Welcome to my home page. Here you'll find various mathematical, historical, and musical topics, in many cases matched with biographical sketches. The mathematical topics include triangle geometry and number theory. The section on triangle geometry contains the world's largest internet collection of triangle centers. The numbertheoretic topics deal mostly with integer sequences and arrays.
The historical collection started out with 19th century scientists associated with the famous town, New Harmony, Indiana, located about 30 miles from Evansville. The collection has grown to include educators, writers, and artists. The music section consists of projects, including links to free downloadable scores and recordings. Ongoing projects include arranging and composing for groups of recorders. Below, you can click and listen to 75 duets and 117 trios performed by Japanese recorder virtuouso Papalin.

ENCYCLOPEDIA OF TRIANGLE CENTERS  ETC
Triangle Centers:
Classical
and
Recent
Euler Line, including 102 triangle centers
Algebraic Highways in Triangle Geometry
Triangle Centers and Central Triangles (book)
INTEGER SEQUENCES AND ARRAYS
Unsolved Problems and Rewards
Interspersions and Dispersions
Continued Fractions
Fractal Sequences
BIOGRAPHICAL STUDIES
New Harmony Scientists, Educators, Writers & Artists
Emmy Noether, Her Mentors & Colleagues
Fibonacci NumberTheorists
Triangle Geometers
MUSIC PROJECTS
Collection 1: AfricanAmerican and Jamaican Melodies
Collection 2: Christmas Carols (available from Mel Bay Publications.)
Collection 3: Irish Melodies
Collection 4: Americana to 1865
Collection 5: Americana after 1865
Collection 6: British Melodies
Collection 7: Melodies by Women Composers
Collection 8: Eastern European and Jewish Melodies
Collection 9: American Indian Melodies
Collection 10: Latin American Melodies
Collection 11: African Melodies
Collection 12: Western European Melodies
Project 2. Compositions for church choirs, available from G.I.A. Publications:
Four Anthems for Mixed Voices and Handbells
Scripture and Season in Song
The Hills Are Hushed This Night of Nights
O God, Beneath Your Hand
O God, Who at the Dawn of Time
Ring Out the Glad Tidings
The King of Love My Shepherd Is
This Easter Morn
Awake with Timbrel and with Dance
Introits and Anthems for Voices and Bells, Volume 1
Introits and Anthems for Voices and Bells, Volume 2
Introits and Anthems for Voices and Bells, Volume 3
Introits and Anthems for Voices and Bells, Volume 4
Introits and Anthems for Voices and Bells, Volume 5
Introits and Anthems for Voices and Bells, Volume 6
Project 3. New hymn tunes, with texts by James Hart Brumm, John Core, Ralph Freeman, Daniel P. Merrick, and John Thornburg. The book is available from Wayne Leupold Editions; click Information.
Project 4. Henry Hart, a composer who lived in Evansville, Indiana (where I live) in the 1870's. Hart published minstrel tunes and music for social dance. The research led to the first of my biographical articles for Wikipedia: Henry Hart. The External Links section at the end will take you to three supplements:
Supplement I. Henry Hart and His Family Orchestra (Including Eli Lilly's Recollections)
Supplement II. Hazel Hart Hendricks and Her School
Supplement III. Myrtle Hart, Harpist
"Myrtle Hart (Louise Kavanaugh, Harpist: A Documentary History", American Harp Journal, 27, no. 4., (2021), 917.
Project 5. Articles in The Hymn, the journal of The Hymn Society in the United States and Canada:
Tunes Lost and Found: BIRSTAL, FOUNDRY, HOTHAM, TRUE ELIJAH, The Hymn 52, no. 1 (Jan., 2001) 2130.
Tunes Lost and Found: HELMSLEY, LEEDS, YORKSHIRE, WEDNESBURY, The Hymn 52, no. 2 (Apr., 2001) 2536.
Tunes Lost and Found: TRUMPET, LOVEFEAST, ATHLONE, DERBY, The Hymn, no. 2 (Apr., 2002) 718.
The Hymn Tunes of Richard Wayne Dirksen, The Hymn 53, no. 4 (Oct., 2002) 1928.
