by David C Keenan, 29-Nov-1999
last updated 3-Nov-2001
http://dkeenan.com
by David C Keenan, 29-Nov-1999
last updated 3-Nov-2001
http://dkeenan.com
Acknowledgements
I pieced together the following scheme for naming 11-limit intervals by taking various people's casual usage on the alternative tuning list, and the English names that Manuel Op de Coul uses in Scala (http://www.tiac.net/users/xen/scala), and then pushing and prodding them until I obtained a reasonably consistent and self-explanatory system. I'm grateful to Paul Erlich for correcting a major misconception in an earlier draft, and Manuel Op de Coul for correcting a minor one (puns intended). Paul also informed me that it was very similar to the system proposed by Adriaan D. Fokker in New Music with 31 Notes (translated 1975), which I then assumed was where most of it originally came from. Manuel sent me a list of (English translations of) Fokker's names for the 31-tET intervals. The system in this document is now, simply a more up-to-date translation of Fokker's system, but applied and extended to 11-limit just intonation. It gives the standard names for the 5-limit intervals of the diatonic scale (and its few ratios of 9), and then extends them in a consistent way to all ratios of 7, 9 and 11, and their inversions. Please let me know if I've made any mistakes, or if you think the system can be improved somehow.
Since informing the alternative tuning list about this document I have learnt that Graham Breed has a similar page at http://www.microtonal.co.uk/31eq.htm. I am also grateful to Graham for suggesting some improvements.
Introduction
Consider the saturated 11-limit otonal chord whose extended ratio is 4:5:6:7:9:11, and consider it as a stack of different kinds of third.
The kinds of third are:
4 : 5 : 6 : 7 : 9 : 11 | major | minor | subminor | supermajor| neutral | | third | third | third | third | third |A purist might say that it should be supramajor, not supermajor, but super and supra mean the same as far as I can tell, and super is more familiar and easier to pronounce.
Now consider every stack of two thirds as a kind of fifth.
| subdiminished | augmented | | fifth | fifth | 4 : 5 : 6 : 7 : 9 : 11 | perfect | perfect | | fifth | fifth |You might be a little surprised to see the 5:7 referred to as a subdiminished fifth, but that's the only way to make it all work. The diminished fifth is its inversion, the 7:10. Of course that means that the 5:7 is also an augmented fourth, but that's ok. Intervals can have more than one name.
Now we look at the sevenths:
| subminor | | seventh | 4 : 5 : 6 : 7 : 9 : 11 | minor | | seventh | | neutral | | seventh |If you are interested in why I use this ratio notation, e.g. 5:7, instead of 7:5 or the fraction notation 7/5, see A note on mathematical notation for musical intervals.
The system
Now we are in a position to consider what I mean by consistent and why the non-diatonic intervals must have the names shown. Consistent means that we want to be able to assign simple index numbers to the various prefixes such as subminor, neutral, diminished etc, in such a way that they obey the ordinary rules of addition when stacking intervals. The first problem is that the terms diminished and augmented have different indexes depending on whether they are applied to intervals that admit of major and minor varieties versus those that admit of perfect varieties. Here's a scheme that copes with that. (In each column, the parenthesised prefix is the one that is implied when there is no prefix.)
Index Prefix for Prefix for unisons, fourths, seconds, thirds, fifths, octaves sixths, sevenths, ninths ----- ----------------- ----------------- -4 double diminished subdiminished -3 subdiminished diminished -2 diminished subminor -1 sub minor 0 (perfect) neutral +1 super (major) +2 augmented supermajor +3 superaugmented augmented +4 double augmented superaugmentedYou can verify that adding the indexes works for the thirds, fifths and sevenths shown above, and gives sensible names for many others. Note that a change of 2 in an index corresponds to a change of a chromatic semitone, e.g. F to F#. This scheme also works for inversions. For an inversion one changes the sign of the index. e.g. The inversion of a subminor third (6:7) is a supermajor sixth (7:12). The inversion of a minor seventh (5:9 or 9:16) is a major second (8:9 or 9:10), also called a whole tone or simply a tone.
Note that, when prefixed to the term tone (or whole tone), the terms major and minor have been usurped by history to distinguish the two varieties differing only by a syntonic comma (8:9 major, 9:10 minor), both of which are major seconds. A similar but more confused situation exists for diatonic semitones and smaller intervals. In every other interval category major and minor differ by a chromatic semitone.
The inversion of a major seventh (8:15) is a minor second (15:16 or 14:15) or diatonic semitone. The octave extension of any interval retains the same prefix. e.g. The octave extension of a super fourth (8:11) is a super eleventh (4:11).
Unfortunately this simple scheme of adding indices comes unstuck when we look at ninths. Ninths are octave extensions of seconds, and so for example, a 4:9 should be called a major ninth, as indeed it has been historically. However a 4:9 can also be constructed as a stack of two 2:3's (perfect fifths), or a 4:5 (major third) and 5:9 (minor seventh) or several other stacks. Note that perfect plus perfect is 0 + 0 = 0, and major plus minor is +1 + -1 = 0, so adding these indexes would have 4:9 as a neutral ninth. This is not correct.
