User:Lhearne/Extra-Diatonic Intervals: Difference between revisions
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=== Common microtonal interval names === | === Common microtonal interval names === | ||
Dating back at least to 1880, after Alexander Ellis and John Land, the interval [[7/6]] has been associated with the label ''subminor third''. in a generalisation of this idea, [[9/7]] is most commonly reffered to as a ''supermajor third,'' [[12/7]] a ''supermajor sixth'', [[14/9]] a ''subminor | Dating back at least to 1880, after Alexander Ellis and John Land, the interval [[7/6]] has been associated with the label ''subminor third''. in a generalisation of this idea, [[9/7]] is most commonly reffered to as a ''supermajor third,'' [[12/7]] a ''supermajor sixth'', [[14/9]] a ''subminor sixth,<!-- plural?! | ||
Thanks, fixed -->'' [[8/7]] a ''supermajor second,'' [[7/4]] a ''subminor seventh'', [[27/14]] a ''supermajor seventh'' and [[28/27]] a ''subminor second.'' This system was further generalised by some theorists and musicians such that an interval a bit smaller than a major is referred to as a ''subminor third'', and an interval a bit larger than a minor third as a ''supraminor third''. Notice that 'supra' is used instead of 'super', but 'sub' is still used. Similarly defined are submajor and supraminor seconds, sixths and sevenths. 'Sub' and 'super' prefixes have also seen occasional application to the perfect scale degrees. Like in the case of the submajor 3rd, etc., super and sub unisons, fourths, fifths and octaves are not associated with particular frequency ratio by all or most microtonal musicians and theorists. In the case of seconds, thirds, sixths and sevenths, intervals half way between major and minor are often called ''neutral''. Finally, in limited use are 'intermediates', where an interval in-between a major 3rd and a perfect fourth, for example, is referred to as a 'third-fourth'. There are undoubtedly other interval naming practice that exists that are not as well known to the author, which are likely to be less commonly used. Although all these microtonal interval naming concepts are in common use, there is not yet a complete system that defined them, only complete systems that depart from them. | |||
=== [[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak] === | === [[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak] === | ||
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== Premise: == | == Premise: == | ||
Extra-diatonic names should be simple, generalisable, widely applicable, backwards compatible with standard diatonic notation and reflecting current informal practice as closely as possible. Extra-diatonic interval names are fifth based; extended from the familiar major, minor and perfect interval names so that diatonic interval arithmetic is conserved. ‘M’, ‘m’, and ‘P’ remain the short-hand for major, minor and perfect. ‘A’ and ‘d’ for Augmented and diminished may also be used in the familiar way. In cases where the chroma (the chromatic semitone, or augmented unison) is represented by multiple steps in the tuning the prefix ‘super’ raises major and perfect intervals by a single step while ‘sub’ lowers minor and perfect intervals, with short-hand ‘S’ and ‘s’. ‘S’ and ‘s’ may also be used to raise minor and lower major intervals respectively, reflecting occasion practice. In this case ‘S’ is short-hand for ‘supra’, and 's' for 'small'. They may also be used to raise or lower diminished and augmented intervals. In this way this scheme is equivalent