New Findings on Dissonance & their

Practical Implications in Music Production

by Pablo Bellinghausen

Abstract

Improvements in neural measurements and recent social psychological research on the origins and nature of aggressive music types in the last 25 years can be used to create a consistent deinition of dissonance, and to study its physiological impact, to help explain empirical indings done by both musicians and sound engineers on how to control it as an artistic tool.

Introduction

There is an very large number of papers that refer to calculations of dissonance yet most of them refer to only 2 tones at the same time. Most researchers somehow hope that calculations will include our empirical information about more complex note combinations, but usually they do not1 . I had tried in vain to make out the relationship between the concepts of dissonance, roughness, beats, dierence tones, intermodulation distortion, AM and FM, but most of the available research I found was both highly specialized and very scattered, and it struck me that no one had already come up with a formally coherent way to connect all of these words. So, reading on all the research that has been done in this century, I decided to attempt to formulate a working deinition, if only a temporary one, for dissonance, and then to assess how these new indings prove themselves useful in explaining practices that are common in the world of music production, but are done more by instinct than by knowledge. he amount of preliminary information gathering I had to do was substantial, and so most of it has been moved into the appendices. I have highlighted the technical terms which are explained in more detail in said appendices, which should preferably be read irst.

A New Definition of Dissonance

Background

here is an undeniable truth to the armation that some sounds are perceptually more attractive than others for the human ear. We can adapt and learn to like sounds because we ind them discordant, yet even the most adamant metal fan or the most progressive Schoenberg lover will agree that they are enjoying sounds that are not naturally pleasant for the human ear, and they know very well which sounds are, if mostly by intuition. his chapter will try to make sense of the seemingly disparate theories that have been proposed

throughout the years. Most have been dismissed, but until not too long ago there was still some debate over some of them. Most theories can be distinguished into four categories: physical, physiological, cognitive, and cultural.

What I want to show is how actually all of these theories are inter-related, something which no one has done in a clear and concise way. his was clearly noted by Richard Parncutt2, which gathered from existing research that an encompassing theory hadnt yet been achieved as of 2006. We will not dwell on those theories that have already been refuted, but well give an overview of the last standing ones, which still explain experimental results reasonably well.

Existing theories

1. Frequency Ratio

Possibly the simplest deinition of consonance has been given by its characterization of simple frequency ratios. Many supporters throughout the last few centuries have advocated it, yet without justiication.3

2. Periodicity Length

Irvine 4 (1946) proposed that consonance and dissonance are related to the length of the periodicity of a cycle. When complex tones are related by simple frequency ratios, the cycle of repetition for the combined signal is relatively short. When tones are not related by simple frequency ratios, the cycle of repetition for the combined signal is long. his cycle of repetition is actually the dierence tone of the fundamentals that are modulating each other and becomes an amplitude-based perception instead of a frequency based one.

3. Difference Tones

here is a strong correlation with perceptually dissonant intervals and harmonic intervals that create unrelated combination tones. Newer studies5 now show that it is a necessary characteristic of the phenomenon, but not a sucient deinition of dissonance.

19 February 2012

1 Mashinter (2006). pp 201-2032 Parncutt (2006). pp.1-63 Galilei (1638)4 Irvine (1946). pp. 21-345 Plomp, Levelt, (1965). pp. 548-560

4. Temporal Dissonance

his theory, accepted as a de facto by many scientists today, was irst proposed by Helmholtz6, and focuses primarily on the avoidance of roughness. It works extremely well to predict our preference for dierent intervals, and some, if not most of its workings, must be true to some degree. Although the deinition of roughness has been reined by major indings in the last four decades7, there are still some major problems with this theory; for one, it does not take into account the dierence in beating frequencies that are considered rough depending on pitch; also, humans with lesions to the auditory cortex lack the ability to experience dissonance. his means that dissonance computation is done by a speciic neural system, and is not just a mechanical characteristic of the cochlea. Most importantly, it does not take into account that basic roughness calculations are always bigger with 3 and 4-note chords than with dyads8, and yet many of them sound more pleasant than their 2 note counterparts (e.g. a major seventh sounds less pleasant than a major seventh chord).

