----------------------------------------------------------------- GENERAL QUESTIONS ----------------------------------------------------------------- Q: What is microtonal music? A: There is no broad consensus on the meaning of the term "microtonal". It may help to look at three popular ways of using the term: 1. literal definition - refers to music having melodic or harmonic intervals smaller than 100 cents This excludes music in temperaments like 7-ET or 10-ET, which is sometimes called "macrotonal". 2. inclusive definition - 'anything other than 12-ET' In this view, only solo instrumental music of piano, guitar, or MIDI keyboard would qualify as NOT microtonal. Some authors even consider guitar and/or piano to be microtonal, due to note bending and octave stretch, respectively. This certainly makes the (valid) point that 12-ET is only an abstraction and that real music gains a lot of its power and subtlety by deviating from it. However, it makes the term "microtonal" almost meaningless by making it apply to everything. 3. exclusive definition - refers to intonation that is difficult to approximate in 12-ET This covers extended just intonation harmony (e.g. 11-limit), nondiatonic music, nonoctave scales, etc. It's worth noting that some authors avoid the term "microtonal" and have suggested alternatives. The most popular of these is "xenharmonic", coined by Ivor Darreg to capture the exclusive definition above: http://en.wikipedia.org/wiki/Xenharmonic ----------------------------------------------------------------- Q: I've read that in microtonal tunings, flats aren't equal to sharps. For instance, Bb isn't the same pitch as A#. Why? A: Some microtonal notation systems don't even use letters for note names, let alone have flats and sharps! So let's just start with standard Western music. Around the year 1500, meantone temperament began to become the standard intonation system in the West. In meantone, sharps are below flats (e.g. C# is lower in pitch than Db). Here is a chain of fifths: Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# All sharps are to the right, and all flats are to the left. Any adjacent pair of notes has to be spelled as a true 5th (five diatonic letter names apart, inclusive). A typical meantone fifth is about 697 cents. That means C# will be 4879 cents, or 4 octaves and 79 cents, above C. Db will be 115 cents above a C three octaves below where we started. Thus, C# is pitched 36 cents below Db in meantone. By about the year 1900, 12-tone equal temperament had fully replaced meantone in the West. But we didn't switch notation systems. Hence, we still have the distinction between C# and Db even though the pitches come out the same in 12-ET. At the end of the day we can say that tuning systems based on a chain of fifths can be notated with standard Western notation. If the fifth is smaller than 700 cents, sharps will be below flats. If the fifth is exactly 700 cents, sharps = flats. And, you guessed it, if the fifth is larger than 700 cents, sharps come out ABOVE flats. ----------------------------------------------------------------- Q: What is "just intonation"? A: Generally, it means tuning instruments so that pitches are related to one another by simple ratios. For example, A = 440 Hz and E = 660 Hz are related by the ratio 3/2. In equal temperament, this A-E "perfect fifth" is very close to 3/2, but differs by a precise amount so that the circle of fifths returns us to the pitch we started on. There are three sources of disagreement over the precise meaning of the term, however. First, What qualifies as a simple ratio? Is 19/16 simple? Historically, only ratios that can be factored with primes no greater than 5 were considered simple. Contemporary authors tend to be more inclusive, but the phrase "extended just intonation" is sometimes used to indicate that primes greater than 5 are being allowed. Second, What tuning accuracy is required? If we have 440 Hz and 660.2 Hz, does it count as 3/2? One answer is that if the tuning error is being applied intentionally (in a systematic way), then it should not be considered just intonation. If instead the error is a result of the limited accuracy of an instrument (and is therefore somewhat random over the range of the instrument), we can still consider the result to be just intonation. Finally, Which intervals should be considered when more than two pitches are involved? The classical "just intonation" scale has the pitches: 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1. It contains chords like 4/3 5/3 2/1, which simplify to 4:5:6 and which are certainly just intonation chords. But it also contains the chord 9/8 4/3 5/3, which simplifies to 27:32:40 and sounds quite harsh if a consonant minor triad is