FKH Extended-diatonic Interval Names: Difference between revisions
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''FKH (Fokker/Keenan/Hearne) Extended-diatonic Interval Names'' is a diatonic-based system for interval naming developed after [[Extended-diatonic interval names|a review of the historical development of Western interval names, and of current proposed schemes]], taking the best and leaving alone the worst aspects of the existing standards. In addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees and the additional qualifiers 'k' and 'K'. Using this scheme almost all small to medium sized [[Equal Temperaments|equal temperaments]] (ETs) can be named such that 'S' and 's' and/or 'K' and 'k' correspond to a displacement of an interval up or down a single degree of the ET, respectively. With the addition of two more qualifiers, 'W' and 'n', FKH can name all intervals from all edos. Many commonly used [[MOS scale|MOS scales]] may also be described using this scheme such that these scales' interval names are expressed consistently in in any tuning that supports them. The scheme, which can also be easily mapped to many of the current interval naming standards, facilitating translation between them, should improve pedagogy and communication in microtonal music. | '''FKH (Fokker/Keenan/Hearne) Extended-diatonic Interval Names''' (formerly known as '''SHEFKHED Interval Names''') is a [[diatonic]]-based system for interval naming developed after [[Extended-diatonic interval names|a review of the historical development of Western interval names, and of current proposed schemes]], taking the best and leaving alone the worst aspects of the existing standards. In addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees and the additional qualifiers 'k' and 'K' (formerly 'c' and 'C'). Using this scheme almost all small to medium sized [[Equal Temperaments|equal temperaments]] (ETs) can be named such that 'S' and 's' and/or 'K' and 'k' correspond to a displacement of an interval up or down a single degree of the ET, respectively. With the addition of two more qualifiers, 'W' and 'n', FKH can name all intervals from all edos. Many commonly used [[MOS scale|MOS scales]] may also be described using this scheme such that these scales' interval names are expressed consistently in in any tuning that supports them. The scheme, which can also be easily mapped to many of the current interval naming standards, facilitating translation between them, should improve pedagogy and communication in microtonal music. | ||
After feedback, the author ([[User:Lhearne]]) has since updated this scheme as [[SKULO interval names#WOFED interval names|WOFED interval names]], and the later re-imagining [[SKULO interval names]]. | |||
==Introduction== | ==Introduction== | ||
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*Consistent mapping to Sagittal and HEWM notation (Sagispeak) | *Consistent mapping to Sagittal and HEWM notation (Sagispeak) | ||
*Consistent mapping to Ups and Downs notation (Ups and Downs) | *Consistent mapping to Ups and Downs notation (Ups and Downs) | ||
Such a system is developed through the extension of Fokker/Keenan Extended-diatonic Interval-names to define prefixes by alterations by specific commas as in Sagispeak, to enable application to Pythagorean | Such a system is developed through the extension of Fokker/Keenan Extended-diatonic Interval-names to define prefixes by alterations by specific commas as in Sagispeak, with the addition of prefix names of Smith and Keenan to enable application to Pythagorean and Superpythagorean systems. Where the prefixes of the Fokker/Keenan system were introduced by Helmholtz/Ellis, attribution to them is added, leading to Smith/Helmholtz/Ellis/Fokker/Keenan/Hearne Extended-diatonic interval names, or SHEFKHED interval names (Hearne is me). | ||
==Prefixes== | ==Prefixes== | ||
*The unison is labelled P1 for perfect unison, and the octave P8 for perfect octave. | *The unison is labelled P1 for perfect unison, and the octave P8 for perfect octave. | ||
*A tuning's best approximation to 3/2 is labelled P5, for perfect fifth, and it's octave-complement labelled P4, for perfect fourth. | *A tuning's best approximation to 3/2 is labelled P5, for perfect fifth, and it's octave-complement labelled P4, for perfect fourth. | ||
*From the Pythagorean diatonic scale | *From the Pythagorean diatonic scale using a tuning's best 3/2 fifth, the two sizes of second, third, sixth and seventh are labelled major, or 'M', for the larger, and minor, or 'm' for the smaller. | ||
*Any perfect or major interval raised by the apotome, the interval between the major and minor intervals of a single interval-class is labelled 'A' for augmented, and any perfect or minor interval lowered by the same is labelled 'd' for diminished. | *Any perfect or major interval raised by the apotome, the interval between the major and minor intervals of a single interval-class is labelled 'A' for augmented, and any perfect or minor interval lowered by the same is labelled 'd' for diminished. | ||
*Any augmented interval may be made doubly augmented, with short-hand 'AA' by the further raising of an apotome and any diminished interval made doubly diminished, with short-hand 'dd' by the further lowering of an apotome. This process may be iterated ad nauseum. At this stage we have simply rigorously defined diatonic interval names. Thankfully what remains of the definition leads to more desirable alternatives for most occasions in which one might find these iteratively diminished and augmented intervals. | *Any augmented interval may be made doubly augmented, with short-hand 'AA' by the further raising of an apotome and any diminished interval made doubly diminished, with short-hand 'dd' by the further lowering of an apotome. This process may be iterated ad nauseum. At this stage we have simply rigorously defined diatonic interval names. Thankfully what remains of the definition leads to more desirable alternatives for most occasions in which one might find these iteratively diminished and augmented intervals. | ||
*Perfect, major and augmented intervals may be given the prefix 'super', with shorthand 'S' which infers an augmentation by the septimal comma, 64/63, whereas perfect, minor and diminished intervals are lowered by the same interval when given the prefix 'sub', with short-form 's'. | |||
