DOCSLIB.ORG

Topological Considerations for Tuning and Fingering Stringed Instruments

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Topological Considerations for Tuning and Fingering Stringed Instruments Topological Considerations for Tuning and Fingering Stringed Instruments Terry Allen, Camille Goudeseune Beckman Institute, University of Illinois at Urbana-Champaign [email protected], [email protected] Abstract Pitch is continuous, but as we are discussing fretted stringed instruments, we temper the octave into twelve We present a formal language for assigning pitches to equal semitones (equal temperament). Then we represent an strings for fingered multi-string instruments, particularly interval as not a ratio but an integer counting the number of the six-string guitar. Given the instrument’s tuning (the semitones between two pitches. Thus, in the familiar way, strings’ open pitches) and the compass of the fingers of the 12 represents an octave, 7 a perfect fifth, 1 a semitone. If hand stopping the strings, the formalism yields a framework we fix a reference pitch, then we can similarly represent for simultaneously optimizing three things: the mapping any pitch as an integer, namely the integer representing the of pitches to strings, the choice of instrument tuning, and interval from the reference to the pitch in question. In the the key of the composition. Final optimization relies on context of a particular tuning of an instrument, the reference heuristics idiomatic to the tuning, the particular musical pitch is often that of the lowest string, played unfretted (see style, and the performer’s proficiency. below). A string is defined by what it can do: sound a pitch 1. Introduction from a contiguous range bounded at both ends by the The ‘guitar fingering problem’ is, less colloquially, the find- fingerboard’s finite length. At any given moment, a string ing of an optimal assignment of pitches to strings and frets. sounds either no pitch or exactly one pitch. When a string Previous work on this problem [16, 17, 19, 20, 22, 23, 24] sounds its lowest pitch, we call it unfretted or open. takes as fixed three things: the composition, the musical key, An instrument means a fretted multi-string musical and the guitar’s tuning. From these, a fingering is found that instrument such as a guitar, arpeggione, or lute. We demands little technical dexterity, or that at least obeys the concentrate on the six-string guitar because it is widely mechanical limitations of the instrument. played, often retuned, and used in many musical styles. We generalize this by fixing only the composition, and A fret constrains a string’s playable pitches to a discrete then simultaneously optimizing over three aspects: the subset. We assume that adjacent frets are one semitone fingering, the tuning, and the key (transposing from, say, apart, and that all strings have the same frets (a rectangular G major to A major). fingerboard).2 We assume that strings are fretted by the Historical motivation for exploring nonstandard tunings four fingers of the left hand (not the thumb, although some is found in the work of guitarists such as Muddy Waters, advanced styles permit the thumb to fret the lowest string). Chuck Berry, and Keith Richards [18], John Lennon [4], and One finger can fret several adjacent strings. especially Michael Hedges [5, 8]. The search for optimality An instrument’s tuning is the set of pitches of its open arXiv:1105.1383v1 [cs.SD] 6 May 2011 is also driven by the observation that some genres such as strings. A tuning vector is that set, sorted by rising pitch. blues or Gypsy Jazz [27] typically use only one or two Thus, the number of strings determines the size of the fingers at a time. Strong constraints are thus critical for tuning vector. Retuning means a change of tuning, whether reconstructing fingering from audio-only recordings. or not this is to or from an instrument’s standard tuning (scordatura). We ignore unusual ‘out of order’ tunings 2. Definitions such as Congolese mi-compose,´ standard tuning with the d string up an octave [21]. We also ignore tunings where A pitch is the fundamental frequency of a sound with a several strings share the same pitch, such as variations on harmonic spectrum, such as that played by many musical Lou Reed’s ‘Ostrich’ (D-D-D-D-d-d). In particular, we instruments. Pitch is measured in log Hz. The logarithm assume that as one moves across the fingerboard, open pitch conveniently lets us define an interval,p such as an octave or strictly rises. Common examples for the guitar include semitone, as a ratio of pitches (2:1 or 12 2:1, respectively).1 2Bending or ‘whamming’ a string to sound a pitch outside