In our favourite chord we have:
| major | | ninth | 4 : 5 : 6 : 7 : 9 : 11 | neutral | | ninth |It turns out that to make it all work the index numbers for the various prefixes need fractional corrections, by different amounts for different interval classes. This involves some messy arithmetic that really isn't necessary to appreciate this system, so feel free to skip the table below and the following paragraph. If you're still with me, it goes like this:
Interval class Correction to index ------------------- ---------- unisons and octaves 0 seconds and ninths -3/7 thirds +1/7 fourths -2/7 fifths +2/7 sixths -1/7 sevenths +3/7These small corrections can accumulate until they make a whole index step. Lets look at stacking the major third and minor seventh again, with these corrections. Now major corresponds to +1 but for thirds we have a +1/7 correction making it 8/7. Minor corresponds to -1 but for sevenths we have a +3/7 correction, making it -4/7. So the index for the corresponding ninth is 8/7 - 4/7 = 4/7. But this corresponds to 1 - 3/7 so we have an uncorrected index of 1 for the ninth, making it a major. The right answer this time.
Despite the context-sensitivity of the terms diminished and augmented, their usage in this system is a serious improvement. Historically it seems they were far more ambiguous. Diminished has always meant reduced in width from the perfect or the minor (whichever exists), and augmented means increased from the perfect or the major, but the actual change in width could be (a) some comma or diesis (index +/-1) (the harmonic or septimal case), or (b) a chromatic semitone (index +/-2) (the classic, meantone or unqualified case), or (c) a diatonic semitone or apotome (index +/-3) (the Pythagorean case). Fokker and I have restricted the usage to case (b) and used the terms super and sub for (a). I suggest superaugmented and subdiminished for (c).
Why does this system work so well? As you may have guessed, it's because it is consistent with 31-tET, as follows. The least-preferred names are in parenthesis. Context will often dictate the use of a less-preferred term, such as the subdiminished fifth in the chord used in the introduction.
31-tET Ratios Names degree ----- ---------- ---------------------------------------------------------------- 0 1:1 ..................... unison 1 48:49 44:45 35:36 32:33 (dimin. second) (super unison) diesis 2 27:28 24:25 20:21 (subminor second) (augmented unison) chromatic semitone 3 15:16 14:15 (minor second) ..................... diatonic semitone 4 11:12 10:11 neutral second 5 9:10 8:9 major second ..................... (whole tone) 6 7:8 supermajor second (diminished third) 7 6:7 (augmented second) subminor third 8 5:6 ..................... minor third 9 9:11 ..................... neutral third 10 4:5 (subdiminished fourth) major third 11 11:14 7:9 (diminished fourth) supermajor third 12 16:21 sub fourth (augmented third) 13 3:4 perfect fourth 14 8:11 super fourth 15 5:7 augmented fourth (subdiminished fifth) 16 7:10 (superaugmented fourth) diminished fifth 17 11:16 ..................... sub fifth 18 2:3 ..................... perfect fifth 19 21:32 (diminished sixth) super fifth 20 9:14 7:11 subminor sixth (augmented fifth) 21 5:8 minor sixth (superaugmented fifth) 22 8:13 11:18 neutral sixth 23 3:5 major sixth 24 7:12 supermajor sixth (diminished seventh) 25 4:7 (augmented sixth) subminor seventh 26 9:16 5:9 ..................... minor seventh 27 11:20 6:11 ..................... neutral seventh 28 8:15 ..................... major seventh 29 14:27 (diminished octave) supermajor seventh 30 18:35 sub octave (augmented seventh) 31 1:2 octave 32 22:45 16:33 super octave (diminished ninth) 33 12:25 10:21 (augmented octave) subminor ninth 34 15:32 7:15 ..................... minor ninth 35 11:24 5:11 ..................... neutral ninth 36 9:20 4:9 ..................... major ninth 37 7:16 (diminished tenth) supermajor ninth 38 3:7 subminor tenth (augmented ninth) 39 5:12 minor tenth 40 9:22 neutral tenth 41 2:5 major tenth (subdiminished eleventh) 42 11:28 7:18 supermajor tenth (diminished eleventh) 43 8:21 (augmented tenth) sub eleventh 44 3:8 ..................... perfect eleventh 45 4:11 ..................... super eleventh 46 5:14 (subdiminished twelfth) augmented eleventh 47 7:20 diminished twelfth (superaugmented eleventh) 48 11:32 sub twelfth 49 1:3 perfect twelfthI have included the interval 8:13 as an "honorary 11-limit interval", because it is certainly a better-known representative of the category of neutral sixth than is 11:18.
You will notice that this simple 31-tET-related scheme does not distinguish certain 11-limit ratios from the inversions of others (or from higher limit ratios). When necessary, they can be further distinguished by the following prefixes which indicate the highest prime factor contained in the ratio.
Highest Prefix Prime ------- ----------- 3 Pythagorean 5 classic 7 septimal 11 undecimal 13 tridecimalWhen the highest prime is the same, we can use the terms small and large to distinguish them, but it's probably best just to give the ratio, e.g. the 20:21 chromatic semitone. In fact it's probably best to give the ratio, as well as the name, for any of these.
