FKH Extended-diatonic Interval Names
FKH (Fokker/Keenan/Hearne) Extended-diatonic Interval Names is a diatonic-based system for interval naming developed after 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 (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 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 WOFED interval names, and the later re-imagining SKULO interval names.
Introduction
It is possible for the best aspects of all interval naming systems to be employed in a single system, namely:
- Backwards compatibility with familiar diatonic interval-names (Fokker/Keenan, Miracle interval naming, Keenan's most recent, size-based systems)
- Conservation of interval arithmetic (Fokker/Keenan, Sagispeak, Ups and Downs)
- Generalisation across all small to medium ETs (Sagispeak, Keenan's most recent, Size-based systems, Ups and Downs)
- Consistency through translation across tunings (Sagispeak)
- Prefixes that imply augmentation and diminution by a single step of an ET (Fokker/Keenan, Ups and Downs)
- Possibility for well-ordered interval name sets (all proposals)
- More than one possible name for intervals (Fokker/Keenan, Miracle interval naming, Sagispeak, Keenan's most recent, Ups and Downs)
- Consistent mapping to Sagittal and HEWM notation (Sagispeak)
- 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, 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
- 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.
- 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 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'.
- 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 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 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.
- 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.
- 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.
| Long-form | Short-form | Mid-form | Alteration |
|---|---|---|---|
| perfect | P | Prf | none |
| major | M | Maj | none |
| minor | m | min | none |
| augmented | A | Aug | up an apotome |
| diminished | d | dim | down an apotome |
| super | S | Sub | up 64/63 (or 65/64) |
| sub | s | spr | down 64/63 (or 65/64) |
| classic | C | Cla | up 81/80 |
| classic | c | cla | down 81/80 |
| comma-wide | C | Co-W | up 81/80 (or 99/98) |
| comma-narrow | c | co-n | down 81/80 (or 99/98) |
| wide | W | Wde | up a single degree |
| narrow | n | nrw | down a single degree |
| Long-form | Short-form | Mid-form | Description |
|---|---|---|---|
| unison-second | 1-2 (unind) | 1-2 (unind) | half of the limma, half way between P1 and m2 |
| second-third | 2-3 (serd) | 2-3 (serd) | half of P4, halfway between M2 and m3 |
| third-fourth | 3-4 (thourth) | 3-4 (thourth) | half of M6, halfway between M3 and P4 |
| fourth-fifth | 4-5 (firth) | 4-5 (firth) | half of the octave, halfway between A4 and d5 |
| fifth-sixth | 5-6 (fixth) | 5-6 (fixth) | half of M10, halfway between P5 and m6 |
| sixth-seventh | 6-7 (sinth) | 6-7 (sinth) | halfway between P5 and P8, halfway between M6 and m7 |
| seventh-octave | 7-8 (sevtave) | 7-8 (sevtave) | halfway between M7 and P8 |
| neutral | N | N | halfway between M and m |
| hemi augmented | hA | h-Aug | halfway between P and A |
| hemi diminished | hd | h-dim | halfway between P and d |
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.
Interval names are ranked in 11 tiers:
- Perfect and neutral
- Major, minor, A4 and d5.
- 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
- Remaining augmented and diminished intervals
- '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 4-8 interval names
- Intervals augmented and diminished more than singularly, and 'n' and 'W' prefixes to these intervals.
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 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.
12edo: P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8
17edo: P1 m2 N2 M2 m3 N3 M3 P4 hA4 hd5 P5 m6 N6 M6 m7 N7 M7 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 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 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
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 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
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
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 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
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
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 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
- 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'.
- 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 '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 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.
