 19\[[24edo|24]]
| 19\[[24edo|24]]
| 950.000
| 950.000
| -
| —
|-
|-
| 23\[[29edo|29]]
| 23\[[29edo|29]]
| 951.724
| 951.724
| -
| —
|-
|-
| [[26/15]]
| [[26/15]]
Line 570: Line 598:
| 4\[[5edo|5]]
| 4\[[5edo|5]]
| 960.000
| 960.000
| -
| —
|-
|-
| 256/147
| 256/147
Line 579: Line 607:
== See also ==
== See also ==
* [[Gentle region]]
* [[Gentle region]]
* [[Equable heptatonic]]
* [[Gallery of just intervals]]
* [[Gallery of just intervals]]


== Notes ==
== Notes ==
<references group="note" />
{{Navbox intervals}}


[[Category:Interseptimal intervals| ]]
[[Category:Interseptimal intervals| ]]
<!-- main article -->
<!-- main article -->
[[Category:Interval category]]
[[Category:Intervals]]
[[Category:Interval naming]]

Latest revision as of 21:55, 11 March 2026

In the theory of Margo Schulter, an interseptimal interval is an interval that belongs in one of four interval regions which are intermediate between two septimal ratios such as 8/7 and 7/6, or 12/7 and 7/4. There are four interseptimal regions given below, with approximate cents ranges from Schulter's essay Regions of the Interval Spectrum:

  • Maj2–min3 – intermediate between 8/7 and 7/6 – 240–260 ¢
  • Maj3–4 – intermediate between 9/7 and 21/16 – 440–468 ¢
  • 5–min6 – intermediate between 32/21 and 14/9 – 732–760 ¢
  • Maj6–min7 – intermediate between 12/7 and 7/4 – 940–960 ¢

Additionally, there are also these 2 interseptimal regions near the unison and octave:

  • 1–min2 – intermediate between 64/63 and 28/27 – 40–60 ¢
  • Maj7-8 – intermediate between 27/14 and 63/32 – 1140–1160 ¢

Interseptimal intervals are well-represented in 24edo at 250 ¢, 450 ¢, 750 ¢, and 950 ¢. They also appear in 19edo and 29edo. As they fall in ambiguous zones between both diatonic and chromatic categories, they are inevitably xenharmonic.

A JI-agnostic synonym is interordinal; here, ordinal refers to the interval classes of the diatonic scale the interordinal intervals lie between, conventionally denoted with ordinal numbers.

See Neutral and interordinal k-mossteps for a partial generalization of interseptimal categories to other mosses.

Categorical and notational approaches

While interseptimals are interesting for falling right in between the typical western interval categories, this also makes them difficult to name and notate: do we classify a 250-cent interval as a second, a third, both, or neither?

Singular, unique names

One option is to give each region a distinct name (analogous to using the word tritone rather than diminished fifth or augmented fourth). Possible names that could be used are:

  • 240¢–260¢ – semifourth – an interval of this size is around half the size of a perfect fourth.
    • The term chthonic (from khthon, an ancient Greek word referring to spirits of the underworld) refers to the 240–260 ¢ region by Zhea Erose.[note 1]
  • 440¢–468¢ – semisixth – an interval of this size is around half the size of a major sixth.
    • The term naiadic (from naiad, a kind of ancient Greek water spirit) refers to the 440–464 ¢ region by Zhea Erose, who uses it frequently.
  • 732¢–760¢ – semitenth – an interval of this size is around half the size of a minor tenth (i. e., an octave plus a minor third). Another possible name is sesquifourth (since this is also about one and a half times the size of a perfect fourth).
    • The term cocytic was proposed by Inthar, by analogy with naiadic.
  • 940¢–960¢ – semitwelfth – an interval of this size is around half the size of a perfect twelfth (i.e. a compound perfect fifth, or tritave). All even edts have a semitwelfth of approximately 951 ¢, analogous to the 600 ¢ tritone shared by all even edos.
    • The term ouranic (by analogy with chthonic, and to match with the other terms) is proposed by Kaiveran.

One might want to use a mixture of above terms. Flora Canou criticizes semisixth and semitenth as they fail to make clear whether the interval to be split is major or minor, and prefers naiadic and cocytic. However, semifourth and semitwelfth are clear enough, so the Greek terms seems practically redundant.

The terminology makes notating these intervals very easy as long as we have an agreed-upon symbol for "semi". By analogy with the "semi" names, the tritone could also be called a semioctave, although the term tritone is so well-established (and so well represented by an unsplit 3-limit) that there seems little reason to change it now. A key difference is that the tritone is intermediate between two septimal ratios separated by a jubilisma (50/49), whereas the other interseptimal ranges listed above are between two septimal ratios separated by a slendro diesis (49/48).