*Major and augmented intervals may be given the prefix 'classic', with short-form 'c', inferring a diminution by the syntonic comma, 81/80, whereas minor and diminished may also be given the prefix 'classic' but with short-hand 'C', inferring an augmentation by 81/80. This results in the labeling of 10/9, 6/5, 5/4, 8/5, 5/3 and 9/5 as classic major second, classic major third, classic minor third, classic minor sixth, classic major sixth and classic minor seventh, as per Keenan's suggestion when a comparison to Pythagorean is needed. | |||
*Perfect intervals may also be given the prefixes 'C' and 'c' to imply augmentation and diminution by the syntonic comma. Where 81/80 is referred to by Smith and Bosanquet simple as 'comma', Smith's interval-naming scheme involves prefixes of ''m''/''n''-comma sharp and ''m''/''n''-comma flat. Following this example but using 'wide' and 'narrow' instead of 'sharp' and 'flat', we associate the long-form 'comma-wide' and 'comma-narrow' to infer movement up or down a syntonic comma from Perfect intervals. | |||
*For seconds, thirds, sixths and sevenths, any interval exactly half-way major and minor is labelled 'neutral', with short-form 'N'. | *For seconds, thirds, sixths and sevenths, any interval exactly half-way major and minor is labelled 'neutral', with short-form 'N'. | ||
*Where neutral intervals split the apotome, to pair with neutral when acting on perfect intervals are 'hemi augmented' and 'hemi diminished', with short form 'hA' and 'hd'. 'Hemi' is used instead of 'semi' of 'half' because 'half diminished' is a type of chord, and 'semi' begins with the letter 's', which has been associated with alterations of 64/63. In all cases it's presence implies neutral temperament and the tempering out of 243/242. Accordingly it implies a diminution from perfect, major, augmented of 33/32, as well as an augmentation from perfect, minor or diminished of 33/32, but may not be used to imply those alterations in any other cases | *Where neutral intervals split the apotome, to pair with neutral when acting on perfect intervals are 'hemi augmented' and 'hemi diminished', with short form 'hA' and 'hd'. 'Hemi' is used instead of 'semi' of 'half' because 'half diminished' is a type of chord, and 'semi' begins with the letter 's', which has been associated with alterations of 64/63. In all cases it's presence implies neutral temperament and the tempering out of 243/242. Accordingly it implies a diminution from perfect, major, augmented of 33/32, as well as an augmentation from perfect, minor or diminished of 33/32, but may not be used to imply those alterations in any other cases. | ||
*To extend to the 13-limit, we add that to | *To extend to the 13-limit, we add that to cP, cM and cA intervals may be added the 'sub' or 's' prefix, in this instance indicating a diminution of [[65/64]], and that to CP, Cm and Cd intervals may be added the 'super' or 'S' prefix, indication an augmentation of the same interval. Accordingly the difference between 65/64 and 64/63, 4096/4095, the ''tridecimal schisma'', is tempered out. 16/13 can then be labelled a 'sub classic major third', or scM3. In tunings where the syntonic comma is tempered out, such that (cP, cM, cA, CP, Cm, Cd) = (P, M, A, P, m, d), the 'c' and 'C' prefixes are dropped in the short-form. | ||
*Similarly, to extend the 11-limit, we add that to SP, SM and SA intervals may be added the 'comma-narrow' or 'c' prefix, in this case indicating a diminution of 99/98, the 7-11 comma, and that to sP, sm and sd intervals may be added the 'comma-wide' of 'C' prefix, in this case indicating an augmentation of the same interval. Accordingly the difference between 81/80 and 99/98, 441/440, is tempered out. 14/11 can then be labelled a 'comma-narrow super major third' or cSM3. In tunings where the septimal comma is tempered out, such that (SP, SM, SA, sP, sm, sd) = (P, M, A, P, m, d), the 'S' and 's' prefixes are dropped in the short form. | |||
*Where N indicates a splitting of the apotome and of the perfect fifth, interval names indicating the splitting of the limma and of the perfect fourth are included for remaining unnamed intervals, reflecting limited, but existing practice. The interval half-way between P1 and m2 is given the short-form '1-2', and long-form 'unison-second' that may be said 'unind'. Similarly the interval half-way between M7 and P8 is given the short-form '1-2' and long-form 'seventh-octave' that may be said 'sevtave'. The interval splitting the fourth, lying half-way between M2 and m3 is given the short-form '2-3', with long-form 'second-third' that may be said 'serd', and it's octave complement, lying half-way between M6 and m7 is given the short-form '6-7', with long-form 'sixth-seventh', that may be said 'sinth'. The interval half-way between M3 and P4 is given the short-form '3-4', with long-form 'third-fourth', that may be said 'thourth', and it's octave-complement, the interval half-way between P5 and m6 has short-form '5-6', with long-form 'fifth-sixth', that may be said 'fixth'. These interval names can be associated with [[The Archipelago|Barbados]] temperament, indicating the tempering out of 676/675, generated by 2-3, half of the fourth, associated with the ratio 15/13. These ''intermediates'' lie 40/39 above major intervals and the perfect unison and fifth, and below minor intervals and the perfect fourth and octave. 3-4, for example, is associated with the ratio 13/10. | *Where N indicates a splitting of the apotome and of the perfect fifth, interval names indicating the splitting of the limma and of the perfect fourth are included for remaining unnamed intervals, reflecting limited, but existing practice. The interval half-way between P1 and m2 is given the short-form '1-2', and long-form 'unison-second' that may be said 'unind'. Similarly the interval half-way between M7 and P8 is given the short-form '1-2' and long-form 'seventh-octave' that may be said 'sevtave'. The interval splitting the fourth, lying half-way between M2 and m3 is given the short-form '2-3', with long-form 'second-third' that may be said 'serd', and it's octave complement, lying half-way between M6 and m7 is given the short-form '6-7', with long-form 'sixth-seventh', that may be said 'sinth'. The interval half-way between M3 and P4 is given the short-form '3-4', with long-form 'third-fourth', that may be said 'thourth', and it's octave-complement, the interval half-way between P5 and m6 has short-form '5-6', with long-form 'fifth-sixth', that may be said 'fixth'. These interval names can be associated with [[The Archipelago|Barbados]] temperament, indicating the tempering out of 676/675, generated by 2-3, half of the fourth, associated with the ratio 15/13. These ''intermediates'' lie 40/39 above major intervals and the perfect unison and fifth, and below minor intervals and the perfect fourth and octave. 3-4, for example, is associated with the ratio 13/10. | ||
