Interseptimal interval: Difference between revisions

(6 intermediate revisions by 3 users not shown)

Line 1:

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}}

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

* 1–min2 – intermediate between [[64/63]] and [[28/27]] – ~~40¢–60¢~~

* 1–min2 – intermediate between [[64/63]] and [[28/27]] – 40–60{{c}}

* Maj7-8 – intermediate between [[27/14]] and [[63/32]] – ~~1140¢–1160¢~~

* Maj7-8 – intermediate between [[27/14]] and [[63/32]] – 1140–1160{{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 both [[5L 2s|diatonic]] and [[chromatic]] categories, they are inevitably xenharmonic.

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.

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.

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:

* 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.

** 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).

** 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]].

Line 37:

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

* 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

* ~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 ===

Line 59:

* 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 ===

Line 68:

Line 78:

=== 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"

Line 204:

Line 214:

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]].

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]].

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 ==

Line 215:

Line 225:

|-

! Interval

! ~~Cents Value~~

! Size (cents)

! Prime ~~Limit~~ (if applicable)

! Prime limit (if applicable)

|-

| [[147/128]]

Line 224:

Line 234:

| 1\[[5edo|5]]

| 240.000

| ~~—~~

| —

|-

| 54/47

Line 252:

Line 262:

| 6\[[29edo|29]]

| 248.276

| ~~—~~

| —

|-

| 5\[[24edo|24]]

| 250.000

| ~~—~~

| —

|-

| [[52/45]]

Line 272:

Line 282:

| 4\[[19edo|19]]

| 252.632

| ~~—~~

| —

|-

| [[22/19]]

Line 284:

Line 294:

| 3\[[14edo|14]]

| 257.143

| ~~—~~

| —

|-

| 297/256

Line 296:

Line 306:

| 5\[[23edo|23]]

| 260.870

| ~~—~~

| —

|}

Line 303:

Line 313:

|-

! Interval

! ~~Cents Value~~

! Size (cents)

! Prime ~~Limit~~ (if applicable)

! Prime limit (if applicable)

|-

| 5\[[88cET]] or 11\[[30edo|30]]

| 440.000

| ~~—~~

| —

|-

| [[40/31]]

Line 316:

Line 326:

| 7\[[19edo|19]]

| 442.015

| ~~—~~

| —

|-

| [[31/24]]

Line 324:

Line 334:

| 10\[[27edo|27]]

| 444.444

| ~~—~~

| —

|-

| [[22/17]]

Line 336:

Line 346:

| 3\[[8edo|8]]

| 450.000

| ~~—~~

| —

|-

| 48/37

Line 348:

Line 358:

| 11\[[29edo|29]]

| 455.172

| ~~—~~

| —

|-

| [[125/96]]

Line 356:

Line 366:

| 8\[[21edo|21]]

| 457.143

| ~~—~~

| —

|-

| 56/43

Line 372:

Line 382:

| 5\[[13edo|13]]

| 461.538

| ~~—~~

| —

|-

| 47/36

Line 392:

Line 402:

| 12\[[31edo|31]]

| 464.516

| ~~—~~

| —

|-

| 7\[[18edo|18]]

| 466.667

| ~~—~~

| —

|-

| [[38/29]]

Line 407:

Line 417:

|-

! Interval

! ~~Cents Value~~

! Size (cents)

! Prime ~~Limit~~ (if applicable)

! Prime limit (if applicable)

|-

| 5\[[13edt]]

| 731.521

| ~~—~~

| —

|-

| [[29/19]]

Line 420:

Line 430:

| 11\[[18edo|18]]

| 733.333

| ~~—~~

| —

|-

| 19\[[31edo|31]]

| 735.484

| ~~—~~

| —

|-

| [[26/17]]

Line 456:

Line 466:

| 13\[[21edo|21]]

| 742.857

| ~~—~~

| —

|-

| [[182/125]]

Line 464:

Line 474:

| 18\[[29edo|29]]

| 744.828

| ~~—~~

| —

|-

| [[20/13]]

