Musical Scales

The Skudučiai, the traditional Lithuanian multi-pipe whistles or pan-pipes, usually contain from 5 to 8 pipes of different lengths with one end closed; made of umbellate plants, wood or bark. The Skudučiai differ from other pan-pipes: they are unbound and blown by several performers. The instruments were used to play specific polyphonic compositions. The pipes of the Skudučiai were tuned “by eye” (i.e. by selecting pipes of different length) and “by ear” (i. e. by playing the parts of the most popular musical composition of the village) successively from the lowest to the highest pitch or vice versa.
We aim to measure and generalize the interval sizes in the tunings of archived Skudučiai, and to reveal their possible psychoacoustical origins.
To complement the earlier findings of Skudučiai tunings, a comprehensive overview of the principles of the tunings based on ethnographical data is presented. Isolated pitches of three sets of Skudučiai preserved in the Lithuanian National Museum as well as several rhythmic patterns were recorded. Each recording was repeated twice (by the two authors). The acoustical measurements revealed the irregularities in the three pitch scales: no correspondence in their intervallic structures was found. The only generalization that can be made is that the scales are designed quite loosely; the intervals range from wide minor seconds to narrow minor thirds. Together with the samples studied earlier, the mode of the distribution of the intervallic steps equals 184 cents and the interquartile range is from 141 to 231 cents. Moreover, the pitches in the recordings by the two authors differ, as do the pitches produced by a single player in isolation and in a rhythmic pattern (presumably because of the differences in blowing force). Often these differences exceed 10-20 cents. It can therefore be concluded that Skudučiai makers were not concerned with precise tuning, in the contemporary sense. In addition, precision better than ~20 cents was hardly possible. Nevertheless, the maintenance of roughly 180-190 cents, on average, suggests that the psychoacoustical phenomenon of roughness was at work: probably the makers intended to achieve considerable roughness in the simultaneous sounding of two neighboring pipes (refer to the abovementioned case of vocal Sutartinės).
The tunings of Skudučiai are loose; they share no resemblance with 12ET and present a peculiar pitch categorization: roughly equal intervals with the tolerance of mistuning considerably higher than that in the usual usage of 12ET. The tunings are centered on the phenomenon of maximum roughness as ideal.
The insights of the present study, possibly, could be extended to the studies of different musical cultures featuring examples of Schwebungsdiaphonie.

Tunings and intonations in the Lithuanian traditional fiddling are analyzed. 14 archival recordings from 1930s are applied for the acoustical analysis. A huge variety in the individual tunings and intonations is found. Although the fiddles were tuned mostly in the standard way, i.e. in perfect fifths, the fifths are quite " rough ". The scales show a strong tendency towards the constituent interval equalizing, or, in other words, certain hybrid scales between the diatonic and equidistant ones are estimated. Concerning the temporal intonations, influence of fiddling motorics and the related influences of bowing force and pitch duration are discussed.

The article is part of the project “People of America sing and dance” of Totolincho Association from
where the relationship of memory and retrieval of traditional technologies is nourished (TECTRA in Spanish),
in this case, from the musical know-how. Therefore, this paper presents an echo of memories about
an ancient and millenary instrument, the Zenú Flutes and its global distribution throughout the north of Colombia (Zenú territory) to the recognition of true expertise and skill of the builders of such instruments.
Thereby, we identify the association between ritual, music, dance and the possibility of complex creations,
which if seen from our contemporary viewpoint, does not let us to realize the natural connection between
the dwellers of the hydraulic complex with their natural environment and the understanding and management
of music through beak and globular flutes. It then reaches for an awakening of the sacred, the daily
life, which is constantly born and dies, the memory of sound from a millennium ago, which, no doubt,
could remain on the shelves of the best museums in the country or, as in this case, turn into a breath of life.
Wondering is the important, acting is the relevant.