Dual "semichromatic" names

Since interseptimal intervals are typically well represented by any EDO or pergen that divides its approximate 3/1 into 2n parts, another option is to repurpose quartertone accidentals to represent them, which is easy as long as we define our "half-sharps" or "half-flats" to be precisely half of a chromatic semitone. With this in mind, we get the following twinned identities for our interseptimals, with the simplest ones (assuming a half-fifth genchain) listed first:

  • semifourth/chthonic = semi-augmented second (+11/2), semi-diminished third (−13/2)
  • semisixth/naiadic = semi-diminished fourth (−9/2), semi-augmented third (+15/2)
  • semitenth/cocytic = semi-augmented fifth (+9/2), semi-diminished sixth (−15/2)
  • semitwelfth/ouranic = semi-diminished seventh (−11/2), semi-augmented sixth (+13/2)

While this does not give the interseptimals a single distinct notational name, it does reflect their ambiguity and flexibility with regards to the surrounding interval categories that many are so fond of. Furthermore, as both identities are exactly 12 notational fifths apart (i.e a direct analogue of the Pythagorean comma), composers can use a mechanism similar to the "po and qu" of Color Notation, or the plus and minus accidentals (+/−) proposed in Rational Comma Notation, to freely switch between the two identities.

Alternatively, one can use the ultra- prefix for sharpening by ~50¢ and infra- for flattening by ~50 ¢, analogous to super- and sub- for modifications by ~30 ¢.

  • semifourth/chthonic = ultramajor second, inframinor third
  • semisixth/naiadic = ultramajor third, infrafourth
  • semitenth/cocytic = ultrafifth, inframinor sixth
  • semitwelfth/ouranic = ultramajor sixth, inframinor seventh

Ultra- and infra- also work for intervals that are very close to 11/8 or 16/11:

  • ~11/8 or ~550 ¢ = ultrafourth, infratritone, infrasemioctave
  • ~16/11 or ~650 ¢ = infrafifth, ultratritone, ultrasemioctave

"Inter" names

Both the "semi-nth" names and the Greek-derived names above are less intuitive than they could be and require some amount of memorization. For this reason, Inthar has proposed the following terms that explicitly name the diatonic interval categories that the interseptimals fall between:

  • semifourth/chthonic = second-inter-third (2×3)
  • semisixth/naiadic = third-inter-fourth (3×4)
  • semitenth/cocytic = fifth-inter-sixth (5×6)
  • semitwelfth/ouranic = sixth-inter-seventh (6×7)

"Plus" names

To combine intuitiveness with conciseness, Kite Giedraitis has proposed using "plus" to indicate interordinals.

  • semifourth = plus-second (+2nd or +2)
  • semisixth = plus-third (+3rd or +3)
  • semitenth = plus-fifth (+5th or +5)
  • semitwelfth = plus-sixth (+6th or +6)

See User:TallKite/Midpoints (work in progress).

Decimal ordinal names

CompactStar has proposed names using decimal ordinals to indicate how these fall between diatonic categories:

  • semifourth/chthonic = 2.5th
  • semisixth/naiadic = 3.5th
  • semitenth/cocytic = 5.5th
  • semitwelfth/ouranic = 6.5th

Within a pentatonic framework

A pentatonic framework, as elucidated in Kite Giedraitis's Alternative Tuning guide, is far more amenable to interseptimal intervals than the traditional Western heptatonic framework. Such a framework is also discussed on the page Pentatonic Functional Just System.

The pentatonic framework
Names Quality Boundaries Heptatonic equivalent
1sn Unison Perfect 1/1 to 64/63 Perfect 1sn
Half-augmented (Interseptimal)
Augmented 28/27 to 16/15 Minor 2nd
(Interpental) Neutral
Penta-2nd Subthird Minor 10/9 to 8/7 Major
Neutral (Interseptimal)
Major 7/6 to 6/5 Minor 3rd
(Interpental) Neutral
Penta-3rd Fourthoid Diminished 5/4 to 9/7 Major
Half-diminished (Interseptimal)
Perfect 21/16 to 27/20 perfect 4th
Half-augmented (Interpental) Half-augmented
Augmented 7/5 to 10/7 Augmented
Penta-4th Fifthoid Diminished Diminished 5th
Half-diminished (Interpental) Half-diminished
Perfect 40/27 to 32/21 Perfect
Half-augmented (Interseptimal)
Augmented 14/9 to 8/5 Minor 6th
(Interpental) Neutral
Penta-5th Subseventh Minor 5/3 to 12/7 Major
Neutral (Interseptimal)
Major 7/4 to 9/5 Minor 7th
(Interpental) Neutral
Hexave Octoid Diminished 15/8 to 27/14 Major
Half-diminished (Interseptimal)
Perfect 63/32 to 2/1 Perfect 8ve

Note the two additional interseptimal regions. The boundary ratios are mostly either 81/80 or 64/63 away from a 3-limit interval. The exceptions are 7/5 and 10/7, which are only a Saruyo comma away from the 3-limit diminished 5th and augmented 4th respectively.