*For completeness, the interval '4-5', long form 'fourth-fifth' that may be said 'firth' is added, though it is separate to the other intermediates, splitting not the limma, but the dieses (between A4 and d5), or the octave. It does not map to any particular ratios and is not needed as a primary interval name, apart from in 16edo, and is included mostly to be used as an optional secondary interval name when there are no others. | *For completeness, the interval '4-5', long form 'fourth-fifth' that may be said 'firth' is added, though it is separate to the other intermediates, splitting not the limma, but the dieses (between A4 and d5), or the octave. It does not map to any particular ratios and is not needed as a primary interval name, apart from in 16edo, and is included mostly to be used as an optional secondary interval name when there are no others. | ||
*In any prefix is used before 'P' then 'P' is removed in both the short-form and long-form names. | *In any prefix is used before 'P' then 'P' is removed in both the short-form and long-form names. | ||
*The prefixes so far take us as far as 53edo (72edo), which is considered a 'commatic' scale by many, and as far as extended-diatonic function, which I hope to reflect with this scheme, could be considered to apply. Keenan's functional names take us to 31edo, after which 'narrow' and 'wide' prefixes are added to differentiate different intervals in medium to large sized edos of the same function. Ups and Downs takes function as far as regular diatonic and mids (equivalent to neutrals), which will give us most of a well-ordered interval name set for 17edo (if mids were extended as I have extended neutrals, all the notes would be obtainable) without up or down prefixes, and only functional names, or all of 19edo or 26edo, since these are meantone edos with the apotome subtended by a single degree and may be given a well-ordered interval names set using only regular diatonic interval names. The up and down prefixes are not functional, and specify movement instead by a single step of an edo. If the naming of systems with more than one interval per function is desired, then 'wide' and 'narrow' prefixes, with short form 'W' and 'n' respectively are to be employed. This also allows the notation of intervals for which intermediates are the only available functional interval name. Note: For regular diatonic intervals, I consider function only to go as far as singly diminished or augmented intervals, and never use multiply diminished or augmented intervals for my interval names. | *The prefixes so far take us as far as 53edo (72edo), which is considered a 'commatic' scale by many, and as far as extended-diatonic function, which I hope to reflect with this scheme, could be considered to apply. Keenan's functional names take us to 31edo, after which 'narrow' and 'wide' prefixes are added to differentiate different intervals in medium to large sized edos of the same function. Ups and Downs takes function as far as regular diatonic and mids (equivalent to neutrals), which will give us most of a well-ordered interval name set for 17edo (if mids were extended as I have extended neutrals, all the notes would be obtainable) without up or down prefixes, and only functional names, or all of 19edo or 26edo, since these are meantone edos with the apotome subtended by a single degree and may be given a well-ordered interval names set using only regular diatonic interval names. The up and down prefixes are not functional, and specify movement instead by a single step of an edo. If the naming of systems with more than one interval per function is desired, then 'wide' and 'narrow' prefixes, with short form 'W' and 'n' respectively are to be employed. This also allows the notation of intervals for which intermediates are the only available functional interval name. Note: For regular diatonic intervals, I consider function only to go as far as singly diminished or augmented intervals, and never use multiply diminished or augmented intervals for my interval names. | ||
*Where 'c', 'a' and 'd' are also note names, in some contexts short-form interval names may be confused as short-form chord names, such as Cm7, which is a minor seventh chord rooted on C. Normally context differentiates between, or it can simple be added 'the interval' or 'the chord', but if alternative abbreviations for interval-names that may not be confused with chord names are desired, such 'mid-form' abbreviations are provided in the following tables, which summarise the prefixes listed in this section. | |||
{| class="wikitable" | {| class="wikitable" | ||
|+'Regular' Prefixes summary | |+'Regular' Prefixes summary | ||
!Long-form | !Long-form | ||
!Short-form | !Short-form | ||
!Mid-form | |||
!Alteration | !Alteration | ||
|- | |- | ||
|perfect | |perfect | ||
|P | |P | ||
|Prf | |||
|none | |none | ||
|- | |- | ||
|major | |major | ||
|M | |M | ||
|Maj | |||
|none | |none | ||
|- | |- | ||
|minor | |minor | ||
|m | |m | ||
|min | |||
|none | |none | ||
|- | |- | ||
|augmented | |augmented | ||
|A | |A | ||
|Aug | |||
|up an apotome | |up an apotome | ||
|- | |- | ||
|diminished | |diminished | ||
|d | |d | ||
|dim | |||
|down an apotome | |down an apotome | ||
|- | |- | ||
|super | |super | ||
|S | |S | ||
|Sub | |||
|up 64/63 (or 65/64) | |up 64/63 (or 65/64) | ||
|- | |- | ||
|sub | |sub | ||
|s | |s | ||
|spr | |||
|down 64/63 (or 65/64) | |down 64/63 (or 65/64) | ||
|- | |- | ||
|classic | |classic | ||
| | |C | ||
|Cla | |||
|up 81/80 | |up 81/80 | ||
|- | |- | ||
|classic | |classic | ||
| | |c | ||
|cla | |||
|down 81/80 | |down 81/80 | ||
|- | |- | ||
|comma-wide | |comma-wide | ||
| | |C | ||
|Co-W | |||
|up 81/80 (or 99/98) | |up 81/80 (or 99/98) | ||
|- | |- | ||
|comma-narrow | |comma-narrow | ||
| | |c | ||
|co-n | |||
|down 81/80 (or 99/98) | |down 81/80 (or 99/98) | ||
|- | |- | ||
|wide | |wide | ||
|W | |W | ||
|Wde | |||
|up a single degree | |up a single degree | ||
|- | |- | ||
|narrow | |narrow | ||
|n | |n | ||
|nrw | |||
|down a single degree | |down a single degree | ||
|} | |} | ||