Hymn Tune Descants, Part 1: 19151934, The Hymn 54, no. 3 (July, 2003) 2027. Reprinted in Journal of the Ralph Vaughan Williams Society 29 (Feb., 2004) 1720.
Hymn Tune Descants, Part 2: 19352001, The Hymn 55, no. 1 (Jan., 2004) 1722.
Three Native American Hymns, The Hymn 56, no. 2 (Spring 2005) 1829. Regarding these hymns, see the Wikipedia article Joseph Renville.
Native Dakota Hymn Tunes, The Hymn 64, no. 3 (Summer 2013) 718.
Three Generations of Works and Their Contributions to Congregational Singing, The Hymn 65, no. 3 (Summer 2014) 1017.
Project 6. Christmas carols for flute and accompanying instrument: Tidings of Comfort and Joy, for flute and keyboard, and 15 Christmas Carols for Flute and Guitar.
Project 7. Duets for Flutes or Recorders for flutes, recorders (bass, tenor, alto, soprano, sopranino), or other instruments: DUETS  Publisher's description.
The duets are performed on various pairs of recorders, from contrabass up to sopranino, by Papalin:
Part 1: Duets 127
Part 2: Duets 2850
Part 3: Duets 5175
Project 8. Trios for Recorders or Flutes. For downloadable free scores, click Online Sheet Music: Recorder Trios.
The 117 trios in this collection are performed on various recorders, from contrabass up to sopranino, by Papalin:
Part 1: Trios 127
Part 2: Trios 2858
Part 3: Trios 5990
Part 4: Trios 91117
For access to the vast collection of Papalin's recordings, IL DIVO "Papalin"
Project 9. Quartets for Recorders. Scored for various ensembles of sopranino, soprano, alto, tenor, and bass recorders. For downloadable free scores, click Online Sheet Music: Recorder Quartets.
Project 10. Ten hymn tunes in In Melody and Songs, Hymns from the Psalm Versions of Isaac Watts, Adrienne Tindall, editor, Darcery Press, 2014.
Project 11. Solos for Soprano Recorder (305 advanced solos), downloadable from IMSLP .
Project 12. 400 Solos for C Instrument, downloadable from IMSLP: .
Project 13. , "The Carol Books of Thomas Hardy (17781837) and Thomas Hardy (18111892)," downloadable from West Gallery Music Association, Articles section. The Thomas Hardys in the title were the grandfather and father of the novelist and poet, Thomas Hardy (18401928).
Project 14. , Indian Melodies by Thomas Commuck (1845), a Narragansett Indian, Harmonized by Clark Kimberling, 2021. Includes historical introduction and five appendices. Downloadable from IMSLP.
Project 15. Piano solos and a few pieces for church choirs (ongoing) IMSLP: .
The Canterbury Dictionary of Hymnology was launched from the University of Bristol in 2013. See Home Page and Editors' Introduction. My interest in hymntune descants, found in 1915 in England with notyetfound French precedents, led to an article in the Journal of the Ralph Vaughan Williams Society, and from there to the Descant article in the CDH. Subsequently, I was invited to write about American hymn tune composers and early American hymn tune collections, such as William Walker's Southern Harmony. Following is a list of the resulting articles (partially downloadable to nonCDHsubscribers):
American Bass Viol
Andrew Adgate, 17621793
George Nelson Allen, 18121877
Ronald Kent Arnatt, 19302018
Samuel Babcock, 17601813
Theodore Baker, 18511934
Nicholas George Julius Ballanta, 18931961(?)
Daniel Bayley, 17291792
R. John Blackley, 1936
Robert Boyd, 17711822?