Line 476:

Line 486:

| 5\[[8edo|8]]

| 750.000

| ~~—~~

| —

|-

| [[54/35]]

Line 488:

Line 498:

| 17\[[27edo|27]]

| 755.556

| ~~—~~

| —

|-

| [[48/31]]

Line 496:

Line 506:

| 12\[[19edo|19]]

| 757.895

| ~~—~~

| —

|-

| [[31/20]]

Line 504:

Line 514:

| 19\[[30edo|30]]

| 760.000

| ~~—~~

| —

|}

Line 511:

Line 521:

|-

! Interval

! ~~Cents Value~~

! Size (cents)

! Prime ~~Limit~~ (if applicable)

! Prime limit (if applicable)

|-

| 18\[[23edo|23]]

| 939.130

| ~~—~~

| —

|-

| [[31/18]]

Line 528:

Line 538:

| 11\[[14edo|14]]

| 942.857

| ~~—~~

| —

|-

| [[50/29]]

Line 544:

Line 554:

| 15\[[19edo|19]]

| 947.368

| ~~—~~

| —

|-

| 64/37

Line 556:

Line 566:

| 19\[[24edo|24]]

| 950.000

| ~~—~~

| —

|-

| 23\[[29edo|29]]

| 951.724

| ~~—~~

| —

|-

| [[26/15]]

Line 588:

Line 598:

| 4\[[5edo|5]]

| 960.000

| ~~—~~

| —

|-

| 256/147

Line 601:

Line 611:

== Notes ==

[[Category:Interseptimal intervals| ]]