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.
| Degree | Primary interval name | Short-form | Secondary interval name | Short-form | Cents | Approximate Ratios |
|---|---|---|---|---|---|---|
| 0 | perfect unison | P1 | super unison | S1 | 0 | 1/1, 64/63 |
| 1 | minor second | m2 | comma-wide unison / sub minor second | C1/sm2 | 54.55 | 33/32, 34/33, 25/24, 81/80 |
| 2 | classic minor second | Cm2 | diminished third | d3 | 109.09 | 18/17, 17/16, 16/15, 15/14 |
| 3 | classic major second | cM2 | augmented unison | A1 | 163.64 | 11/10, 10/9 |
| 4 | major second | M2 | super major second | SM2 | 218.18 | 9/8, 8/7, 17/15 |
| 5 | minor third | m3 | super minor third | sm3 | 272.73 | 7/6, 20/17 |
| 6 | classic minor third | Cm3 | diminished fourth | d4 | 327.27 | 6/5, 17/14, 11/9 |
| 7 | classic major third | cM3 | augmented second | A2 | 381.82 | 5/4 |
| 8 | major third | M3 | super major third | SM3 | 436.36 | 9/7, 14/11, 22/17 |
| 9 | perfect fourth | P4 | sub fourth | s4 | 490.91 | 4/3, 21/16 |
| 10 | comma-wide fourth | C4 | diminished fifth | d5 | 545.45 | 11/8, 15/11, 27/20 |
| 11 | classic augmented fourth
classic diminished fifth |
cA4
Cd5 |
fourth-fifth | 4-5 (firth) | 600 | 7/5, 17/12, 45/32
10/7, 24/17, 64/45 |
| 12 | comma-narrow fifth | c5 | augmented fourth | A4 | 654.55 | 16/11, 22/15, 40/27 |
| 13 | perfect fifth | P5 | super fifth | S5 | 709.09 | 3/2, 32/21 |
| 14 | minor sixth | m6 | sub minor sixth | sm6 | 763.64 | 11/7, 14/9, 17/11 |
| 15 | classic minor sixth | Cm6 | diminished seventh | d7 | 818.18 | 8/5 |
| 16 | classic major sixth | cM6 | augmented fifth | A5 | 872.73 | 5/3, 18/11, 28/17 |
| 17 | major sixth | M6 | super major sixth | SM6 | 927.27 | 12/7, 17/10 |
| 18 | minor seventh | m7 | sub minor seventh | sm7 | 981.82 | 7/4, 16/9, 30/17 |
| 19 | classic minor seventh | Cm7 | diminished octave | d8 | 1036.36 | 20/11, 9/5 |
| 20 | classic major seventh | cM7 | augmented sixth | A6 | 1090.91 | 15/8, 32/17, 17/9, 28/15 |
| 21 | major seventh | M7 | super major seventh / comma-narrow octave | SM7/c8 | 1145.45 | 33/17, 64/33, 48/25, 160/81 |
| 22 | perfect octave | P8 | sub octave | s8 | 1200 | 2/1, 63/32 |
2.3.5.7.11.17 patent val means using the best approximations to the 2nd, 3rd, 5th, 7th, 11th and 17th partials/harmonics for the interval names and approximated ratios. Only the 7-limit ratios are needed in this case for the interval names. This interval names in this table tell us what the 7-limit ratios do, that 64/63 in tempered out, meaning it is a superpythagorean tuning and that 81/80 and 25/24 are represented by a single degree. They also show us that the chromatic semitone or apotome is 3 degrees wide.
