# 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.

## Contents

## 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, to enable application to Pythagorean, Superpythagorean and other systems.

## 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 generated by 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.
- 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.
- 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' or 'klassisch', with short-form 'k', inferring a diminution by the syntonic comma, 81/80, whereas minor and diminished may also be given the prefix 'classic' or 'klassisch' but with short-hand 'K', 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. 'K' and 'k' are used instead of 'C' and 'c', and the German translation 'klassisch' invoked in order that interval names like Cm7 that are equivalent to common chord names are avoided.
- Perfect and neutral intervals (including hA and hd) may also be given the prefixes 'K' and 'k' 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. Translation of 'comma' into the German 'komma' is invoked as in 'klassische'.

- To extend to the 13-limit, we add that to kP, kM and kA 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 skM3. In tunings where the syntonic comma is tempered out, such that (kP, kM, kA, KP, Km, Kd) = (P, M, A, P, m, d), the 'k' and 'K' 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.

Long-form | Short-form | Alteration |
---|---|---|

perfect | P | none |

major | M | none |

minor | m | none |

augmented | A | up an apotome |

diminished | d | down an apotome |

super | S | up 64/63 (or 65/64) |

sub | s | down 64/63 (or 65/64) |

classic/klassisch | K | up 81/80 |

classic/klassisch | k | down 81/80 |

comma-wide/komma-wide | K | up 81/80 (or 99/98) |

comma-narrow/komma-wide | k | down 81/80 (or 99/98) |

wide | W | up a single degree |

narrow | n | 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
- Major, minor, A4 and d5.
- 'S', 's', 'K', 'K', 'SK' and 'sk' prefixes to tier 1-2 intervals
- Neutrals (including hAs and hds)
- 'S', 's', 'K', 'K', 'SK' and 'sk' prefixes to tier 4 intervals
- Intermediates
- Remaining augmented and diminished intervals
- 'S', 's', 'K', 'k', 'SK' and 'sk' 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 k4 or s4, s4 is preferred and where the same interval may be named K4 or S4, K4 is preferred. Similarly, where the same interval may be named K5 or S5, S5 is preferred and where the same interval may be named k5 or s5, k5 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 Km2 kM2 M2 m3 Km3 kM3 M3 P4 K4 kA4/Kd5 k5 P5 m6 Km6 kM6 M6 m7 Km7 kM7 M7 P8

24edo: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 hA4 A4/d5 N5 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 hA1 N2 kM2 M2 m3 Km3 N3 kM3 M3 P4 hA4 skA4/Kd5 kA4/SKd5 hd5 P5 m6 Km6 N6 kM6 M6 m7 Km7 N7 hd8 M7 P8

29edo: P1 K1/S1/sm2 m2 Km2 kM2 M2 SM2/sm3 m3 Km3 kM3 M3 SM3/s4 P4 K4 kA4/d5 A4/Kd5 k5 P5 S5/sm6 m6 Km6 kM6 M6 SM6/sm7 m7 Km7 kM7 M7 SM7/S8/k8 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 K1 m2 Km2 N2 kM2 M2 2-3 m3 Km3 N3 kM3 M3 3-4 P4 K4 N4/d5 kA4/Kd5 A4/N5 k5 P5 5-6 m6 Km6 N6 kM6 M6 6-7 m7 Km7 N7 kM7 M7 k8 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 K1/S1 sm2 m2 Km2 N2 kM2 M2 SM2 sm3 m3 Km3 N3 kM3 M3 SM3 s4 P4 K4 hA4 kA4 Kd5 hd5 k5 P5 S5 sm6 m6 Km6 N6 kM6 M6 SM6 sm7 m7 Km7 N7 kM7 M7 SM7 k8/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 K1/S1 sm2 m2 Km2 SKm2 skM2 sM2 M2 SM2 sm3 m3 Km3 SKm3 skM3 kM3 M3 SM3 s4 P4 K4 SK4 skA4/d5 kA4/Kd5 A4/SKd5 SA4/sk5 k5 P5 S5 sm6 m6 Km6 SKm6 skM6 sM6 M6 SM6 sm7 m7 Km7 SKm7 skM7 kM7 M7 SM7 k8/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 K1/S1 1-2 sm2 m2 Km2 SKm2 skM2 kM2 M2 SM2 2-3 sm3 m3 Km3 SKm3 skM3 kM3 M3 SM3 3-4 s4 P4 K4 SK4 skA4 kA4 Kd5 SKd5 sk5 k5 P5 S5 5-6 sm6 m6 Km6 SKm6 skM6 kM6 M6 SM6 6-7 sm7 m7 Km7 SKm7 skM7 kM7 M7 SM7 7-8 k8/s8 P8

72edo: P1 K1 S1 hA1 sm2 nm2 m2 Km2 kN2 N2 KN2 kM2 M2 WM2 SM2 2-3 sm3 nm3 m3 Km3 kN3 N3 KN3 kM3 M3 WM3 SM3 3-4 s4 k4 P4 K4 S4 hA4 KhA4 kA4 A4/d5 Kd5 khd5 hd5 s5 k5 P5 K5 S5 5-6 sm6 nm6 m6 Cm6 kN6 N6 KN6 kM6 M6 WM6 SM6 6-7 sm7 nm7 m7 Km7 kN7 N7 KN7 kM7 M7 WM7 SM7 hd8 s8 k8 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 'K' and 'k'.
- 29edo, 41edo, 46edo, 53edo and 72edo are Pythagorean tunings through the use of both 'S' and 's'; and 'K' and 'k'.
- 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.

