TAMNAMS/Appendix

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This page is an appendix to TAMNAMS.

Reasoning for step ratio names

Derivation of the step ratio names

The idea is to start with the simplest ratios (L/s = 1/0 and L/s = 1/1) and derive more complex ratios through repeated application of the mediant (aka Farey addition) to adjacent fractions.

  • Applying the mediant to the starting intervals 1/0 and 1/1 gives (1+1)/(1+0) = 2/1, and as this is the simplest possible ratio where the large and small step are distinguished and nonzero, it is called the basic tuning. (Note that if applying the mediant to 1/0 seems confusing, think of it as equivalent to applying the mediant to 0/1 and 1/1 and the ratios as flipped, thus representing s/L rather than L/s when written this way.)
  • As L/s = 1/1 represents L and s being equal in size, it is called equalized.
  • As L/s = 1/0 represents s = 0, it is called collapsed, as the small scale steps collapse to zero cents and disappear.
  • The mediant of 1/1 and 2/1 is 3/2, thus making the scale sound mellower/softer, and as this is the simplest (in the sense of lowest integer limit) ratio to represent such a property, it is simply called the soft tuning.
  • Analogously, the mediant of 2/1 and 1/0, 3/1, is called the hard tuning. Thus you can say that a step ratio tuning is hard of or soft of another step ratio tuning.
  • To get something between soft and basic we take the mediant again and get 5/3 for semisoft, and analogously 5/2 for semihard. To get something more extreme we take the mediant of 1/0 with 3/1 for a harder-than-hard tuning, giving us 4/1 for superhard and analogously 4/3 for supersoft.

There are also tertiary names beyond the above:

  • Anything softer than supersoft is ultrasoft, and anything harder than superhard is ultrahard. Something between soft and supersoft is parasoft, as para- means both beyond and next to. Something between hard and superhard is parahard.
  • Something between soft and basic is hyposoft as it is less soft than soft. Something between hard and basic is hypohard for the same reason. Between semisoft and basic is minisoft and between semihard and basic is minihard.
  • Finally, between soft and semisoft is quasisoft as such scales may potentially be mistaken for soft or semisoft while not being either - hence the use of the prefix quasi-, and between hard and semihard is quasihard for the same reason.

The reasoning for the para- super- ultra- progression (note that super- is the odd one out as it refers to an exact ratio) is it mirrors naming for shades of musical intervals and because parapythagorean is between pythagorean and superpythagorean.

This results in the central spectrum - an elegant system which names all exact L/s ratios in the 5-integer-limit excepting only 5/1 and 5/4 which are disincluded intentionally for a variety of reasons: to keep the maximum corresponding notes per period in an equal pitch division low, because it keeps the 'tree' of mediants complete to a certain number of layers, and because their disinclusion gives a roughly-equally-spaced set of ratios, with the regions between 4/3 and 1/1 and between 4/1 and 1/0 being the only exceptions - corresponding to extreme tunings. Note that filling in those extreme regions is the purpose of the extended spectrum.

Extending the spectrum's edges

Extending the spectrum builds on the central spectrum and relies on a few key observations.

Firstly, as periods and mosses come in wildly different shapes and sizes, and as we want to represent a somewhat representative variety of simple tunings for the step ratio for a given mos pattern and period, the notion of simple used will correspond to the number of equally-spaced tones per period required. This is expressed as [number of large steps in pattern]*L + [number of small steps in pattern]*s, where L and s are from the step ratio itself, L/s, and are assumed to be coprime. Then, in order to not introduce bias to mos patterns with more L's or more s's, we should assume that both are equally likely and thus weight both equally, which means that the resulting minimum number of tones per period for a ratio L/s is L+s.

The next observation is that the large values of L/s can be a lot more consequential than the ones close to 1/1 due to the fact that small steps are guaranteed to be smaller than large steps and that we don't know how many small steps there are compared to large steps, and therefore the hard end of the spectrum is more vast, and analogously, L/s values close to 1/1 will tend to be inconsequential and for very close values likely impractical to distinguish - in the extremes only serving small tuning adjustments rather than melodic properties. This leads to another observation: mos patterns with periods tuned to step ratios, while related to temperaments, are not temperaments - instead forming a sort of amalgamative superset of temperaments if you want to force a temperament interpretation, and thus their main function is in melodic structure, with temperaments informing potential harmonies and microtunings. Thus, the spectrum should be kept minimal and simple so that it is both generally hearable and not too specific.