| Degrees | Interval names | Short-form | Cents | Approximated Ratios |
|---|---|---|---|---|
| 0 | perfect unison | P1 | 0.00 | 1/1 |
| 1 | comma-wide unison/super unison | C1/S1 | 29.27 | 81/80, 64/63 |
| 2 | subminor second, hemi-augmented unison | sm2, hA1 | 58.54 | 25/24, 28/27, 33/32 |
| 3 | minor second | m2 | 87.80 | 21/20, 22/21 |
| 4 | classic minor second, augmented unison | Cm2, A1 | 117.07 | 16/15, 15/14 |
| 5 | neutral second | N2 | 146.34 | 12/11 |
| 6 | classic major second, diminished third | cM2, d3 | 175.61 | 10/9, 11/10 |
| 7 | major second | M2 | 204.88 | 9/8 |
| 8 | super major second | SM2 | 234.15 | 8/7 |
| 9 | sub minor third | sm3 | 263.41 | 7/6, 32/25 |
| 10 | minor third | m3 | 292.68 | 32/27 |
| 11 | classic minor third, augmented second | Cm3, A2 | 321.95 | 6/5 |
| 12 | neutral third | N3 | 351.22 | 11/9, 27/22 |
| 13 | classic major third, diminished fourth | cM3, d4 | 380.49 | 5/4 |
| 14 | major third | M3 | 409.76 | 14/11, 81/64 |
| 15 | super major third | SM3 | 439.02 | 9/7 |
| 16 | sub fourth | s4 | 468.29 | 21/16 |
| 17 | perfect fourth | P4 | 497.56 | 4/3 |
| 18 | comma-wide fourth, augmented third | C4, A3 | 526.83 | 15/11, 27/20 |
| 19 | hemi-augmented fourth | hA4 | 556.10 | 11/8 |
| 20 | classic augmented fourth, diminished fifth | cA4, d5 | 585.37 | 7/5, 45/32 |
| 21 | classic diminished fifth, augmented fourth | Cd5, A4 | 614.63 | 10/7, 64/45 |
| 22 | hemi-diminished fifth | h5 | 643.90 | 16/11 |
| 23 | comma-narrow fifth, diminished sixth | c5, d6 | 673.17 | 22/15, 40/27 |
| 24 | perfect fifth | P5 | 702.44 | 3/2 |
| 25 | super fifth | S5 | 731.71 | 32/21 |
| 26 | sub minor sixth | sm6 | 760.98 | 14/9, 25/16 |
| 27 | minor sixth | m6 | 790.24 | 11/7, 128/81 |
| 28 | classic minor sixth, augmented fifth | Cm6, A5 | 819.51 | 8/5 |
| 29 | neutral sixth | N6 | 848.78 | 18/11, 44/27 |
| 30 | classic major sixth, diminished seventh | cM6, d7 | 878.05 | 5/3 |
| 31 | major sixth | M6 | 907.32 | 27/16 |
| 32 | super major sixth | SM6 | 936.59 | 12/7 |
| 33 | sub minor seventh | sm7 | 965.85 | 7/4 |
| 34 | minor seventh | m7 | 995.12 | 16/9 |
| 35 | classic minor seventh, augmented sixth | Cm7, A6 | 1024.39 | 9/5, 20/11 |
| 36 | neutral seventh | N7 | 1053.66 | 11/6 |
| 37 | classic major seventh, diminished octave | cM7, d8 | 1082.93 | 15/8 |
| 38 | major seventh | M7 | 1112.20 | 40/21, 21/11 |
| 39 | super major seventh, hemi-diminished octave | SM7, hd8 | 1141.46 | 48/25, 27/14, 64/33 |
| 40 | comma-narrow octave/sub octave | c8/s8 | 1170.73 | 160/81, 63/32 |
| 41 | perfect octave | P8 | 1200 | 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.