Musicologists no longer argue about the fact that the scale of Georgian folk music is neither major-minor nor does it coincide with medieval European modes, though what exactly this sequence of sounds concretely represents is the issue modern researchers are interested in. At different times this problem has been studied by Georgian and foreign scholars – Kakhi Rosebashvili, Malkhaz Erkvanidze, Stewart Gelzer and others – who made quite a few suggestions. The present paper is another attempt at presenting the results about the Georgian scale produced on the basis of the two-year-long practical research. In the rst stage of research the hymns-troparia, recorded by Artem Erkomaishvili, were selected as the subject of the study (as it is known in Erkomaishvili's recordings only six voices of the troparion are present: the third and fourth voices are missing). In order to reveal their sequence of sounds by means of the spectral analysis provided in the programme we measured the sonic frequency of the top voice 1. Having measured the medium frequency of each sound in Hz (in notated examples it is indicated with numbers above the notes) and the distance between them in cents (below the notes), we determined the scale of sounds of the whole hymn by means of the corresponding programs. The measurement showed that the voice of absolutely every troparion has its own musical scale (which will be explained below). After that we measured the top voice (the lead voice, i.e. Dimitri Patarava's voice) in the eleven hymns recorded in three parts by Dimitri Patarava, Varlam Simonishvili and Artem Erko-maishvili (the CD is called " Eleven Pearls " 2). A count of the average medium of the sound height of each scale step of the given hymns yielded very strange and unexpected results: the top voice (i.e. the lead part) of every hymn occurred in the same scale, in spite of the fact that all eleven hymns are of different genres, voices and moods. And the sound scale is as follows: a seven-degree octave, the distance between the degrees being equal, i.e. approximately 170–175 cents. " Approximately " because the deviation of 10, 20 and even 30 cents does not change anything, as during the sounding of the song or hymn the human ear is virtually unable to perceive it. An example is the hymn Dghes Tskhovrebisa (On the Day of Life) recorded by Artem Erko-maishvili (the troparion of the fourth voice of the Annunciation). The rest four sounds are the same. After measuring these four sounds it was discovered that the deviation between these sounds is 40-50 cents, but the human ear perceives them as being on the same pitch (ex. 1, audio ex. 1). The situation is the same in the hymn Sitqvisa Ghvtisa (God's Word), St Nino's troparion of the fourth voice (recorded by Artem Erkomaishvili): here too the rst four sounds are the same, though the uctuation between them is about 60 cents (ex. 2, audio ex. 2). The result indicates that in the top voice of the " Eleven Pearls " uses only one musical scale, and within the octave the distance between the neighboring sounds is the same (a so-called equidistant scale).

La gamme est un condensé de mélodie qu'on peut appréhender sans connaître le solfège, par l'approche sensorielle. La gamme, c'est ce qui change sur les instruments à notes fixes lorsqu'on change la hauteur de la mélodie. La notion de mode est naturelle et reliée à la hauteur relative. Un mode, c'est ce qui reste fixe lorsqu'on change la hauteur de la mélodie. C'est une façon de diviser l'octave en échelons intermédiaires. Cette distinction est apparue récemment dans l'histoire musicale, en même temps que l'idée de hauteur absolue. Ceux qui, conditionnés par le gavage scolaire, s'imaginent que l'univers musical repose en tout et pour tout sur les modes majeur et mineur, devront réviser leur point de vue et reculer les frontières de leur esprit. La construction des modes devient un jeu surprenant et amusant. De la manière d'arranger des intervalles à l'intérieur d'une octave. Découverte de nombreux types de modes produits par des cultures créatives de notre vaste monde.

Pourquoi les interrogations sur la définition du ton et des intervalles constitutifs des gammes ont-elles préoccupé tant de compositeurs et de musicologues depuis l'antiquité grecque? Le ton, la tierce, la quinte, l'octave sont-ils des données absolues? Non, bien au contraire. Au cours du temps, et cela au moins depuis l'antiquité grecque, les intervalles ont été déterminés sur des bases mathématiques et techniques, en faisant appel à la science des nombres, tout en cherchant à s'adapter aux instruments existants. La gamme tempérée a succédé à d'autres gammes, telles que la gamme pythagoricienne, la gamme de Zarlino, et divers systèmes de tempéraments. Elle est une convention passagère, correspondant à la mentalité d'une époque. L'essentiel réside dans l'effet sonore qui résulte de la gamme, et dans la façon dont elle résonne et agit sur le corps et sur l'âme. Des recherches récentes tentent de trouver une intonation qui soit juste pour le corps et pour l'Être. Mais est-ce possible de façon définitive et universelle?

Les éléments harmoniques et géométriques qui touchent notre âme et notre corps: Résonances corporelles; Modes et gammes; Sons créateurs de géométrie; Évolution historique de l'expression musicale; Caractéristiques physiques des sons; Controverses sur les diapasons

As we take a look at the history of Polish music, we can see, that the tendencies popular in Western Europe in the beginning of the 20 th century, e.g. impressionism, expressionism or dodecaphony, were developed much more lately. This was mostly because of the political situation in Poland but also because of sentimentalism to the Romantic Era and audience's conservatism. However, there were many excellent examples of the approach to modernism in Polish composers' works. The author of this paper examines two of them. The first one can be seen in a composition by Karol Szymanowski entitled Mity, Op. 30 (1915) for violin and piano, in which the composer created a new musical style-violin impressionism. The second one is seen in a musical piece by Ludomir Michał Rogowski, Caprices (1922?) for viola, soprano or clarinet and piano, in which the composer tried to break the rules of classical harmony by usage of, as he said, "natural" scales. The aim of the article is to show a new approach to musical material by the aforementioned composers that is-their way of modernizing Polish music.