expected. Nevertheless, it is often implied that whatever combination of pitches we pluck from a scale like the one above, we are still in just intonation. An alternative is to consider just intonation to be a property of chords (simultaneous pitches) only. Scales can then be discussed in terms of how many justly intoned chords they contain, what portion of their triads are justly intoned, etc. Except for the harmonic series itself (and only the first few octaves thereof), all rational scales with more than a handful of pitches/octave contain chords that fail classical tests of just intonation, such as being easy to tune by ear. ----------------------------------------------------------------- Q: I keep seeing stuff like "5-limit" or "9-limit". What is this nomenclature and why have I never heard of it before? A: Please see: http://en.wikipedia.org/wiki/Limit_(music) It may be new to you because music theory is almost always taught under the assumption that you will forever and always be hearing, performing, and composing exclusively in 12-ET. Q: Can I get a quick summary of how to calculate these? A: OK. For a ratio n/d in lowest terms, to find its prime limit, take the product n*d and factor it. Then report the largest prime you used in the factorization. To find its odd limit, simply divide n by 2 until you can no longer divide it without a remainder, then do the same for d. Then report the larger of the two numbers left over. To find the prime limit or odd limit of a list of ratios (such as a scale), simply calculate it for each of them individually and report the maximum. To find the prime or odd limit of a chord, first compute its table of dyads, e.g. for major triad C-E-G the dyads are C-E, E-G, and C-G. Then apply the procedure for a list of ratios given above. ----------------------------------------------------------------- Q: I see the terms "tuning", "temperament", and "scale" used interchangeably. Is there any distinction between them? A: There is no consensus distinction between these terms. In traditional Western keyboard practice, a "temperament" is usually what defines how the keys are tuned, and a "scale" is a more abstract thing, often thought of in terms of patterns of 2nds (e.g. LLsLLLs for the diatonic scale) or subsets of whatever temperament is used. Microtonal music theory needs much more general terms, since the scales and consonances being targeted are not taken as given. Gene Ward Smith has proposed distinct definitions for each of these terms in order to make discourse about microtonal music more precise. His definitions have seen adoption on the tuning lists, but it falls short of consensus. Here they are: scale- An ordered list of intervals, which may be applied to a given concert pitch to generate an ordered list of pitches that can be used to tune an instrument. Scala's scale file format (.scl extension) is an embodiment of this definition. temperament- A homomorphic mapping from just intonation to an abstract free group of smaller rank. This algebraic definition basically means that for every interval in just intonation, which is composed of some combination of primes, we express it instead as a combination of generators (called the generators of the temperament) such that there are fewer generators than there were primes. For example, in 5-limit just intonation every interval can be expressed as a product of 2s, 3s, and 5s. In meantone temperament, every interval can be expressed as a product of octaves and fifths. 5-limit JI is therefore a rank 3 intonation, and meantone a rank 2 temperament. In 12-ET, every interval can be expressed in semitones. It is therefore a rank 1 temperament. tuning- A choice of intervals for the generators of a temperament. For example, 1200 cents & 697 cents is a good tuning for meantone. So is 1200 cents & 696 cents. They lead to different scales, but they are both good meantone tunings because they produce good approximations of 5-limit JI. Sometimes tunings are named after the optimization they solve. For example, "RMS optimal meantone" refers to the tuning that minimizes the RMS error from JI under the constraint of the meantone mapping. ----------------------------------------------------------------- Q: I've heard the terms "homophonic" and "monophonic". What's the difference? A: # of voices # of rhythms harmony? monophony * 1 1 N heterophony many many N homophony many 1 Y polyphony many many Y * Harry Partch called his music style "monophony", partly in contradiction to the usual meaning, above. His early work does indeed feature the "intoning voice" in monophonic settings, but he soon undertook the use of more complex textures. ----------------------------------------------------------------- CONSONANCE AND