| Degree | Interval names | Short-form | Cents | Approximate Ratios |
|---|---|---|---|---|
| 0 | perfect unison | P1 | 0.00 | 1/1 |
| 1 | comma-narrow unison/super unison | C1/S1 | 22.64 | 81/80, 64/63, 65/64, 50/49 |
| 2 | unison-second | 1-2 | 45.28 | 49/48, 36/35, 33/32, 128/125 |
| 3 | sub minor second | sm2 | 67.92 | 28/27, 27/26, 26/25, 25/24, 22/21 |
| 4 | minor second | m2 | 90.57 | 21/20, 256/243 |
| 5 | classic minor second, augmented unison | Cm2, A1 | 113.21 | 16/15, 15/14 |
| 6 | super classic minor second | SCm2 | 135.85 | 14/13, 13/12, 27/25 |
| 7 | sub classic minor second | scM2 | 158.49 | 12/11, 11/10, 800/729 |
| 8 | classic minor second, diminished third | cM2, d3 | 181.13 | 10/9 |
| 9 | major second | M2 | 203.77 | 9/8 |
| 10 | super major second | SM2 | 226.42 | 8/7, 256/225 |
| 11 | second-third | 2-3 | 249.06 | 15/13, 144/125 |
| 12 | sub minor third | sm3 | 271.70 | 7/6, 75/64 |
| 13 | minor third | m3 | 294.34 | 13/11, 32/27 |
| 14 | classic minor third, augmented second | Cm3, A2 | 316.98 | 6/5 |
| 15 | super classic minor third | SCm3 | 339.62 | 11/9, 243/200 |
| 16 | sub classic major third | scM3 | 362.26 | 16/13, 100/81 |
| 17 | classic major third, diminished fourth | cM3, d4 | 384.91 | 5/4 |
| 18 | major third | M3 | 407.55 | 81/64 |
| 19 | super major third | SM3 | 430.19 | 9/7, 14/11 |
| 20 | third-fourth | 3-4 | 452.83 | 13/10, 125/96 |
| 21 | sub fourth | s4 | 475.47 | 21/16, 675/512, 320/243 |
| 22 | perfect fourth | P4 | 498.11 | 4/3 |
| 23 | comma-wide fourth, augmented third | C4, A3 | 520.75 | 27/20 |
| 24 | super comma-wide fourth | SC4 | 543.40 | 11/8, 15/11 |
| 25 | sub classic augmented fourth | scA4 | 566.04 | 18/13 |
| 26 | classic augmented fourth, diminished fifth | cA4, d5 | 588.68 | 7/5, 45/32 |
| 27 | classic diminished fifth, augmented fourth | Cd5, A4 | 611.32 | 10/7, 64/45 |
| 28 | super classic diminished fifth | SCd5 | 633.96 | 13/9 |
| 29 | sub comma-narrow fifth | sc5 | 656.60 | 16/11, 22/15 |
| 30 | comma-narrow fifth | c5 | 679.25 | 40/27 |
| 31 | perfect fifth | P5 | 701.89 | 3/2 |
| 32 | super fifth | S5 | 724.53 | 32/21, 243/160, 1024/675 |
| 33 | fifth-sixth | 5-6 | 747.17 | 20/13, 192/125 |
| 34 | sub minor sixth | sm6 | 769.81 | 14/9, 25/16, 11/7 |
| 35 | minor sixth | m6 | 792.45 | 128/81 |
| 36 | classic minor sixth, augmented fifth | Cm6, A5 | 815.09 | 8/5 |
| 37 | super classic minor sixth | SCm6 | 837.74 | 13/8, 81/50 |
| 38 | sub classic major sixth | scM6 | 860.38 | 18/11, 400/243 |
| 39 | classic major sixth, diminished seventh | cM6, d7 | 883.02 | 5/3 |
| 40 | major sixth | M6 | 905.66 | 22/13, 27/16 |
| 41 | super major sixth | SM6 | 928.30 | 12/7 |
| 42 | sixth-seventh | 6-7 | 950.94 | 26/15, 125/72 |
| 43 | sub minor seventh | sm7 | 973.58 | 7/4 |
| 44 | minor seventh | m7 | 996.23 | 16/9 |
| 45 | classic minor seventh, augmented sixth | Cm7, A6 | 1018.87 | 9/5 |
| 46 | super classic minor seventh | SCm7 | 1041.51 | 11/6, 20/11, 729/400 |
| 47 | sub classic major seventh | scM7 | 1064.15 | 13/7, 24/13, 50/27 |
| 48 | classic major seventh, diminished octave | cM7, d8 | 1086.79 | 15/8 |
| 49 | major seventh | M7 | 1109.43 | 40/21, 243/128 |
| 50 | super major seventh | SM7 | 1132.08 | 48/25, 27/14 |
| 51 | seventh-octave | 7-8 | 1154.72 | 125/64 |
| 52 | comma-narrow octave/sub octave | c8/S8 | 1177.36 | 160/81, 63/32, 128/65 |
| 53 | perfect octave | P8 | 1200 | 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
5n-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:
P1/m2 M2/m3 M3/P4 P5/m6 M6/m7 M7/P8
Our primary interval names are P1 M2/m3 P4 P5 M6/m7 P8.