Here are some 11-limit intervals that need to be distinguished in this manner. Let me know if I've left out your favourite interval. They are listed in order of increasing width. (The words in parenthesis are optional.)
Ratio Name Alternative name ----- --------------------------------- -------------------------- 48:49 small septimal diesis (It's essential to give 44:45 small undecimal diesis the ratio as well, 35:36 (large) septimal diesis when "small" or "large" 32:33 (large) undecimal diesis are involved.) 27:28 (small) septimal chromatic semitone 24:25 (classic) chromatic semitone 20:21 large septimal chromatic semitone 15:16 (classic) diatonic semitone classic minor second 14:15 septimal diatonic semitone septimal minor second 11:12 (small) (undecimal) neutral second 10:11 large (undecimal) neutral second 9:10 classic (whole) tone classic major second 8:9 (Pythagorean) (whole) tone Pythagorean major second 11:14 undecimal supermajor third 7:9 (septimal) supermajor third 9:14 (septimal) subminor sixth 7:11 undecimal subminor sixth 8:13 tridecimal neutral sixth 11:18 undecimal neutral sixth 9:16 (Pythagorean) minor seventh 5:9 classic minor seventh 11:20 small (undecimal) neutral seventh 6:11 (large) (undecimal) neutral seventh 22:45 small (undecimal) super octave 16:33 (large) (undecimal) super octave 12:25 (classic) subminor ninth 10:21 septimal subminor ninth 15:32 (classic) minor ninth 7:15 septimal minor ninth 11:24 (small) (undecimal) neutral ninth 5:11 large (undecimal) neutral ninth 9:20 classic major ninth 4:9 (Pythagorean) major ninthOf course the point of this system isn't great long lists of interval names like the ones above, it's the fact that if you know the system you can work out the name from the ratio, or conversely work out the ratio given the name. This is really quite difficult for many of those in the table above, so one should always give the ratio as well.
The rule I have used for which interval can omit the qualifier (classic, septimal, small etc.) is that it is the one whose greatest prime factor (GPF) is the lowest, and if two have the same greatest prime factor, then it is the one whose greatest prime factor is raised to the lowest power, and if two have the same GPF to the same power then it is the one whose second-greatest prime factor is the lowest, and so on. I feel however, that we should make an exception for the 8:13 and 11:18 neutral sixths and insist that they both keep their qualifier.
Perhaps a better (and simpler) rule is that it is the one for which the two sides of the ratio sum to the lowest value. These two rules disagree for the chromatic semitones, diatonic semitones, neutral seconds, subminor sixths, minor sevenths, subminor ninths, minor ninths and neutral ninths. Let me know what you think. Can you think of a simple rule that corresponds more closely to common usage?
For a completely different (and perhaps more natural) way of distinguishing
nearby intervals see
http://dkeenan.com/Music/Miracle/MiracleIntervalNaming.txt
The term "tritone" really has no place in 11-limit just intonation or it's approximations, but if it is used I think it should apply to both 5:7 and 7:10 and nothing outside that range, although "sub tritone" or "super tritone" might make sense.
To distinguish the two we use:
5:7 small (septimal) tritone 7:10 large (septimal) tritone7:10 is also called Euler's tritone, but this is not self-explanatory so we do not use this name, just as we do not use the name "Ptolemy's second" for 10:11.
If we wish to refer to the actual tempered intervals in 31-tone equal temperament, we would use the prefix 31-tET unless it is made clear by the context.. This should also apply to approximations of these intervals in other temperaments.
Relationship to staff notation
Manuel suggested investigating how this system relates to the JI notation to be introduced in version 1.6 of Scala. Here are the modifiers needed for 11-limit.
/ comma sharp, 81/80 \ comma flat, 80/81 ) diesis sharp, 128/125 ( diesis flat, 125/128 7 septimal comma sharp, 64/63 L septimal comma flat, 63/64 ^ undecimal diesis sharp, 33/32 v undecimal diesis flat, 32/33The relationship is simply that 81/80 does not register at all and the other three all correspond to a change of 1 in the index of the prefix (e.g. minor to subminor). This is a result of the step size of 31-tET and the fact that it is a meantone.
To make each of the above modifiers correspond to a different number of steps and have the system be 11-limit consistent, we need to go to at least 80-tET. Better accuracy is obtained at 121-tET. But these would be overkill, if you simply want a name that tells you what the interval sounds like.
Further investigation
It remains to be seen how the system of this document relates to equal temperaments with more than 31 tones, such as 41, 53 and 72-tET and non-meantone equal temperaments with fewer tones, such as 22-tET. Certainly in the case of non-meantones (Pythagorean or super-Pythagorean) it does not give names that correspond to the names derived from the sharps and flats in a notation based on the chain of fifths.
References
Fokker, Adriaan D., translated by Leigh Jardine, New Music with 31 Notes, Verlag für Systematische Musikwissenschaft GmbH, Bonn, 1975.