thus far to Ups and Downs notation, where ‘^’ or ‘up’ corresponds to ‘S’, ‘super’ or ‘supra’ and ‘v’ or ‘down’ to ‘sub’ or 'small' . | Extra-diatonic names should be simple, generalisable, widely applicable, backwards compatible with standard diatonic notation and reflecting current informal practice as closely as possible. Extra-diatonic interval names are fifth based; extended from the familiar major, minor and perfect interval names so that diatonic interval arithmetic is conserved. ‘M’, ‘m’, and ‘P’ remain the short-hand for major, minor and perfect. ‘A’ and ‘d’ for Augmented and diminished may also be used in the familiar way. In cases where the chroma (the chromatic semitone, or augmented unison) is represented by multiple steps in the tuning the prefix ‘super’ raises major and perfect intervals by a single step while ‘sub’ lowers minor and perfect intervals, with short-hand ‘S’ and ‘s’. ‘S’ and ‘s’ may also be used to raise minor and lower major intervals respectively, reflecting occasion practice. In this case ‘S’ is short-hand for ‘supra’, and 's' for 'small'. They may also be used to raise or lower diminished and augmented intervals similarly. In this way this scheme is equivalent thus far to Ups and Downs notation, where ‘^’ or ‘up’ corresponds to ‘S’, ‘super’ or ‘supra’ and ‘v’ or ‘down’ to ‘sub’ or 'small' . | ||
== Additions and examples: == | == Additions and examples: == | ||
''Neutrals'' and ''intermediates'' are also included, where neutrals occur between opposing sizes of a single generic interval the intermediates between each generic interval and the next. | ''Neutrals'' and ''intermediates'' are also included, where neutrals occur between opposing sizes of a single generic interval the intermediates between each generic interval and the next. | ||
Interval names for equal tunings are ranked in five tiers. | Interval names for equal tunings are ranked in five tiers. | ||
Interval names for equal tunings are ranked in eight tiers. | |||
# Perfect | |||
# Neutral | |||
# Major, minor, A4 and d5 and, if the chroma is subtended by a single (positive) step of the edo, other augmented and diminished intervals | |||
# Super, sub, supra and small prefixes to major, minor, perfect interval and to A4 and d5 | |||
# Intermediates | |||
# Augmented and diminished intervals (for when the chroma is subtended by more than a single (positive) step of the edo) | |||
# Super, sub, supra and small prefixes to augmented and diminished intervals | |||
# Intervals augmented and diminished more than singularly | |||
When more than one interval name corresponds to a specific interval, the names are privileged in order of the tiers. By this ordering, the first available name is the ‘primary’ for that interval, the second available ‘secondary’ and third 'tertiary'. | |||
I initially intended for Intermediates to lay in the second tier with neutrals, but that led to primary intervals results that were not preferred by my colleagues over other options and I was persuaded to avoid using intermediates unless they are necessary. | |||
=== Neutrals === | === Neutrals === | ||
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The same names give the primary interval names for [[7edo]], whose secondary intervals names are: | The same names give the primary interval names for [[7edo]], whose secondary intervals names are: | ||
N1 m2/M2 m3/M3 | N1 m2/M2 m3/M3 N4 N5 m6/M6 m7/M7 N8. | ||
The secondary interval names show that the chroma is equivalent to a unison in 7edo. | The secondary interval names show that the chroma is equivalent to a unison in 7edo. | ||