DISSONANCE ----------------------------------------------------------------- Q: I've come up with a simple formula to measure the consonance of just intonation dyads. Is it original? A: Probably not. A multitude of consonance ranking formulas have been proposed and investigated on this mailing list over the years. Of these, the product of numerator and denominator, often written n*d, for a ratio n/d in lowest terms, has been found to best agree with informal rankings by listeners as well as published psychoacoustic data. Of course, no such simple formula can be perfect or apply in all contexts, but the n*d rule seems to work reasonably well in typical musical settings. The n*d rule is also called Tenney height, after James Tenney^1, who is one of the theorists to propose it (along with Galileo Galilei^2, Denny Genovese^3, and countless others). Tenney height generalizes to geomean(a*b*c...) for a chord a:b:c..., which gives sqrt(n*d) for dyads. ^1 James Tenney (1983). John Cage and the Theory of Harmony, Soundings 13: The Music of James Tenney, Ed. Peter Garland. Santa Fe: SOUNDINGS Press, 1984. ^2 Galileo Galilei (1638). Discorsi e dimostrazioni matematiche interno a due nuove scienze attenenti alla mecanica ed i movimenti locali. Leiden: Elsevier, 1638. Trans. H. Crew and A. de Salvio as Dialogues concerning Two New Sciences. New York: McGraw-Hill, 1963. ^3 Denny Genovese (1991). The Natural Harmonic Series as a Practical Approach to Just Intonation. Unpublished Thesis, New College of Florida. ----------------------------------------------------------------- Q: Tenney height is a 'small whole numbers' rule. Didn't Plomp/Levelt/Sethares show that such rules are only valid for timbres containing harmonic partials? A: No. Plomp & Levelt tested simultaneities of two sine tones only. Had they tested 3-, 4-, or 5-tone interactions, they would have discovered that 'major' chords are more consonant than 'minor' chords. This difference is mild but well-known in the case of the 5-limit triads, and quickly becomes overwhelming with increasing harmonic limit (e.g. in 11-limit just intonation). Sethares applies the P&L results to many-tone complexes, but does so in a pairwise fashion. No pairwise model can explain the difference between major and minor chords, since they contain the the same intervals! Moreover, timbres whose partials stray too far from the harmonic series will cease to evoke any single sensation of pitch (unless one partial is vastly louder than the others). Matching timbre to tuning to minimize critical band interactions is a powerful technique that can remove beating from tempered music, but there is only so much temperament error it can disguise. Timbres created by Bill Sethares for temperaments like 10-ET often sound bell-like for this reason. ----------------------------------------------------------------- Q: Critical band interactions are strongly supported by physiological and psychoacoustic evidence. What's the evidence for an innate affinity towards simple rational intervals? A: Sounds with inharmonic spectra do not evoke well-resolved pitches, as do sounds with harmonic spectra. This "virtual pitch" phenomemon has been studied in pyschoacoustics: http://www.mmk.ei.tum.de/persons/ter/top/virtualp.html As social animals, humans are highly adapted to extract information from speech sounds. Human vocal folds produce rich spectra with perfectly harmonic overtones, and vowels sounds in all natural languages are defined by selectively boosting or cutting regions of those spectra by using the vocal tract as a resonant filter: http://en.wikipedia.org/wiki/Formant Thus, a hearing system that is able to identify harmonic spectra as single sources and continuously characterize their spectral balance over time has high adaptive significance for humans. It is noted that Western tonal music produces spectra with similar characteristics to that of speech. Recent research on primates (using in vivo single-cell recording) reveals "combination sensitive" neurons in the auditory cortex that can detect when two frequencies resolved by the cochlea are harmonically related: http://jn.physiology.org/cgi/content/abstract/89/3/1603 Imaging of human subjects locates the work of pitch perception in the analagous tissue: http://www.cell.com/neuron/retrieve/pii/S0896627302010607 http://cercor.oxfordjournals.org/content/13/7/765.full Additionally, Tramo et al cite evidence from subjects with legions in this area: http://www.ncbi.nlm.nih.gov/pubmed/11458869 ----------------------------------------------------------------- Q: If Tenney height works, why do we need harmonic entropy? A: Tenney height applies only to rational numbers, and only to simple ones at that (where the product n*d is below 100 or