| Line 103: | Line 121: | ||
|2-3 (serd) | |2-3 (serd) | ||
|2-3 (serd) | |2-3 (serd) | ||
|half of P4, halfway between M2 and m3 | |half of P4, halfway between M2 and m3 | ||
|- | |- | ||
|third-fourth | |third-fourth | ||
| Line 118: | Line 136: | ||
|5-6 (fixth) | |5-6 (fixth) | ||
|5-6 (fixth) | |5-6 (fixth) | ||
|half of M10, halfway between P5 and m6 | |half of M10, halfway between P5 and m6 | ||
|- | |- | ||
|sixth-seventh | |sixth-seventh | ||
| Line 145: | Line 163: | ||
|halfway between P and d | |halfway between P and d | ||
|} | |} | ||
==Name privileging== | ==Name privileging== | ||
As in Keenan/Fokker and Ups and Downs, intervals may be given multiple names. The following details the order to which certain names are privileged above others. | As in Keenan/Fokker and Ups and Downs, intervals may be given multiple names. The following details the order to which certain names are privileged above others. | ||
Interval names are ranked in 11 tiers: | Interval names are ranked in 11 tiers: | ||
#Perfect | #Perfect and neutral | ||
#Major, minor, A4 and d5. | #Major, minor, A4 and d5. | ||
#'S', 's', ' | #hA4 and hd5 | ||
# | #'S', 's', 'C', 'c', 'SC' and 'sc' prefixes to major, minor, perfect intervals and to A4 and d5 | ||
#hA1 and hd8 (plus any other hAs and hds if needed) | |||
#Intermediates | #Intermediates | ||
#Remaining augmented and diminished intervals | #Remaining augmented and diminished intervals | ||
#'S', 's', ' | #'S', 's', 'C', 'c', 'SC' and 'sc' prefixes to augmented and diminished intervals | ||
#'n' and 'W' prefixes to tier 1-3 interval names | #'n' and 'W' prefixes to tier 1-3 interval names | ||
#'n' and 'W' prefixes to tier 4-8 interval names | #'n' and 'W' prefixes to tier 4-8 interval names | ||
| Line 162: | Line 181: | ||
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'. | 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'. | ||
Where the same interval may be named | Where the same interval may be named c4 or s4, s4 is preferred and where the same interval may be named C4 or S4, C4 is preferred. Similarly, where the same interval may be named C5 or S5, S5 is preferred and where the same interval may be named c5 or s5, c5 is preferred. This is to ensure the interval is named after the simpler ratio. On top of this, well-ordered interval-name sets are desired, leading to interval names in lower tires being used in preference to higher-tier names in some cases. | ||
==Application in Regular diatonic edos== | |||
All ''regular diatonic'' edos (edos whose best fifth is greater than 4 degrees of 7edo and less than 3 degrees of 5edo, such that the diatonic scale has 5 large and 2 small steps) up to 46 can be simply given primary well-ordered interval names. All of those that I've seen used have their primary well-ordered interval-names below, with the addition of 53edo which is about as far as this scheme's functional interval names can go, and should, by my opinion. Using he 'function-less' prefixes, 'n' and 'W', 50edo may be named. | == Application in Regular diatonic edos == | ||
All ''regular diatonic'' edos (edos whose best fifth is greater than 4 degrees of 7edo and less than 3 degrees of 5edo, such that the diatonic scale has 5 large and 2 small steps) up to 46 can be simply given primary well-ordered interval names. All of those that I've seen used have their primary well-ordered interval-names below, with the addition of 53edo which is about as far as this scheme's functional interval names can go, and should, by my opinion. Using he 'function-less' prefixes, 'n' and 'W', 50edo may be named. | |||
12edo: P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8 | 12edo: P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8 | ||
| Line 172: | Line 192: | ||
19edo: P1 S1/sm2 m2 M2 SM2/sm3 m3 M3 SM3/s4 P4 A4 d5 P5 S5/sm6 m6 M6 SM6/sm7 m7 M7/s8 P8 | 19edo: P1 S1/sm2 m2 M2 SM2/sm3 m3 M3 SM3/s4 P4 A4 d5 P5 S5/sm6 m6 M6 SM6/sm7 m7 M7/s8 P8 | ||
22edo: P1 m2 | 22edo: P1 m2 Cm2 cM2 M2 m3 Cm3 cM3 M3 P4 C4 cA4/Cd5 c5 P5 m6 Cm6 cM6 M6 m7 Cm7 cM7 M7 P8 | ||
24edo: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 hA4 A4/d5 | 24edo: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 hA4 A4/d5 hd5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8 | ||
26edo: P1 S1 sm2 m2 M2 SM2 sm3 m3 M3 SM3 s4 P4 A4 SA4/sd5 d5 P5 S5 sm6 m6 M6 SM6 sm7 m7 M7 SM7 s8 P8 | 26edo: P1 S1 sm2 m2 M2 SM2 sm3 m3 M3 SM3 s4 P4 A4 SA4/sd5 d5 P5 S5 sm6 m6 M6 SM6 sm7 m7 M7 SM7 s8 P8 | ||
27edo: P1 m2 N1 N2 | 27edo: P1 m2 N1 N2 cM2 M2 m3 Cm3 N3 cM3 M3 P4 N4 scA4/Cd5 cA4/SCd5 N5 P5 m6 Cm6 N6 cM6 M6 m7 Cm7 N7 N8 M7 P8 | ||
29edo: P1 | 29edo: P1 C1/S1/sm2 m2 Cm2 cM2 M2 SM2/sm3 m3 Cm3 cM3 M3 SM3/s4 P4 C4 cA4/d5 A4/Cd5 c5 P5 S5/sm6 m6 Cm6 cM6 M6 SM6/sm7 m7 Cm7 cM7 M7 SM7/S8/c8 P8 | ||
31edo: P1 S1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 hA4 A4 d5 hd5 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 s8 P8 | 31edo: P1 S1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 hA4 A4 d5 hd5 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 s8 P8 | ||
34edo: P1 | 34edo: P1 C1 m2 Cm2 N2 cM2 M2 2-3 m3 Cm3 N3 cM3 M3 3-4 P4 C4 N4/d5 cA4/Cd5 A4/N5 c5 P5 5-6 m6 Cm6 N6 cM6 M6 6-7 m7 Cm7 N7 cM7 M7 c8 P8 | ||
36edo: P1 S1 sm2 m2 Sm2 sM2 M2 SM2 sm3 m3 Sm3 sM3 M3 SM3 s4 P4 S4 sA4 A4/d5 SA4 s5 P5 S5 sm6 m6 Sm6 sM6 M6 SM6 sm7 m7 Sm7 sM7 M7 SM7 s8 P8 | 36edo: P1 S1 sm2 m2 Sm2 sM2 M2 SM2 sm3 m3 Sm3 sM3 M3 SM3 s4 P4 S4 sA4 A4/d5 SA4 s5 P5 S5 sm6 m6 Sm6 sM6 M6 SM6 sm7 m7 Sm7 sM7 M7 SM7 s8 P8 | ||
| Line 190: | Line 210: | ||
38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 hA4 A4 SA4/sd5 d5 hd5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8 | 38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 hA4 A4 SA4/sd5 d5 hd5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8 | ||
41edo: P1 | 41edo: P1 C1/S1 sm2 m2 Cm2 N2 cM2 M2 SM2 sm3 m3 Cm3 N3 cM3 M3 SM3 s4 P4 C4 hA4 cA4 Cd5 hd5 c5 P5 S5 sm6 m6 Cm6 N6 cM6 M6 SM6 sm7 m7 Cm7 N7 cM7 M7 SM7 c8/s8 P8 | ||
43edo: P1 S1 1-2 sm2 m2 Sm2 sM2 M2 SM2 2-3 sm3 m3 Sm3 sM3 M3 SM3 3-4 s4 P4 S4 sA4 A4 d5 Sd5 s5 P5 S5 5-6 sm6 m6 Sm6 sM6 M6 SM6 6-7 sm7 m7 Sm7 sM7 M7 SM7 7-8 s8 P8 | 43edo: P1 S1 1-2 sm2 m2 Sm2 sM2 M2 SM2 2-3 sm3 m3 Sm3 sM3 M3 SM3 3-4 s4 P4 S4 sA4 A4 d5 Sd5 s5 P5 S5 5-6 sm6 m6 Sm6 sM6 M6 SM6 6-7 sm7 m7 Sm7 sM7 M7 SM7 7-8 s8 P8 | ||
46edo: P1 | 46edo: P1 C1/S1 sm2 m2 Cm2 SCm2 scM2 cM2 M2 SM2 sm3 m3 Cm3 SCm3 scM3 cM3 M3 SM3 s4 P4 C4 SC4 scA4/d5 cA4/Cd5 A4/SCd5 SA4/sc5 c5 P5 S5 sm6 m6 Cm6 SCm6 scM6 cM6 M6 SM6 sm7 m7 Cm7 SCm7 scM7 cM7 M7 SM7 c8/s8 P8 | ||
