{"id":4276,"date":"2026-03-21T08:00:35","date_gmt":"2026-03-21T12:00:35","guid":{"rendered":"https:\/\/mathvoices.ams.org\/mathmedia\/?p=4276"},"modified":"2026-03-21T15:27:28","modified_gmt":"2026-03-21T19:27:28","slug":"tonys-take-january-february-2026","status":"publish","type":"post","link":"https:\/\/mathvoices.ams.org\/mathmedia\/tonys-take-january-february-2026\/","title":{"rendered":"Tony&#8217;s Take January-February 2026"},"content":{"rendered":"<h2>This column&#8217;s topics:<\/h2>\n<ul>\n<li><a href=\"#one\">Combinatorics in <i>The New York Times<\/i> <\/a><\/li>\n<li><a href=\"#two\">The algebra of melody<\/a><\/li>\n<\/ul>\n<p><a name=\"one\"><\/a><\/p>\n<h3>Combinatorics in <i>The New York Times<\/i>.<\/h3>\n<p><a href=\"https:\/\/arxiv.org\/pdf\/2511.15864\">Recent work<\/a> by Neil J. A. Sloane and David Cutler addresses the &#8220;pancake-cutting&#8221; problem: If you cut a pancake $n$ times, how many pieces can you get? Of course, since this is mathematics, the pancake is infinitely large and the knife infinitely long. For a straight-edged knife the answer has been known for some time\u2014$n$ cuts produce at most $(n^2+n+2)\/2$ pieces. The new results examine &#8220;exotic&#8221; knives, made up of a finite number of straight edges. The only rule is that the whole cutting surface is connected. Siobhan Roberts <a href=\"https:\/\/www.nytimes.com\/2026\/01\/20\/science\/infinite-pancake-math-puzzle.html\">reported on the work<\/a> for <em>The New York Times<\/em>.<\/p>\n<figure id=\"attachment_4299\" aria-describedby=\"caption-attachment-4299\" style=\"width: 231px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"wp-image-4299 size-medium\" src=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/straight-knives.png?resize=231%2C300&#038;ssl=1\" alt=\"\" width=\"231\" height=\"300\" srcset=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/straight-knives.png?resize=231%2C300&amp;ssl=1 231w, https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/straight-knives.png?w=300&amp;ssl=1 300w\" sizes=\"auto, (max-width: 231px) 100vw, 231px\" \/><figcaption id=\"caption-attachment-4299\" class=\"wp-caption-text\">The most efficacious $n$ slices of a straight-edged knife for $n= 1, 2, 3, 4, 5$ result in $2, 4, 7, 11$ and $16$ pieces, respectively. The arrows represent knife edges going off to infinity. Image courtesy of Neil Sloane.<\/figcaption><\/figure>\n<p>One of Sloane and Cutler&#8217;s exotic knives is shaped like a long-legged ${\\sf A}$, with two blades going off to infinity and one strut between them. A single ${\\sf A}$-knife divides the planar pancake into three parts, while two (once positioned and affinely adjusted) can yield 14 separate pieces.<\/p>\n<figure id=\"attachment_4300\" aria-describedby=\"caption-attachment-4300\" style=\"width: 278px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"wp-image-4300 size-full\" src=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/A-slices-e1773848890665.png?resize=278%2C316&#038;ssl=1\" alt=\"Two A-knife cuts overlaid cut the plane into polygonal regions.\" width=\"278\" height=\"316\" srcset=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/A-slices-e1773848890665.png?w=278&amp;ssl=1 278w, https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/A-slices-e1773848890665.png?resize=264%2C300&amp;ssl=1 264w\" sizes=\"auto, (max-width: 278px) 100vw, 278px\" \/><figcaption id=\"caption-attachment-4300\" class=\"wp-caption-text\">Two ${\\sf A}$-knives (inset) partition the plane into 14 distinct pieces. Drawing courtesy of Neil Sloane, with added colors.<\/figcaption><\/figure>\n<p>According to Sloane and Cutler&#8217;s preprint, the ${\\sf A}$-knife research began last summer, when Sloane spoke to undergraduates at Rutgers about the pancake-slicing problem. Cutler was in attendance, along with two other students, Edward Xiong and Jonathan Pei. Inspired by Sloane&#8217;s talk, the three students solved a version of the problem in which the knife is a <strong>tripod<\/strong>: three infinite blades radiating from one point. They proved that $n$ tripods can cut the plane into at most $(9n^2-5n+2)\/2$ pieces.