Since the 1960's, a kind of musicological gospel claims that Amhara secular music is based,
supposedly from time immemorial, on a so-called closed system of four pentatonic modes or
q!ñ!t called ambasäl (አምባሰል), anči hoye (አንቺ ሆዬ), bati (ባቲ) and t!z!ta (ትዝታ), usually
presented as a set of tempered or quasi-tempered intervals such as major and minor seconds
and thirds. It is presented as an established, accepted and reliable norm, in scholarly as well
as in general publications. Today urbanized azmaris (the wandering minstrels of the Amhara
and Christian highlands) present it as the basic of traditional music – whereas azmaris of the
deep countryside have simply no knowledge about it. This study aimed to re-explore in detail
the Amhara secular scales, and to tackle in depth q!ñ!t historicity and its musicological
relevance. To this end, a corpus of recordings dating back to 1939 have been analyzed and
confronted with written sources. A control methodology was developed, in order to verify
incoherent results due to age and quality of some of the recordings. A comparison with the
intervals measured in songs recorded in 1976 (by Cynthia Tse Kimberlin) and 2005 (by
Stéphanie Weisser) was completed, covering therefore a period of about 70 years.
Keywords: Ethiopia, q!ñ!t, tuning, scale, intervals, historicity, music

This dissertation presents the theory of well-formed scales in the context of the mathematics of distribution modulo 1. Deleted: volume descriptions. Chapter III presents proofs for the assertions of the theory. Three entirely new aspects of the theory are asserted and proven in Chapter III (Theorems 1, 4, and 5). These concern a definition of well-formedness, the nature of the step sequence of the well-formed scale, and the conditions that govern coherence, or generic ordering in well-formed scales and constitute and significant new contribution to the theory. This is also true of the nine corollaries to Theorem 2. Applications and extensions of the theory are the subject of Chapter IV. The theory of well-formed scales is applied to compositional issues including microtonalism, and an extension of the theory into the realm of rhythmic analysis, in which I show that hierarchies of well-formed scales are an appropriate model for the study of rhythm of Bartók. The first two chapters review recent diatonic theory and reconstruct the theory of well-formed scales. Appendix A presents the mathematical concepts and notation that are used throughout the dissertation, particularly in the third chapter. Appendix B illustrates mappings between mathematical and musical realms. Appendix C presents the code for " ScaleDesigner, " a computer program I wrote in order to explore the microtonal possibilities of well-formed scales. ScaleDesigner is another new contribution in this dissertation. Appendix D is the score and analysis of a song I composed, " Amarantha, " which exploits the contrapuntal advantages of a non-triadic diatonic [2] harmony. Appendix E is a glossary. 2

Pythagoras developed an eight-note musical scale in which ratios of small whole numbers were used to define the length of the strings. While Pythagoras committed nothing to writing, later Pythagoreans wrote a good deal about these matters, and these principles were used to tune musical instruments until fairly modern times. In the seventeenth century, equal-tempered chromatic scales were introduced in which the frequencies of successive notes differ by a common ratio. In the early days of Rosicrucian activity in America, H. Spencer Lewis published a system of " natural harmonics " in which the notes of the chromatic scale were correlated with the colors of visible light and certain vowel sounds. Dr. Lewis used a value of middle C (C4), 273 Hz, that was based on a specific wavelength of green light in the middle of the visible spectrum. Modern scales, on the other hand, use values such as 256 Hz and 262 Hz for middle C. As the physiological and psychic effects attributed to the musical notes may be a function of the scale used, this paper examines and compares different musical scales. Based on this examination, it is believed that no one frequency for middle C is best for all occasions. Résumé Pythagore a développé une gamme musicale de huit notes (accord pythagoricien) qui utilisait les rapports de petits nombres entiers pour définir l'accord. Tandis que Pythagore ne cocha guère ses théories par écrit, des pythagoriciens ont ultérieurement écrit de nombreux traités sur cet argument et ses principes ont été utilisés pour accorder les instruments de musique jusqu'à la fin du Moyen-Age. Le XVIIe siècle vit l'introduction des gammes tempérées (tempérament égal), dans lesquelles les fréquences des notes successives sont reliées par une relation commune (intervalles égaux). Dans les premiers temps d'activité de la Rose-Croix en Amérique, le Docteur H. Spencer Lewis publia un système d' « harmoniques naturels », dans lequel les notes de la gamme chromatique sont en corrélation avec les couleurs de la lumière visible et avec certaines voyelles. Le Dr. Lewis utilisa à cet effet une valeur pour le do central de 273 Hz, qui correspond à une longueur d'onde spécifique de la lumière verte du milieu du spectre visible. Les gammes actuelles, en revanche, utilisent des valeurs d'environ 262 Hz. Du moment que les effets physiologiques et psychiques attribués aux notes musicales peuvent différer selon les fréquences utilisées, cet article examine et compare les différentes gammes musicales. On en déduit qu'aucune fréquence unique pour le do central n'est idoine pour toutes les circonstances.