The most obvious adjustment to the edges is to draw a distinction between ultrasoft and pseudoequalized by adding a step ratio corresponding to semiequalized, and between ultrahard and pseudocollapsed by adding a step ratio corresponding to semicollapsed. Thus:

Ultrasoft is between supersoft and semiequalized and pseudoequalized is between semiequalized and equalized.

Ultrahard is between superhard and semicollapsed, and pseudocollapsed is between semicollapsed and collapsed.

Then all that's left is to decide what the step ratios for semicollapsed and semiequalized should be. In order to keep the spacing (of the s/L ratios when graphed, or to a lesser extent the L/s ratios if you see the roughly gradual increase in spacing in that form) roughly consistent with all the other ratios, semiequalized should be L/s = 6/5 rather than L/s = 5/4. Then note the complexity of L/s = 6/5 is 6+5=11, so to find the corresponding complexity for semicollapsed we use L/s = 10/1 as 10+1=11 too. Then finally, to preserve some of the symmetry, we include L/s = 6/1 as extrahard. Although L/s = 10/1 for semicollapsed may seem a little extreme of a boundary, L/s = 12/1 would actually be what is the most equally spaced continuing on from 6/1 for the same reason that L/s = 6/5 is the most equally spaced. Note that while the range from superhard to semicollapsed is ultrahard, the region may be split into two sub-ranges:

superhard (L/s=4/1) to extrahard (L/s=6/1) is hyperhard (4 < L/s < 6).

extrahard (L/s=6/1) to semicollapsed (L/s=10/1) is clustered (6 < L/s < 10).

With the inclusion of these 3 new L/s ratios nearer the edges of the spectrum and names for the range divisions they create, we get the extended spectrum, summarised and detailed above, just for the regions affected to avoid repetition.

Extended spectrum

Extended spectrum of step ratio ranges and specific step ratios
Central ranges Extended ranges Specific step ratios Notes
1:1 (equalized) Trivial/pathological
1:1 to 2:1 (soft-of-basic) 1:1 to 4:3 (ultrasoft) 1:1 to 6:5 (pseudoequalized)
6:5 (semiequalized)
6:5 to 4:3 (ultrasoft)
4:3 (supersoft) Nonextreme range, as detailed by central spectrum
4:3 to 3:2 (parasoft) 4:3 to 3:2 (parasoft)
3:2 (soft)
3:2 to 2:1 (hyposoft) 3:2 to 5:3 (quasisoft) 3:2 to 5:3 (quasisoft)
5:3 (semisoft)
5:3 to 2:1 (minisoft) 5:3 to 2:1 (minisoft)
2:1 (basic)
2:1 to 1:0 (hard-of-basic) 2:1 to 3:1 (hypohard) 2:1 to 5:2 (minihard) 2:1 to 5:2 (minihard)
5:2 (semihard)
5:2 to 3:1 (quasihard) 5:2 to 3:1 (quasihard)
3:1 (hard)
3:1 to 4:1 (parahard) 3:1 to 4:1 (parahard)
4:1 (superhard)
4:1 to 1:0 (ultrahard) 4:1 to 10:1 (ultrahard) 4:1 to 6:1 (hyperhard)
6:1 (extrahard)
6:1 to 10:1 (clustered)
10:1 (semicollapsed)
10:1 to 1:0 (pseudocollapsed)
1:0 (collapsed) Trivial/pathological

Terminology and final notes

A ratio of L/s = k/1 can be called k-hard and a ratio of L/s = k/(k-1) can analogously be called k-soft, so the simplest ultrasoft tuning is 5-soft or pentasoft, the simplest hyperhard tuning is 5-hard or pentahard, the simplest clustered tuning is 7-hard or heptahard, 8-hard is octahard, 9-hard is nonahard, and finally, the characteristic simple ultrahard tuning is 6-hard or extrahard, as previously discussed, which can be seen to be similar to hexahard - hopefully helping with memorisation.

A perhaps useful (or otherwise mildly amusing) mnemonic is 2-soft is too soft to be hard and 2-hard is too hard to be soft, representing that 2-soft = 2-hard = 2/1 = basic.

