DOCSLIB.ORG

About the Tuning

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
About the Tuning About the Tuning This tuning was originally devised by Lou Harrison and Bill Slye for the National Reso-Phonic Just Intonation Guitar, used in Scenes from Nek Chand, Lou’s last completed composition before his death in 2003. I originally composed these three movements for that same guitar. The guitar tuning is in D, but I have transposed it to G for these keyboard arrangements. The tuning comprises a twelve-tone-per-octave scale in 11-limit Just Intonation. In particular, it features two harmonic series segments (1–3–5–7–9–11)—what Harry Partch termed otonality hexads—on fundamentals separated by a perfect fifth (3:2), C and G. Two additional tones needed to complete a twelve-tone scale, A and E , are not part of either harmonic series, but were added by chaining fifths below B , the b b b seventh harmonic of C. Although the harmonics of G are themselves harmonics of C (3–9–15–21–27–33), I have named G 1/1 in the diagrams on this and the following page because of the primary role it plays in these pieces. (Those interested in a detailed explanation of the Just Intonation terminology used here should consult my Just Intonation Primer.) Although the scale has twelve tones per octave, it bears little resemblance Cents deviation from 12-equal to an equally tempered chromatic scale. Step sizes between adjacent tones vary from as little as 53 cents (slightly larger than a tempered quartertone) G 1/1 = 0 Normalized to A to as much as 151 cents (more than a tone and one-half), with many other G ±0.00 G –3.91 sizes in between. The scale provides a great variety of modal possibilities, A –37.04 A –40.95 both historical and novel, but if one desires consonant harmony, one is limited b b A +3.91 A ±0.00 mainly to structures centered on C and G. Most of the intervals in the scale can be easily tuned by ear by an experienced B b –33.13 Bb –37.04 tuner, though the seventh and especially the eleventh harmonic may prove B –13.69 B –17.6 difficult for those unfamiliar with these intervals. The harmonic seventh, C –1.96 C –5.87 7:4, can be tuned by zero beating. It is 27 cents flatter than a Pythagorean minor seventh. Probably the best procedure is to start from the Pythagorean C # –48.68 C # –52.59 interval and then lower the pitch until the desired beat-free position is found. D +1.95 D –1.96 The harmonic tritone, 11:8, is harder to tune by ear, as the eleventh harmonic E b –35.08 E b –38.99 may be too weak for its beating to be easily detected. The 11:8 tritone has a E –15.64 E –19.55 width of approximately 551 cents, almost exactly the midpoint between the 12-equal perfect fourth and augmented fourth. It is probably best to use a F –31.17 F –35.08 programmable electronic tuner to set this interval until it becomes familiar to F +49.36 F +45.45 the ear. The composer will provide a CD with reference tones on request. G 0.00 G –3.91 The absolute pitch of the tuning is not crucial; the scale may be tuned based on A 440 or on any other historical A that your instrument may be accustomed to. The Tuning Represented as a Lattice 5 5 E 3 B 4 Major Third/ 28 14 7 7 Minor Sixth Ab 27 Eb 9 Bb 6 F 4 C 4 G 1 D 3 A 9 3 1 2 8 Harmonic Seventh/ Perfect Supermajor Second FB 11 11 Fifth/Fourth 6 CB 8 Harmonic Augmented Fourth/ Diminished Fifth Tuning Schema Keyboard Suite Tuning cents from 0 63 204 267 386 498 551 702 765 884 969 1049 1200 G 1/1: 1 28 9 7 5 4 11 3 14 5 7 11 2 1 27 8 6 4 3 8 2 9 3 4 6 1 ww# w bww n ww# w &w28 bww 243 n 28 bww 15 n 16 33 12 28 15 21 22 12 27 224 27 14 15 32 11 27 14 20 21 11 cents between 63 141 63 119 112 53 151 63 119 84 81 151 degrees: Tuning Sequence Harmonic 7th Harmonic Tritone (7:4) (11:8) Pure Fifths/Fourths Pure Fifth Pure Fifths/Fourths Pure Fifth w bw #w & w w w w bw bw w #w w Purew Majorw Third Checks (Consonant Chords) w w w w &ww w w w bw bw nw w Major Minorw bw Subminorw n wDominant Ninth (very narrow minor third, (maximally consonant but quite consonant) harmonic pentad) As Harmonic Series Degrees 4567911 4567911 Pure Fourth/Fifth w #w #w w w bw & w bw w w w bw bw w w (1) ? (1) W W.