Sastra lisan bagi masyarakat adat Lampung merupakan bagian penting sebagai simbol budaya. Oleh karena itu,keberadaannya perlu dilestarikan dengan pelbagai upaya, termasuk melalui pendidikan. Di tengah kemajuan teknologi dan perubahan iklim sosial yang semakin bebas, keberadaan sastra seperti menunggu‘bom waktu’ karena tergerus oleh budaya global. Pisaan sebagai salah satu bentuk sastra lisan Lampung semakin jarang ditemui karena kurangnya upaya-upaya pelestarian, fungsi media media,serta kemampuan untuk mengajarkan mewariskannya melalui pengajaran yang baik. Di lingkungan sekolah para guru perlu untuk mempelajari Pisaan dan mengetahui metodologi pengajarannya. Tujuannya agar mereka menjadi agen pendidikan dalam mempertahankan tradisi lisan Lampung. Kenyataannya tidak banyak guru-guru yang menguasai sastra lisan (termasuk Pisaan). Kalaupun ada yang mampu melantunkan, tetapi umumnya tidak banyak yang bisa menularkan kemampuannya. Jika kondisi ini dibiarkan, maka keberadaan Pisaan dan sastra lisan Lampung akan mengalami kepunahan. Pada konteks itulah mengapa buku ajar ini hadir untuk menjawab persoalan-persoalan di atas. Penulis 1 buku ini adalah dosen sekaligus Ketua Program Studi Magister Pendidikan Bahasa Lampung di lingkungan FKIP Universitas Lampung. Bidang keahlian yang dikuasai meliputi kajian linguistik, budaya, dan sastra daerah. Penulis 2 adalah seorang dosen pendidikan musik di lingkungan FKIP Universitas Lampung. Keduanya aktif meneliti budaya Lampung dengan kesadaran bahwa kajian budaya dapat direfleksikan dalam berbagai ilmu terapan, termasuk sastra lisan. Melalui perpaduan kedua disiplin ilmu-linguistik dan musikdiharapkan mampu memperkaya kajian sastra lisan Lampung, khususnya pada bidang metode pembelajaran Pisaan. Penulis berharap buku ini dapat dikonsumsi oleh banyak kalangan pembaca. Buku ini disusun untuk para guru dan calon guru yang pengajarannya berkaitan dengan sastra lisan, khususnya Pisaan. Buku ini memberikan pengalaman belajar Pisaan secara musikal. Di dalam Pisaan terdapat melodi yang merupakan unsur penting dalam musik. Melalui pendekatan belajar secara musikal, Pisaan tidak hanya dipelajari secara teks. Tetapi juga dipelajari dari aspek musikal yang melengkapi estetika penyajiannya. Materi dalam buku ini berasal dari data-data penelitian awal yang dilakukan pada tahun 2017-2018. Data berupa rekaman yang didapat dari para pelantun Pisaan perempuan yang berasal dari etnis Pepadun. Melalui data tersebut penulis melakukan penelitian dan mulai mengembangkan metode musikal untuk pembelajaran Pisaan. Selain itu, data didapat berdasarkan buku dan pengalaman penulis 1 yang juga merupakan seorang budayawan Lampung. Istilah dan tata bahasa yang digunakan dalam buku lebih banyak menggunakan istilah-istilah musik. Penulis berusaha menyederhanakannya agar mudah dipahami. Untuk pembaca yang belum terlalu akrab dengan istilah-istilah di dalamnya akan disedikan keterangan. Buku ini terdiri dari lima bab, diantaranya: (1) sastra lisan Lampung; (2) model pembelajaran Pisaan; (3) menentukan teks dan notasi; (4) tonalitas; dan (5) latihan Pisaan. Materimateri yang disusun dalam bab-bab tersebut mengulas tentang pendekatan musik untuk menganalisis dan mengajarkan Pisaan. Pada akhirnya penulis menyadari bahwa buku ini masih memiliki banyak kekurangan. Oleh karena penulis terbuka untuk menerima kritik dan saran untuk menyempurnakan materi yang ada di dalamnya. Semoga buku ini dapat memberikan pengetahuan baru bagi para pembacanya. Selain itu menjadi catatan amal bagi penulis dan tercatat sebagai bentuk ibadah kepada Allah SWT.