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=== Intermediates === | === Intermediates === | ||
To provide native support for | To provide native support for Barbados temperament, intermediates are also added to the system. It should be noted immediately that intermediates are not as common to common microtonal interval naming as neutrals and though are a useful addition to this scheme, may be left out if desired. The appendix includes the MOS scales and edos from 'lists of all edos and MOS Scales', but without any intermediates. | ||
‘2-3’ lies exactly half-way between M2 and m3 and divides the P4 in half. It may be read ‘second-third’ or ‘serd’. ‘6-7’, it’s octave-inverse lies exactly half-way between M6 and m7 and may be read | ‘2-3’ lies exactly half-way between M2 and m3 and divides the P4 in half. It may be read ‘second-third’ or ‘serd’. ‘6-7’, it’s octave-inverse lies exactly half-way between M6 and m7 and may be read ‘sixth<!-- plural?! | ||
Yeah I dunno why I kept doing that -->-seventh’ or ‘sinth’. | |||
‘1-2’ lies exactly half-way between P1 and m2, dividing the m2 in half. It may be read ‘unison-second’ or ‘unicond’. Its octave-inverse, ‘7-8’, lies exactly half-way between M7 and P8 and may be read ‘seventh-octave’ or ‘sevtave’. | ‘1-2’ lies exactly half-way between P1 and m2, dividing the m2 in half. It may be read ‘unison-second’ or ‘unicond’. Its octave-inverse, ‘7-8’, lies exactly half-way between M7 and P8 and may be read ‘seventh-octave’ or ‘sevtave’. | ||
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Gareth: Hi Piotr Cool use of comments for 'soft edits'. Good place to talk here about it. My use of 1-2 is more aligned with 2-3 and 3-4 (common uses) than your is, where it splits the limma, the Pythagorean diatonic semitone. Half of the whole-tone, as you are using it isn't really ambiguously a unison or a second or both, it's very close to a minor second, and in 19edo, equivalent to it. | Gareth: Hi Piotr Cool use of comments for 'soft edits'. Good place to talk here about it. My use of 1-2 is more aligned with 2-3 and 3-4 (common uses) than your is, where it splits the limma, the Pythagorean diatonic semitone. Half of the whole-tone, as you are using it isn't really ambiguously a unison or a second or both, it's very close to a minor second, and in 19edo, equivalent to it. | ||
Piotr: The difference is that my systems include the augmented unison, while yours don't. With the 100 cents being both in the 12edo, 24edo and 36edo meantones, I settled on unison–second as the name for 4\50, which is 96 cents. 3\50 is augmented unison, and 5\50 is diminished second. The unison–second and fourth–fifth could be perceived as splits of the diesis in half. They're in the middle of semitones and tritones respectively. Many systems seem to abuse (no offense) the name "minor second" for any semitone, when in fact minor second and augmented unison are separate intervals in the circle of fifths. An octave is made of 7 minor seconds and 5 augmented unisons. While it can be said that three octaves is 8 major thirds and 4 diminished fourths, I excluded the diminished fourth from my notation because it's considered wolf in 5–limit meantone and 9/7 in septimal meantone, which is a supermajor third. --> | Piotr: The difference is that my systems include the augmented unison, while yours don't. With the 100 cents being both in the 12edo, 24edo and 36edo meantones, I settled on unison–second as the name for 4\50, which is 96 cents. 