50edo: P1 W1 S1 sm2 nm2 m2 Sm2 sM2 M2 WM2 SM2 sm3 nm3 m3 Sm3 sM3 M3 WM3 SM3 s4 n4 P4 W4 S4 A4 WA4/nd5 d5 s5 n5 P5 WS5 S5 sm6 nm6 m6 Sm6 sM6 M6 WM6 SM6 sm7 nm7 m7 Sm7 sM7 M7 WM7 SM7 s8 n8 P8 | 50edo: P1 W1 S1 sm2 nm2 m2 Sm2 sM2 M2 WM2 SM2 sm3 nm3 m3 Sm3 sM3 M3 WM3 SM3 s4 n4 P4 W4 S4 A4 WA4/nd5 d5 s5 n5 P5 WS5 S5 sm6 nm6 m6 Sm6 sM6 M6 WM6 SM6 sm7 nm7 m7 Sm7 sM7 M7 WM7 SM7 s8 n8 P8 | ||
53edo: P1 | 53edo: P1 C1/S1 1-2 sm2 m2 Cm2 SCm2 scM2 sM2 M2 SM2 2-3 sm3 m3 Cm3 SCm3 scM3 cM3 M3 SM3 3-4 s4 P4 C4 SC4 scA4 cA4 Cd5 SCd5 SA4/sc5 c5 P5 S5 5-6 sm6 m6 Cm6 SCm6 scM6 sM6 M6 SM6 6-7 sm7 m7 Cm7 SCm7 scM7 cM7 M7 SM7 7-8 c8/s8 P8 | ||
72edo: P1 | 72edo may also be given 'functional names' by allowing application of 'comma-wide' and 'comma-narrow' to the neutrals and in the other direction for M, m, A and d. Regular 'wide' and 'narrow' function-less prefixes may be used alternatively. For application to the neutrals, 'c' and 'C' give associations with ratios that are not too complex. | ||
72edo: P1 C1 S1 hA1 sm2 nm2 m2 Cm2 cN2 N2 CN2 cM2 M2 WM2 SM2 2-3 sm3 nm3 m3 Cm3 cN3 N3 CN3 cM3 M3 WM3 SM3 3-4 s4 n4 P4 C4 S4 hA4 ChA4 cA4 A4/d5 Cd5 chd5 hd5 s5 c5 P5 C5 S5 5-6 sm6 nm6 m6 Cm6 cN6 N6 CN6 cM6 M6 CM6 SM6 6-7 sm7 cm7 m7 Cm7 cN7 N7 CN7 cM7 M7 CM7 SM7 hd8 s8 c8 P8 | |||
We can see that | We can see that | ||
*17edo, 24edo, 27edo, 31edo, 34edo (through 17edo), 38edo, 41edo and 72edo (through 24edo) are neutral tunings from the use of 'N'. We can find the MOS scale Neutral[10] 5|4: P1 N2 M2 N3 P4 hd5 P5 N6 m7 N7 P8 in all of these edos. | *17edo, 24edo, 27edo, 31edo, 34edo (through 17edo), 38edo, 41edo and 72edo (through 24edo) are neutral tunings from the use of 'N'. We can find the MOS scale Neutral[10] 5|4: P1 N2 M2 N3 P4 hd5 P5 N6 m7 N7 P8 in all of these edos. | ||
*19edo, 24edo (through 12edo), 26edo, 31edo, 36edo (through 12edo), 38edo (through 19edo) and 43edo are meantone tunings through the use of 'S' and 's'. | *19edo, 24edo (through 12edo), 26edo, 31edo, 36edo (through 12edo), 38edo (through 19edo) and 43edo are meantone tunings through the use of 'S' and 's'. | ||
*22edo, 27edo and 34edo (through 17edo) are superpythagorean tunings from the use of ' | *22edo, 27edo and 34edo (through 17edo) are superpythagorean tunings from the use of 'C' and 'c'. | ||
*29edo, 41edo, 46edo, 53edo and 72edo are Pythagorean tunings through the use of both 'S' and 's'; and ' | *29edo, 41edo, 46edo, 53edo and 72edo are Pythagorean tunings through the use of both 'S' and 's'; and 'C' and 'c'. | ||
*34edo, 43edo, 53edo and 72edo (through 24edo) are barbados tunings through the use of intermediates. We can find the scale Barbados[9] 4|4, P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8 in all of those edos, but not necessarily in the primary interval names. | *34edo, 43edo, 53edo and 72edo (through 24edo) are barbados tunings through the use of intermediates. We can find the scale Barbados[9] 4|4, P1 1-2 M2 2-3 3-4 P4 P5 5-6 6-7 m7 7-8 P8 in all of those edos, but not necessarily in the primary interval names. | ||
Every edo in which we see SM2/sm3 also supports barbados, where this interval is the generator, at half a fourth, however rather than 15/13 the generator is more simple represented as 8/7~7/6. The temperament generated by the semi-fourth wherein it represent both SM2 and sm3 (tempering out [[49/48]]) is called Semaphore. | Every edo in which we see SM2/sm3 also supports barbados, where this interval is the generator, at half a fourth, however rather than 15/13 the generator is more simple represented as 8/7~7/6. The temperament generated by the semi-fourth wherein it represent both SM2 and sm3 (tempering out [[49/48]]) is called Semaphore. | ||
Semaphore[9] 4|4 has primary interval names P1 M2 SM2/sm3 SM3/s4 P4 P5 S5/sm6 | Semaphore[9] 4|4 has primary interval names P1 S1/sm2 M2 SM2/sm3 SM3/s4 P4 P5 S5/sm6 m7 SM7/s8 P8, which can be seen in 19edo, 24edo and 29edo, so we know they are Semaphore tunings. | ||
Well-ordered primary and their secondary interval names for 22edo, 41edo and 53edo are shown below in more detail. | Well-ordered primary and their secondary interval names for 22edo, 41edo and 53edo are shown below in more detail. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+22edo (patent 2.3.7.11.17 val) | |+22edo (patent 2.3.5.7.11.17 val) | ||
!Degree | !Degree | ||
!Primary interval name | !Primary interval name | ||
| Line 226: | Line 248: | ||
|perfect unison | |perfect unison | ||
|P1 | |P1 | ||
|super | |super unison | ||
|S1 | |S1 | ||
|0 | |0 | ||
| Line 234: | Line 256: | ||
|minor second | |minor second | ||
|m2 | |m2 | ||
|comma-wide unison / | |comma-wide unison / sub minor second | ||
| | |C1/sm2 | ||
|54.55 | |54.55 | ||
|33/32, 34/33, 25/24, 81/80 | |33/32, 34/33, 25/24, 81/80 | ||
| Line 241: | Line 263: | ||
|2 | |2 | ||
|classic minor second | |classic minor second | ||
| | |Cm2 | ||
|diminished third | |diminished third | ||
|d3 | |d3 | ||
| Line 249: | Line 271: | ||
|3 | |3 | ||
|classic major second | |classic major second | ||
| | |cM2 | ||
|augmented unison | |augmented unison | ||
|A1 | |A1 | ||
| Line 273: | Line 295: | ||
|6 | |6 | ||
|classic minor third | |classic minor third | ||
| | |Cm3 | ||
|diminished fourth | |diminished fourth | ||
|d4 | |d4 | ||
| Line 281: | Line 303: | ||
|7 | |7 | ||
|classic major third | |classic major third | ||
| | |cM3 | ||
|augmented second | |augmented second | ||
|A2 | |A2 | ||
| Line 305: | Line 327: | ||
|10 | |10 | ||
|comma-wide fourth | |comma-wide fourth | ||
| | |C4 | ||
|diminished fifth | |diminished fifth | ||
|d5 | |d5 | ||
| Line 314: | Line 336: | ||
|classic augmented fourth | |classic augmented fourth | ||
classic diminished fifth | classic diminished fifth | ||
| | |cA4 | ||
Cd5 | |||
|fourth-fifth | |fourth-fifth | ||
|4-5 (firth) | |4-5 (firth) | ||
| Line 324: | Line 346: | ||
|12 | |12 | ||
|comma-narrow fifth | |comma-narrow fifth | ||
| | |c5 | ||
|augmented fourth | |augmented fourth | ||
|A4 | |A4 | ||
| Line 348: | Line 370: | ||
|15 | |15 | ||
|classic minor sixth | |classic minor sixth | ||
| | |Cm6 | ||
|diminished seventh | |diminished seventh | ||
|d7 | |d7 | ||
| Line 356: | Line 378: | ||
|16 | |16 | ||
|classic major sixth | |classic major sixth | ||
| | |cM6 | ||
|augmented fifth | |augmented fifth | ||
|A5 | |A5 | ||
| Line 380: | Line 402: | ||
|19 | |19 | ||
|classic minor seventh | |classic minor seventh | ||
| | |Cm7 | ||
|diminished octave | |diminished octave | ||