Carrie Breck, 18551934
Lee Hastings Bristol, Jr., 19231979
Oliver Brownson, 17461815
Allen Carden, 17921859
Amzi Chapin, 17681835
Edwin Chapin, 18141880
Lucius Chapin, 17601842
Christian Harmony, 1867 
Norman CokeJephcott, 18931962 
The Colored Sacred Harp, 1934
Thomas Commuck, 18041855
Harry Coopersmith, 19021975
Henry Stephen Cutler, 18251902
Daniel ben Judah, fourteenth century
Elkanah Kelsay Dare, 17821826
Ananias Davisson, 17801857
Lizzie DeArmond, 18471936
Descant, 1915
Richard Dirksen, 19212003
Josiah Flagg, 17371794
Stephen Foster, 18261864
Philip Frazier, 18921964
Lewis Freeman, 17801859
Jacob French, 17541817
Harold William Friedell, 19051958
Clement Gale, 18601934
Louis Moreau Gottschalk, 18291869
Nathaniel Duren Gould, 17811864
Jean M. T. Gower, 18681957
John Henry Gower, 18551922
James Martin Gray, 18511935
Henry Wellington Greatorex, 18131858
Gerre Hancock, 19342012
Harp (as a title)
Karl P. Harrington, 18611953
William. C. Hauser, 18121880
Wilbur Caldwell Held, 19142015
Edward Hodges, 17961867
John Sebastian Bach Hodges, 18301895
Henry Houseley, 18511925
Matilda Durham Hoy, 18151901
Richard Hulan, 1939 
Charles Lewis Hutchins, 18381920
Hymns and Poems by Elizabeth Scott
Hymntune Index and Related Hymn Materials
I WONDER AS I WANDER, 1934
Indian Melodies, 1845
Jeremiah Ingalls, 17641838
Simeon Jocelin, 17461823
Artemas N. Johnson, 18171892
Julia H. Johnston, 18491919
Kentucky Harmony and Supplement, 1816 
Elisha J. King, 1821?1844
Karl Douglas Kroeger, 1932 
Chaunc(e)y Langdon, 17631830
Andrew Law, 17491821
John Leland, 17541841
James Lyon, 17351794
Albert Hay Malotte, 1895, 1964
Malotte's Lord's Prayer, 1935
Elias Mann, 17501825
Many and great, O God, are thy works, 1846
Cotton Mather, 16631728
Abraham Maxim, 17731829
Jesse Mercer, 17691841
Mercer's 'Cluster'
Arthur H. Messiter, 18341916
Missouri Harmony, 1820 
Mary Whittle Moody, 18701963
James R. Murray, 18411905
John Murray, 18401915
Judith Murray, 17511820
John Jacob Niles, 18921980
Charles S. Nutter, 18421928
Philip Phillips, 18341895
Harriet H. Pierson, 18651921
Amos Pilsbury, 17721812
George Frederick Root, 18121895
The Sacred Harp, 1844 
Alexander Schreiner, 19011987
Elizabeth Scott, 17081776
Setpiece
Shenandoah Harmony, 2012
Mary Shindler, 18101883
Nehemiah Shumway, 17611843
Southern Harmony, 1835 
Harriet Spaeth, 18451925
John Stickney, 17441827
Timothy Swan, 17581843
Elizabeth Miller Tangeman, 19051996
Reubern Archer Torrey, 18561928
Daniel B. Towner, 18501919
David Elton Trueblood, 19001994
James Ireland Tucker, 18191895
William Tuckey, 17081781
John Tufts, 16891750
Grant Colfax Tullar, 18691950
Samuel Wakefield, 17991895
William Walker, 18091875
Thomas Walter, 16961725
William Henry Walter, 18251893
Samuel Augustus Ward, 18471903
Henry Ware, Jr., 17941843
Donald DeWitt Wasson, 19212009
George James Webb, 18031887
George Whelpton, 18471930
Benjamin Franklin White, 18001879
David McK. Williams, 18871978
Frederick Jerome Work, 1878?1942
John Wesley Work (I), 1848?1923
John Wesley Work (II), 1872?1923
John Wesley Work (III), 19011967
Wyeth's Repository of Sacred Music, 1813
YISRAEL V'ORAITA (TORAH SONG), (1937)
Neil Sloane's Online Encyclopedia of Integer Sequences (OEIS) is a treasury of mathematical patterns, including references, links, and comments. Many contributors have developed prototypical sequences that introduce newly discovered relationships among numbers, some proved and others conjectural. Here are several:
Accumulation arrays and chains, weight arrays:
A144112,
A185906,
A185778
Adjectivenoun sequences and conjectures:
A055168 A055185 and A055186 A055191
Adjusted joint rank sequences:
A186219,
A186221,
A186350
Algorithm ("rules 1,2,3", conjectured permutations of N):
A257705,
A257905,
A258046
Arrays defined by recurrences:
A026519,