Note that often the central spectrum will be sufficient for exploring a mos pattern-period combination, and the extended spectrum is intended more for (literally) edge cases where it may be useful. Often if a temperament interpretation doesn't seem to show up for a mos pattern-period combination, it just means the temperament needs a more complex mos pattern to narrow down the generator range. An example of this phenomena is the highly complex mos pattern of 12L 17s represents near-Pythagorean tunings well due to having a generator of a fourth or a fifth bounded between those of 12edo and those of 29edo, which are roughly equally off but in opposite directions, and many important near-Pythagorean systems show up in just the ratios of the central spectrum alone.

Reasoning for mos interval names

Reasoning for 0-indexed intervals

Note that a unison is a 0-mosstep, rather than a mos1st; likewise, the term 1-mosstep is used rather than a mos2nd. One might be tempted to generalize diatonic 1-indexed ordinal names: In 31edo's ultrasoft mosh scale, the perfect mosthird (aka Pmosh3rd) is a neutral third and the major mosfifth (aka Lmosh5th) is a perfect fifth. The way intervals are named above (and in 12edo theory) has a problem. An interval that's n steps wide is named (n+1)th. This means that adding two intervals is more complicated than it should be. Stacking two fifths makes a ninth, when naively it would make a tenth. We're used to this for the diatonic scale, but when dealing with unfamiliar scale structures, it can be very confusing.

To overcome this, TAMNAMS uses a 0-indexed name system for non-diatonic mos intervals, which makes the arithmetic needed to understand mos intervals much smoother. Going up a 0-mosstep means to go up zero steps, and stacking two 4-mossteps produces an 8-mosstep, rather than stacking two mos5ths to produce a mos9th. The use of ordinal indexing is generally discouraged when referring to non-diatonic mos intervals.

Deriving the names for mossteps

To find what mos interval sizes are found in a mos, start with the patterns of large and small steps that represents the mos in its brightest mode and its darkest mode (which is the reverse pattern for the mos's brightest mode). The idea of mos recursion may be of help with finding the generators of a mos. Likewise, the idea of modal brightness and UDP may be of help for a mos's modes.

Given the mos xL ys, the following algorithm is used to find the brightest mode for that mos.

  1. If either x or y is equal to 1 (base cases):
    1. If both x and y are equal to 1, then the final scale is "Ls".
    2. If only x is equal to 1, then the final scale is L followed by y s's.
    3. If only y is equal to 1, then the final scale is x L's followed by s.
  2. If neither x nor y is equal to 1 (recursive cases):
    1. Let k be the greatest common factor of x and y.
    2. If x and y share a common factor k, where k is greater than 1, then recursively call this algorithm to find the scale for (x/k)L (y/k)s; the final scale will be (x/k)L (y/k)s duplicated k times.
    3. If x and y don't share a common factor that is greater than 1 (if x and y are coprime), then:
      1. Let m1 = min(x, y) and m2 = max(x, y).
      2. Let z = m2 mod m1 and w = m1 - z.
      3. Let prescale be the mos string for zL ws. Recursively call this algorithm to find the scale for zL ws; the final scale will be based on this.
      4. If x < y, reverse the order of characters in the prescale. This is only needed if there are more L's than s's in the final scale.
      5. To produce the final scale, the L's and s's of the prescale must be replaced with substrings consisting of L's and s's. Let u = ceil(m2/m1) and v = floor(m2/m1).
        1. If x > y, every instance of an L in prescale is replaced with one L and u s's, and every s replaced with one L and v s's. This produces the final scale in its brightest mode.
        2. If y > x, every instance of an L in prescale is replaced with u L's and one s, and every s replaced with v L's and one s. This produces the final scale in its brightest mode.

Using 3L 4s as an example, this is LsLsLss (brightest) and ssLsLsL (darkest). To find the large sizes of each k-mosstep, consider the first k mossteps that make up the mos pattern for the brightest mode. Repeat this process with the mos pattern for the darkest mode to find each k-mosstep's small size. To make these sizes more clear, we can denote the mos intervals as a sum of large and small steps iL+js, where i and j are the number of L's and s's in the interval's step pattern; this is to say that the order of L's and s's doesn't matter, rather the amount of each step size. The large and small sizes should differ by replacing one L in the large size with an s.