|d8 | |d8 | ||
| Line 388: | Line 410: | ||
|20 | |20 | ||
|classic major seventh | |classic major seventh | ||
| | |cM7 | ||
|augmented sixth | |augmented sixth | ||
|A6 | |A6 | ||
| Line 398: | Line 420: | ||
|M7 | |M7 | ||
|super major seventh / comma-narrow octave | |super major seventh / comma-narrow octave | ||
|SM7/ | |SM7/c8 | ||
|1145.45 | |1145.45 | ||
|33/17, 64/33, 48/25, 160/81 | |33/17, 64/33, 48/25, 160/81 | ||
| Line 425: | Line 447: | ||
|- | |- | ||
|1 | |1 | ||
|comma-wide unison/super | |comma-wide unison/super unison | ||
| | |C1/S1 | ||
|29.27 | |29.27 | ||
|[[81/80]], 64/63 | |[[81/80]], 64/63 | ||
|- | |- | ||
|2 | |2 | ||
| Line 437: | Line 459: | ||
|- | |- | ||
|3 | |3 | ||
|minor second | |minor second | ||
|m2 | |m2 | ||
|87.80 | |87.80 | ||
| Line 443: | Line 465: | ||
|- | |- | ||
|4 | |4 | ||
|classic minor | |classic minor second, augmented unison | ||
| | |Cm2, A1 | ||
|117.07 | |117.07 | ||
|[[16/15]], [[15/14]] | |[[16/15]], [[15/14]] | ||
| Line 455: | Line 477: | ||
|- | |- | ||
|6 | |6 | ||
|classic major | |classic major second, diminished third | ||
| | |cM2, d3 | ||
|175.61 | |175.61 | ||
|[[10/9]], [[11/10]] | |[[10/9]], [[11/10]] | ||
| Line 485: | Line 507: | ||
|- | |- | ||
|11 | |11 | ||
|classic minor | |classic minor third, augmented second | ||
| | |Cm3, A2 | ||
|321.95 | |321.95 | ||
|[[6/5]] | |[[6/5]] | ||
| Line 497: | Line 519: | ||
|- | |- | ||
|13 | |13 | ||
|classic major | |classic major third, diminished fourth | ||
| | |cM3, d4 | ||
|380.49 | |380.49 | ||
|[[5/4]] | |[[5/4]] | ||
| Line 527: | Line 549: | ||
|- | |- | ||
|18 | |18 | ||
|comma-wide | |comma-wide fourth, augmented third | ||
| | |C4, A3 | ||
|526.83 | |526.83 | ||
|[[15/11]], [[27/20]] | |[[15/11]], [[27/20]] | ||
| Line 539: | Line 561: | ||
|- | |- | ||
|20 | |20 | ||
|classic augmented fourth, diminished fifth | |classic augmented fourth, diminished fifth | ||
| | |cA4, d5 | ||
|585.37 | |585.37 | ||
|[[7/5]], 45/32 | |[[7/5]], 45/32 | ||
|- | |- | ||
|21 | |21 | ||
|classic diminished fifth, augmented fourth | |classic diminished fifth, augmented fourth | ||
| | |Cd5, A4 | ||
|614.63 | |614.63 | ||
|[[10/7]], 64/45 | |[[10/7]], 64/45 | ||
| Line 552: | Line 574: | ||
|22 | |22 | ||
|hemi-diminished fifth | |hemi-diminished fifth | ||
| | |h5 | ||
|643.90 | |643.90 | ||
|[[16/11]] | |[[16/11]] | ||
|- | |- | ||
|23 | |23 | ||
|comma-narrow | |comma-narrow fifth, diminished sixth | ||
| | |c5, d6 | ||
|673.17 | |673.17 | ||
|[[22/15]], [[40/27]] | |[[22/15]], [[40/27]] | ||
| Line 587: | Line 609: | ||
|- | |- | ||
|28 | |28 | ||
|classic minor | |classic minor sixth, augmented fifth | ||
| | |Cm6, A5 | ||
|819.51 | |819.51 | ||
|[[8/5]] | |[[8/5]] | ||
| Line 599: | Line 621: | ||
|- | |- | ||
|30 | |30 | ||
|classic major | |classic major sixth, diminished seventh | ||
| | |cM6, d7 | ||
|878.05 | |878.05 | ||
|[[5/3]] | |[[5/3]] | ||
| Line 629: | Line 651: | ||
|- | |- | ||
|35 | |35 | ||
|classic minor | |classic minor seventh, augmented sixth | ||
| | |Cm7, A6 | ||
|1024.39 | |1024.39 | ||
|[[9/5]], [[20/11]] | |[[9/5]], [[20/11]] | ||
| Line 641: | Line 663: | ||
|- | |- | ||
|37 | |37 | ||
|classic major | |classic major seventh, diminished octave | ||
| | |cM7, d8 | ||
|1082.93 | |1082.93 | ||
|[[15/8]] | |[[15/8]] | ||
|- | |- | ||
|38 | |38 | ||
|major seventh | |major seventh | ||
|M7 | |M7 | ||
|1112.20 | |1112.20 | ||
| Line 659: | Line 681: | ||
|- | |- | ||
|40 | |40 | ||
|comma-narrow octave/sub | |comma-narrow octave/sub octave | ||
| | |c8/s8 | ||
|1170.73 | |1170.73 | ||
|[[160/81]], 63/32 | |[[160/81]], 63/32 | ||
|- | |- | ||
|41 | |41 | ||
| Line 669: | Line 691: | ||
|1200 | |1200 | ||
|2/1 | |2/1 | ||
|}Secondary interval names are not available for every note without going into double and triple augmented and diminished intervals. I include only up to singly augmented and diminished and leave most secondary interval names out. Accordingly I do not write them in a separate column. We can see from the interval names that 64/63 and 81/80 are represented both by a single degree and the augmented unison by three, that it is a neutral tuning, and that it is a [[Schismatic]] tuning, where the diminished fourth approximates 5/4, and that it is at least an 11-limit tuning. | |} | ||
Secondary interval names are not available for every note without going into double and triple augmented and diminished intervals. I include only up to singly augmented and diminished and leave most secondary interval names out. Accordingly I do not write them in a separate column. We can see from the interval names that 64/63 and 81/80 are represented both by a single degree and the augmented unison by three, that it is a neutral tuning, and that it is a [[Schismatic]] tuning, where the diminished fourth approximates 5/4, and that it is at least an 11-limit tuning. | |||
{| class="wikitable" | {| class="wikitable" | ||
|+53edo (13-limit patent val) | |+53edo (13-limit patent val) | ||
| Line 686: | Line 711: | ||
|1 | |1 | ||
|comma-narrow unison/super unison | |comma-narrow unison/super unison | ||
| | |C1/S1 | ||
|22.64 | |22.64 | ||
|81/80, 64/63, 65/64, 50/49 | |81/80, 64/63, 65/64, 50/49 | ||
| Line 710: | Line 735: | ||
|5 | |5 | ||
|classic minor second, augmented unison | |classic minor second, augmented unison | ||
| | |Cm2, A1 | ||
|113.21 | |113.21 | ||
|16/15, 15/14 | |16/15, 15/14 | ||
| Line 716: | Line 741: | ||
|6 | |6 | ||
|super classic minor second | |super classic minor second | ||
| | |SCm2 | ||
|135.85 | |135.85 | ||
|14/13, 13/12, 27/25 | |14/13, 13/12, 27/25 | ||
| Line 722: | Line 747: | ||
|7 | |7 | ||
|sub classic minor second | |sub classic minor second | ||
| | |scM2 | ||
|158.49 | |158.49 | ||
|12/11, 11/10, 800/729 | |12/11, 11/10, 800/729 | ||
| Line 728: | Line 753: | ||
|8 | |8 | ||
|classic minor second, diminished third | |classic minor second, diminished third | ||
| | |cM2, d3 | ||
|181.13 | |181.13 | ||
|10/9 | |10/9 | ||
| Line 764: | Line 789: | ||
|14 | |14 | ||
|classic minor third, augmented second | |classic minor third, augmented second | ||
| | |Cm3, A2 | ||
|316.98 | |316.98 | ||
|6/5 | |6/5 | ||
| Line 770: | Line 795: | ||
|15 | |15 | ||
|super classic minor third | |super classic minor third | ||
| | |SCm3 | ||
|339.62 | |339.62 | ||
|11/9, 243/200 | |11/9, 243/200 | ||
| Line 776: | Line 801: | ||
|16 | |16 | ||
|sub classic major third | |sub classic major third | ||
| | |scM3 | ||