A026568,
A027926,
A246694 (every positive integer exactly twice),
A246696 (every positive integer exactly once)
Augmentation arrays:
A193091
Averaging arrays:
A197737
Beatty arrays:
A181661,
A181886,
A183988
Beatty discrepancy, complementary equations:
A136496,
A138253,
A138330
Beatty sequences (intersections):
A346308,
A347469,
A347793
Beatty trees and backstepping:
A258212
Best remaining lower approximates to golden ratio:
A143514
Changeofbase functions (geometric fractals):
A214969
Coefficient arrays from recursively defined polynomials:
A249057,
A249100,
A249128,
A249250
Complementary equations:
A022424,
A293076,
A293358,
A294381,
A294414,
A294532,
A295053,
A295357,
A295862,
A296000,
A296245,
A297830,
A298000,
A299634,
A304799 (advanced subscripts),
A309156,
A335999
Constant associated with fractal changeofbase functions:
A188593,
Constants associated with special shapes:
A188593,
A188618,
A188635,
A195284 (Philo distance, incenter),
A195304 (Philo distance, centroid),
A248965,
A248967,
A248968,
A336043,
A336073,
Constants associated with extreme values:
A196500
Constant associated with continued fraction and Hofstadter eta sequence:
A245215
Constants associated with convergent sums of reciprocals:
A228040,
A228044
Constants associated with the Lambert W funciton (PowerLog):
A299613,
A228044
Constants associated with tangencies:
A196625,
A197016
Constants that solve transcendental equations:
A196396,
A196401,
A196407,
A197737,
A197739
Constants defined by limits:
A298512
Continued fractions, accelerated:
A228667
Continued fractions, delayed:
A228825
Continued fractions (including conjectures):
A013659 (initialized continued fractions)
A096654 A096658 (selfgenerating continued fractions)
A143515 (best remaining upper approximates)
A143527 (best lower remaining approximates)
A229287 (primes)
A229348 (Fibonacciratio terms)
A229351 (fractional terms)
A229981 (associated with divergent continued fraction)
A229983 (associated with divergent continued fraction)
A229985 (associated with divergent continued fraction)
A229987 (associated with divergent continued fraction)
A230000 (continued fraction [1,1/x,1/x^2,...])
A259588 ("otherside convergents")
A259911 (discriminants of purely periodic continued fractions)
A265288 (sum{x  c(2n1), n=1,2,...}, where c = convergents to (x = golden ratio))
A265762 (coefficients in minimal polynomial of [1^n,2,1,1,1...])
Continued fraction transform:
A229594
Convolution arrays:
A213500
Convolutory inverses:
A077607,
A078140,
A225127
Convolutory roots:
A255811
Digits in various bases (pits and peaks, rises and falls, etc.)_
A296712
A296882,
A297030,
A297146,
A297330
Dispersions:
A114537 (the primes),
A191426,
A191735,
A248513
Divided difference sequences:
A204892
Double interspersion:
A161179
Egyptian fractions (denominates, convergent series):
A226049
Euclidean algorithm, modified, accelerated (see also Continued fractions, accelerated, delayed):
A228247,
A228468 (definition of trace of (x,y)),
A228486 (definition of trace of a number, in particular, pi),
A228487,
A228488,
A269993
Iteration from natural numbers to lower Wythoff sequence:
A181108
Factorialnested interval sequences:
A269970,
A269982(definition of factorial fractility)
Fibonaccirelated iterations:
A252230
Fibonacci regression array:
A097367
Fibonacci (minimal alternating representations of nonnegative integers, using Fibonacci numbers):
A256655
Fission arrays (of polynomial sequences, infinite matrices)
A193842,
A193844 (Pascal selffission array),
A202503 (Fibonacci selffission array)
Flooralgorithm (steps from n to 1):
,
A257698A257704
Floorsum sequence:
A182653
Floorrelated recurrence sequences:
A298338
Fractal sequences:
A003602,
A194029,
A194832,
A248514
A272900
Fractalization:
A194959