Load more

Recommended publications
  • The Misunderstood Eleventh Amendment
    UNIVERSITY of PENNSYLVANIA LAW REVIEW Founded !"#$ Formerly AMERICAN LAW REGISTER © !"!# University of Pennsylvania Law Review VOL. !"# FEBRUARY $%$! NO. & ARTICLE THE MISUNDERSTOOD ELEVENTH AMENDMENT WILLIAM BAUDE† & STEPHEN E. SACHS†† The Eleventh Amendment might be the most misunderstood amendment to the Constitution. Both its friends and enemies have treated the Amendment’s written † Professor of Law, University of Chicago Law School. †† Colin W. Brown Professor of Law, Duke University School of Law. We are grateful for comments and suggestions from Sam Bray, Tom Colby, Scott Dodson, Jonathan Gienapp, Tara Grove, Julian Davis Mortenson, Caleb Nelson, Jim Pfander, Richard Re, Amanda Schwoerke, Eric Segall, Jonathan Urick, Ingrid Wuerth, and from participants in the Fall Speaker Series of the Georgetown Center for the Constitution, the Hugh & Hazel Darling Foundation Originalism Works-in-Progress Conference at the University of San Diego Law School, and Advanced Topics in Federal Courts at the University of Virginia School of Law. Thanks as well to Brendan Anderson, Kurtis Michael, and Scotty Schenck for research assistance, and to the SNR Denton Fund and Alumni Faculty Fund at Chicago for research support. (!"#) !"# University of Pennsylvania Law Review [Vol. "!$: !#$ text, and the unwritten doctrines of state sovereign immunity, as one and the same— reading broad principles into its precise words, or treating the written Amendment as merely illustrative of unwritten doctrines. The result is a bewildering forest of case law, which takes neither the words nor the doctrines seriously. The truth is simpler: the Eleventh Amendment means what it says. It strips the federal government of judicial power over suits against states, in law or equity, brought by diverse plaintiffs.
    [Show full text]
  • Ninth, Eleventh and Thirteenth Chords Ninth, Eleventh and Thirteen Chords Sometimes Referred to As Chords with 'Extensions', I.E
    Ninth, Eleventh and Thirteenth chords Ninth, Eleventh and Thirteen chords sometimes referred to as chords with 'extensions', i.e. extending the seventh chord to include tones that are stacking the interval of a third above the basic chord tones. These chords with upper extensions occur mostly on the V chord. The ninth chord is sometimes viewed as superimposing the vii7 chord on top of the V7 chord. The combination of the two chord creates a ninth chord. In major keys the ninth of the dominant ninth chord is a whole step above the root (plus octaves) w w w w w & c w w w C major: V7 vii7 V9 G7 Bm7b5 G9 ? c ∑ ∑ ∑ In the minor keys the ninth of the dominant ninth chord is a half step above the root (plus octaves). In chord symbols it is referred to as a b9, i.e. E7b9. The 'flat' terminology is use to indicate that the ninth is lowered compared to the major key version of the dominant ninth chord. Note that in many keys, the ninth is not literally a flatted note but might be a natural. 4 w w w & #w #w #w A minor: V7 vii7 V9 E7 G#dim7 E7b9 ? ∑ ∑ ∑ The dominant ninth usually resolves to I and the ninth often resolves down in parallel motion with the seventh of the chord. 7 ˙ ˙ ˙ ˙ & ˙ ˙ #˙ ˙ C major: V9 I A minor: V9 i G9 C E7b9 Am ˙ ˙ ˙ ˙ ˙ ? ˙ ˙ The dominant ninth chord is often used in a II-V-I chord progression where the II chord˙ and the I chord are both seventh chords and the V chord is a incomplete ninth with the fifth omitted.
    [Show full text]
  • On the Perception of Pitch Accents by Kevin L
    On the Perception of Pitch Accents by Kevin L. Landel B.S., Electrical Engineering and Computer Science University of California, Berkeley Berkeley, California 1983 SUBMITTED TO THE MEDIA ARTS AND SCIENCES SECTION IN PARTIAL FULFILLMENT OF THE REQUIREMENTS OF THE DEGREE OF MASTER OF SCIENCE AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 1988 @Massachusetts Institute of Technology 1988 All Rights Reserved. Signature of the Author ....................................................................... Kevin L. Landel Media Arts and Sciences Section June 6, 1988 //7 Certified by ....................................................................... Chris Schmandt Principal Research Scientist Thesis Supervisor Accepted by ..... .j~f/V.V"'.. f ..... .. ....... Stephen A. Benton Chairman Departmental Committee on Graduate Students MASSACHUSETS INSTiTUTE nr TVf"I4mfl 01Wy NOV 0 31988 LIBRARIES On the Perception of Pitch Accents by Kevin L. Landel Submitted to the Media Arts and Sciences Section on June 6, 1988 in partial fulfillment of the requirements of the degree of Master of Science at the Massachusetts Institute of Technology Abstract A series of experiments was performed to determine the thresholds at which pertur- bations to pitch contours became important to the semantic interpretation of utterances. The perception of pitch accents was controlled by manipulating the accents' relative mag- nitudes and surrounding prosodic environment. The perturbations investigated were: the separation of H and L pitch accents, the presence vs. the absence of pitch accents, and the effects of pitch range and prominence on the perception of pitch accents. The intent is to suggest the pertinent information which needs to be addressed for the implementation of speech systems that make better use of prosody. Thesis Supervisor: Chris Schmandt Title: Principal Research Scientist This work was supported by the Defense Advance Research Projects Agency (DARPA) contract number N00039-86-C-0406.