so). Harmonic entropy is a smooth, differentiable function of interval size, i.e. 301:200 and 3:2 should have similar entropies. Near the simple rationals, harmonic entropy is proportional to sqrt(n*d). Triadic harmonic entropy has never been fully computed, but we do know something about how it must turn out. In 2002, Paul Erlich showed that the area of the Voronoi cells of rational triads in 2-D triadspace is strongly correlated to the generalized Tenney height of the triads: http://groups.yahoo.com/group/harmonic_entropy/files/triadic.gif ----------------------------------------------------------------- Q: Is there a 'harmonic entropy for dummies' somewhere? A: Harmonic entropy is based on the hypothesis that the brain tries to interpret what it hears in terms of a harmonic series. To do this it must be able to recognize rational intervals. The rational numbers are infinite in extent, but probably we have evolved the ability to recognize only the simplest of them, since the spectra of human voices have most of their energy in the first several partials. Pick any limit you like on the complexity of the rationals. Plot everything below it on a number line and you'll find that the simpler the fraction, the greater the distance on the number line between it and the next point you plotted. Now the ear, like any measuring device, will have noise in its output. If a 440 Hz sine tone is coming in your ear, the signal arriving in the brain will fluctuate in a range around 440 Hz. How does the brain assign a ratio to a pair of these noisy signals? Let's say the ear is presented with a pair of sine tones 697 cents apart. We can draw a Gaussian representing the noise in the measurement of these tones, above the number line where we plotted our recognizable fractions: _ . . _.' '._ -|--------|-------|---- 4/3 3/2 5/3 Clearly, it's a 3/2. But what if it looked like this? __ . . .' '. .' '. _.' '._ -|----|-----|-------|---- 4/3 7/5 3/2 5/3 OK, let's cut the Gaussian up into slices above each of the possible ratios: __ . . .' : '. .' : :'. _.: : : '._ -|-:--|--:--|---:---|---- 4/3 7/5 3/2 5/3 Now we can compute the area of each slice, and the slice with the largest area should win -- looks like 3/2 again. Harmonic entropy measures the ambiguity of this contest between possible ratios. In the first curve we drew there was no contest, so the entropy would be zero. In the second example, 7/5 gave 3/2 a run for its money, but 3/2 won by a decent margin. We still think we are hearing 3/2. But if the Gaussian were cut into several slices of similar area, the contest would be harder to call. In a political election that is too close to call, the country has an uneasy feeling until the new leader is determined. Similarly, we feel uneasy when the pitch or root of what we're hearing is undetermined. ----------------------------------------------------------------- Q: Which is the correct JI tuning for the minor triad: 10:12:15, 16:19:24, or 6:7:9? A: There is no single "correct" tuning -- it's ultimately a matter of musical taste. But we can examine the question using speculative psychoacoustics... When presented with an auditory stimulus, the brain will attempt to assign it a pitch. It does this by extracting harmonic components from its spectrum. The stimulus need not be periodic for this to work, but even if it is, the hearing system is limited in its ability to: 1. perform the spectral analysis -- failure here results in psychoacoustic roughness 2. detect periodicity -- failure here results in high harmonic entropy For clean stimuli with periodicities in the range the brain is good at detecting (such as 5-limit major triads), the extracted pitch will be fairly unambiguous (sometimes it vacillates by an octave or two). For other stimuli, there may be several pitches competing for the 'answer'. Harmonic entropy measures this uncertainty. In the case of 10:12:15, the probability distribution for the fundamental will tend to include the pitches 4, 5, 8, and 10. If the tones of the chord are complex tones, the timbre will influence their relative likelihoods of winning the contest (since the hearing system considers all partials). If 4 or 8 win, the brain is interpreting 10:12:15 literally, as a segment of harmonics relatively high in the series. It is somewhat more likely that 5 or 10 will win; the brain hears 10:15 = 3:2 and dismisses 12 as an artifact. The same tradeoff exists in the case of 16:19:24, but here the literal interpretation is less likely because it relies on even higher harmonics. However, this time both the literal and 'outer fifth only' interpretations point to the same pitch, namely 16 (or 4 or 8). Therefore, it dominates the probability distribution and 16:19:24 tends to sound more stable than 10:12:15 (though its critical band roughness is greater). Many listeners rate 6:7:9 as more consonant than either of the chords just discussed. But in has two things against it: 1. 