Where the M6/m7 represents both 7/4 and 12/7, we know that 5edo is a superpythagorean tuning, tempering out 64/63, and a semaphore tuning, tempering out 49/48. It is therefore also a barbados tuning, tempering out 676/675. We may write 5edo then as
P1 SM2/sm3 P4 P5 SM6/m7 P8 to express it as a semaphore tuning (equivalent to Semaphore[5] 2|2), or
P1 2-3 P4 P5 6-7 P8 to express it as a barbados tuning (equivalent to Barbados[5] 2|2), where secondary names for P1 are sm2 and 1-2 respectively, etc.
Up to 30edo, for all 5n-edos the 3\5 fifth (3 degrees of 5edo) is the best fifth. 10edo and 15edo may be easily named:
10edo: P1 N2 M2/m3 N3 P4 N4/N5 P5 N6 M6/m7 N7 P8
15edo: P1 C1/Cm2 cM2 M2/m3 Cm3 cM3 P4 C4 c5 P5 Cm6 cM6 M6/m7 Cm7 cM7/c8 P8
The remaining 5n-edos are difficult, however.
In the 13-limit, 20edo's patent val maps only to notes of 10edo, so only half the notes are available, while the fifths of 10edo are very sharp and 5/4 rather flat, we might wonder if using the less well approximated sharp third might be better. We can test this most simply by finding the 7-odd limit interval (interval consisting of no odd number greater than 7) with the highest error for either mapping. For the patent mapping of 5 and the second best mapping of five, the error associated with the intervals of the 7-odd limit are as follows: (only the intervals in the first half of the octave are included, as the intervals in the top half of a purely tuned octave contain exactly the same error as their octave-inverses.
| Interval | Error patent (degrees) | Error alternative (degrees) |
|---|---|---|
| 4/3 | 0.30 | 0.30 |
| 5/4 | 0.44 | 0.66 |
| 6/5 | 0.74 | 0.36 |
| 7/5 | 0.29 | 0.71 |
| 7/6 | 0.45 | 0.45 |
| 8/7 | 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).
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.
| Interval | Error 25p (degrees) | Error 25b (degrees) | Error 25d (degrees) |
|---|---|---|---|
| 4/3 | 0.38 | 0.62 | 0.38 |
| 5/4 | 0.05 | 0.05 | 0.05 |
| 6/5 | 0.42 | 0.58 | 0.42 |
| 7/5 | 0.14 | 0.14 | 0.76 |
| 7/6 | 0.56 | 0.44 | 0.44 |
| 8/7 | 0.18 | 0.18 | 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:
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.
7n-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:
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:
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:
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
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.
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 cMn = dn+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:
P1 M2 M3 m3 P4 P5 M6 m6 m7 P8.
9edo has diatonic interval names:
P1 M2 m2/M3 m3 P4 P5 M6 m6/M7 m7 P8.
Applying our enharmonic equivalences our primary well-ordered interval names for Mavila[9] 4|4 and 9edo are:
P1 M2 Sm3 sM3 P4 P5 Sm6 sM6 m7 P8.
11edo has diatonic interval names:
P1 M2 M3 m2 m3 P4 P5 M6 M7 m6 m7 P8.
Adding neutrals and applying enharmonic replacements our primary well-ordered interval names are:
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:
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 = SM = cM
It's primary well-ordered interval names are:
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:
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.