3\50 is augmented unison, and 5\50 is diminished second. The unison–second and fourth–fifth could be perceived as splits of the diesis in half. They're in the middle of semitones and tritones respectively. Many systems seem to abuse (no offense) the name "minor second" for any semitone, when in fact minor second and augmented unison are separate intervals in the circle of fifths. An octave is made of 7 minor seconds and 5 augmented unisons. While it can be said that three octaves is 8 major thirds and 4 diminished fourths, I excluded the diminished fourth from my notation because it's considered wolf in 5–limit meantone and 9/7 in septimal meantone, which is a supermajor third. | ||
Gareth: My system does include the Augmented unison in my system, it just doesn't show up much because in most edos that number of steps has other possible names which are prioritized. It still does come up a few times in this article. I hope you weren't implying that the scheme I am proposing doesn't observe the difference between a minor second and an augmented unison, although if you are perhaps just know that it does. I see now you have chosen to label the interval splitting the Augmented unison and minor second as the first-second. I suppose that's an equally valid choice to the one I made. --> | |||
‘3-4’ lies exactly half-way between M3 and P4, dividing the M6 in half. It may be read ‘third-fourth’ or ‘thourth’. It’s octave-inverse, ‘5-6’, lies exactly half-way between P5 and m6 and may be read ‘fifth-sixth’ or ‘fixth’. <!-- plural?! | ‘3-4’ lies exactly half-way between M3 and P4, dividing the M6 in half. It may be read ‘third-fourth’ or ‘thourth’. It’s octave-inverse, ‘5-6’, lies exactly half-way between P5 and m6 and may be read ‘fifth-sixth’ or ‘fixth’. <!-- plural?! | ||
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1-2 2-3 3-4 5-6 6-7 7-8. | 1-2 2-3 3-4 5-6 6-7 7-8. | ||
The primary interval names for 5edo | The primary interval names for 5edo do not include them however, as they may be described by diatonic intervals: | ||
P1/ | P1 M2/m3 P4 P5 M6/m7 P8 | ||
Barbados[5] 2|2 may be described as | |||
P1 2-3 P4 P5 6-7 P8, | P1 2-3 P4 P5 6-7 P8, | ||
and | and Barbados[9] 4|4 as | ||
P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8. | P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8. | ||
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The final intermediate, ‘4-5’ lies exactly between P4 and P5 and divides the octave in half. It may be read ‘fourth-fifth’ or ‘firth’. | The final intermediate, ‘4-5’ lies exactly between P4 and P5 and divides the octave in half. It may be read ‘fourth-fifth’ or ‘firth’. | ||
The primary interval names for 12edo are | The primary interval names for 12edo are as we are familiar: | ||
P1 m2 M2 m3 M3 P4 | P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8, | ||
but the interval labeled A4/d5 has the secondary name '4-5'. | |||
=== [[10edo]], Pajara and a problem === | === [[10edo]], Pajara and a problem === | ||
10edo may be written as only neutrals and intermediates: | |||
N1/1-2 N2 2-3 N3 3-4 N4/4-5/N5 5-6 N6 6-7 N7 7-8/N8 | |||
However it much more sense to write is using the primary interval name set: | |||
P1/ | P1 N2 M2/m3 N3 P4 N4/N5 P5 N6 M6/m7 N7 P8 | ||
It's secondary intervals as follows: | |||
m2 Sm2/sM2 | m2 Sm2/sM2 2-3 Sm3/sM3 M3 S4/s5 m6 Sm6/sM6 6-7 Sm7/sM7 M7 | ||
We can see that 10edo supports [[Neutral third scales]], given that we can make the interval names for Neutral[10] using the primary | We can see that 10edo supports [[Neutral third scales]], given that we can make the interval names for Neutral[10] using the primary interval names for 10edo. | ||