<\/p>\n<figure id=\"attachment_4301\" aria-describedby=\"caption-attachment-4301\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"wp-image-4301 size-full\" src=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/tripods.png?resize=300%2C262&#038;ssl=1\" alt=\"Two tripod knife cuts overlaid, one with the tripod point oriented to the left and the other with the point oriented up.\" width=\"300\" height=\"262\" \/><figcaption id=\"caption-attachment-4301\" class=\"wp-caption-text\">Last summer, Xiong, Pei and Cutler proved that $n$ tripods can cut the pancake into at most $(9n^2-5n+2)\/2$ pieces. For $n=2$ this gives 14, as illustrated here. Image credit: Tony Phillips.<\/figcaption><\/figure>\n<p>Later, Cutler and Sloane experimentally studied ${\\sf A}$-knives; their results exactly matched Xiong, Pei and Cutler&#8217;s formula for tripods. &#8220;This was not a coincidence!&#8221; they write. They went on to prove that for a given number of cuts, tripods and ${\\sf A}$-knives always achieve the same maximum number of pancake pieces. Their paper also generalizes Xiong, Pei and Cutler&#8217;s result to <strong>$k$-pods<\/strong>, where the knife has $k$ infinite blades instead of 3. With $n$ cuts of a $k$-pod, one can create $\\binom{n}{2}k^2 + n(k-1) +1$ pancake pieces.<\/p>\n<p><a name=\"two\"><\/a><\/p>\n<h3>The algebra of melody.<\/h3>\n<p>Last summer, the University of Waterloo&#8217;s Olga Ibragimova and Chrystopher Nehaniv published &#8220;<a href=\"https:\/\/link.springer.com\/chapter\/10.1007\/978-3-031-84869-8_5\">Algebraic Applications in Investigation of Musical Symmetry<\/a>,&#8221; a study on the tonal structure of melody. (February 19, <i>Phys.org<\/i> posted a <a href=\"https:\/\/phys.org\/news\/2026-02-secret-math-catchy-melodies.html\">press release<\/a>, whose headline &#8220;Why some tunes stick: Mathematical symmetry helps explain catchy melodies&#8221; is not completely accurate: Ibragimova and Nehaniv&#8217;s article does not mention stickiness or catchiness.)<\/p>\n<p>In this preliminary study, the authors do not consider the length of notes nor their exact pitches. Instead, each note is represented by a number locating its <i>pitch class<\/i> in the middle octave on a piano (C$=1$, C#$=2$, D$=3$, &#8230; , B$=12$). Mathematically, they think of an $n$-note melody as a function $m \\colon L \\mapsto C$. Here, $L$ is the space of locations <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5C%7B1%2C+%5Cdots%2C+n%5C%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;{1, &#92;dots, n&#92;}\" class=\"latex\" \/>, while $C$ is the set of pitch classes <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5C%7B1%2C+%5Cdots%2C+12%5C%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;{1, &#92;dots, 12&#92;}\" class=\"latex\" \/>.<\/p>\n<figure id=\"attachment_4303\" aria-describedby=\"caption-attachment-4303\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"wp-image-4303 size-full\" src=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/keyboard.png?resize=300%2C241&#038;ssl=1\" alt=\"Top: Octave on a piano, with keys labeled by number. Bottom: In sheet music notation, the pitch class sequence A B C D E F G A B C D.\" width=\"300\" height=\"241\" \/><figcaption id=\"caption-attachment-4303\" class=\"wp-caption-text\">Numerical encoding of melody. All the notes in a melody are taken to be the same length (here, a quarter-note) and each is represented by a number locating its pitch-class representative in the middle octave. Credit for all images and sound samples in this item: Tony Phillips.<\/figcaption><\/figure>\n<p>The following familiar 8-note melody, with its numerical encoding, will be used to illustrate the musical transformations considered by Ibragimova and Nehaniv. Click &#8220;Play&#8221; on the sound-bar to hear it.<\/p>\n<div style=\"margin-left: auto;margin-right: auto;width: 300px\">\n<audio class=\"wp-audio-shortcode\" id=\"audio-4276-1\" preload=\"none\" style=\"width: 100%;\" controls=\"controls\"><source type=\"audio\/mpeg\" src=\"https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica.mp3?