@@ Line 1: / Line 1: @@
 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}}
 Additionally, there are also these 2 interseptimal regions near the unison and octave:
-* 1–min2 – intermediate between [[64/63]] and [[28/27]] – 40¢–60¢
+* 1–min2 – intermediate between [[64/63]] and [[28/27]] – 40–60{{c}}
-* Maj7-8 – intermediate between [[27/14]] and [[63/32]] – 1140¢–1160¢
+* Maj7-8 – intermediate between [[27/14]] and [[63/32]] – 1140–1160{{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 both [[5L 2s|diatonic]] and [[chromatic]] categories, they are inevitably xenharmonic.
+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.
 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.
@@ Line 22: / 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:
 * 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.
-** 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).
 ** 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]].
@@ Line 37: / Line 37: @@
 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
 * 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
+* ~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 ===
@@ Line 59: / Line 59: @@
 * 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 ===
@@ Line 68: / Line 78: @@
 === 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"
@@ Line 204: / Line 214: @@
 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]].
+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 ==
@@ Line 215: / Line 225: @@
 |-
 ! Interval
-! Cents Value
+! Size<br />(cents)
-! Prime Limit (if applicable)
+! Prime limit<br />(if applicable)
 |-
 | [[147/128]]
@@ Line 224: / Line 234: @@
 | 1\[[5edo|5]]
 | 240.000
-| &mdash;
+| —
 |-
 | 54/47
@@ Line 252: / Line 262: @@
 | 6\[[29edo|29]]
 | 248.276
-| &mdash;
+| —
 |-
 | 5\[[24edo|24]]
 | 250.000
-| &mdash;
+| —
 |-
 | [[52/45]]
@@ Line 272: / Line 282: @@
 | 4\[[19edo|19]]
 | 252.632
-| &mdash;
+| —
 |-
 | [[22/19]]
@@ Line 284: / Line 294: @@
 | 3\[[14edo|14]]
 | 257.143
-| &mdash;
+| —
 |-
 | 297/256
@@ Line 296: / Line 306: @@
 | 5\[[23edo|23]]
 | 260.870
-| &mdash;
+| —
 |}
@@ Line 303: / Line 313: @@
 |-
 ! Interval
-! Cents Value
+! Size<br />(cents)
-! Prime Limit (if applicable)
+! Prime limit<br />(if applicable)
 |-
 | 5\[[88cET]] or 11\[[30edo|30]]
 | 440.000
-| &mdash;
+| —
 |-
 | [[40/31]]
@@ Line 316: / Line 326: @@
 | 7\[[19edo|19]]
 | 442.015
-| &mdash;
+| —
 |-
 | [[31/24]]
@@ Line 324: / Line 334: @@
 | 10\[[27edo|27]]
 | 444.444
-| &mdash;
+| —
 |-
 | [[22/17]]
@@ Line 336: / Line 346: @@
 | 3\[[8edo|8]]
 | 450.000
-| &mdash;
+| —
 |-
 | 48/37
@@ Line 348: / Line 358: @@
 | 11\[[29edo|29]]
 | 455.172
-| &mdash;
+| —
 |-
 | [[125/96]]
@@ Line 356: / Line 366: @@
 | 8\[[21edo|21]]
 | 457.143
-| &mdash;
+| —
 |-
 | 56/43
@@ Line 372: / Line 382: @@
 | 5\[[13edo|13]]
 | 461.538
-| &mdash;
+| —
 |-
 | 47/36
@@ Line 392: / Line 402: @@
 | 12\[[31edo|31]]
 | 464.516
-| &mdash;
+| —
 |-
 | 7\[[18edo|18]]
 | 466.667
-| &mdash;
+| —
 |-
 | [[38/29]]
@@ Line 407: / Line 417: @@
 |-
 ! Interval
-! Cents Value
+! Size<br />(cents)
-! Prime Limit (if applicable)
+! Prime limit<br />(if applicable)
 |-
 | 5\[[13edt]]
 | 731.521
-| &mdash;
+| —
 |-
 | [[29/19]]
@@ Line 420: / Line 430: @@
 | 11\[[18edo|18]]
 | 733.333
-| &mdash;
+| —
 |-
 | 19\[[31edo|31]]
 | 735.484
-| &mdash;
+| —
 |-
 | [[26/17]]
@@ Line 456: / Line 466: @@
 | 13\[[21edo|21]]
 | 742.857
-| &mdash;
+| —
 |-
 | [[182/125]]
@@ Line 464: / Line 474: @@
 | 18\[[29edo|29]]
 | 744.828
-| &mdash;
+| —
 |-
 | [[20/13]]
@@ Line 476: / Line 486: @@
 | 5\[[8edo|8]]
 | 750.000
-| &mdash;
+| —
 |-
 | [[54/35]]
@@ Line 488: / Line 498: @@
 | 17\[[27edo|27]]
 | 755.556
-| &mdash;
+| —
 |-
 | [[48/31]]
@@ Line 496: / Line 506: @@
 | 12\[[19edo|19]]
 | 757.895
-| &mdash;
+| —
 |-
 | [[31/20]]
@@ Line 504: / Line 514: @@
 | 19\[[30edo|30]]
 | 760.000
-| &mdash;
+| —
 |}
@@ Line 511: / Line 521: @@
 |-
 ! Interval
-! Cents Value
+! Size<br />(cents)
-! Prime Limit (if applicable)
+! Prime limit<br />(if applicable)
 |-
 | 18\[[23edo|23]]
 | 939.130
-| &mdash;
+| —
 |-
 | [[31/18]]
@@ Line 528: / Line 538: @@
 | 11\[[14edo|14]]
 | 942.857
-| &mdash;
+| —
 |-
 | [[50/29]]
@@ Line 544: / Line 554: @@
 | 15\[[19edo|19]]
 | 947.368
-| &mdash;
+| —
 |-
 | 64/37
@@ Line 556: / Line 566: @@
 | 19\[[24edo|24]]
 | 950.000
-| &mdash;
+| —
 |-
 | 23\[[29edo|29]]
 | 951.724
-| &mdash;
+| —
 |-
 | [[26/15]]
@@ Line 588: / Line 598: @@
 | 4\[[5edo|5]]
 | 960.000
-| &mdash;
+| —
 |-
 | 256/147
@@ Line 601: / Line 611: @@
 == Notes ==
+<references group="note" />
+{{Navbox intervals}}
 [[Category:Interseptimal intervals| ]]