5. Synchrony of Neural Firings

Boomsliter and Creel (1961)9 proposed the theory that consonance arises when neural irings in the auditory system are synchronized. Fishman (2001)10 found, however, that phase-locking happens on dissonant intervals in neurons of monkeys as well as those in humans. Apparently, phase-locking happens for the amplitude modulation of the whole signal; if the neural iring synchronization frequency is mathematically related to the fundamental frequencies of both sounds, we feel consonance; but if the amplitude modulation (namely the dierence frequency between the 2 tones) is unrelated, we feel dissonance.

6. Tonotopic Dissonance

he organ of Corti, deep inside the cochlea, contains about 3500 inner hair cells, each of which is connected to around 10 auditory ibers. heyre placed in a single row, and, due to varying tension of the basilar membrane, which vibrates more strongly to low or high frequency sounds on the same axis as that row, they are able to ire up only when very speciic frequencies are making the membrane vibrate. his mechanism is very accurate ( 3 Hz at 2000 Hz) but this happens only when one sound is making the membrane vibrate at any given time. Research carried out by Plomt and Levelt (1965)11 links perceptual roughness of pure sine waves to 24 critical bands, of about 1/3 octave each. If two tones are separated by less than 30-40% of their critical band, then the frequencies interact in the same place of basilar membrane and their pitch height remains unresolved.

One can already see that many of these theories are strongly correlated: periodicity length, dierence tones and frequency ratio can be logically derived from each other (although

focusing on dierent acoustic phenomena) and are based on the same perceptual experiences. heir physiological justiication is not given, though, and that is where the last three theories come in. he problem is, I stress, these last three havent been clearly linked in a coherent fashion until now. Reliable information about the auditory nerve has only started to be gathered in the last half century, and the inner workings of hair cells, in the last two decades. Ater several hours of correlating all the data, I postulate a working deinition that both agrees with existing research and supersedes previous incomplete theories in a consistent and coherent approach. his is far from a deinite explication and obviously requires further research but it is an improvement. I have marked the number of the theory that each sentence refers to in brackets.

A new working definition of dissonance

Combinations of complex tones with a large integer ratio (1) will have harmonics that will create beats with each other (4). Most tonal sounds, when combined, form very speciic phase relationships (due to the high mathematical relationship between harmonics), so the total amplitude variation of the resulting sound relates highly to the beating of the clashing harmonics, which determines periodicity length (2) and the beatings of these harmonics will be highly correlated, creating roughness (new indings supporting 4). Since the ear is capable of iltering sounds by means of band-pass ilters in the cochlea, this wouldn't be a problem (iltering out tones that create amplitude variations eliminates the latter), but the bandwidth of those ilters is quite large, so roughness arises between tones in the same cochlea's critical bands (6), which causes synchrony of neural irings (5) and is read as the brain as dissonance. Intermodulation products, more speciically the third-order dierence tones 2f1-f2 and 2f2-f1, can also fall within the same critical band, becoming audible and thus exacerbating the problem (3). When more than 2 complex notes are played, say a triad, the number of harmonics that fall in the same critical band is greater, but the periodicity of the resulting roughness across the dierent critical bands is somewhat randomized, and so amplitude variation coherence tends to decrease. Statistic masking of most rough harmonics can also come into place, particularly if two of the notes are at a very consonant interval (e.g. a ith or an octave). he brain will then focus on the resolved harmonics to restore the pitch perception and the total perceived dissonance will be similar to a triad, as long as the interval relationship between these three notes is as simple as, or simpler, than the one between the previously perceived dyad, so even if absolute roughness increases, dissonance remains similar. hus, many triads can sound almost as consonant as many dyads (for example comparing a major triad to a major third), which is something that had eluded dissonance calculations so far. A lot more research is needed in the actual neural workings of the

6 Helmholtz. (1877)7 Fleischer, H. (1976). pp. 202-2098 Richard Parncutt (2006). pp.1-69 Boomsliter, Creel, (1961). pp. 2-3010 Fishman, (2001)11 Greenwood, (1961); Plomp, Levelt, (1965); Kameoka, Kuriyagawa, (1969)

auditory cortex, since it is not exactly clear how a tendency for synchrony in neural irings is interpreted by the brain into separate notes, so that question is let open for the moment.