    [Show full text]
  • View of Related Literature
    THE EFFECT OF OCTAVE AND TIMBRE COMBINATIONS ON UNDERGRADUATE BAND MEMBERS’ PERCEPTION OF PITCH by CLINTON M. STEINBRUNNER Submitted in partial fulfillment of the requirements for the degree of Master of Arts Thesis Advisor: Dr. Nathan B. Kruse Department of Music CASE WESTERN RESERVE UNIVERSITY August 2019 2 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the thesis of Clinton M. Steinbrunner candidate for the degree of Master of Arts. Committee Chair Dr. Nathan B. Kruse Committee Member Dr. Kathleen A. Horvath Committee Member Dr. Ryan V. Scherber Date of Defense June 29, 2019 *We also certify that written approval has been obtained for any proprietary material contained therein. 3 TABLE OF CONTENTS LIST OF TABLES .......................................................................................................... 5 ABSTRACT ................................................................................................................... 6 CHAPTER 1: INTRODUCTION .................................................................................... 7 Tuning as an Individual Process .............................................................................. 8 Instrument Timbre ................................................................................................... 9 Ensemble Tuning Procedures and Octave Displacement ........................................ 10 Need for the Study .................................................................................................... 11 Purpose
    [Show full text]
  • A LIGHTWEIGHT NOTATION for PLAYING PIANO for FUN by R.J.A
    A LIGHTWEIGHT NOTATION FOR PLAYING PIANO FOR FUN by R.J.A. Buhr 1. INTRODUCTION Music notation gets in the way of the joy of making music on the piano that can be had without first spending thousands of hours at the keyboard to develop piano “wizardry,” meaning a combination of keyboard skill and mastery of music notation. The message of this document is the joy can be had, without any loss of musical substance, with no more wizardry than is required to read and play single melody lines. The vehicle is a lightweight notation I call PKP (for Picturing Keyboard Patterns) for annotating the core flow of harmony in interval terms above the staff. I developed the notation as an adult beginner with no training in music theory after becoming frustrated with the way music notation obscures the keyboard simplicity of harmony. That the notation has substance in spite of this humble origin is indicated by the following opinion of PKP provided by a music theorist specializing in improvisational music (see supplementary pages): “The hook ... , at least in my opinion, is that it's possible to attain a deep understanding of chords (and their constituent intervals) without recourse to Western notation. This has direct consequences for physical patterning, fingerings, etc. Essentially, your method combines the utility of a play-by-ear approach with the depth of a mathematically-informed theory of music.” PKP follows from two observations that anyone can make. One is the piano keyboard supplies only 12 piano keys to play all the notes of an octave that music notation describes in many more than 12 ways by key signatures and accidentals (sharps, flats, naturals).
    [Show full text]
  • The Consecutive-Semitone Constraint on Scalar Structure: a Link Between Impressionism and Jazz1
    The Consecutive-Semitone Constraint on Scalar Structure: A Link Between Impressionism and Jazz1 Dmitri Tymoczko The diatonic scale, considered as a subset of the twelve chromatic pitch classes, possesses some remarkable mathematical properties. It is, for example, a "deep scale," containing each of the six diatonic intervals a unique number of times; it represents a "maximally even" division of the octave into seven nearly-equal parts; it is capable of participating in a "maximally smooth" cycle of transpositions that differ only by the shift of a single pitch by a single semitone; and it has "Myhill's property," in the sense that every distinct two-note diatonic interval (e.g., a third) comes in exactly two distinct chromatic varieties (e.g., major and minor). Many theorists have used these properties to describe and even explain the role of the diatonic scale in traditional tonal music.2 Tonal music, however, is not exclusively diatonic, and the two nondiatonic minor scales possess none of the properties mentioned above. Thus, to the extent that we emphasize the mathematical uniqueness of the diatonic scale, we must downplay the musical significance of the other scales, for example by treating the melodic and harmonic minor scales merely as modifications of the natural minor. The difficulty is compounded when we consider the music of the late-nineteenth and twentieth centuries, in which composers expanded their musical vocabularies to include new scales (for instance, the whole-tone and the octatonic) which again shared few of the diatonic scale's interesting characteristics. This suggests that many of the features *I would like to thank David Lewin, John Thow, and Robert Wason for their assistance in preparing this article.