7:6 is getting close to the critical band -- even in the middle of the piano, beating between some of its lower partials can be heard, and this lends it a somewhat 'pinched' sound. 2. It is found low enough in the harmonic series that its virtual fundamental more likely to be 4 (or 2) and the 7 less likely to be dismissed. This gives it the same disadvantage that 10:12:15 had to 16:19:24, only moreso. That its virtual fundamental doesn't clearly coincide with its root may make it less suitable for minor-key tonal harmony, which needs to be able to resolve to a 'finished' minor tonic. In summary, 6:7:9 is an excellent minor triad and can often be used in performances of classical music. But generally either 10:12:15 or 16:19:24 are to be preferred, because of their greater tendency to evoke a fundamental that is the same as their lowest tone (or octave extension thereof). ----------------------------------------------------------------- Q: Why did Bach favor minor keys? A: It may just have been aesthetic preference. But we may also speculate that counterpoint works better in minor keys, partly because the consonance of the tonic chord is weaker, and therefore the voices are more free and less likely to be heard as overtones of it. ----------------------------------------------------------------- TEMPERAMENT ----------------------------------------------------------------- Q: I was wondering about the progress of tunings over time. AFAIK, the oldest records are of Pythagorean tuning, which is a rank 2 system. Many centuries later, there came meantone temperament, which is also rank 2. Then centuries again until very recently, when a huge variety of tuning systems is suddenly available. Was all really quiet between these times? A: There are two intertwined histories here: the evolution of just intonation harmony and the evolution of the temperaments used to support it. We know most musical traditions simply didn't employ harmony, at least not in an organic fashion. Outside of Europe and prior to the 12th century, music was primarily monophonic or heterophonic. Organum was the birth of 3-limit harmony, which is rank 2. http://en.wikipedia.org/wiki/Organum All really does seem quiet until about 250 years later, when the 5-limit (rank 3) sprang onto the scene: http://en.wikipedia.org/wiki/Contenance_Angloise Though Euler and others theorized about the 7-limit (rank 4), tetradic harmony didn't emerge in practice until the late 19th century, in the French impressionist and African-American forms.* They treat the most consonant tetrads in 12-ET (approximations of 8:10:12:15, 10:12:15:18, 4:5:6:7, 5:6:7:9 and others) on roughly equal footing -- rank 4 down to rank 1. Barbershop quartet singing (1940s) is closer to a true 7-limit style, as it favors the 4:5:6:7 chord and uses pure intonation, though it often employs rank 2 adaptive JI to support diatonic melodies and chord progressions. Harry Partch is apparently the first musician to systematically employ the 11-limit (rank 5). As for temperament, certain African and Eastern traditions may have been using 5-, 7-, and 12-ET from the time of the European middle ages. This is a reduction from rank 2 to rank 1, but as already mentioned these forms did not have polyhponic harmony. In Europe, the first extant description of a systematic tempering of the 5ths dates from the early 1500s; about a century after the 5-limit was introduced in song. I expect a number of theorists noticed non-meantone temperings of the chain of 5ths (Newton ... Helmholtz), but the first thorough inventory of linear temperaments with 5th-based generators I know of is due to Bosanquet (1876). Fokker was apparently the first to understand tempering in general as reducing the rank of just intonation. Erv Wilson was apparently the first to explore linear temperaments with non-5th generators. Then, the recent explosion you mention, which took place on this mailing list, in which the full universe of temperaments was delineated (no pun intended). Paul Erlich, Paul Hahn, Graham Breed, Dave Keenan, and Gene Ward Smith were especially instrumental in this between 1997 and 2006. Most of these systems remain untested in musical practice, though more than a few have been demonstrated. * I'm unaware of direct cross-pollination between these schools prior to Gershwin. It may have been taking place from the outset, or it may be a case of parallel evolution. ----------------------------------------------------------------- Q: The other day, somebody posted > name: beatles > comma: [19 -9 -2> 524288/492075 > mapping: <1 1 5] > <0 2 -9] > TOP period: 1197.1 > TOP generator: 354.7 > MOS: 1+1, 1+2, 3+1, 3+4, 7+3, 10+7 and my eyes glazed over. What does all this mean? A: It identifies a temperament called "beatles". You may be familiar with meantone temperament, which is generated by a chain of approximate 3:2 "fifths" reduced by 2:1 "octaves". These two intervals are said to be the generator and period, respectively, of meantone temperament. Actually there is a range of sizes of approximate fifths and octaves that will produce meantone. You may have heard of "quarter-comma meantone", which refers to the meantone tuning obtained when octaves are pure and fifths are one quarter of a syntonic comma flat. Above, we are told that beatles temperament has a "TOP" period of 1197.1 cents (an approximate octave) and TOP generator of 354.7 cents (a neutral third). TOP stands for Tenney OPtimal. It's an optimization procedure that's supposed to spit out the best generator and period for any temperament. TOP meantone has a fifth of 697.6 cents and an octave of 1201.7 cents. We know that in meantone, the pentatonic and diatonic scales are usually preferred over the hexatonic or octatonic scales. We also know that 5 + 7 = 12, 7 + 12 = 19, and 12 + 19 = 31, and that these are all equal temperaments which support meantone. This isn't a coincidence. It's caused by a recurrence relation characteristic of meantone temperament. Every temperament has a recurrence relation that tells us which ETs, and scales within those ETs, are likely to be fruitful. The scales in these patterns were dubbed MOS (Moments Of Symmetry) by Erv Wilson.* You may know that the syntonic comma, 81/80, vanishes in meantone temperament. If you perform a chord progression that has a net modulation of 81/80 in just intonation, it will bring you back to unison in meantone. Well, beatles temperament does the same thing to 524288/492075. That comma's a bit big as a fraction, so it's easier to factor it as 2^19 * 3^-9 * 5^-2 and write it as [19 -9 -2>. This shorthand is borrowed from "braket" notation in physics, and is called a "monzo" after Joe Monzo, who promoted the use of prime-factor notation. The thing to remember is that monzos represent JI intervals and they always point to the right. Beatles is a 5-limit rank 2 temperament, and all 5-limit rank 2 temperaments can be uniquely identified by a single comma that they temper out (things get more complicated above the 5-limit). The mapping rows give us two vals for beatles -- this is another way to uniquely identify it. Vals are like monzos except they always point to the left, and instead of representing JI intervals in terms of prime factors, they represent tempered intervals in terms of periods or generators. In this case, to get an octave (2/1) out of beatles we need to travel +1 period and +0 generators. In other words, the period of beatles is an octave. To get a 3/1, we must travel +1 period and +2 generators. So the generator must be some kind of half-fifth. Lo and behold, that agrees with the TOP period and generator we looked at earlier. For comparison, here's the entry for meantone: > name: meantone > comma: [-4 4 -1> 81/80 > mapping: <1 2 4] > <0 -1 -4] > TOP period: 1201.7 > TOP generator: 504.1 > MOS: 1+1, 2+1, 2+3, 5+2, 7+5, 12+7, 19+12 * Aspects of this concept were independently discovered by each of the following theorists: Balzano, Carey, Clampitt, Clough, Myhill, and Yasser, but only Yasser's work predated Wilson's and none of them seem to have understood the phenomenon as deeply. ----------------------------------------------------------------- Q: What is "meantone"? Is 24-ET a meantone temperament? A: "Meantone" is the 5-limit rank 2 temperament with 81/80 in its kernel. 24-ET, as a temperament, is rank 1. So it's not meantone. And even if we made an exception, the meantone-like val you'd use in 24 has torsion so it's not valid temperament anyway. As a scale, however, there may be an acceptable meantone tuning within 24-ET. For a scale to support meantone temperament, it must satisfy the map < 1 2 4 ] < 0 -1 -4 ] And a good scale should satisfy it so that the most accurate approximation in the scale for each prime is the one mapped. In 24-ET, if we choose 24 steps & 10 steps as generators, the above mapping reduces to < 24 38 56 ] That is, we get the 700 cents for 3:2 and 400 cents for 5:4, which are the best approximations available. So as a scale, 24-ET supports meantone temperament. ----------------------------------------------------------------- Q: What are comma shift, comma drift, and comma pumps?? A: Comma pumps are chord progressions that require either comma drift or comma shifts to perform in strict just intonation. Drift occurs when concert pitch is