We know 10edo and 12edo both support Pajara temperament. Pajara[10] 2|2 (2) consists of: | We know 10edo and 12edo both support Pajara temperament. Pajara[10] 2|2 (2) consists of: | ||
P1 Sm2 M2 sM3 P4 | P1 Sm2 M2 sM3 P4 sA4/Sd5 P5 Sm6 m7 sM7 P8, | ||
and [[Pajara12|Pajara[12]]] 3|2 (2) of | and [[Pajara12|Pajara[12]]] 3|2 (2) of | ||
P1 Sm2 M2 Sm3 sM3 P4 | P1 Sm2 M2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 m7 sM7 P8. | ||
We can see Pajara[10] in 10edo, but in 12edo, wouldn’t sM3 be m3? | We can see Pajara[10] in 10edo, but in 12edo, wouldn’t sM3 be m3? | ||
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=== [[14edo]] and Injera === | === [[14edo]] and Injera === | ||
Like 10edo, 14edo may be written using all the neutrals and intermediates, but without any intervals described both as a neutral and as an intermediate: | |||
N1 1-2 N2 2-3 N3 3-4 N4 4-5 N5 5-6 N6 6-7 N7 7-8 N8. | |||
The | The primary interval names for 14edo are as follows: | ||
P1 sm2 N2 SM2/sm3 N3 SM3/s4 P4 SA4/sd5 P5 S5/sm6 N6 SM6/sm7 N7 SM7 P1 | |||
We know that 12edo and 14edo support Injera, where Injera[12] 3|2 (2) may be labelled | We know that 12edo and 14edo support Injera, where Injera[12] 3|2 (2) may be labelled | ||
P1 sm2 M2 sm3 SM3 P4 | P1 sm2 M2 sm3 SM3 P4 SA4/sd5 P5 sm6 M6 m7 SM7 P8, | ||
and Injera[14] 3|3 (2) labelled | and Injera[14] 3|3 (2) labelled | ||
P1 sm2 M2 sm3 m3 SM3 P4 | P1 sm2 M2 sm3 m3 SM3 P4 SA4/sd5 P5 sm6 M6 SM6 m7 SM7 P8. | ||
We can see both in 14edo, and to get 12edo from Injera[12], as with Pajara, we remove all the ‘s’ and ‘S’ prefixes. | We can see both in 14edo, and to get 12edo from Injera[12], as with Pajara, we remove all the ‘s’ and ‘S’ prefixes. | ||
=== [[Blacksmith]] and further extension === | === [[Blacksmith]] and further extension === | ||
10edo also | 10edo also supports Blacksmith temperament, and we may think to write Blacksmith[10] 1|0 (5) as: | ||
P1 | P1 sM2 M2/m3 sM3 P4 s5 P5 sM6 M6/m7 sM7 P8 | ||
But we have now added mappings, but are yet to define the use of ‘S’ and ‘s’ for perfect intervals. In Blacksmith, the interval we might call ‘s5’ is 81/80 below P5, however, more commonly ‘s5’ is used to refer to [[16/11]], and S4 [[11/8]]. Since these intervals have above been labelled N4 and N5 above however, we do not need to worry about that, and can add that s5, a 'small 5th', is 81/80 below 3/2, and S4, a 'supra 4th' lies 81/80 above 4/3. where ‘s4’ has been typically been used to refer to [[21/16]], and ‘S5’ to [[32/21]], we add that s4 is lower than P4 by 64/63 and that S5 is higher than P5 by 64/63. | But we have now added mappings, but are yet to define the use of ‘S’ and ‘s’ for perfect intervals. In Blacksmith, the interval we might call ‘s5’ is 81/80 below P5, however, more commonly ‘s5’ is used to refer to [[16/11]], and S4 [[11/8]]. Since these intervals have above been labelled N4 and N5 above however, we do not need to worry about that, and can add that s5, a 'small 5th', is 81/80 below 3/2, and S4, a 'supra 4th' lies 81/80 above 4/3. where ‘s4’ has been typically been used to refer to [[21/16]], and ‘S5’ to [[32/21]], we add that s4 is lower than P4 by 64/63 and that S5 is higher than P5 by 64/63. | ||
Blacksmith[15] 1|1 (5) | The primary intervals names of Blacksmith[15] 1|1 (5) then are as follows: | ||
P1 | P1 Sm2 sM2 M2/m3 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 M6/m7 Sm7 sM7 P8, | ||