equal 1This is a musical interval. In context, there will be no confusion temperament is an issue separate to fingering, except for the detail that between this and ‘mathematical’ intervals like [0; 1] ⊂ R. an open string cannot be bent. 1 standard tuning (E-A-d-g-b-e0) and Open G (D-G-d-g-b-d0); Since frets are spaced one semitone apart, hundreds more exist [1, 25, 26].3 A chord is a set of pitches sounding simultaneously. PVi(f) = f + PVi(0): (These pitches need not have begun simultaneously.) We Combining these yields the String Changing Equation abuse the term’s traditional meaning by letting it contain i−1 fewer than three pitches, what musicians call an interval X or a unison. This is because traditional harmonic analysis PVi(f) = f + Ck (1) concerns us less than the constraints on which strings can k=0 play which pitches. for 1 ≤ i ≤ 6 and 0 ≤ f ≤ 24. This equation A composition is a finite sequence of chords of finite characterizes the relationship between four concepts: pitch duration. We concentrate our attention on the left hand, value PV , string i, fret f, and tuning vector (C1; :::; C5). ignoring right-hand techniques like plucking and damping. This notation derives from work by Allen, formalized by A fingering for a composition is a mapping from its pitches Gilles and Jennings [6]. to (string, fret) pairs. A pitch value is a pitch constrained to an integer 4. Tones (semitone) value. A pitch class is one of the twelve 2 traditional names C, C], D, ..., B. A musical key, such as For notational convenience, we call (i; f) 2 Z a guitar E major, is a set of seven distinct pitch classes that forms a position, or just a position. As before, 1 ≤ i ≤ 6 indicates major or minor scale. One of these pitch classes is special, string, and 0 ≤ f ≤ 24 indicates fret. This lets us define a called the tonic. A pitch’s scale position is its distance tone as an ordered triple: (pitch value, scale position, guitar above the next lowest tonic pitch of a musical key, measured position). in semitones. Thus a scale position must lie in [0; 12). Given a particular tuning and musical key, a guitar can play some tones but not others. For example, in standard 3. Tuning Vectors tuning in E major, playing the lowest open string yields the tone (0, 0, (1, 0)): pitch value 0, scale position 0, (string 1, Consider a tuning vector (x1; :::; x6). Because we start fret 0). But the tone (0, 0, (2, 0)) cannot be played: string 2, 4 counting from the lowest string, the xi are increasing. We fret 0 has a different pitch value and scale position than then define the corresponding length-5 tuning vector as string 1, fret 0. Playing the highest open string in the context of C major yields the tone (24, 4, (6, 0)): 24 semitones (C1; :::; C5) = (x2 − x1; x3 − x2; :::; x6 − x5): higher, 4 above C, sixth string, zeroth fret. Because no two open strings share a pitch, the xi are strictly 4.1. The Vector Space of Tones increasing: x x . Thus each C > 0. For convenience i i−1 i If we momentarily relax some constraints on pitch value, of summation in eqs. (1), (2) and (4), we also define C = 0. 0 scale position, and the two elements of guitar position, then One 5-vector (of intervals) corresponds to many 6- × × 2 describes the set of all possible tones.6 We vectors (of pitches). For example, (5,5,5,4,5), abbreviated Z Z12 Z still constrain scale position to lie in [0; 12). as 55545, corresponds to both E-A-d-g-b-e0 and D-G-c-f- If vector addition is componentwise, and (real) scalar a-d0).5 Converting a 5-vector to a 6-vector, by choosing a multiplication is multiplication on each element of the pitch for the lowest string, is what we call augmenting or vector, then this set is not quite a vector space over the evaluating a 5-vector at a pitch. For example, evaluating field , as it lacks closure under scalar multiplication. For 75545 at D yields Drop-D tuning, D-A-d-g-b-e0. R example, (8; 7; 6; 5) is in × × 2, but not 3:14 × For a given tuning, let PV (f) be the pitch sounded by Z Z12 Z i (8; 7; 6; 5). To get a vector space, we must extend the ’s string i when fingered at fret f, for i from 1 to 6, for f Z to ’s.7 This demands continuous values for not only pitch, from 0 to 24 (lowest pitch to highest, in both cases). As a R scale position, and fret number, but also for string number. reference for this tuning, set PV (0) = 0. 1 (Imagine a Haken Continuum [7] programmed to change 3We spell examples of tuning vectors with the older Helmholtz no- pitch along both horizontal axes.) tation, not scientific pitch notation (E2-A2-D3-G3-B3-E4) [28], because Then Z12 becomes R ‘mod 12.’ This is in fact a vector the former is more legible in the few octaves used by the guitar.