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 SM6/sm7 m7 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 octave | S1 | 0 | 1/1, 64/63 |

1 | minor second | m2 | comma-wide unison / super minor second | K1/sm2 | 54.55 | 33/32, 34/33, 25/24, 81/80 |

2 | classic minor second | Km2 | diminished third | d3 | 109.09 | 18/17, 17/16, 16/15, 15/14 |

3 | classic major second | kM2 | 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 | Km3 | diminished fourth | d4 | 327.27 | 6/5, 17/14, 11/9 |

7 | classic major third | kM3 | 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 | K4 | diminished fifth | d5 | 545.45 | 11/8, 15/11, 27/20 |

11 | classic augmented fourth
classic diminished fifth |
kA4
Kd5 |
fourth-fifth | 4-5 (firth) | 600 | 7/5, 17/12, 45/32
10/7, 24/17, 64/45 |

12 | comma-narrow fifth | k5 | 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 | Km6 | diminished seventh | d7 | 818.18 | 8/5 |

16 | classic major sixth | kM6 | 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 | Km7 | diminished octave | d8 | 1036.36 | 20/11, 9/5 |

20 | classic major seventh | kM7 | augmented sixth | A6 | 1090.91 | 15/8, 32/17, 17/9, 28/15 |

21 | major seventh | M7 | super major seventh / comma-narrow octave | SM7/k8 | 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, comma-narrow hemi-augmented unison | K1/S1, khA1 | 29.27 | 81/80, 64/63, 55/54 |

2 | subminor second, hemi-augmented unison | sm2, hA1 | 58.54 | 25/24, 28/27, 33/32 |

3 | minor second, comma-wide hemi-augmented unison | m2 | 87.80 | 21/20, 22/21 |

4 | classic minor second, comma-narrow neutral second, augmented unison | Km2, kN2, A1 | 117.07 | 16/15, 15/14 |

5 | neutral second | N2 | 146.34 | 12/11 |

6 | classic major second, comma-wide neutral second, diminished third | kM2, KN2, 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, comma-narrow neural third, augmented second | Km3, kN3, A2 | 321.95 | 6/5 |

12 | neutral third | N3 | 351.22 | 11/9, 27/22 |

13 | classic major third, comma-high neutral third, diminished fourth | kM3, KN3, 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, comma-narrow hemi-augmented fourth, augmented third | K4, khA4, A3 | 526.83 | 15/11, 27/20 |

19 | hemi-augmented fourth | hA4 | 556.10 | 11/8 |

20 | classic augmented fourth, diminished fifth, comma-wide hemi-augmented fourth | kA4, d5 | 585.37 | 7/5, 45/32 |

21 | classic diminished fifth, augmented fourth, comma-narrow hemi-diminished fifth | Kd5, A4 | 614.63 | 10/7, 64/45 |

22 | hemi-diminished fifth | hd5 | 643.90 | 16/11 |

23 | comma-narrow fifth, comma-wide hemi-diminished fifth, diminished sixth | k5, Khd5 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, comma-narrow neutral sixth, augmented fifth | Km6, kN6, A5 | 819.51 | 8/5 |

29 | neutral sixth | N6 | 848.78 | 18/11, 44/27 |

30 | classic major sixth, comma-wide neutral sixth, diminished seventh | kM6, KN6, 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, comma-narrow neutral seventh, augmented sixth | Km7, kN7, A6 | 1024.39 | 9/5, 20/11 |

36 | neutral seventh | N7 | 1053.66 | 11/6 |

37 | classic major seventh, comma-wide neutral seventh, diminished octave | kM7, KN7, d8 | 1082.93 | 15/8 |

38 | major seventh, comma-narrow hemi-diminished octave | 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, comma-wide hemi-diminished octave | k8/s8, Khd8 | 1170.73 | 160/81, 63/32, 108/55 |

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 | K1/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 | Km2, A1 | 113.21 | 16/15, 15/14 |

6 | super classic minor second | SKm2 | 135.85 | 14/13, 13/12, 27/25 |

7 | sub classic minor second | skM2 | 158.49 | 12/11, 11/10, 800/729 |

8 | classic minor second, diminished third | kM2, 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 | Km3, A2 | 316.98 | 6/5 |

15 | super classic minor third | SKm3 | 339.62 | 11/9, 243/200 |

16 | sub classic major third | skM3 | 362.26 | 16/13, 100/81 |

17 | classic major third, diminished fourth | kM3, 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 | K4, A3 | 520.75 | 27/20 |