_=1\" \/><a href=\"https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica.mp3\">https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica.mp3<\/a><\/audio>\n<\/div>\n<figure id=\"attachment_4305\" aria-describedby=\"caption-attachment-4305\" style=\"width: 281px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"wp-image-4305 size-full\" src=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica.png?resize=281%2C102&#038;ssl=1\" alt=\"Pitch class sequence C E C B D F D C.\" width=\"281\" height=\"102\" \/><figcaption id=\"caption-attachment-4305\" class=\"wp-caption-text\">This sample melody would be numerically encoded as $[1~5~1~12~3~6~3~1]$. Here each note is represented by an octave-related pair to reflect the authors&#8217; working definition of melody as a sequence of pitch <i>classes<\/i>.<\/figcaption><\/figure>\n<p>The authors consider four operations that transform a melody without changing what they call its &#8220;inherent properties.&#8221; These are <i>transposition, translation, inversion,<\/i> and <i>retrograde.<\/i><\/p>\n<ul>\n<li><i>Transposition<\/i> shifts all the notes in a melody up or down by a specified number of steps.\n<div style=\"margin-left: auto;margin-right: auto;width: 300px\">\n<audio class=\"wp-audio-shortcode\" id=\"audio-4276-2\" preload=\"none\" style=\"width: 100%;\" controls=\"controls\"><source type=\"audio\/mpeg\" src=\"https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-F.mp3?_=2\" \/><a href=\"https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-F.mp3\">https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-F.mp3<\/a><\/audio>\n<\/div>\n<p><figure id=\"attachment_4312\" aria-describedby=\"caption-attachment-4312\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"wp-image-4312 size-medium\" src=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-F.png?resize=300%2C123&#038;ssl=1\" alt=\"Pitch class sequence F A F E G B\u266d G F.\" width=\"300\" height=\"123\" srcset=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-F.png?resize=300%2C123&amp;ssl=1 300w, https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-F.png?w=331&amp;ssl=1 331w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-4312\" class=\"wp-caption-text\">In this transposition, the notes in the sample melody are shifted up by 5 steps: $[1~5~1~12~3~6~3~1]$ becomes $[6~10~6~5~8~11~8~6]$. Notice that the addition is done in modular &#8220;clock arithmetic&#8221; where $12+5=5$.<\/figcaption><\/figure><\/li>\n<li><i>Translation<\/i> shifts all the notes in a melody backwards or forwards by a specified number of steps.\n<div style=\"margin-left: auto;margin-right: auto;width: 300px\">\n<audio class=\"wp-audio-shortcode\" id=\"audio-4276-3\" preload=\"none\" style=\"width: 100%;\" controls=\"controls\"><source type=\"audio\/mpeg\" src=\"https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-T.mp3?_=3\" \/><a href=\"https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-T.mp3\">https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-T.mp3<\/a><\/audio>\n<\/div>\n<p><figure id=\"attachment_4317\" aria-describedby=\"caption-attachment-4317\" style=\"width: 269px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"wp-image-4317 size-full\" src=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-T.png?resize=269%2C102&#038;ssl=1\" alt=\"Pitch class sequence B D F D C C E C.\" width=\"269\" height=\"102\" \/><figcaption id=\"caption-attachment-4317\" class=\"wp-caption-text\">This translation shifts the notes in the sample melody back by 3 steps: $[1~5~1~12~3~6~3~1]$ becomes $[12~3~6~3~1~1~5~1]$. For melodies of length $n$ this translation is done <i>modulo<\/i> $n$, so that notes pushed off one end reappear at the other.<\/figcaption><\/figure><\/li>\n<li><i>Inversion<\/i> flips a melody upside-down. In this study, inversion moves pitch-class $k$ to pitch-class $13-k$.\n<div style=\"margin-left: auto;margin-right: auto;width: 300px\">\n<audio class=\"wp-audio-shortcode\" id=\"audio-4276-4\" preload=\"none\" style=\"width: 100%;\" controls=\"controls\"><source type=\"audio\/mpeg\" src=\"https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-I.mp3?