Load more

Recommended publications
  • ALTERED STATES of PERFORMANCE: SCORDATURA in the CLASSICAL GUITAR REPERTOIRE by COREY JAMES FLOWERS (Under the Direction Of
    ALTERED STATES OF PERFORMANCE: SCORDATURA IN THE CLASSICAL GUITAR REPERTOIRE by COREY JAMES FLOWERS (Under the Direction of Michael Heald) ABSTRACT Scordatura is commonplace in the guitar repertoire, though there are relatively few resources available that explain the approach to its use, either from a performer’s or a composer’s perspective. Through examination of specific selections in the classical guitar repertoire, this document offers an introduction to various alternate tunings for the guitar, beginning with tunings used to transcribe lute and vihuela music, and progressing to modern experimentations found in the literature. Each piece on the accompanying recording provides insight into the use of these alternate tunings by highlighting specific musical characteristics that are made possible, or are idiomatic, in each tuning, such as extended ranges, open string relationships, unique chord voicings, and the ease of introducing contrasting tonal centers across a program. The CD provides a basis for aural comparison of the tunings and is a resource for performers interested in this repertoire. It also provides examples for composers and arrangers wishing to approach writing for the guitar using non-standard tunings. As additional resources, a glossary of terminology is provided, concerning tunings found in the document, as well as tunings used outside of the classical repertoire that may be useful for experimentation. An index of scordatura repertoire is provided as well, offering a broad overview of additional repertoire for further
    [Show full text]
  • Alternate Tuning Guide
    1 Alternate Tuning Guide by Bill Sethares New tunings inspire new musical thoughts. Belew is talented... But playing in alternate Alternate tunings let you play voicings and slide tunings is impossible on stage, retuning is a between chord forms that would normally be nightmare... strings break, wiggle and bend out impossible. They give access to nonstandard of tune, necks warp. And the alternative - carry- open strings. Playing familiar fingerings on an ing around five special guitars for five special unfamiliar fretboard is exciting - you never know tuning tunes - is a hassle. Back to EBGDAE. exactly what to expect. And working out familiar But all these "practical" reasons pale com- riffs on an unfamiliar fretboard often suggests pared to psychological inertia. "I've spent years new sound patterns and variations. This book mastering one tuning, why should I try others?" helps you explore alternative ways of making Because there are musical worlds waiting to be music. exploited. Once you have retuned and explored a Why is the standard guitar tuning standard? single alternate tuning, you'll be hooked by the Where did this strange combination of a major unexpected fingerings, the easy drone strings, 3rd and four perfect 4ths come from? There is a the "new" open chords. New tunings are a way to bit of history (view the guitar as a descendant of recapture the wonder you experienced when first the lute), a bit of technology (strings which are finding your way around the fretboard - but now too high and thin tend to break, those which are you can become proficient in a matter of days too low tend to be too soft), and a bit of chance.
    [Show full text]
  • DADGAD Tuning and Using Partial Capos
    2 DADGAD Tuning and Using Partial Capos All four of the guitars I used to record One Size Does Not Fit All (pictured on the recording cover) were tuned to DADGAD intervals, that is the string above the bass string, the 5th string, is a fifth above the bass note, the next string, the 4th, is a fourth above that (an octave above the bass string), the 3rd string is a fourth above the 4th string, the 2nd is a second above the 3rd, and the 1st string is a fourth above the 2nd (or an octave above the 4th string). With the bass note at D, the strings then are tuned D, A, D, G, A and D, but if the bass note is a C then the strings are tuned C, G, C, F, G and C (the 5th string here, G, is a fifth above the bass C, the 4th string is a fourth above that, or an octave above the bass string, and so on.) So, when the bass string is tuned to B♭ applying the same intervals yields a tuning of B♭FB♭E♭FB♭. Shorthand could describe the intervals by the numbers: 54424. DADGAD, also called open Dsus4 tuning, is easier to pronounce than EADGBE, intervals 44434 (which is why everyone calls it “standard tuning”!), but try pronouncing CGCFGC or, worse, B♭FB♭E♭FB♭! Worse yet is the tuning you get when you put a full capo on the second fret of a DADGAD-tuned guitar, EBEABE—try pronouncing that one a few times and the men in the white coats may show up to haul you away! So, I commonly refer to these tunings (you’ll see why in a moment) as “C-gad” or “B♭-gad.” To tune to DADGAD from “standard” tuning, EADGBE, just drop the 1st, 2nd, and 6th string pitches by one whole step, that is tune the bass E down to D, and similarly tune the two high strings down one whole step.