allowed to rise (or fall) each time the pump repeats. Shift is where, instead, common tones between adjacent chords are adjusted to keep the concert pitch stable, introducing tiny melodic steps. In adaptive JI, the shifts are allowed to be irrational (example: 1/4 syntonic comma). It is essentially the comma shift solution, but we are allowed to divide the shifts into even smaller steps (which may be inperceptible) and distribute them more widely. ----------------------------------------------------------------- Q: What's a "pun"? A: A pun is a surjection http://en.wikipedia.org/wiki/File:Surjection.svg The farther apart 3 & 4 are, the greater the pun. In music theory, the thing on the left is JI, the thing on the right is a temperament, and 3 & 4 would be far apart in cents. In comedy, 3 & 4 would be far apart in sense. Nyark. ----------------------------------------------------------------- Q: Why do good temperaments tend to temper out superparticular commas? Meantone tempers out 81/80, miracle 225/224, and so on. A: Temperaments are good if they provide many consonances with few notes, without introducing a lot of tuning error. Tempering out a comma makes it a "unison vector", which cuts the infinite prime-limit lattice into periodic sections. The simpler a comma's ratio, the closer it lies to the lattice's origin and the narrower the periodic slices it will define. So "few notes" is satisfied by simple commas because they provide access to the entire lattice via periodic slices or tiles containing fewer lattice points. Low tuning error, on the other hand, is satisfied by commas that are small in magnitude, since it is their magnitude that will vanish in the temperament. It's straightforward that superparticular ratios tend to have the smallest magnitude for a given complexity. ----------------------------------------------------------------- SCALES AND MELODY ----------------------------------------------------------------- Q: I've played around with creating my own scales, but none of them quite sound as 'simple' as the diatonic scale in 12-ET. Why is that? A: There are two primary kinds of simplicity worth mentioning when it comes to scales. First, there's something about the number of total number of intervals in a scale. Usually one can just look at all the 2nds of the scale (e.g. 9/8 10/9 etc.) and get a good idea of the total number of unique intervals. A scale with only one kind of 2nd can only have one kind of anything else (is an ET). With two kinds of 2nd the order matters; LsLsLs... gives only 2 kinds of 3rd, but LLssLLss gives three kinds of 3rd (2L, L+s, and 2s). Scales with fewer unique intervals seem to me to have a more 'coherent' sound, for lack of a better word. Several authors on the tuning list have noticed this over the years -- such scales seem to be more 'singable'. On the other hand, having a high number of different intervals (as many JI scales do) also imparts a unique sound that is certainly valid fodder for music-making. The second kind of simplicity has to do with symmetry at the 3:2. Octave-equivalent scales are fully "closed" at the octave. This means that if you sing a note an octave away from a pitch in the scale you'll wind up on another pitch in the scale. There's some evidence that a similar effect may be important for the next most powerful consonance after the octave, 3:2. But no scale can be fully closed with respect to both the octave and the 3:2 (this would mean some number of fifths came out equal to some number of octaves, which we know is impossible). So octaves get first dibs, then we do the best we can with 3:2s. Pythagorean chains will be optimal here. After that, classical tetrachordal scale construction gives good results. ----------------------------------------------------------------- Q: What's "Rothenberg propriety"? What's a "rank-order matrix"? A: David Rothenberg is a mathematician who published a series of papers on the perception of melody and the construction of musical scales in the 1970s. He starts with the notion that a melody is not only a series of pitches (e.g. E3, D3 etc) but also a series of intervals. And not only specific intervals (5:4, 9:8 etc) but also scalar intervals (3rd, 2nd etc). The scale is the language of the melody, in other words. Evidence for this notion includes the fact that diatonically transposing a melody (e.g. to the relative minor) preserves a great deal of its essence. He further assumes that listeners do not carry out something like: "I'm hearing a 5:4. I know all 5:4s are 3rds, therefore I am hearing a 3rd." The process is assumed to be something closer to: "This interval is smaller than the previous one. Therefore if the previous interval was a 4th, this ought to be a 3rd." R. claims that