Winner of the Alt Pibroch Club and Altpiborch.com “Modern Pibroch Library” Competition, 2018.

Program Notes and Piece Mechanics:

Shrouded in myth and mystery, a tradition that has been passed from master to pupil for unknown centuries, piobaireachd (Scots Gaelic for “piping” and anglicized as “pibroch”) is the “classical music” of the Great Highland Bagpipe, originating sometime during the Middle Ages. Piobaireachd is the oldest and perhaps most austere of all the Scottish bagpiping genres, focusing on simple tone progressions with slight elaborations over a series of variations. Crucial to this style is tone and timbre, and thus highly refined and particular tunings were constructed around these pieces, generally emphasizing tuning by Pythagorean ratios, particularly focusing on the 7th and 11th partials.

This piece returns the modern Scottish bagpipe to these medieval traditions, however in unconventional ways through building a scale of 15 micro-tonal notes based on octave transpositions of the prime partials of an implied fundamental. Along with the 7th and 11th partials, the scale of this piece incorporates all the prime partials from 2 to 31. This resulting piece is both Medieval in aesthetic and form, but modern in its expansion of Pythagorean tuning and extended techniques on the bagpipe, including unconventional droning, alternative fingerings, and multi-phonics.

The scale is based on octave transpositions of all the prime partials (up to 31) of the fundamental note, which is (approximately) 60 Hz. This fundamental was chosen for convenience, since most Great Highland Bagpipes are tuned to about 480Hz on the lowest “A” of the chanter. The three drones form the second, third, and fifth partials above the non-sounding, implied fundamental of 60Hz. The primary chanter notes (melodic notes), given on the first page of the score instructions, form the other octave-transposed prime partials up to the 31st – 7, 11, 13, 17, 19, 23, 29, 31 – barring the low-A, which is simply a multiple of the 60Hz fundamental (Note: the tuning of these partials is as close as possible, +/- 5 Hz or so, pretty much indistinguishable to most human ears). The additional microtonal inflections or embellishments are totally for color and are not based on any mathematical proportion. They are simply created through alternative fingerings. The primary melodic content, however, is solely based on these prime partials.

APC and Altpibroch.com:

The Alt Pibroch Club and altpibroch.com are sites dedicated to the pursuit of bringing new approaches to bagpipe performance informed by ancient scores. While we support reconstructions of old playing, they are also interested in bringing new musicality to piobaireachd long since forgotten, much to the detriment of audiences and performers alike. AltPibroch.com is the definitive resource for information on pre-1841 pibroch manuscripts, their interpretation and history. These are the original, unexpurgated primary sources of our art, and the different musical perspective, interpretation and insight they bring is a revelation to those accustomed to the modern pibroch scene. They encourage performers to explore the full musical depth and creativity of the folk music of piobaireachd, from as many sources of instrumentation and inspiration as they can find.

Jordan Alexander Key:

Jordan Alexander Key (b. 1990) currently pursues his PhD in composition and musicology at the University of Florida. He earned bachelor’s degrees in music composition, mathematics, and Eastern Asian philosophy from The College of Wooster (2013), as well as a master’s degree in music composition from the University of Arizona (2015). His significant recent projects include his European premier with Vienna’s PHACE Ensemble performing his Octet, “Threnody on the Death of Children;” a performance by Boston String Quartet of his String Quartet No. 1; a concert with Bold City Contemporary Ensemble performing his Quintet, “Discursus Anachronismus;” a collaboration with the Vancouver Queer Arts Festival for his Art Song, "God Ourselves;” and the display of his recent audio-visual projects as part of the Wolfsburg Kunstmuseum’s new exhibit, “Never Ending Stories: The Loop in Art, Film, Architecture, and Music,” in Germany during the Winter and Spring of 2018. Jordan gives regular concerts on his two instruments: the pipe organ and bagpipes. His performance repertoire focuses primary on music from the Middle Ages, Early Renaissance, and Early Baroque. He has been a bagpiper for 15 years as of 2018. For more information on Jordan Alexander Key and for recent recordings of his works, see his website, www.jordanalexanderkey.com, or search for his work on YouTube (channel, “Jordan Alexander Key”)