    [Show full text]
  • Advanced Music Theory Handbell Musicians of America Certification Course C3 Course Outline Instructor: Michael J
    Advanced Music Theory Handbell Musicians of America Certification Course C3 Course Outline Instructor: Michael J. Glasgow I. Pitch names and how handbells use them a. C1 through C8 (handbell designation C2 through C9) b. Transposing reminder II. Rhythmic values of notes and rests a. British/international nomenclature (from Renaissance period, if not before) i. Thirty-second (demisemiquaver) b. Tuplets: beyond basic triplets i. Ties in tuplets (implied and overt) 1. “swung” eighth notes ii. Sometimes they expand (“stretch”) the note value 1. Example: eighth-note duplets in compound meter iii. Often – especially in simple meters – they “compress” the notes into a tighter ratio. The number on the beam/bracket indicates that that number of notes should be played in the time traditionally occupied by the next lower “standard grouping” of them that space. 1. Example: in 4/4 time, a quintuplet over one beat would be shown as five sixteenth-notes, since the next lower “standard grouping of notes in one beat” from five is four, and four sixteenth notes usually take up one beat. 2. Example: a nonuplet of nine sixteenth-notes in 4/4 time would be like saying that those nine sixteenth-notes should be played in the space normally occupied by eight sixteenth-notes (two beats). iv. Nested tuplets v. Ambiguity: many systems, many guides, no panacea for all situations 1. Use basic math and intuition. Interpret the printed page with context clues; create the printed page with common sense. 2. Example: if we wanted to indicate seven notes played evenly over an entire measure of 4/4 time, we could use a “compressed” ratio of 7:4, and bracket together seven quarter-notes; however, it would also be acceptable to use a “stretched” ratio of 7:8, and show seven eighth-notes.
    [Show full text]
  • Prohibition Versus Apotheosis of a Tritone: Rhetorical Aspects
    GESJ: Musicology and Cultural Science 2019|No.2(20) ISSN 1512-2018 UDC 78 PROHIBITION VERSUS APOTHEOSIS OF A TRITONE: RHETORICAL ASPECTS Božena Čiurlionienė Lithuanian Academy of Music and Theatre Abstract: This paper explores the change of the concept of a tritone – from its prohibition to tritonic apotheosis as well as draws attention to the acoustic peculiarities of the „Devil‘s interval“ (Diabolus in musica1). The tritone is a very small object but features a portentous and unique sound as well as problematic intensity. The peculiarity of the discussed interval has evoked multifarious creative expressions and has also provoked controversial theoretical speculations throughout the music history. This paper aims to draw a historical perspective providing a comprehensive representation of both theoretical and practical approach to the tritone in relation to its acoustic intensity and rhetorical aspects. Critical consideration of the most remarkable theoretical declarations as well as analytical inquiry of particular musical examples reveal the significance and functionality of a tritone in different historical contexts. The paper reveals the constant change of conceptions, canons of usage, composition techniques and harmonic systems although referring to the same object on discourse. Keywords: tritone, dissonance, diabolus in musica, compositional process, musical symmetry, contemporary music. INTRODUCTION: PSYCHOACOUSTIC PECULIARITIES OF THE TRITONE AND ITS MUSICAL PERCEPTION The uniqueness of a tritone, in terms of both theory and practice of composition, can be explained by its physical acoustic features. Such descriptions as rough, harsh, etc. can be easily explained by the results of cognitive music psychology of the 20th century. Tritone’s physical parameters vary in different harmonies: in the Pythagorean theory, increased fourth is 729:512, reduced fifth is 1024:729, and in the tempered combination is √2 = 1.414.