these being identical to the the primary interval names of 15edo. | |||
We may further add that ‘S’ (supra) and ‘s’ (small) may raise diminished, and lower augmented intervals by 81/80 as they do to minor and major respectively and that when ‘S’ (super) raised an augmented interval, or ‘s’ (sub) lowers it, the change is by 64/63. | We may further add that ‘S’ (supra) and ‘s’ (small) may raise diminished, and lower augmented intervals by 81/80 as they do to minor and major respectively and that when ‘S’ (super) raised an augmented interval, or ‘s’ (sub) lowers it, the change is by 64/63. | ||
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[[22edo]]: P1 m2 Sm2 sM2 M2 m3 Sm3 sM3 M3 P4 S4 4-5 s5 P5 m6 Sm6 sM6 M6 m7 Sm7 sM7 M7 P8 | [[22edo]]: P1 m2 Sm2 sM2 M2 m3 Sm3 sM3 M3 P4 S4 4-5 s5 P5 m6 Sm6 sM6 M6 m7 Sm7 sM7 M7 P8 | ||
[[24edo]]: P1 | [[24edo]]: P1 N1 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 N4 A4/d5 N5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 N8 P8 | ||
[[26edo]]: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 | [[26edo]]: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 SA4/sd5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8 | ||
[[27edo]]: P1 m2 Sm2 N2 sM2 M2 m3 Sm3 N3 sM3 M3 P4 N4 d6 A3 N5 P5 m6 Sm6 N6 sM6 M6 m7 Sm7 N7 sM7 M7 P8 | [[27edo]]: P1 m2 Sm2 N2 sM2 M2 m3 Sm3 N3 sM3 M3 P4 N4 d6 A3 N5 P5 m6 Sm6 N6 sM6 M6 m7 Sm7 N7 sM7 M7 P8 | ||
[[29edo]]: P1 | [[29edo]]: P1 sm2 m2 Sm2 sM2 M2 SM2/sm3 m3 Sm3 sM3 M3 SM3/s4 P4 S4 d5 A4 s5 P5 sm6 m6 Sm6 sM6 M6 SM6/sm7 m7 Sm7 sM7 M7 SM7 P8 | ||
[[31edo]]: P1 N1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 N4 A4 d5 N4 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 N8 P8 | [[31edo]]: P1 N1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 N4 A4 d5 N4 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 N8 P8 | ||
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[[34edo]]: P1 1-2 m2 Sm2 N2 sM2 M2 2-3 m3 Sm3 N3 sM3 M3 3-4 P4 S4 N4 4-5 N5 s5 P5 5-6 m6 Sm6 N6 sM6 M6 6-7 m7 Sm7 N7 sM7 M7 7-8 P8 | [[34edo]]: P1 1-2 m2 Sm2 N2 sM2 M2 2-3 m3 Sm3 N3 sM3 M3 3-4 P4 S4 N4 4-5 N5 s5 P5 5-6 m6 Sm6 N6 sM6 M6 6-7 m7 Sm7 N7 sM7 M7 7-8 P8 | ||
[[38edo]]: P1 S1 1-2 sm2 m2 N2 M2 SM2 2-3 sm3 m3 N3 M3 SM3 3-4 s4 P4 N4 A4 | [[38edo]]: P1 S1 1-2 sm2 m2 N2 M2 SM2 2-3 sm3 m3 N3 M3 SM3 3-4 s4 P4 N4 A4 SA4/sd5 d5 N5 P5 S5 5-6 sm6 m6 N6 M6 SM6 6-7 sm7 m7 N7 M7 SM7 7-8 s8 P8 | ||
[[41edo]]: P1 N1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 s4 P4 S4 N4 d5 A4 N5 sM5 P5 S5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 N8 P8 | [[41edo]]: P1 N1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 s4 P4 S4 N4 d5 A4 N5 sM5 P5 S5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 N8 P8 | ||
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[[43edo]]: P1 S1 1-2 A1/sm2 m2 dd3 AA1 M2 SM2/d3 2-3 A2/sm3 m3 dd4 AA2 M3 SM3/d4 3-4 A3/s4 P4 dd5 AA3 A4 d5 dd6 AA4 P5 S5/A6 5-6 A5/sm6 m6 dd7 AA5 M6 SM6/d7 6-7 A6/sm7 m7 dd8 AA6 M7 SM7/d8 7-8 s8 P8 | [[43edo]]: P1 S1 1-2 A1/sm2 m2 dd3 AA1 M2 SM2/d3 2-3 A2/sm3 m3 dd4 AA2 M3 SM3/d4 3-4 A3/s4 P4 dd5 AA3 A4 d5 dd6 AA4 P5 S5/A6 5-6 A5/sm6 m6 dd7 AA5 M6 SM6/d7 6-7 A6/sm7 m7 dd8 AA6 M7 SM7/d8 7-8 s8 P8 | ||
[[46edo]]: P1 S1 sm2 m2 Sm2 A1 d3 sM2 M2 SM2 sm3 m3 Sm3 A2 d4 sM3 M3 SM3 s4 P4 S4 sd5 d5 | [[46edo]]: P1 S1 sm2 m2 Sm2 A1 d3 sM2 M2 SM2 sm3 m3 Sm3 A2 d4 sM3 M3 SM3 s4 P4 S4 sd5 d5 sA4/Sd5 A4 SA4 s5 P5 S5 sm6 m6 Sm6 A4 d7 sM6 M6 SM6 sm7 m7 Sm7 A6 d8 sM7 M7 SM7 s8 P8 | ||