Interseptimal intervals are now easily named. However there are now hard-to-name "interpental" intervals which would be neutral intervals in the heptatonic framework, containing such ratios as 12/11, 11/9, etc. This is because interseptimal intervals are the neutral intervals with respect to the parent mos 2L 3s of the diatonic mos 5L 2s.

Thus composing in a pentatonic framework may allow interseptimal intervals to play much more pivotal roles than usual.

Examples

Some interseptimal intervals in all four ranges, both just and tempered, are listed below.

Maj2–min3 (semifourth/chthonic)

Interval Size
(cents) 		Prime limit
(if applicable)
147/128 239.607 7
1\5 240.000 —
54/47 240.358 47
23/20 241.961 23
1152/1001 243.238 13
38/33 244.240 19
144/125 244.969 5
15/13 247.741 13
6\29 248.276 —
5\24 250.000 —
52/45 250.304 13
37/32 251.344 37
81/70 252.680 7
4\19 252.632 —
22/19 253.805 19
29/25 256.950 29
3\14 257.143 —
297/256 257.183 11
36/31 258.874 31
5\23 260.870 —

Maj3–4 (semisixth/naiadic)

Interval Size
(cents) 		Prime limit
(if applicable)
5\88cET or 11\30 440.000 —
40/31 441.278 31
7\19 442.015 —
31/24 443.081 31
10\27 444.444 —
22/17 446.363 17
35/27 449.275 7
3\8 450.000 —
48/37 450.611 37
13/10 454.214 13
11\29 455.172 —
125/96 456.986 5
8\21 457.143 —
56/43 457.308 43
43/33 458.245 43
30/23 459.994 23
5\13 461.538 —
47/36 461.597 47
64/49 462.348 7
98/75 463.069 7
17/13 464.428 17
12\31 464.516 —
7\18 466.667 —
38/29 467.936 29

5–min6 (semitenth/cocytic)

Interval Size
(cents) 		Prime limit
(if applicable)
5\13edt 731.521 —
29/19 732.064 29
11\18 733.333 —
19\31 735.484 —
26/17 735.572 17
49/75 736.931 7
49/32 737.652 7
72/47 738.403 47
23/15 740.006 23
66/43 741.755 43
43/28 742.692 43
13\21 742.857 —
182/125 743.014 5
18\29 744.828 —
20/13 745.786 13
37/24 749.389 37
5\8 750.000 —
54/35 750.725 7
17/11 753.637 17
17\27 755.556 —
48/31 756.919 31
12\19 757.895 —
31/20 758.722 31
19\30 760.000 —

Maj6–min7 (semitwelfth/ouranic)

Interval Size
(cents) 		Prime limit
(if applicable)
18\23 939.130 —
31/18 941.126 31
512/297 942.817 11
11\14 942.857 —
50/29 943.050 29
19/11 946.195 19
140/81 947.320 7
15\19 947.368 —
64/37 948.656 37
45/26 949.696 13
19\24 950.000 —
23\29 951.724 —
26/15 952.259 13
125/72 955.031 5
33/19 955.760 19
1001/576 956.762 13
40/23 958.039 23
47/27 959.642 47
4\5 960.000 —
256/147 960.393 7

See also

Notes

  1. ↑ As per Primodal Archive.


View • Talk • EditInterval classification
Interval regions
Unison and octave Unison • Comma and diesis • Octave
Seconds Minor second • Neutral second • Major second
Thirds Minor third • Neutral third • Major third
Fourths and fifths Perfect fourth • Superfourth • Tritone • Subfifth • Perfect fifth
Sixths Minor sixth • Neutral sixth • Major sixth
Sevenths Minor seventh • Neutral seventh • Major seventh
Interseptimal intervals Interseptimal 2nd-3rd • Interseptimal 3rd-4th • Interseptimal 5th-6th • Interseptimal 6th-7th
Interval qualities
Diatonic qualities Diminished • Minor • Perfect • Major • Augmented
Tuning ranges Neutral (interval quality) • Submajor and supraminor • Pental major and minor • Novamajor and novaminor • Neogothic major and minor • Supermajor and subminor • Ultramajor and inframinor
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