24 | super comma-wide fourth | SK4 | 543.40 | 11/8, 15/11 |

25 | sub classic augmented fourth | skA4 | 566.04 | 18/13 |

26 | classic augmented fourth, diminished fifth | kA4, d5 | 588.68 | 7/5, 45/32 |

27 | classic diminished fifth, augmented fourth | Kd5, A4 | 611.32 | 10/7, 64/45 |

28 | super classic diminished fifth | SKd5 | 633.96 | 13/9 |

29 | sub comma-narrow fifth | sk5 | 656.60 | 16/11, 22/15 |

30 | comma-narrow fifth | k5 | 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 | Km6, A5 | 815.09 | 8/5 |

37 | super classic minor sixth | SKm6 | 837.74 | 13/8, 81/50 |

38 | sub classic major sixth | skM6 | 860.38 | 18/11, 400/243 |

39 | classic major sixth, diminished seventh | kM6, 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 | Km7, A6 | 1018.87 | 9/5 |

46 | super classic minor seventh | SKm7 | 1041.51 | 11/6, 20/11, 729/400 |

47 | sub classic major seventh | skM7 | 1064.15 | 13/7, 24/13, 50/27 |

48 | classic major seventh, diminished octave | kM7, 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 | k8/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

### 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:

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 5*n*-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 K1/Km2 kM2 M2/m3 Km3 kM3 P4 K4 k5 P5 Km6 kM6 M6/m7 Km7 kM7/k8 P8

The remaining 5*n-*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 K1/Km2 N2 kM2 M2/m3 Km3 N3 kM3 P4 K4 N4/N5 k5 P5 Km6 N6 kM6 M6/m7 Km7 N7 kM7/k8 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 Km2 kM2 nM2 M2/m3 Wm3 Km3 kM3 nM3/n4 P4 W4 K4 k5 n5 P5 W5/nm6 Km6 kM6 nM6 M6/m7 Wm7 Km7 kM7 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.

### 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 therefore with neutral:

7edo has primary interval names: P1 m2/M2 m3/M3 P4 P5 m6/M6 m7/M7 P8, but may alternatively be written as Neutral[7] 3|3: P1 N2 N3 P4 P5 N6 N7 P8.

It is easy to apply our scheme to 14edo:

P1 S1/sm2 m2/M2 SM2/sm3 m3/M3 SM3/s4 P4 SA4/sd5 P5 S5/sm6 m6/M6 SM6/sm7 m7/M7 SM7/s8 P8

We can see that 14edo in a Semaphore tuning, and therefore also a barbados tuning. From our secondary interval names:

A1 1-2 N2 2-3 N3 3-4 A4 4-5 d5 5-6 N6 6-7 N7 7-8 d8, 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 m2/M2 SM2 sm3 m3/M3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 m6/M6 SM6 sm7 m7/ 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 use neutrals (hA4 and hd5) then we can still build a well-ordered interval names set using alterations of 81/80:

28edo: P1 kA1 SA1/sm2 Km2 N2 kM2 SM2/sm3 Km3 N3 kM3 SM3/s4 K4 hA4 kA4 SA4/sd5 Kd5 hd5 k5 S5/sm6 Km6 N6 kM6 SM6/sm7 Km7 N7 kM7 SM7/sd8 Kd8 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 sk _ ≤ s/k_ ≤ _ ≤ S/K_ ≤ SK _ (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 kM = m, where meantone can be defined by kM = M, and schismatic by kM*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:

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 Km3 kM3 P4 P5 Km6 kM6 m7 P8.

11edo has diatonic interval names:

P1 M2 M3 m2 m3 P4 P5 M6 M7 m6 m7 P8.

Even though major is now below minor, we may still give the name 'neural' to an interval half way between major and minor with an interval class. Adding neutrals and applying enharmonic replacements our primary well-ordered interval names are:

P1 Km2 N2 N3 kM3 P4 P5 Km6 N6 N7 kM7 P8, in which we can see Neutral[7] 3|3.

Where 11edo's best fifth is 47c flat, most would not really call it a P5. Accordingly we may wish instead to label 11edo as every second step of 22edo, where we still get major seconds and thirds, and minor sixths and sevenths:

P1 Km2 M2 Km3 M3 K4 k5 m6 kM6 m7 kM7 P8.

Since this labeling is based on a much more accurate fifth, the interval names mach their size much more closely, e.g. using 11edo's fifth, 2\11 was labelled a N2, but at 218c, is much closer to a M2, as it is labeled using 22edo's fifth. Since 11edo is abnormally flat even for Mavila, I suggest labeling using 22edo's fifth.

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 = SM

It's primary well-ordered interval names are:

P1 S1 Km2 kM2 Km3 kM3 s4 P4 S4/s5 P5 S5 Km6 kM6 Km7 kM7 s8 P8

Similarly, the primary well-ordered interval names for 23edo are:

P1 S1 1-2 Km2 kM2 2-3 Km3 kM3 3-4 s4 P4 S4 s5 P5 S5 5-6 Km6 kM6 6-7 Km7 kM7 7-8 s8 P8, from which we can see it is a Barbados tuning.