Research on Psychological Factors

Memorability and preference of melodies in infants

It has long been established that there is a hierarchy of preference of intervallic relationship in adults, relating directly with their consonance, but for a long time there had been major doubts about whether these preferences were due to inborn characteristics of our species or to enculturation. Recent studies have shown, though, that infants as young as 2 months old show preferences for consonant over dissonant music, and that their brains can transpose music subconsciously, showing an innate human ability to understand musical language12. It is interesting that not only do they they fuss, fret and turn away from dissonant versions of the same Mozart that bemused them (instruments, volume and intention remaining identical), but they are apparently a lot less able to make sense of dissonant intervals, and particularly melodies that are rich in these. his means that infants do not really understand them, and as such, theyre easily forgotten13.

Another interesting aspect of infants musical development is natural maternal contribution. It may sound irrelevant at irst sight, but maternal singing has happened naturally in all cultures and ages.14 he fact that these harmonious sounds are the most harmonious ones for them is very justiiable in an evolutionary sense. Maternal singing is high-pitched (as are the noises created by small animals, more likely to be inoensive), simple, and highly consonant, qualities that one might then associate with safety and comfort15. We are hard-wired by evolution to be moved by these sounds, and one can arguably say that these three characteristics correlate very well with what we could qualify as pretty music. he most performed tune in the whole western world, is Happy Birthday16. A song that is short, simple, predictable, and, well, happy. Its no surprise that it was written by two women, because lyrical content aside, it could very well have been a lullaby.

Why do people listen to music styles that contravene natural tendencies?

But a large number of people dont generally experience, create or consume art because it is pretty, although it is evidently a valid reason to like something! Some people crave for powerful, meaningful, inspiring and enriching experiences,

and they demand art to serve this purpose. In a way, a lot of artistic behaviour relates to these demands, be it music, architecture, cinema, literature, culinary and scenic arts. So, aesthetic judgement cannot be taken without a context, and dierent parts in an artistic work evoke real life experiences and feelings. In that way, scary music could be equated to a horror ilm. A grave, solemn symphony could be equated to an epic late-romantic poem. And, in a comparison a bit more far-fetched, dissonance could be thought as serving a similar purpose to hot spices; it adds richness to the general piece when added tastefully, and can be used in more violent ways, but by itself it is perceptually annoying.

Liking dissonance is a cultural phenomenon, since were not naturally inclined towards it. Reasons for this are as numerous as the theories, but most of the latter tend to relate to revolve around reward and emotion neural systems17 , in-group versus out-group social identiication with subcultures18 , and in some cases, just sheer elitism (minority inluence)19 . Several new studies have revealed, though, that there is a strong correlation between preference for sonically aggressive music styles, and strong sensation-seeking. Strong musical experiences are caused by the same reasons than other strong experiences (e.g. extreme sports), namely, a high arousal brain state20. In musical terms, high arousal states can be caused by: high volume, an arousing social environment, memorability, sentimental compatibility, and complexity; all of these characteristics can be found at a hard rock concert. he fact that hard rock and heavy metal elicits strong perceptual and emotional experiences in fans is quite obvious, but particularly conirmed by a study conducted by Arnett (1991)21 in which he found that 61% of metal fans count at least a musician as one of their top three most admired people, whereas only 13% of non-fans did so. his is obviously related to the listeners personality, but also in the arousing potential of the music. And for music to be arousing, it needs to be complex, yet memorable (there are types of metal that do not care about memorability, but their fans tend to care more about sheer violence than quality). Musical complexity can be in structure (musical language or note density) or in texture (sounds with a higher harmonic and/or percussive content). A complex musical structure is, above a certain threshold, always in detriment to memorability (dissonant chords and lightning fast scales are just not catchy), but textural complexity seems to preserve it; the tradeos are sonic aggressiveness and dissonance, which are no problem for a heavy metal audience.

12 Trehub, (2003)13 Zentner, Kagan, (1998)14 Brakeley, (1950). pp. 65365415 Trehub, Hill, Kamenetsky, (1997). pp. 38539616 D. Huron (2001), pp. 43-6117 Blood, Zatorre, (2001)18 Tarrant, Hargreaves, and North, (2001). pp. 565-58119 Aebischer, Hewstone, Henderson, (1984)20 Konecni. (2005). pp.27-4421 Arnett, J. (1991). pp. 76-98