A new theory of musical consonance discovered by Jacques Dudon through the creation of photosonic disks.
J.Dudon found that all traditional musical scales of the world show a high degree of differential coherence, that can be experienced in their acoustic qualities. Differential coherence completes the usual consonance theories, showing the importance of specific small variations for certain notes within musical scales, often considered as single notes. Differential coherence helped J.Dudon to create also many new scales and systems as well as new "fractal" sounds optimising tuning and timbre adequations.

A teoria das redes complexas tem se tornado cada vez mais em uma poderosa teoria computacional capaz de representar, caracterizar e examinar sistemas com estrutura não trivial, revelando características intrínsecas locais e globais que facilitam a compreensão do comportamento e da dinâmica de tais sistemas. Nesta tese são exploradas as vantagens das redes complexas na resolução de problemas relacionados com tarefas do âmbito musical, especificamente, são estudadas três abordagens: reconhecimento de padrões, mineração e síntese de músicas. A primeira abordagem é desempenhada através do desenvolvimento de um método para a extração do padrão rítmico de uma peça musical de caráter popular. Nesse tipo de peças coexistem diferentes espécies de padrões rítmicos, os quais configuram uma hierarquia que é determinada por aspectos funcionais dentro da base rítmica. Os padrões rítmicos principais são caracterizados por sua maior incidência dentro do discurso musical, propriedade que é refletida na formação de comunidades dentro da rede. Técnicas de detecção de comunidades são aplicadas na extração dos padrões rítmicos, e uma medida para diferenciar os padrões principais dos secundários é proposta. Os resultados mostram que a qualidade da extração é sensível ao algoritmo de detecção,
ao modo de representação do ritmo e ao tratamento dado às linhas de percussão na hora de gerar a rede. Uma fase de mineração foi desempenhada usando medidas topológicas sobre a rede obtida após a remoção dos padrões secundários. Técnicas de aprendizado supervisionado e não-supervisionado foram aplicadas para discriminar o gênero musical segundo os atributos calculados na fase de mineração. Os resultados revelam a eficiência da metodologia proposta, a qual foi constatada através de um teste de significância estatística. A última abordagem foi tratada mediante o desenvolvimento de modelos para a composição de melodias através de duas perspectivas, na primeira perspectiva é usada uma caminhada controlada por critérios sobre redes complexas predefinidas e na segunda redes neurais recorrentes e sistemas dinâmicos caóticos. Nesta última perspectiva, o modelo é treinado para compor uma melodia com um valor preestabelecido de alguma característica tonal subjetiva através de uma estratégia de controle proporcional que modifica a complexidade de uma melodia caótica, melodia que atua como entrada de inspiração da rede.

Because of an absence of a fixed referential tonality, musical scales can experience certain gradual changes in the course of a capella vocal performance. For example, the phenomenon of floating tonality or gradual transposition (usually a gradual rise of pitch up to several semitones) common in traditional, unaccompanied singing has been discussed (Niemi & Jouste, Greene, Ambrazevičius). Alexeyev studied some peculiar instances of so-called " evolving " scales in Yakut singing, i.e. the intervals between adjacent scale degrees tend to widen in the course of performance. In the present study, we aim to identify types of gradual changes of musical scales in traditional singing without accompaniment and to discuss possible causes for the changes. Praat-aided acoustical measurements of pitches in traditional unaccompanied vocal performances were carried out. The sample was composed of 20 song records exemplifying the Lithuanian vocal tradition, both monophony and homophony. Several phenomena were revealed. First, the gradual transposition (rise) from the beginning to the end of the song performances can be identified as typical case. Second, the phenomenon of " evolving " scales was found to range from negligible to striking, for different vocal performances. Third, sometimes an opposite phenomenon of gradual shrinking of the musical scales, i.e. microtonal narrowing of intervals, takes place. The rise and widening might be attributed to warming-up, timbral brightening, and mastering of range, whereas the gradual shrinking manifests as the upper scale degrees reach gradually the upper part of a vocal range not comfortable for voice production.