Fractional parts (counts involving inequalities, intervals):
A194285,
A194368,
A194738,
A194813,
A194832
Fractional parts (difference between):
A194368
Fusion arrays (of polynomial sequences, infinite matrices)
A118800 (Pascal selffusion array),
A193722,
A202453 (Fibonacci selffusion array)
Golden ratio:
A020959,
A252229,
A347065,
A347068,
A350526,
A350527
Golden triangle:
A152149,
A178988,
A188635
Greatest cube that divides n!:
A248762
Greedy sequences:
A270744
Greedy residue sequences:
A241833 (greedy residue sequence of squares),
A241834 (of triangular numbers),
A242252 (of odd primes),
A242293 (of cubes)
Harmonic mean: A227012 (includes conjectures)
Harmonic numbers:
A088907 (least k such that mean(1,1/2,...,1/k)<=1/n)
A224820 (array, includes open question about linear recurrence)
A225918 (partial sums of harmonic series),
A226894 (joint ranking of {H(k)} and {log k})
A227581 (closeness of H(m*n) to H(m) + H(n))
A227631 (definition of least splitter, a conjecture)
Infinite Fibonacci word (related sequences):
A241422,
A245920,
A245975,
A246104,
A246127,
A246354,
A339949,
A339950
Inflection point of product{1 + x^k, k >=1}:
A257394
Jointrank arrays:
A184912 A184927,
A182801,
A182869
Jointly generated arrays (by paired polynomial recurrences):
A208510
Jointly ranked sequences (with a conjecture):
A206911
Jump sequences (involving floor and ceiling, with conjectures):
A219085
Kimberling expulsion array:
A006852,
A007963,
A035486
Knight's moves on infinite chessboard:
A183049
Kolakoski fan conjectures:
A144586,
A143589
Lagrange pairs:
A194508
Lattice points in a region, counted:
A227347
Lazy Fibonacci representations:
A095791,
A095903
Least k such that e  (1 + 1/k)^k < 1/n:
A247778
Least k such that 1/e  (1  1/k)^k < 1/n:
A247781
Least k such that e  1/((k!)^(1/k)) < 1/n:
A247988
Least k such that k*sin(1/k)1 < 1/n^2:
A248515
Least number of (or not of) a prescribed form:
A305328,
A325417
Lexicographic ordering of all pairs i,j:
A349520,
A349521
Limit involving complex numbers and continued fractions:
A248749
Limitreverse of words:
A245920 (of infinite Fibonacci word)
A245933 (of Beatty sequence for sqrt(2))
A245936 (of OldenburgerKolakoski sequence)
A245939 (of ThueMorse sequence)
Limiting block extension: A246127
Limiting counting procedure (Theorems 1 and 2):
A225660,
A225669
Matrix counts:
A194508
Minimal alternating representations of nonnegative integers, using various bases:
A256655 (Fibonacci numbers),
A256696 (powers of 2),
A256974 (triangular numbers),
A256789 (square numbers),
Morphism sequences:
A189576,
A189628,
A287663,
A287385
Multinomial coefficients:
A248686
Mysterious Bsequence (array):
A186158
Natural number array:
A000027,
A185787
Number of diophantine solutions (w,x,y,n), etc.:
A211266,
A211422,
A211795,
A212959
Order arrays:
A163254,
A163255,
A340245
Orderings of all 01words:
A076478,
A341256,
A341258,
A341334,
A342753,
A342910
Orderings of rational numbers and other sets; see also "Trees generated by various sequences" and "Rulesgenerated sequences" :
A226080 (Fibonacci  or rabbit  ordering, all positive integers)
A226130 (all rationals)
A226247 (all rationals)
A232559 (all positive rationals)
A232723 (all integers)
A274455 (all integers)
A232895 (all positive integers; linear recurrence conjecture)
A232890 (all rationals)
A233694 (all Gaussian rationals)
PINVERT of a sequence:
A289780,
A290890
A291000,
A291219,
A291382,
A291728,
A292479
A292480
Palindromic linear recurrences; array R(n,k) = n+[n/2]+...[n/k]:
A211701
Palindromic words (infinite):
A260390
Partial complements, polygonal numbers:
A086270,
A086272
Partitions of the set of positive integers ("generalized Beatty sequences", joint rankings):
A184812,
A184928
Partitions of positive integers:
A184812
A209082 (least power separator)
A209122 (# partitions into k parts is k for some k)