Increasing dissonance whilst retaining memorability

he second third of the 20th century was characterized by the race towards dissonance in classical music, that culminated in the serial and dodecaphonic music of Schoenberg and his disciples. But, strangely enough, instead of setting us free of the tyranny of tonality, this era of music seems like an oddball, which fades away in silence and without further protest. Most major works of that time are notoriously absent in auditoriums around the globe, no major music composers are writing serial music, and the ones of that period that are still active (e.g. Ligeti, Penderecki) have returned to more tonal, conservative musical languages. Why is that so? What did that whole batch of talented musicians forget? We listen to music because we like to do so, all other things considered; but, in order for it to be meaningful, it needs to be likable in the irst place. All the fuss about the so-called problem music (namely rock, metal and hip-hop) in the last decades has somehow overlooked the fact that, people actually enjoy said music, and they actually feel particularly strongly about it as well (although hip hop does not involve the use of dissonance, and as such is outside the main topic of this assignment). his is because the music is both memorable and arousing, and evokes feelings and ideas which the audience can relate to. Now, dissonance that impairs cognitive processing is bad for any type of music that deems itself as memorable; dissonance that does not can become an acquired taste, particularly on people wishing to run away from the bland and meaningless.

Practical Applications

Sharpness: both problem and tool

he notion of sharpness is somewhat vague, but its existence is both empirically veriiable and already actively used by sound engineers around the world. he easiest way to realize this is when we boost high-mid frequencies to certain sounds to make them cut through the mix. What were actually doing is increasing the sharpness of the sound to make it more noticeable, whilst occupying a small place in the frequency spectrum. Another, more novel, use of sharpness is a new design of de-esser calculation algorithm, devised by Martin Wolters and Markus Sapp in 200122 , and it has already been implemented in their Omnia brand of hardware processors for broadcast. he annoyance of sharp high frequency transients in recorded vocals has long been recognized, and use of sharpness-aware algorithms present an elegant tool to ix it.

Just intonation and electronic music applications

For about a hundred years, equal temperament has reigned supreme, irst in classical music with the rise of twelve tone and serial music, and then in popular music. he reason why has never been because of any degree of sonic superiority23.

he truth is, every chord played on a 21st century keyboard is in fact out of tune. he justiication has always been practical; to play mechanical instruments with ixed notes (be it keyboard-based, like a piano or an organ, or fretted, like bass and guitars) means either to play in only one key, or to use tempered tunings, and endure roughness in many common chords. With the advent of DAWs and electronic-based instruments however, the practical limitations quickly fade away. Any MIDI based melodic instrument can theoretically easily be programmed so that the pitch of the notes change in time, so that most intervallic relationships in every chord are perfect. But with the increase in the culturally acceptable threshold of dissonance and the tuning inconsistencies of the most common instruments (hear appendices 2 and 3) makes those dierences almost negligible. Still, there are musical contexts in which it might be useful, and now possible and even easy, to try variable note pitch, just intonation scales.

Controlled dissonance as a useful parameter for sonic aggressiveness

As we have seen, dissonance that does not hamper note recognition can bring both richness and aggressiveness to the sound. But what kind of processing, or eect, would create it? Well concern mainly with guitar and voice, since theyre the two main melodic instruments in aggressive music types. First, electric guitars have had, since they were invented, more than their share of distortion, caused by the non-linearities of ampliiers and speakers. he main eect this distortion has that is of interest to us is intermodulation distortion. Since it creates sidebands that are the sum and dierence of dierent multiples of the input tones, feeding it back with just one tone and its harmonics does nothing but recreate more harmonics. But the most common use in guitar music is chords, and thats where dissonance comes in. he roughness of that intermodulation distortion created by tempered chords, even a power chord (hear appendices 2 and 3) can actually be desirable, as long as the chord is simple, since it does not detract from note recognition. If a complex chord is played, then the sidebands overtake the harmonics and the pitch of the notes is unresolved, basically sounding like an amorphous buzz. If a complex chord needs to be played, one must rely on overdubbing each note separately (hear appendices 4 and 5). he result can be somewhat artiicial sounding, but it has been used with great success by artists like Brian May of Queen.

Vocal sound is another thing altogether, because the actual instrument (that is, the human body) provides us with an eect that is unrivaled in the electric world; this eect is the very elaborate set of amplitude modulations that are the main sonic characteristic of screaming. Although ring modulators can provide a bit of that raspiness (hear appendices 6 and 7), the complex interaction between throat muscles, vocal folds, and forced air creates subharmonics and numerous sidebands,

22 Wolters, Sapp, (2001)23 Dun (2006). pp. 103-118

most of which tend to be related to the screamed note, but some who are not. he singer actually can control how much of these unrelated sidebands are produced as a whole, increasing the feeling of roughness in the voice at will24 (hear appendices 9 and 10). In the most extreme musical styles, some techniques (e.g. black metal growling and hardcore pig squeals) are completely atonal, but since they do not contribute melodically, they are not memorable by themselves.