In 43edo we encounter the first time we have to use double Augmented and diminished intervals. 43edo marks the first instance in which Jones' alternative ups and downs interval names do not match those from this system. In his system, for example, AA2 would simply be sM3, but in this system since sM3 implies an approximation to 5/4 and the M3 already represents 5/4, and therefore is equivalent to sM3, we cannot do this. This tells us however that no simple ratio is approximated by the interval, and perhaps it is better understood as an AA2. Larger edos contain unlabeled intervals (without resorting to extended diatonic interval names). The association of 'super' and 'sub' with 64/63 and with 'supra' and 'small' with 81/80 may effect the assignment of primary interval names, but for all of these edos, as well as all those mentioned before, when 'S' and 's' are used, they still signify a raising or lowering by a single step of the edo, and thus appear equivalent to the Ups and Downs version. The comma associations add that, though use of enharmonic equivalences and secondary interval names may be necessary, intervals from MOS scales may be spelled in a consistent way across tuning to different edos. | In 43edo we encounter the first time we have to use double Augmented and diminished intervals. 43edo marks the first instance in which Jones' alternative ups and downs interval names do not match those from this system. In his system, for example, AA2 would simply be sM3, but in this system since sM3 implies an approximation to 5/4 and the M3 already represents 5/4, and therefore is equivalent to sM3, we cannot do this. This tells us however that no simple ratio is approximated by the interval, and perhaps it is better understood as an AA2. Larger edos contain unlabeled intervals (without resorting to extended diatonic interval names). The association of 'super' and 'sub' with 64/63 and with 'supra' and 'small' with 81/80 may effect the assignment of primary interval names, but for all of these edos, as well as all those mentioned before, when 'S' and 's' are used, they still signify a raising or lowering by a single step of the edo, and thus appear equivalent to the Ups and Downs version. The comma associations add that, though use of enharmonic equivalences and secondary interval names may be necessary, intervals from MOS scales may be spelled in a consistent way across tuning to different edos. | ||
=== Other rank-2 temperaments' MOS scales === | === Other rank-2 temperaments' MOS scales === | ||
On top of those discussed thus far, other temperaments generated by the P5 or a fraction of it are also supported to some extent, where their MOS scales may be represented, including [[Augmented]], [[Porcupine]], [[Diminished]], [[Negri]], [[Tetracot]] and [[Slendric]]. | On top of those discussed thus far, other temperaments generated by the P5 or a fraction of it are also supported to some extent, where their MOS scales may be represented, including [[Semaphore]], [[Augmented]], [[Porcupine]], [[Diminished]], [[Negri]], [[Tetracot]] and [[Slendric]]. | ||
Semaphore[5] 2|2: P1 SM2/sm3 P4 P5 SM6/sm7 P8 | |||
Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8 | Semaphore[9] 4|4: P1 M2 SM2/sm3 SM3/s4 P4 P5 S5/sm6 SM6/sm7 m7 P8 | ||
Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8 | |||
Augmented[9] 1|1 (3): P1 Sm2 Sm3 sM3 P4 P5 Sm6 SM6 sM7 P8 | Augmented[9] 1|1 (3): P1 Sm2 Sm3 sM3 P4 P5 Sm6 SM6 sM7 P8 | ||
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Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8 | Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8 | ||
Diminished[8] 1|0 (4): P1 sM2 Sm3 P4 | Diminished[8] 1|0 (4): P1 sM2 Sm3 P4 sA4/Sd5 Sm6 sM6 sM7 P8 | ||