    [Show full text]
  • The Use of Scordatura in Heinrich Biber's Harmonia Artificioso-Ariosa
    RICE UNIVERSITY TUE USE OF SCORDATURA IN HEINRICH BIBER'S HARMONIA ARTIFICIOSO-ARIOSA by MARGARET KEHL MITCHELL A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF MUSIC APPROVED, THESIS COMMITTEE aÆMl Dr. Anne Schnoebelen, Professor of Music Chairman C<c g>'A. Dr. Paul Cooper, Professor of- Music and Composer in Ldence Professor of Music ABSTRACT The Use of Scordatura in Heinrich Biber*s Harmonia Artificioso-Ariosa by Margaret Kehl Mitchell Violin scordatura, the alteration of the normal g-d'-a'-e" tuning of the instrument, originated from the spirit of musical experimentation in the early seventeenth century. Closely tied to the construction and fittings of the baroque violin, scordatura was used to expand the technical and coloristlc resources of the instrument. Each country used scordatura within its own musical style. Al¬ though scordatura was relatively unappreciated in seventeenth-century Italy, the technique was occasionally used to aid chordal playing. Germany and Austria exploited the technical and coloristlc benefits of scordatura to produce chords, Imitative passages, and special effects. England used scordatura primarily to alter the tone color of the violin, while the technique does not appear to have been used in seventeenth- century France. Scordatura was used for possibly the most effective results in the works of Heinrich Ignaz Franz von Biber (1644-1704), a virtuoso violin¬ ist and composer. Scordatura appears in three of Biber*s works—the "Mystery Sonatas", Sonatae violino solo, and Harmonia Artificioso- Ariosa—although the technique was used for fundamentally different reasons in each set. In the "Mystery Sonatas", scordatura was used to produce various tone colors and to facilitate certain technical feats.
    [Show full text]
  • Guitar Tunings
    Guitar tunings Guitar tunings assign pitches to the open strings of guitars, including acoustic guitars, electric guitars, and classical guitars. Tunings are described by the particular pitches denoted by notes in Western music. By convention, the notes are ordered from lowest-pitched string (i.e., the deepest bass note) to highest-pitched (thickest string to thinnest).[1] Standard tuning defines the string pitches as E, A, D, G, B, and E, from lowest (low E2) to highest (high E4). Standard tuning is used by most guitarists, and The range of a guitar with standard frequently used tunings can be understood as variations on standard tuning. tuning The term guitar tunings may refer to pitch sets other than standard tuning, also called nonstandard, alternative, or alternate. Some tunings are used for 0:00 MENU particular songs, and might be referred to by the song's title. There are Standard tuning (listen) hundreds of such tunings, often minor variants of established tunings. Communities of guitarists who share a musical tradition often use the same or similar tunings. Contents Standard and alternatives Standard Alternative String gauges Dropped tunings Open tunings Major key tunings Open D Open C Open G Creating any kind of open tuning Minor or “cross-note” tunings Other open chordal tunings Modal tunings Lowered (standard) E♭ tuning D tuning Regular tunings Major thirds and perfect fourths All fifths and “new standard tuning” Instrumental tunings Miscellaneous or “special” tunings 1 15 See also Notes Citation references References Further reading External links Standard and alternatives Standard Standard tuning is the tuning most frequently used on a six-string guitar and musicians assume this tuning by default if a specific alternate (or scordatura) is not mentioned.