For aggressive music types, overtly dissonant passages help the perceived power and impact of the track (hear appendix 11), but they tend to be short, tend to avoid complete atonality in all instruments at the same time, and they are seldom in the memorable part of the song (namely the hook, usually the chorus).

Taming unwanted roughness

Sometimes, actually most oten, roughness is undesired in a recording. It can arise because of tuning inaccuracies, unwanted clipping in a recording, or the consistently avoided combination of high harmonic distortion and complex chords. Sometimes, the recorded dissonance needs to be tamed, in order for it to blend into the track, or to mask the recording or performance imperfection. Since the sensation of roughness arises not only from pairs of clashing harmonics in the same critical band, but mostly from their coherence25, randomizing their phase can help reduce said roughness. he best way to do this is through reverberation (hear appendices 8 and 10). Reverb also reduces the high frequency content, and high-end units also have a modulation parameter, which creates a great number of relections that are so slightly out of tune, and are in such great number, that roughness cannot arise between them. A similar, yet simpler event happens when employing the use of vibrato, both as a processing eect or a performance one; What vibrato eectively does is supersede roughness with its own, low frequency modulation.

Conclusions and Future Directions

In the end, most practical applications for the new indings in the subject turn out to have already existed, albeit without theoretical justiication. his intuitive evolution both in musical styles and recording techniques is actually quite obvious, since most modern music is meant to be liked, whether this motivation is admitted and conscious or not. Modern technology will keep advancing, making more discoveries and allowing both engineer and musician to accomplish sounds that are still either impossible or impractical, and the physics and psychological theories will eventually come to explain them, but these can still be of practical relevance every once in a while. In any case, there is still a lot of room for research, particularly in the domains of social psychology and neurology, and in the still largely, but unjustly, unexplored ield of modern popular music.

24 Titze, (1992)25 Terhardt, (2000)

Appendices

Physiological and Perceptual Principles

Sharpness

he most basic property about the sound quality of a sound that has to do with dissonance is sharpness (see igure 1). It is measured in Acums (latin for sharpness) and it seldom gets a mention in music production because it is mostly used for noise in industrial applications. It is proportional to the spectral center of gravity and increases with loudness. It could be deined as spectral unpleasantness26 , which is something that we experience, for example, when we hear music we like through horrible tiny speakers: no bass, and highly exaggerated upper midrange and treble. Another example would be sibilance, which can be a result of incorrect recording methods but can actually be down to the singers speaking habits.

Beats

Beats are periodic variations in amplitude that arise when two sounds of slightly dierent frequencies are being produced at the same time. his natural amplitude modulation is caused by the physics of sinusoidal waveforms: when two waveforms are in phase, they add up, when theyre out of phase, the cancel out. he frequency of this beating is equal to the dierence between the frequencies in play (see igure 2). Its important to note that beats between more than two sounds of complex waveforms can be extremely dicult to measure (since beats will be created between all the harmonics as well at dierent rates), and in most cases it generates a sensation of richness, as long as the beats are not too pronounced and as long as their frequency is low. his purely acoustic phenomenon was irst described by Mersenne in 163627 , but it had arguably been well known long before this to organ tuners, who actually used it in their advantage to tune strings in unison, or by a distance of a calculated number of beats, with the harmonics of strings of lower frequency.

Intermodulation Distortion

Most physical systems that transmit or reproduce sound are at least slightly (and sometimes very) non-linear, meaning that the input is not completely proportional to the output. Such a system such as these adds harmonics to every single reproduced, or transmitted, frequency. When several tones are added through it, the mathematical sum implies the creation of extraneous tones, called intermodulation products, which are sums and dierences of every multiple of the original frequencies. If the mathematical relationship between the original tones is not simple, then the intermodulation products are unrelated to these, and in most cases, will produce beats at unpredictable rates28. his is a very important fact, since it dictates that complex tones with a simple mathematical ratio (e.g. 1/4, 3/5) will tend to create intermodulation products that do not produce beats, whereas those with a complex ratio will tend to produce intermodulation products that do.