Interval Large size (LsLsLss) Small size (ssLsLsL)
Step pattern Sum Step pattern Sum
0-mosstep (unison) none 0 none 0
1-mosstep L L s s
2-mosstep Ls L+s ss 2s
3-mosstep LsL 2L+s ssL 1L+2s
4-mosstep LsLs 2L+2s ssLs 1L+3s
5-mosstep LsLsL 3L+2s ssLsL 2L+3s
6-mosstep LsLsLs 3L+3s ssLsLs 2L+4s
7-mosstep (octave) LsLsLss 3L+4s ssLsLsL 3L+4s

Given the mos xL ys, the following algorithm is used to find the bright generator and its complement.

  1. If either x or y is equal to 1 (base cases):
    1. If both x and y are equal to 1, then the generator is "L" and its complement is "s".
    2. If only x is equal to 1, then the generator is "L" followed by y-1 s's, and the complement is "s".
    3. If only y is equal to 1, then the generator is "L" and the complement is x-1 L's followed by "s".
  2. If neither x nor y is equal to 1 (recursive cases):
    1. Let k be the greatest common factor of x and y.
    2. If x and y share a common factor k, where k is greater than 1, then recursively call this algorithm to find the generator and complement for (x/k)L (y/k)s; the intervals returned this way will apply to the period rather than the octave.
    3. If x and y don't share a common factor that is greater than 1 (if x and y are coprime), then:
      1. Let m1 = min(x, y) and m2 = max(x, y).
      2. Let z = m2 mod m1 and w = m1 - z.
      3. Let gen be the scale's generator and comp be the generator's octave complement for the mos zL ws. Recursively call this algorithm to find these intervals for zL ws; the final scale's generator and complement will be based on this.
      4. If x < y, reverse the order of characters in gen and comp, then swap gen and comp. This is only needed if there are more L's than s's in the scale.
      5. To produce the scale's generator and complement, the L's and s's of both intervals must be replaced with substrings consisting of L's and s's. Let u = ceil(m2/m1) and v = floor(m2/m1).
        1. If x > y, every instance of an L in both intervals is replaced with one L and u s's, and every s replaced with one L and v s's. This produces the final scale's generator and complement.
        2. If y > x, every instance of an L in both intervals is replaced with u L's and one s, and every s replaced with v L's and one s. This produces the final scale's generator and complement.

The length of gen is the number of mossteps for the bright generator, and the length of comp is the number of mossteps in the dark generator. For our example of 3L 4s, the algorithm returns the step pattern Ls as the bright generator and LsLss as its complement, which are 2 and 5 mossteps wide, respectively. Since the large size of a bright generator is perfect and its small size diminished, and the small size of a dark generator is perfect and its small size perfect, the scale's generators can be identified as shown in the table.

Interval Specific mos interval Abbreviation Interval size
0-mosstep (unison) Perfect unison P0ms 0
1-mosstep Minor mosstep (or small mosstep) m1ms s
Major mosstep (or large mosstep) M1ms L
2-mosstep Diminished 2-mosstep d2ms 2s
Perfect 2-mosstep P2ms L+s
3-mosstep Minor 3-mosstep m3ms 1L+2s
Major 3-mosstep M3ms 2L+s
4-mosstep Minor 4-mosstep m4ms 1L+3s
Major 4-mosstep M4ms 2L+2s
5-mosstep Perfect 5-mosstep P5ms 2L+3s
Augmented 5-mosstep A5ms 3L+2s
6-mosstep Minor 6-mosstep m6ms 2L+4s
Major 6-mosstep M6ms 3L+3s
7-mosstep (octave) Perfect octave P7ms 3L+4s

Reasoning for mos pattern names

The goal of TAMNAMS mos names is to choose memorable but aesthetically neutral names.

All names ending in -oid refer to an exotemperament which, when including extreme tunings, covers the entire range of the corresponding octave-period mos, such that many edos with simple step ratios for that mos will correspond to valid tunings, if not by patent val, then with a small number of warts.

All names for mosses with five or less notes - excluding (mono)wood and biwood (which like all n-wood mosses are specific to octave tuning) - require that some small integer multiple of the period is equal to an octave, under the reasoning that mosses with five or less notes are common and broad in tuning enough that they are much more likely to find interest in non-octave contexts. Because of this, their names were chosen to be extremely general, both to avoid bias/being too flavorful and (correspondingly) so that the terms could validly be reused for any mos for which the period is not equal to a (potentially tempered) octave.