In contrast to Western art music, the dissonance-like sonorities in Schwebungsdiaphonie-cultures are at the core of the tonal structures. These cultures, although not abundant, are found in different locations all over the world (Cazden, Brandl, Messner, etc.). Sutartinės are a Lithuanian type of Schwebungsdiaphonie (Račiūnaitė-Vyčinienė, Ambrazevičius & Wiśniewska, etc.). On the one hand, the studies on psychoacoustic roughness and sensory dissonance are really big in number. The notions of roughness and sensory dissonance are usually considered as synonyms. On the other hand, it was proposed that ideal sounding of Schwebungsdiaphonie conforms to a maximum dissonance / roughness (Brandl, the diaphony in the Balkans and elsewhere; Ambrazevičius, the Lithuanian Sutartinės). In the present study, we analyze the occurrences of the notions of roughness and sensory dissonance in the psychoacoustic studies and define the case of Sutartinės in this context. The review of the experimental findings on the intervals corresponding to the maximum values of roughness / sensory dissonance reveals certain discrepancies between the concepts of roughness and sensory dissonance. It seems that, at least for a substantial frequency range, roughness is associated with larger interval sizes (Plomp & Levelt, Kameoka & Kuriyagawa, Terhardt, Zwicker & Fastl, Hutchinson & Knopoff, Vassilakis, etc.). Collation of these results and the findings of acoustical measurements of Sutartinė performances leads to the conclusion that the ideal vocal " clash " in Sutartinės most probably corresponds to psychoacoustic roughness, but not to sensory dissonance.

Acoustical measurements of pitches in a dozen songs exemplifying the Lithuanian traditional vocal homophony were carried out. Several phenomena were revealed. First, the entire scales experience gradual transposition (rise) from the beginning to the end of the song performances. Second, the transposition is supplemented with the gradual shrinking of the musical scales (the intervals become narrower). Third, the intonations of the scale degrees are dynamic, i.e. they depend on the musical (both melodic and harmonic) contexts. Fourth, the versions of musical scales work as certain markers for the idiolects (further studies could show if this might be extrapolated to the realm of dialects). All these insights raise issues about the perceptual qualities of the musical scales and their manifestations in the performance.

Background in ethnomusicology. The term Schwebungsdiaphonie ('beat diaphony') refers to two-part musical (usually vocal) styles with a lot of dissonant (beating) intervals such as seconds. In contrast to Western tonal music, the dissonant sonorities in Schwebungsdiaphonie lie at the core of their tonal structures. These musical cultures, although not abundant, are found in different locations all over the world (Cazden, 1945; Brandl, 1989; Messner, 1989; etc.). Sutartinės are a Lithuanian type of Schwebungsdiaphonie (Račiūnaitė-Vyčinienė, 2002; Ambrazevičius & Wiśniewska, 2009). Background in psychoacoustics. Roughness is considered either synonymous with sensory dissonance or its main component. It is caused by fast beating (amplitude modulations). Schwebungsdiaphonie singers adjust interval sizes to maximize sensory dissonance (Brandl, 1989; referring to the diaphony in the Balkans and elsewhere; Ambrazevičius, 2008a, referring to Lithuanian Sutartinės). Aims. We aim to differentiate between roughness and sensory dissonance (as defined in psychoacoustic studies) and consider the case of Sutartinės in this context. A supplementary purpose is to discuss the phenomena of tonality perception in Sutartinės. Main contribution. The psychoacoustic studies were overviewed and discrepancies between the concepts of roughness and sensory dissonance were noted. The experimental findings on the intervals corresponding to the maximum values of roughness / sensory dissonance were collated and significant disparities were found. It seems that, at least for a substantial frequency range, maximum roughness is associated with larger interval sizes than maximum sensory dissonance. Comparing these results with the findings of acoustical measurements of Sutartinė performances suggests that the ideal vocal " clash " in Sutartinės involves maximum roughness, but not maximum sensory dissonance. The tonal hierarchies in Sutartinės diverge substantially from Krumhansl's tonal hierarchy profiles, anchoring on a central nucleus and dissipating towards more peripheral pitches. Nevertheless, some associations with the tension-relaxation patterns characteristic of Western tonal music are apparent. Implications. Studies on sensory dissonance and roughness should define these notions more specifically and differentiate them. Findings may contribute to ethnomusicological courses on Sutartinės (and probably Schwebungsdiaphonie, in general).