    [Show full text]
  • Correlation Between the Frequencies of Harmonic Partials Generated from Selected Fundamentals and Pitches of Supplied Tones
    University of Montana ScholarWorks at University of Montana Graduate Student Theses, Dissertations, & Professional Papers Graduate School 1962 Correlation between the frequencies of harmonic partials generated from selected fundamentals and pitches of supplied tones Chester Orlan Strom The University of Montana Follow this and additional works at: https://scholarworks.umt.edu/etd Let us know how access to this document benefits ou.y Recommended Citation Strom, Chester Orlan, "Correlation between the frequencies of harmonic partials generated from selected fundamentals and pitches of supplied tones" (1962). Graduate Student Theses, Dissertations, & Professional Papers. 1913. https://scholarworks.umt.edu/etd/1913 This Thesis is brought to you for free and open access by the Graduate School at ScholarWorks at University of Montana. It has been accepted for inclusion in Graduate Student Theses, Dissertations, & Professional Papers by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. CORRELATION BETWEEN THE FREQUENCIES OF HARMONIC PARTIALS GENERATED FROM SELECTED FUNDAMENTALS AND PITCHES OF SUPPLIED TONES by CHESTER ORLAN STROM BoMo Montana State University, 1960 Presented in partial fulfillment of the requirements for the degree of Master of Music MONTANA STATE UNIVERSITY 1962 Approved by: Chairman, Board of Examine Dean, Graduate School JUL 3 1 1902 Date UMI Number: EP35290 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion.
    [Show full text]
  • La Monte Young's the Well-Tuned Piano Kyle Gann Perspectives Of
    La Monte Young's The Well-Tuned Piano Kyle Gann Perspectives of New Music, Vol. 31, No. 1. (Winter, 1993), pp. 134-162. Stable URL: http://links.jstor.org/sici?sici=0031-6016%28199324%2931%3A1%3C134%3ALMYTWP%3E2.0.CO%3B2-3 Perspectives of New Music is currently published by Perspectives of New Music. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/pnm.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Sat Jan 12 10:29:55 2008 A MONTEYOUNG'S The Well-Tuned Piano may well be the most Limportant American piano work since Charles Ives's Concord Sonata-in size, in influence, and in revolutionary innovation.
    [Show full text]
  • The Prehistory of Chinese Music Theory
    ELSLEY ZEITLYN LECTURE ON CHINESE ARCHAEOLOGY AND CULTURE The Prehistory of Chinese Music Theory ROBERT BAGLEY Princeton University The division of the octave into twelve semitones and the transposition of scales have also been discovered by this intelligent and skillful nation. But the melodies transcribed by travellers mostly belong to the scale of five notes. Helmholtz, On the Sensations of Tone (1863)1 THE EARLIEST TEXTS ABOUT MUSIC THEORY presently known from China are inscriptions on musical instruments found in the tomb of Marquis Yi of Zeng, the ruler of a small state in the middle Yangzi region, who died in 433 BC. Excavated in 1978, the tomb was an underground palace of sorts, a timber structure furnished with an astonishing array of lordly posses- sions (Fig. 1).2 In two of its four chambers the instruments of two distinct musical ensembles were found. Beside the marquis’s coffin in the east chamber was a small ensemble for informal entertainments, music for the royal bedchamber perhaps, consisting of two mouth organs, seven zithers, and a small drum. In the central chamber, shown under excavation in Figure 2, a much larger ensemble for banquets and rituals was found. Composed of winds, strings, drums, a set of chime stones, and a set of bells, it required more than twenty performers. The inscriptions about music theory appear on the chime stones and the bells. They concern Read at the Academy on 26 October 2004. 1 Quoted from Alexander Ellis’s 1885 translation (p. 258) of the fourth German edition (1877). 2 The excavation report, in Chinese, is Zeng Hou Yi mu (1989).
    [Show full text]
  • Music Theory Contents
    Music theory Contents 1 Music theory 1 1.1 History of music theory ........................................ 1 1.2 Fundamentals of music ........................................ 3 1.2.1 Pitch ............................................. 3 1.2.2 Scales and modes ....................................... 4 1.2.3 Consonance and dissonance .................................. 4 1.2.4 Rhythm ............................................ 5 1.2.5 Chord ............................................. 5 1.2.6 Melody ............................................ 5 1.2.7 Harmony ........................................... 6 1.2.8 Texture ............................................ 6 1.2.9 Timbre ............................................ 6 1.2.10 Expression .......................................... 7 1.2.11 Form or structure ....................................... 7 1.2.12 Performance and style ..................................... 8 1.2.13 Music perception and cognition ................................ 8 1.2.14 Serial composition and set theory ............................... 8 1.2.15 Musical semiotics ....................................... 8 1.3 Music subjects ............................................. 8 1.3.1 Notation ............................................ 8 1.3.2 Mathematics ......................................... 8 1.3.3 Analysis ............................................ 9 1.3.4 Ear training .......................................... 9 1.4 See also ................................................ 9 1.5 Notes ................................................
    [Show full text]