Any multiperiod mos with more than five notes was given a name that wasn't reliant on the name of a mos with five or less notes as such names were based on those mos names formerly requiring an octave tuning (which is to say some small integer multiple of their period must be equal to a (potentially slightly tempered) octave).

Former names like "orwelloid" and "sensoid" were abandoned because the names were too temperament-specific in the sense that even considering extreme tunings did not cover the whole range of the mos. The remaining temperament-based names have been abstracted or altered heavily, namely "pine", "hyrulic", "jaric", "ekic" and "lemon".

The inclusion of mos names for "multiperiod" mosses was from a desire to have all ten-note-and-under mosses named for completeness, which is also what prompted some of the reconsiderations mentioned earlier. Similarly, the inclusion of mosses of the form 1L ns using the "anti-" prefix (or an- for less-than-six-note mosses) was also for a practical consideration; although the tuning range is very unhelpful for knowing what such a mos will sound, it is nonetheless useful for describing structure in situations where one does not want to use the mathematical name, especially given that in such situations the tuning will likely be specified somewhere already. Jaric and taric specifically were chosen over bipedal and bimanual because of this, and to a lesser extent, lemon and lime were chosen over antibipentic and bipentic respectively (and for consistency with that their parent MOSS, 4L2s, is named citric).

The distinction between using the prefixes "anti-" vs "an-" for reversing the number of large vs. small steps is also not as trivial as it may sound. In the case of mosses with six or more notes, as the period is always an octave, there is a very large tuning range for the 1L ns mosses (hence the original reason for omitting such mosses), but the "anti-" prefix shows that what is significant is that it has the opposite structure to the corresponding nL 1s mos while pointing out the resulting ambiguity of range. In the case of mosses with five or less notes, as the period is not known and therefore could be very small, this is not as much of a concern as fuller specification is likely required anyway, especially in the case of larger periods, so the name should not be tediously long as the name refers to a very simple mos pattern, and for related reasons, the name shouldn't give as much of a sense of one 'orientation' of the structure being more 'primary' than the other, while with mosses with more than five notes, this suggestion of sense is very much intended, because it will almost always make more sense to talk about the (n+1)L 1s child mos of whatever 1L ns mos you want to speak of.

Name-specific reasonings

Pedal (1L 4s)

Pedals are operated with feet, which have one large toe and four small toes. Also comes from words like "bipedal", where in TAMNAMS, "bipedal" would literally mean a pedal scale with a period equal to half of some chosen interval, although such a scale would have either two right feet or two left feet depending on orientation chosen. If you think "car"/"vehicle" when you think "pedal" and don't think (or want to think) much about feet then you can think about "beeping" (as beep is the 7-limit 4&5 exotemperament). Because this name relies so heavily and fundamentally on there being 1 large and 4 small steps per period, it is appropriate to generalise for any size of period you would want. In that regard, same goes for manual, pentic and anpentic.

Malic (2L 4s) and citric (4L 2s)

Malic derives from Latin malus 'apple'. An apple has two concave ends, and large steps in a scale with more small steps are hole-like, hence the two large steps in malic. Citric (4L 2s) is named after the child mosses of citric, namely lemon (6L 4s) and lime (4L 6s). Unlike apples, lemons have two convex pointy ends, and small steps in a scale with more large steps are pointy, hence the two small steps. Malic and citric acids are both ubiquitous in food and biology, thus justifying their use for fairly small mos scales.

Machinoid (5L 1s)

Machine is the 5&6 temperament in the 2.9.7.11 subgroup with a comma list of 64/63 and 99/98.

This temperament is supported by 5, 6, 11, 12, 16, 17, 22, 23, 27, 28 and 33 equal divisions, with non-patent val tunings including 5+5=10e, 5+10e+12=21be, 5+5+5+5+6=26qe, which are mentioned here for demonstrating virtual completeness of the tuning range, and the unusually large 33edo tuning being to show 11edo's strength as a tuning.