in order for this to work, the scale has to have the property that no 2nd is larger than any 3rd, and so on. Such scales are called "proper". Improper scales are predicted to be unsuitable for diatonic-like music, and instead must be used against a drone. We can show all the intervals of a scale by listing all modes of the scale on consecutive rows of a matrix, such that scale degrees are shown in columns. Doing this for the diatonic scale, we indeed find that all 5:4s are in the 3rds column. But since only relative sizes matter in Rothenberg's regime, we can replace each interval with its size rank among all the intervals in the matrix. You can see this in Scala by choosing: View > Interval Ranking Matrix. Rothenberg claims that if two scales have the same rank-order matrix, they will tend to be heard as different tunings of the same thing. Rank-order matrices thus function like scale classes. It's a powerful idea whose implications remain largely unexplored. ----------------------------------------------------------------- Q: I've read that melodies are easier to sing in scales of low Rothenberg mean variety. Is this true? A: There's a plausible reason why it may be true. Humans are generally able to recognize transposition. As the number of different intervals in a scale goes down, the number of excerpts of a melody that are identical under transposition goes up. In 12-ET, the phrase D-E-F-A-B-C consists of congruent sections D-E-F and A-B-C. In the 5-limit diatonic scale this symmetry is broken (10/9 - 16/15 vs. 9/8 - 16/15). Generally, it should be easier to sing something if you've sung it before. ----------------------------------------------------------------- Q: Why is MOS important? A: The simplest answer is that it may not be terribly important. Nevertheless, the majority of music is made with MOS scales like the pentatonic and diatonic/natural minor, rather than non-MOS scales like the harmonic minor. One explanation is that MOS may just happen to have the 'right' amount of symmetry, for both harmony and melody. Approaching from the harmony side of things, MOS scales are embodiments of rank 2 temperaments. Given a just intonation of rank n, we can temper down to rank n-1, n-2... n = 1. So we might ask, why is rank 2 important? That raises the question, why is temperament important? And again, maybe it isn't. But it is useful. It reduces the number of notes one needs to do certain things musically. That makes notation easier to think about, instruments easier to build and play, and effects like puns possible. Temperament also seems quite natural in many cases... for example, choirs tend to spontaneously ignore 81/80 when performing 5-limit homophony. There is thought to be a limit to the number of tones that can be used in a melody. George Miller famously suggested it should be about 9 notes/oct. But ETs < 10 don't get us very far in the tuning accuracy dept. The obvious thing is to try higher-rank systems, but then we start to lose the benefits of temperament mentioned above. Rank 2 seems to strike a balance. 5-limit JI is itself rank 3, so rank 2 is the least degree of temperament available without adding more primes. One could draw a melody from a small scale and harmonize it chromatically. But aside from the fact that it can be hard to keep the chromatic steps out of the melody, there are purely melodic arguments why the small scale should be MOS anyway. From the melody side of things, MOS scales are scales with Myhill's property -- every interval class comes in exactly two sizes. Again, this seems to be a good amount of symmetry. ETs can sound too symmetrical. It can be hard to keep one's place in the scale, the modes tending to blend together. Already though, rank 3 scales can be hard to sing, because one has to articulate different steps of similar size in a certain order. Motivically, it can be too *difficult* to lose one's place in the scale, to establish a new mode... Imagine failing to recognize a motif because it appears in a mode too dissimilar from the one in which it was first presented. In the harmonic minor for instance, the upper tetrachord is quite different from the lower one (which may be why the melodic minor was developed). Most of the popular scales in common practice music that aren't MOS are 'near-Myhill'. The harmonic minor, neglected as it is, still has only two sizes of 3rd/6th. Likewise, the octatonic scale is slightly MORE symmetrical than a MOS, since some interval classes have only one size. This leads to Rothenberg's "mean variety", which is the average number of sizes per interval class. It appears that, for music using both tonal and modal effects, values near 2 are best. -----------------------------------------------------------------