|362.26 | |362.26 | ||
|16/13, 100/81 | |16/13, 100/81 | ||
| Line 782: | Line 807: | ||
|17 | |17 | ||
|classic major third, diminished fourth | |classic major third, diminished fourth | ||
| | |cM3, d4 | ||
|384.91 | |384.91 | ||
|5/4 | |5/4 | ||
| Line 818: | Line 843: | ||
|23 | |23 | ||
|comma-wide fourth, augmented third | |comma-wide fourth, augmented third | ||
| | |C4, A3 | ||
|520.75 | |520.75 | ||
|27/20 | |27/20 | ||
| Line 824: | Line 849: | ||
|24 | |24 | ||
|super comma-wide fourth | |super comma-wide fourth | ||
| | |SC4 | ||
|543.40 | |543.40 | ||
|11/8, 15/11 | |11/8, 15/11 | ||
| Line 830: | Line 855: | ||
|25 | |25 | ||
|sub classic augmented fourth | |sub classic augmented fourth | ||
| | |scA4 | ||
|566.04 | |566.04 | ||
|18/13 | |18/13 | ||
| Line 836: | Line 861: | ||
|26 | |26 | ||
|classic augmented fourth, diminished fifth | |classic augmented fourth, diminished fifth | ||
| | |cA4, d5 | ||
|588.68 | |588.68 | ||
|7/5, 45/32 | |7/5, 45/32 | ||
| Line 842: | Line 867: | ||
|27 | |27 | ||
|classic diminished fifth, augmented fourth | |classic diminished fifth, augmented fourth | ||
| | |Cd5, A4 | ||
|611.32 | |611.32 | ||
|10/7, 64/45 | |10/7, 64/45 | ||
| Line 848: | Line 873: | ||
|28 | |28 | ||
|super classic diminished fifth | |super classic diminished fifth | ||
| | |SCd5 | ||
|633.96 | |633.96 | ||
|13/9 | |13/9 | ||
| Line 854: | Line 879: | ||
|29 | |29 | ||
|sub comma-narrow fifth | |sub comma-narrow fifth | ||
| | |sc5 | ||
|656.60 | |656.60 | ||
|16/11, 22/15 | |16/11, 22/15 | ||
| Line 860: | Line 885: | ||
|30 | |30 | ||
|comma-narrow fifth | |comma-narrow fifth | ||
| | |c5 | ||
|679.25 | |679.25 | ||
|40/27 | |40/27 | ||
| Line 896: | Line 921: | ||
|36 | |36 | ||
|classic minor sixth, augmented fifth | |classic minor sixth, augmented fifth | ||
| | |Cm6, A5 | ||
|815.09 | |815.09 | ||
|8/5 | |8/5 | ||
| Line 902: | Line 927: | ||
|37 | |37 | ||
|super classic minor sixth | |super classic minor sixth | ||
| | |SCm6 | ||
|837.74 | |837.74 | ||
|13/8, 81/50 | |13/8, 81/50 | ||
| Line 908: | Line 933: | ||
|38 | |38 | ||
|sub classic major sixth | |sub classic major sixth | ||
| | |scM6 | ||
|860.38 | |860.38 | ||
|18/11, 400/243 | |18/11, 400/243 | ||
| Line 914: | Line 939: | ||
|39 | |39 | ||
|classic major sixth, diminished seventh | |classic major sixth, diminished seventh | ||
| | |cM6, d7 | ||
|883.02 | |883.02 | ||
|5/3 | |5/3 | ||
| Line 950: | Line 975: | ||
|45 | |45 | ||
|classic minor seventh, augmented sixth | |classic minor seventh, augmented sixth | ||
| | |Cm7, A6 | ||
|1018.87 | |1018.87 | ||
|9/5 | |9/5 | ||
| Line 956: | Line 981: | ||
|46 | |46 | ||
|super classic minor seventh | |super classic minor seventh | ||
| | |SCm7 | ||
|1041.51 | |1041.51 | ||
|11/6, 20/11, 729/400 | |11/6, 20/11, 729/400 | ||
| Line 962: | Line 987: | ||
|47 | |47 | ||
|sub classic major seventh | |sub classic major seventh | ||
| | |scM7 | ||
|1064.15 | |1064.15 | ||
|13/7, 24/13, 50/27 | |13/7, 24/13, 50/27 | ||
| Line 968: | Line 993: | ||
|48 | |48 | ||
|classic major seventh, diminished octave | |classic major seventh, diminished octave | ||
| | |cM7, d8 | ||
|1086.79 | |1086.79 | ||
|15/8 | |15/8 | ||
| Line 992: | Line 1,017: | ||
|52 | |52 | ||
|comma-narrow octave/sub octave | |comma-narrow octave/sub octave | ||
| | |c8/S8 | ||
|1177.36 | |1177.36 | ||
|160/81, 63/32, 128/65 | |160/81, 63/32, 128/65 | ||
| Line 1,001: | Line 1,026: | ||
|1200 | |1200 | ||
|2/1 | |2/1 | ||
|}We can see from the interval names that 64/63, 81/80 and 65/64 are represented all by a single degree and the augmented unison by five, that it is a barbados tuning, and that it is a [[Schismatic]] tuning, where the diminished fourth approximates 5/4 and that it is at least a 2.3.5.7.13 tuning. | |} | ||
==Application in other edos== | We can see from the interval names that 64/63, 81/80 and 65/64 are represented all by a single degree and the augmented unison by five, that it is a barbados tuning, and that it is a [[Schismatic]] tuning, where the diminished fourth approximates 5/4 and that it is at least a 2.3.5.7.13 tuning. | ||
===5''n-''edos=== | |||
== Application in other edos == | |||
=== 5''n-''edos === | |||
On the limit for a diatonic scale of 5 large and 2 small steps is 5edo, where the small steps, the diatonic semitones are reduced to unisons. The diatonic interval names for 5edo are as follows: | On the limit for a diatonic scale of 5 large and 2 small steps is 5edo, where the small steps, the diatonic semitones are reduced to unisons. The diatonic interval names for 5edo are as follows: | ||
| Line 1,020: | Line 1,047: | ||
10edo: P1 N2 M2/m3 N3 P4 N4/N5 P5 N6 M6/m7 N7 P8 | 10edo: P1 N2 M2/m3 N3 P4 N4/N5 P5 N6 M6/m7 N7 P8 | ||
15edo: P1 | 15edo: P1 C1/Cm2 cM2 M2/m3 Cm3 cM3 P4 C4 c5 P5 Cm6 cM6 M6/m7 Cm7 cM7/c8 P8 | ||
The remaining 5''n-''edos are difficult, however. | The remaining 5''n-''edos are difficult, however. | ||
| Line 1,054: | Line 1,081: | ||
|0.15 | |0.15 | ||
|0.15 | |0.15 | ||
|}The second best mapping of 5 is better by this measure, so we may notate 20edo using this mapping: 20c ('c' here is called a wart, indicating the use of the second best approximations of the third prime, 5). | |} | ||
The second best mapping of 5 is better by this measure, so we may notate 20edo using this mapping: 20c ('c' here is called a wart, indicating the use of the second best approximations of the third prime, 5). | |||
20edo (20c): P1 | 20edo (20c): P1 C1/Cm2 N2 cM2 M2/m3 Cm3 N3 cM3 P4 C4 N4/N5 c5 P5 Cm6 N6 cM6 M6/m7 Cm7 N7 cM7/c8 P8. | ||
In 25edo 5/4 is two degrees below the M3, so the interval in-between does not have a separate function using the patent val in the 7-limit. In 25edo the approximation of 5 is excellent, so we check the second best approximations of 7 and 3. | In 25edo 5/4 is two degrees below the M3, so the interval in-between does not have a separate function using the patent val in the 7-limit. In 25edo the approximation of 5 is excellent, so we check the second best approximations of 7 and 3. | ||
| Line 1,095: | Line 1,123: | ||
|0.18 | |0.18 | ||
|0.82 | |0.82 | ||
|}The patent val, 25p performs best here. We may still use either 25b or 25d if we desire, however if we want to use 25p, we may narrow and wide prefixes to name the intervals that do not carry a separate function under this mapping: | |} | ||
The patent val, 25p performs best here. We may still use either 25b or 25d if we desire, however if we want to use 25p, we may narrow and wide prefixes to name the intervals that do not carry a separate function under this mapping: | |||