Onyx (1L 6s)

"1Ln-ic's" and "nL1-ic's (like, the -ic suffix applied to MOSS names, collectivised for 1Lns and nL1s) sounds like "one-el-en-ics" or "en-el-one-ics" which abbreviated sort of sounds like "one-ics" => "onyx". Then "onyx" sounds sort of like "one-six". Furthermore the onyx mineral comes in many colours and types, which seems fitting given this is the parent scale for a wide variety of MOSSes; specifically of interest being 7L 1s (pine), 8L 1s (subneutralic) and 9L 1s (sinatonic). Finally, the name "onyx" is also supposed to be vaguely reminiscent of "anti-archaeotonic" as "chi" (the greek letter) is written like an "x" (this is related to why "christmas" is abbreviated sometimes as "X-mas") and other than that, the letters "o" and "n" and their sounds are also present in "archaeotonic", and "x" is vaguely reminiscent of negation and multiplication. There is also something like a "y" sound in "archaeotonic" in the "aeo" part (depending partially on your pronounciation).

Subaric (2L 6s), jaric (2L 8s), and taric (8L 2s)

The name "subaric" alludes to the fact that 2L 6s is the largest proper subset mos of both jaric (2L 8s) and taric (8L 2s).

The name "jaric" alludes to a few highly notable and generally inaccurate (with the exception of diaschismic) temperaments that exist in the tuning range of this MOSS. Specifically, notice how the letters and sound of "jaric" has (or is intended to have) a lot of overlap with pajara, diaschismic and injera (listed in order of increasingly sharp fourths; note that diatonic fourths and 4-jarasteps are equated in jaric, a notable property).

The name "taric" was named based on it being the only octave-tuned TAMNAMS pattern with a basic tuning of 18edo (because 7L 4s has more than 10 notes so is out of the scope of TAMNAMS, although not necessarily out of the scope of extensions) and it was also named based on rhyming with jaric (as they share the parent mos 2L 6s).

Sephiroid (3L 7s)

Sephiroth is the 3&10 temperament in the 2.5.11.13.17.21 subgroup with commas including 65/64, 85/84, 105/104, 169/168, 170/169, 221/220, 273/272, 275/273.

This temperament is supported by 3, 10, 13, 16, 23 and 26 equal divisions, with non-patent val tunings including 6eg, 7e*, 19eg, 20e, 29g, 32egq, 33ce, 36c.

* Extreme tunings even occasionally go outside of this range like with 7e, but this would never be considered a good tuning.

(Note that q in the above is a placeholder symbol meaning that the generator 21 is warted.)

Note therefore how practically a full range of tunings is covered both in breadth and depth.

Dicoid (7L 3s)

Dichotic is the 7&10 temerament in the 11-limit with commas including 25/24, 45/44, 55/54, 56/55, 64/63 and is an extension of the 5-limit exotemperament dicot which tempers 25/24, equating 5/4 and 6/5 into a neutral third sized interval, which is the generator. To help justify using these temperament for inspiration for the name, note that:

This temperament is supported by 7, 10 and 17 equal divisions, with non-patent val tunings including 14cd(=7+7), 20e(=10+10), 24cd(=17+7), 27ce(=17+10).

Note there are many more warted tunings than this with even more extreme tunings, which makes it reasonable to loosely associate the exotemperament with the range of vaguely saner tunings.

Armotonic (7L 2s)

The name "superdiatonic" has seen some precedent of use on the Xen Wiki to refer to the mos pattern 7L 2s, so is accepted as a possible name, but "armotonic" is preferred due to its clarity as "superdiatonic" could reasonably be confused as describing sharp-fifth diatonic scales. This mos is part of a series of mos patterns (5+2k)L 2s, which starts with diatonic (5L 2s, k=0) and superdiatonic (7L 2s, k=1), hence the reasoning for that name; like 5L 2s, 7L 2s is also a fifth-generated scale and has a structure similar to diatonic in some ways, but with more large steps. Because of the ambiguity, the name "armotonic", in reference to Armodue theory, is TAMNAMS' recommended name, but "superdiatonic" is allowed in contexts where it's truly unambiguous if the writer prefers it.

On the term diatonic

In TAMNAMS, diatonic exclusively refers to 5L 2s. This is because while diatonic has accrued a variety of exact meanings over time, it has a clear choice of referent when talking about MOS scales: 5L 2s with an octave or tempered-octave period.