[[Diminished12|Diminished[12]]] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 | [[Diminished12|Diminished[12]]] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 Sm7 sM7 P8 | ||
Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8 | Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8 | ||
| Line 254: | Line 275: | ||
Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8 | Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8 | ||
Negri[9] 4|4: P1 Sm2 | Negri[9] 4|4: P1 Sm2 SM2/sm3 sM3 P4 P5 Sm6 SM6/sm7 sM7 P8 | ||
Negri[10] 4|4: P1 Sm2 | Negri[10] 4|4: P1 Sm2 SM2/sm3 sM3 P4 A4 P5 Sm6 SM6/sm7 sM7 P8 | ||
Tetracot[6] 3|2: P1 sM2 N3 S4 N6 Sm7 P8 | Tetracot[6] 3|2: P1 sM2 N3 S4 N6 Sm7 P8 | ||
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P1 M2 M3 m3 P4 P5 M6 m6 m7 P8, | P1 M2 M3 m3 P4 P5 M6 m6 m7 P8, | ||
which is also the primary interval name set for 9edo. | |||
If we don't want to have major being below minor, we can hide it with some secondary interval names: | If we don't want to have major being below minor, we can hide it with some secondary interval names: | ||
| Line 341: | Line 364: | ||
We might think that the primary interval names of [[9edo]] are the same as our first spelling of Mavila[9] 4|4 above, however we note that the third step is half-way between M2 and m3, and so is primarily a serd. The fourth step, half-way between M3 and P4 is a thourth. If the M3 was a serd, it is also a m2, meaning that the second step is half-way between P1 and m2, and is primarily a unicond. Our primary interval names for 9edo are as follows: | We might think that the primary interval names of [[9edo]] are the same as our first spelling of Mavila[9] 4|4 above, however we note that the third step is half-way between M2 and m3, and so is primarily a serd. The fourth step, half-way between M3 and P4 is a thourth. If the M3 was a serd, it is also a m2, meaning that the second step is half-way between P1 and m2, and is primarily a unicond. Our primary interval names for 9edo are as follows: | ||
P1 M2 M3 m3 P4 P5 M6 m6 m7 P8 | |||
P1 1-2 2-3 3-4 P4 P5 5-6 6-7 7-8 P8. | P1 1-2 2-3 3-4 P4 P5 5-6 6-7 7-8 P8. | ||
| Line 598: | Line 623: | ||
Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8 | Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8 | ||
Barbados[5] 2|2: P1 2-3 P4 P5 6-7 P8 | |||
Barbados[9] 4|4: P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8 | |||
Blackwood[10] 1|0 (5): P1/1-2 sM2 2-3 sM3 3-4/P4 s5 P5/ 5-6 sM6 6-7 sM7 7-8/P8 | Blackwood[10] 1|0 (5): P1/1-2 sM2 2-3 sM3 3-4/P4 s5 P5/ 5-6 sM6 6-7 sM7 7-8/P8 | ||
| Line 645: | Line 674: | ||
Neutral[17] 8|8: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8 | Neutral[17] 8|8: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8 | ||
Pajara[10] 2|2 (2): P1 Sm2 M2 sM3 P4 | Pajara[10] 2|2 (2): P1 Sm2 M2 sM3 P4 sA4/Sd5 P5 Sm6 m7 sM7 P8 | ||
Pajara[12] 3|2 (2): P1 Sm2 M2 Sm3 sM3 P4 | Pajara[12] 3|2 (2): P1 Sm2 M2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 m7 sM7 P8 | ||
Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8 | Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8 | ||
| Line 655: | Line 684: | ||
Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8 | Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8 | ||
Semaphore[5] 2|2: P1 | Semaphore[5] 2|2: P1 SM2/sm3 P4 P5 SM6/sm7 P8 | ||
Semaphore[9] 4|4: P1 M2 | Semaphore[9] 4|4: P1 M2 SM2/sm3 SM3/s4 P4 P5 S5/sm6 SM6/sm7 m7 P8 | ||
Slendric[5] 2|3: P1 SM2 s4 S5 sm7 P8 | Slendric[5] 2|3: P1 SM2 s4 S5 sm7 P8 | ||