A209354 (max  min)
A209423 (# odd parts minus # even parts)
A209489 (partitions of 1 into rational parts)
A230025 (outliers in partitions)
A235130 (partitions that include partitions)
A236129 (constant reversal sums)
A236292 (distinct cyclic permutations of partitions of n)
A236360 (numerator, mean of all parts of partitions of n)
A236543 (least part and greatest part, frequencies)
A237665 (string of consecutive integers)
A237668 (sums of parts is a part)
A237685 (depth of a partition)
A237980 (square partitions)
A237981 (directional partitions; definition of Ferrers matrix)
A238325 (antidiagonal partitions)
A238326 (diagonal partitions)
A238616 (partitions with standard deviation < 1)
A238883 (upper triangular partitions)
A238885 (lower triangular partitions)
A238943 (size of Ferrers matrix of partitions)
A239312 (condensed partitions)
A239319 (condensed partitions)
A239482 (separable partitions)
A241150 (partition graph, G(n))
A241815 (strict partition graph, S(n))
A241900 (components of disconnected partition graph, S(n))
A349552 (halving partitions)
Partitions of zero:
A209535,
A236429
Pathcounting arrays:
A047040,
A047089,
A049600,
A051288,
A247049,
A247352,
A247049,
A247352
Permutations of integers (conjectures):
A131388 A131395
Polynomial reduction:
A192175,
A192232,
A192744,
A192872,
A192904,
A192951
Polynomials defined by recurrences:
A184049,
A240755,
A240758,
A248664,
A248669,
A248670
Power ceiling functions:
A214992 (several open questions)
Primes (conjectures):
A174349,
A184774,
A185510,
A219096,
A258036,
A255909
Prime divisor subtractions iterated until reaching 0:
A233341
Primerelated arrays:
A192175,
A229608
Primes, a limit:
A260458 (limit of GCD(PP(n)  k, PP(n) + k) as k > oo, where PP(n) is the product of the first n primes.
Primes, minimal nested palindromic primes:
A261881,
A262627,
A260444 (infinite palindromic word)
Primes in various bases:
A262727 (2,3,5)primes,
A262728 (2,3,5,7)primes,
A262729 strong (2,3,5,7)primes
Primesonly best approximates (POBAs) to 1:
A265759
Pythagorean fraction approximations:
A195500
kPythagorean triples:
A195770
Qresidues of polynomials:
A193649
Quartersquares, greedy representations of positive integers (with theorem):
A257018
Rank transforms:
A187224
Ratiosums:
A296469
Reciprocity arrays:
 A259572A259583
Repetitionresistant sequence:
A007061,
A079101
Rulesgenerated sequences:
A074049,
A147991,
A183544,
A190803,
A191106
Selfgenerating arrays:
A203300,
A203301
Shiftsums of zeroone sequences:
A187950,
A188014
Signature sequences:
A007337,
A084531
Subsets counted by maximal differences:
A255874
Swappage (lower even swappage of upper Wythoff sequence):
A141104
Symmetric matrices and interlacing roots of polynomials:
A202605,
A204016
Symmetric matrices and permanents:
A204263,
A204269,
A204435,
A204551
Trees generated by rules about edges:
A228853
Trees generated by various sequences or rules:
A074049 (generated by Wythoff sequences)
A183079 (generated by triangular numbers)
A183169 (generated by squares)
A183231 (1st of two complementary trees generated by triangular numbers)
A183542 (1st of two complementary trees generated by Wythoff sequences
A233135 (shortest (x+1,2x)code of n)
A252864 (a tree of ordered pairs)
A255809 (generation n has ntuples with gapsizes <=2)
A257569 (Fibonaccirelated tree)
A262841 (Fibonacci zerotree)
A264799 (every integer exactly once)
Unimodular 2x2 matrices:
A210000
Vandermonde permanents:
A203468
Wythoff array, W:
A035513,
A114579 (transposition sequence of W),
A185778,
A186007,
A259556,
A349550 (metaWythoff array)
WythoffB array:
A340244
Wythoff ceiling array:
A341260
Wythoff difference array:
A080164
sWythoff sequences:
A184117
Zeckendorf distance between m and n:
A226207
Zeroone sequences (selfgenerating):
A188967