25edo: P1 WP1/Wm2 | 25edo: P1 WP1/Wm2 Cm2 cM2 nM2 M2/m3 Wm3 Cm3 cM3 nM3/n4 P4 W4 C4 c5 n5 P5 W5/nm6 Cm6 cM6 nM6 M6/m7 Wm7 Cm7 cM7 nM7/nP8 P8 | ||
In 25edo 81/80 is represented by 2 degrees rather than by a single degree, so our scheme doesn't completely work for 25edo, but our scheme is based on the diatonic scale, which in 25edo has pretty much completely broken down. | In 25edo 81/80 is represented by 2 degrees rather than by a single degree, so our scheme doesn't completely work for 25edo, but our scheme is based on the diatonic scale, which in 25edo has pretty much completely broken down. | ||
7edo | === 7''n''-edos === | ||
At the other limit, in 7edo the large and small steps of the diatonic scale are the same size, and the apotome is tempered out and therefore major and minor are equated with each other, and therefor with neutral: | |||
7edo: P1 N2 N3 P4 P5 N6 N7 P8, equivalent to Neutral[7] 3|3. | |||
It is easy to apply our scheme to 14edo: | It is easy to apply our scheme to 14edo: | ||
P1 S1/sm2 | P1 S1/sm2 N2 SM2/sm3 N3 SM3/s4 P4 SA4/sd5 P5 S5/sm6 N6 SM6/sm7 N7 SM7/s8 P8 | ||
We can see that 14edo in a Semaphore tuning, and therefore also a barbados tuning. From our secondary interval names: | We can see that 14edo in a Semaphore tuning, and therefore also a barbados tuning. From our secondary interval names: | ||
N1 1-2 m2/M2 2-3 m3/M3 3-4 N4 4-5 N5 5-6 m6/M6 6-7 m7/M7 7-8 N8, along with our first, we can see Samaphore[9] and Barbados[9] as subsets of 14edo. | |||
In 21edo, 81/80 is subtended by a single degree, but in the wrong direction. We use alterations of 64/63 to name the intervals below m and above M just as we do normally, however as these intervals are equivalent, and are also neutral, they are labelled neutral: | In 21edo, 81/80 is subtended by a single degree, but in the wrong direction. We use alterations of 64/63 to name the intervals below m and above M just as we do normally, however as these intervals are equivalent, and are also neutral, they are labelled neutral: | ||
21edo: P1 S1 sm2 | 21edo: P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8 | ||
In 28edo, 81/80 is also subtended by -1 degrees, but since 64/63 is subtended by 2 degrees we cannot label all of our intervals using 'S' and 's'. If we relabel P4 'hA4', and P5 'hd5', we can still build a well-ordered interval names set using alterations of 81/80: | |||
28edo: P1 cA1 SA1/sm2 Cm2 N2 cM2 SM2/sm3 Cm3 N3 cM3 SM3/s4 C4 hA4 cA4 SA4/sd5 Cd5 hd5 c5 S5/sm6 Cm6 N6 cM6 SM6/sm7 Cm7 N7 cM7 SM7/sd8 Cd8 P8 | |||
=== Super-flat edos === | |||
===Super-flat edos=== | There are edos whose best fifth is flatter even than 4\7. In such edos major intervals are smaller than minor intervals, augmented smaller than major and diminished larger than minor. We expand our definition of well-ordered intervals to include that within each degree... ≤ dd ≤ d ≤ m ≤ M ≤ A ≤ AA ≤ ... or ... ≤ dd ≤ d ≤ P ≤ A ≤ AA ≤ ..., and where sc _ ≤ s/c_ ≤ _ ≤ S/C_ ≤ SC _ (where '_' represents any of ... dd, d, m, (P), M, A, AA ...). In order to obtain well-ordered interval-name sets, we use enharmonic equivalences, replacing diatonic intervals with altered intervals. | ||
There are edos whose best fifth is flatter even than 4\7. In such edos major intervals are smaller than minor intervals, augmented smaller than major and diminished larger than minor. We expand our definition of well-ordered intervals to include that within each degree... ≤ dd ≤ d ≤ m ≤ M ≤ A ≤ AA ≤ ... or ... ≤ dd ≤ d ≤ P ≤ A ≤ AA ≤ ..., and where | |||
In super flat edos, the fifths are so flat that the major third, from four fifths approximates the classic minor third, 6/5 and the minor third approximates the classic major third, 5/4, tempering out 135/128, resulting in [[Mavila temperament]]. Mavila temperament can be defined in the 5-limit using the enharmonic equivalence cla M = m, where meantone can be defined by cM = M, and schismatic by cM''n'' = d''n+1'' (superpyth in 2.3.7 can be defined by SM = M). Mavila[7] 3|3 reads the same as Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8, however Mavila[9] 4|4 has diatonic interval names: | In super flat edos, the fifths are so flat that the major third, from four fifths approximates the classic minor third, 6/5 and the minor third approximates the classic major third, 5/4, tempering out 135/128, resulting in [[Mavila temperament]]. Mavila temperament can be defined in the 5-limit using the enharmonic equivalence cla M = m, where meantone can be defined by cM = M, and schismatic by cM''n'' = d''n+1'' (superpyth in 2.3.7 can be defined by SM = M). Mavila[7] 3|3 reads the same as Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8, however Mavila[9] 4|4 has diatonic interval names: | ||
| Line 1,133: | Line 1,164: | ||
Applying our enharmonic equivalences our primary well-ordered interval names for Mavila[9] 4|4 and 9edo are: | Applying our enharmonic equivalences our primary well-ordered interval names for Mavila[9] 4|4 and 9edo are: | ||
P1 M2 | P1 M2 Sm3 sM3 P4 P5 Sm6 sM6 m7 P8. | ||
11edo has diatonic interval names: | 11edo has diatonic interval names: | ||
| Line 1,141: | Line 1,172: | ||
Adding neutrals and applying enharmonic replacements our primary well-ordered interval names are: | Adding neutrals and applying enharmonic replacements our primary well-ordered interval names are: | ||
P1 | P1 Cm2 N2 N3 cM3 P4 P5 Cm6 N6 N7 cM7 P8, in which we can see Neutral[7] 3|3. | ||
16edo has diatonic interval names: | 16edo has diatonic interval names: | ||
| Line 1,147: | Line 1,178: | ||
P1 d1 M2 m2 M3 m3 A4 P4 d4/A5 P5 d5 M6 m6 M7 m7 A8 P8. | P1 d1 M2 m2 M3 m3 A4 P4 d4/A5 P5 d5 M6 m6 M7 m7 A8 P8. | ||
In 16edo 81/80 is represented by -1 degrees and 64/63 by 1 degree, so m = | In 16edo 81/80 is represented by -1 degrees and 64/63 by 1 degree, so m = SM = cM | ||
It's primary well-ordered interval names are: | It's primary well-ordered interval names are: | ||
P1 S1 | P1 S1 Cm2 cM2 Cm3 cM3 s4 P4 S4/s5 P5 S5 Cm6 cM6 Cm7 cM7 s8 P8 | ||
Similarly, the primary well-ordered interval names for 23edo are: | Similarly, the primary well-ordered interval names for 23edo are: | ||
P1 S1 1-2 | P1 S1 1-2 Cm2 cM2 2-3 Cm3 cM3 3-4 s4 P4 S4 s5 P5 S5 5-6 Sm6 sM6 6-7 Sm7 sM7 7-8 s8 P8, from which we can see it is a Barbados tuning. | ||
[[Category:Interval naming]] | |||
[[Category:Equal divisions of the octave]] | |||