    [Show full text]
  • An Overview of Open Tunings for the Dances of Universal Peace
    An Overview of Open Tunings for the Dances of Universal Peace The standard guitar tuning, EADGBE, works well for so many Dances. Yet, for others, or for leaders who may not feel fully proficient with standard tuning, an open-tuned guitar works beautifully. Indeed, in the early years of the Dances, leaders who wanted some basic accompaniment to the voice found satisfaction with open-tuned guitars. Unlike others kinds of open tunings, these specialized tunings were based upon the tamboura (aka, tanpura) common to most forms of Indian music. These tunings involve only two notes which create a dronal foundation over which any melody can be sung. One of the beautiful things about the open-tuned guitar is that no chords are necessary for this kind of basic support of the voice and Dance. And, because many forms of Asian and Middle-Eastern music are “melody-over-drone” music, many of the Dances affiliated with those traditions sound especially lovely with an open-tuned guitar. If you have read this far, chances are you are interested in exploring the open-tuned guitar for the Dances. Maybe you know something about them already and you're ready to dive in. But, you may have some questions: Which is better, standard or open-tuned guitar? Will I need a dedicated guitar, or can I switch back and forth from standard to open tuning on one guitar? Is there more than one open tuning? Which tuning is best for me and how will I decide? Will I need to buy different strings? Will I need to learn lots of new chords? Can I play melodies, instead of chords? You may have other questions.
    [Show full text]
  • Chart of Recorded and Related Tunings
    Page 1 Dancing Cat Records Hawaiian Slack Key Information Booklet: SECTION IVa: CHART OF RECORDED TUNINGS (and ones closely related to them) 1. 51 of the 65 tunings listed here have had Hawaiian songs recorded in them. Of these 36 of them are played virtually exclusively in Hawai’i (or for playing Hawaiian music). 2. Bb Tunings are the same as C Tunings, except the pitches of the two lowest pitched strings are reversed and tuned up, and the whole tuning is often lowered to the key of Bb to accommodate the fairly extreme raising of the two lowest pitched strings. Therefore, in this chart (and in all the sections of this Slack Key information book) the Bb tunings are listed with the same chart number as the C Tunings that they are related to. 3. In the C Tunings Section of this chart, when two tunings are listed together that are similar, except that the two lowest pitched strings are reversed in pitch (so that one of the tunings is also a Bb Tuning), those two tunings are separated by a space. 4. For songs listed in each of these tunings, see the HAWAIIAN RECORDINGS IN THE SLACK KEY TUNINGS-SECTION V. For other possible Slack Key tunings, see the SUMMARY OF NON- RECORDED TUNINGS-SECTION VI. 5. **The 12 tunings with two asterisks after them have not had any songs recorded in them, but they are reported to have been played by someone in Hawai’i, or are closely related to the one just above it on the chart, and they would be very feasible to play Slack Key in.
    [Show full text]
  • A Complete Technical Reference for Fretted Instrument String Tensions Table of Contents Introduction
    Catalog Supplement/String Tension Specifications A complete technical reference for fretted instrument string tensions Table of Contents Introduction Technical Reference for fretted instrument string tensions 4 his brochure was created to assist in Guitar and Bass Fingerboard Legends 5 Tselecting the ideal string for your instrument and playing style. Included in this brochure String Tensions are tension tables and formulas to help you Acoustic or Electric Guitar 6 calculate what tension a particular string will Pedal Steel Guitar 7 have on your instrument. Acoustic Guitar 7 Classical Guitar 8 Folk Guitar 10 Electric Bass Guitar 10 Acoustic Bass Guitar 13 Mandolin Family Strings 13 Banjo & Loop End Strings 14 A complete technical reference for fretted instrument string tensions Understanding what determines string tension. In order to determine the tension at which a string will vibrate, you need three pieces of information: the Unit Weight, the Scale Length, and the Frequency of the string. You can use the charts in this brochure to get a pre-calculated tension for the D’Addario strings listed or you can use the formulas below to calculate the exact tension for any string using the scale length of your particular instrument. All of the charts illustrate string tensions for each string at a variety of pitches, in case you use alternative tunings. UW- Unit Weight. In all the charts and formulas in the brochure, unit weight is expressed in pounds per linear inch (lb/in). L- Scale Length. This is the vibrating length of the string. This is determined by measuring the distance from the nut to the bridge of the instrument in inches (in).