O E-book Diários de Estudo surgiu da vontade de dividir minhas
reflexões sobre o estudo do instrumento, mais especificamente sobre a
minha trajetória no estudo de improvisação.
Ele foi feito com a intenção de auxiliar pessoas interessadas em estudo
de música e improvisação, que toquem instrumentos com alturas
definidas.
Dividi o E-book em 2 partes: A primeira parte contém exercícios que
compus e exploram padrões melódicos sobre a sonoridades das escalas. Na segunda parte, o foco são exercícios que criei partindo das
transcrições de solos que fiz.

Musical scales play an important role in melodies, since its properties are reflected to the melodic essence. The extraction and understanding of scales are essential in both analysis and composition of music. However, the scale identification is a nontrivial task. Consequently, classic algorithms for identifying scales have been developed based on the most popular scales, such as major and minor scales. In this paper, we propose a comprehensive method for identifying musical scales, which allows to detect a wide range of scales beyond the traditional ones. Our method uses a deterministic walk through the nodes of a graph, where each node represents a valid interval structure. The transition between nodes is performed following a validation rule that governs the fragmentation of intervals. Moreover, if the scale is incomplete, possible structures can be determined and the scale is estimated according to the harmonic similarity percentage measure. The proposed method has been tested using a database of Finnish folk melodies and a data set of random melodies composed using rarely used scales. Experimental results show good performance of the proposed technique.

The study discusses features of musical scales in Lithuanian traditional vocal performances. Manifestation of the equidistant model of musical scale and its interplay with diatonic thinking are considered. Three samples representing three different eth-nomusical dialects (54 songs, in total) are examined: the methods of diatonic contrast and clustering are applied in the processing of results of the acoustical pitch measurements. The evaluation of diatonic contrast shows that the song examples resembling the theoretical equidistant scale prevail over those resembling diatonics. The cluster analysis reveals certain groups of the scales characteristic of the examined dialects.

A lot is said about the relationships between music and mathematics. However very few times those relationships are given in clear and applied cases. At the Luthiers School (Escuela de Laudería – INBA, Querétaro, Qro.) these connections between science and art have been useful. Understanding mathematics and music theory as a building theory that shows links and connections with art and science implies understanding lutherie as an applied art, related to science and history. At the beginning of the career, the relationships between science and music stands out as part of instrument design, involving therefore theories of physical shapes and mathematical resources. In this field I identified the easiest way for students to understand set theory and mathematical functions related with music theory. Another case is by building modal scales and their relatives by the function previously given; fine connections can be made by associating major scales with their minor relatives.

By presenting of some recommendations to practice, this article considers some other issues regarding on metronome, tempo, sonority and scales.

در اين گفتار توصيه‌هايي براي تمرين هر چه بهتر براي نوازند‌گان، ارائه مي‌شود و در ارتباط با مترونم، تمپو، سونوريته، گام‌هاي موسيقي و غيره اشاراتي موجود است؛ مطالعه اين مقاله براي اكثر هنرجويان و نوازند‌گان، بسيار مفيد خواهد بود.

Diatonic set theory, as established by John Clough and others (see Clough 1979, 1980, Carey and Clampitt 1989, Carey 1998), applies the tools of standard set theory of 12-tone ET to the heptatonic set of seven tones of the diatonic scale. The two universes differ from each other in a number of ways other than simple cardinality. Although 12 is nearly twice as big as 7, the fact that 7 is prime and 12 composite contributes to a number of subtle differences between them. Every positive integer less than 7 is a unit mod 7, thus every diatonic interval generates the entire set. In the set of 12 tones, there are only four units, 1, 5, 7, and 11. Further, because of tuning, the geographies, if you will, of the sets also differ. In the equal-tempered 12-tone landscape, every place looks like everyplace else. In the diatonic scale, each generic span is inhabited by several different specific intervals. Because of this, the terrain is everywhere distinct, contributing to the phenomenon of gravitational asymmetry and of tonality.

Because of an absence of a fixed referential tonality, musical scales can experience certain gradual changes in the course of a capella vocal performance. For example, the phenomenon of floating tonality or gradual transposition (usually a gradual rise of pitch up to several semitones) common in traditional, unaccompanied singing has been discussed (Niemi & Jouste, Greene, Ambrazevičius). Alexeyev studied some peculiar instances of so-called “evolving” scales in Yakut singing, i.e. the intervals between adjacent scale degrees tend to widen in the course of performance. In the present study, we aim to identify types of gradual changes of musical scales in traditional singing without accompaniment and to discuss possible causes for the changes. Praat-aided acoustical measurements of pitches in traditional unaccompanied vocal performances were carried out. The sample was composed of 20 song records exemplifying the Lithuanian vocal tradition, both monophony and homophony. Several pheno...