Ear Non-Linearity and Combination Tones

he human ear is capable of hearing an impressive dynamic range. he dierence in power between 1 and 120 dB is staggering, but yet we dont hear sounds as thousands of times louder than others. his is due to our cochlea, inside our inner ear, not being a linear system. Its outer hair cells act as a real-time compression system, and a very good one at that, but it does induce some intermodulation distortion. So, when two tones are heard, we can sometimes hear what are called combination tones, namely intermodulation sidebands. Most of the time they are masked by the original sounds, since it is an rather uncommon thing to hear sine waves in nature. But when the sounds are very tonal in nature, then those combination tones create sonic illusions. he

Figure 1: Sharpness vs. requency and bandwidth

Figure 2: sine wave interference.

he top graph shows both of them in

parallel; the bottom one shows its sum.

26 Fastl, Eberhard, Zwicker (2007). p 239.27 Mersenne, (1636), p. 362.28 Jlicher, Andor, Duke, (2001)

violinist Giuseppe Tartini noted in 1754 that he could create the illusion of hearing a note an octave down if he played a perfect ith on some scales. Its actually not as dependent on volume as some people thought29(hear appendix 1). So, to sum it up, note combinations whose fundamentals have a simple mathematical ration will share many combination tones, and they will be in tune with said fundamentals. If the ratio is complex, then beatings will occur between them; and depending on the exact waveforms, notes and phase, they can be extremely noticeable.

Auditory Roughness

If the frequency of perceived beating increases above approximately 30 Hz, and if the two tones are of a similar amplitude, the amplitude variations become increasingly hard to follow independently. A new sonic attribute arises, called roughness. his attribute does not need to be caused by wave interference; any sonic event, played in rapid succession, will generate it. Roughness is an auditory percept representing a sounds time structure, though in a time integrated manner.30 In other words, it is our brains way of telling us its the same bit played over and over again, so quickly i cant count it. Roughness increases with an increase in pulse frequency, with sharpness and level of harmonic content, and coherence of these luctuations. his means that, if the amplitudes and phases of Fourier components are randomized, (incoherence is introduced) then roughness is decreased. his is an important fact, because that is precisely what reverb does, and thus it can help diminish the sensation of roughness when used as an eect, for example. New research in this last century has refuted the widely spread notion that modulation is necessary for roughness. Terhard indicates that for a multi-component complex signal, though its auditory excitation covers a wider area of the cochlear partition, it is the luctuation of temporal envelope that must be regarded as responsible for roughness. he luctuations of envelopes, however, are practically coherent in a wide area of the cochlear partition. Whatever, it is inadequate to think that roughness is an auditory sensation that is exclusively associated with modulated sounds and even is a kind of auditory representation of a sound's modulation (i.e., either AM or FM).31

Although both AM and FM (beats are a form of AM, and intermodulation distortion is a mixture of both32) are the most common ways of experiencing roughness, the fact that the phenomenon is not dependent on modulation alone is important, since there are then ways to reduce the roughness in what we previously though were inherently rough frequency intervals, namely for chords that we classify as dissonant.

Tuning and Temperament

Western music has always relied on certain sonic intervals that sound more musical than others. Since Pythagoras more than 2000 years ago, the basic musical note relationships have been based on simple frequency relationships. he octave (1/2), the ith (2/3), the fourth (3/4), the major third (4/5) and the minor third (5/6) are the basis for both the pentatonic scale and the diatonic scale, which is overwhelmingly the one used for centuries.

Interestingly enough, these relationships can only relate to one fundamental note, or tonic, at any given time. Modulating into another scale based on another note changes the exact note positions, so the ones that sounded good for the irst scale sound out of tune on another note. his dichotomy is most easily explained since pitch structure is logarithmic and harmonic content is linear; and for centuries, musicians have had to compromise on tuning accuracy, just to be able to play melodies and chords that modulate into dierent keys. hese deviations are called temperaments, and there have been literally dozens of dierent tuning systems in the last two millennia. he one in use this day, called equal temperament, was discovered hundreds of years ago, but most people didnt like it at all, since the major third is very out of tune compared to the ideal one. Our ears have grown used to it, but some people are still against it to this day.

29 Chris Darwin, (2008)30 Ernst Terhard, (2000), pp. 1-331 ibid., p. 432 Suzuki, Hideo; Shibata, Shigenori, (1983)

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