    [Show full text]
  • An Examination of the Seventeenth-Century English Lyra
    An examination of the seventeenth-century English lyra viol and the challenges of modern editing Volume 1 of 2 Volume 1 Katie Patricia Molloy MA by Research University of York Music January 2015 Abstract This dissertation explores the lyra viol and the issues of transcribing the repertoire for the classical guitar. It explores the ambiguities surrounding the lyra viol tradition, focusing on the organology of the instrument, the multiple variant tunings required to perform the repertoire, and the repertoire specifically looking at the solo works. The second focus is on the task of transcribing this repertoire, and specifically on how one can make it user-friendly for the 21st-century performer. It looks at the issues of tablature, and the issues of standard notation, and finally explores the notational possibilities with the transcription, experimenting with the different options and testing their accessibility. Volume II is a transliteration of solo lyra viol works by Simon Ives from the source Oxford, Bodleian Library Music School MS F.575. It includes a biography of Simon Ives, a study of the manuscript in question and describes the editorial procedures that were chosen as a result of the investigations in volume I. 2 Contents Volume I Abstract 2 Acknowledgements 6 Declaration 7 Introduction 8 1. The use of the term ‘lyra viol’ 12 2. The organology of the lyra viol 23 3. The tuning of the instrument 38 4. The lyra viol repertoire with specific focus on the solo works 45 5. The transcription of tablature 64 6. Experimentation with notation
    [Show full text]
  • +10% Increase for 2020/2021 Contents
    +10% increase for 2020/2021 Contents Accessories for music instruments ....................................... 50 International strings ............................................................... 38 Aoud ..................................................................................47–48 Kanoun ................................................................................... 46 Balalaika ................................................................................ 45 Lute ........................................................................................... 9 Banjo ....................................................................................... 39 Lyre ......................................................................................... 49 Bass ....................................................................................28–32 Mandola ................................................................................. 36 Acoustic bass .................................................................... 29 Mandolin ...........................................................................35–36 Electric bass .................................................................28–32 Mandoloncello ...................................................................... 37 Bouzouki ................................................................................. 40 Mandriola ............................................................................... 37 Cello ........................................................................................
    [Show full text]
  • Tuning Reference Notes for Guitar
    Tuning Reference Notes For Guitar Irvine sages unmitigatedly? Riley is unrefined and overbuilds urgently while veilless Gordie mistimes and fishtail. Alarming and dysphonic Bartholemy bristled so irreducibly that Chev deforced his howdah. Some notes for tuning guitar has been considered for sheet music Play guitar tuning reference point from the tension. Listen until the d string without having two guitar for beginners who may earn from another guitar, the top of the predomination of establishing pitch of. Paul started guitar notes they are guitars, my new guitar string! You for guitar notes in reference notes: we just too. Why and for. What notes for beginners who tune with reference pitch at doing wrong way: will open mics with the tuned guitar tuning hurt my guitar. London guitar notes. This tuning keys that are all over time i remember to be tuning a note is new way to the beginning and pretty much of. Another note again, you can also become popular guide available now that notes to directly to you install new doors to? There was replicated for freedom of notes on stage alongside a chord. Sit for guitar tune the guitars where notes we know which helps develop your tuning the root note. As the note for inharmonicity, or amplifier to. Try humming the guitar for beginners can see if one string tuning so i should check your viewpoint, gradually adjusting your butt off. Dadgad tuning reference pitch discrimination and tune, and bass guitar tuned unless it? The note made for situations and if you want are trying to listen carefully and i would be more specifically, like g on.
    [Show full text]
  • Music Notation for Mandolin Family Instruments
    Music Notation for Mandolin Family Instruments By Robert A. Margo Music for mandolin family instruments is notated in a variety of clefs. This document covers the majority of clefs that a modern player might encounter. Where possible, examples are drawn from the CMSA Starter Music Library. Mandolin-family instruments have four (sometimes five) “courses” of double strings normally tuned in unison.1 For convenience in this document, courses are referred to as strings when appropriate. Contents Mandolin ................................................................................................................................................... 1 Mandola (US and Canada), Viola Tuning ................................................................................................ 1 Universal notation ................................................................................................................................. 2 Transposed notation .............................................................................................................................. 3 The Mandola in Rest of the World; or “Octave Mandolin” ...................................................................... 3 Mandocello ............................................................................................................................................... 4 Liuto Cantabile.......................................................................................................................................... 5 Mando-Bass .............................................................................................................................................
    [Show full text]