_=4\" \/><a href=\"https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-I.mp3\">https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-I.mp3<\/a><\/audio>\n<\/div>\n<p><figure id=\"attachment_4319\" aria-describedby=\"caption-attachment-4319\" style=\"width: 286px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"wp-image-4319 size-full\" src=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-I.png?resize=286%2C109&#038;ssl=1\" alt=\"Pitch class sequence B G B C A F# A B.\" width=\"286\" height=\"109\" \/><figcaption id=\"caption-attachment-4319\" class=\"wp-caption-text\">The rule $k \\rightarrow 13-k$ changes our sample melody $[1~5~1~12~3~6~3~1]$ to its inversion $[12~8~12~1~10~7~10~12]$.<\/figcaption><\/figure><\/li>\n<li><i>Retrograde<\/i> plays the original melody backwards.\n<div style=\"margin-left: auto;margin-right: auto;width: 300px\">\n<audio class=\"wp-audio-shortcode\" id=\"audio-4276-5\" preload=\"none\" style=\"width: 100%;\" controls=\"controls\"><source type=\"audio\/mpeg\" src=\"https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-R.mp3?_=5\" \/><a href=\"https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-R.mp3\">https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-R.mp3<\/a><\/audio>\n<\/div>\n<p><figure id=\"attachment_4321\" aria-describedby=\"caption-attachment-4321\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"wp-image-4321 size-medium\" src=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-R-1.png?resize=300%2C116&#038;ssl=1\" alt=\"Melody C D F D B C E C.\" width=\"300\" height=\"116\" srcset=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-R-1.png?resize=300%2C116&amp;ssl=1 300w, https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/eeroica-R-1.png?w=319&amp;ssl=1 319w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-4321\" class=\"wp-caption-text\">Under the <i>retrograde<\/i> transformation, the sample melody $[1~5~1~12~3~6~3~1]$ becomes $[1~3~6~3~12~1~5~1]$.<\/figcaption><\/figure><\/li>\n<\/ul>\n<p>Since each melody $m$ is a map from the locations <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=L+%3D+%5C%7B1%2C+%5Cdots%2C+n+%5C%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"L = &#92;{1, &#92;dots, n &#92;}\" class=\"latex\" \/> to <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=C+%3D+%5C%7B1%2C+%5Cdots%2C+12+%5C%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"C = &#92;{1, &#92;dots, 12 &#92;}\" class=\"latex\" \/>, the possible melodies of length $n$ form the Cartesian product<\/p>\n<p style=\"text-align: center\"><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=C%5En+%3D+%5C%7B1%2C%5Cdots%2C+12%5C%7D%5En.&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"C^n = &#92;{1,&#92;dots, 12&#92;}^n.\" class=\"latex\" \/><\/p>\n<p>Each of our four operations is a permutation of $C^n$ which preserves two things: the size of the jumps between notes, and their relative position in the sequence. These must be the &#8220;inherent properties&#8221; that the authors have in mind. If you compose two of these transformations, for example transposing and then reflecting, the new melody will retain its inherent properties. The four types of operations, and all their possible combinations, form a group ${\\bf G}$, which we can think of as the group of <i>musical<\/i> permutations of $C^n$.<\/p>\n<p>Ibragimova and Nehaniv point out that translation and retrograde keep the same notes, but rearrange the locations $L$, while transposition and inversion preserve $L$ and change the pitch-classes $C$. They explain that ${\\bf G}$ is structurally the <i>direct product<\/i> of $G_L$ and $G_C$,$${\\bf G} = G_L \\times G_C,$$where $G_L$ is the group of symmetries of <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=L%3D+%5C%7B1%2C%5Cdots%2C+n%5C%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"L= &#92;{1,&#92;dots, n&#92;}\" class=\"latex\" \/> generated by translations and applications of retrograde, and $G_C$ is the group of symmetries of <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=C%3D+%5C%7B1%2C%5Cdots%2C+12%5C%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"C= &#92;{1,&#92;dots, 12&#92;}\" class=\"latex\" \/> generated by transpositions and inversions. This means that every element ${\\bf g}$ of ${\\bf G}$ can be written as a pair of elements ${\\bf g}=(g_L, g_C)$, one from each group, and that the product of ${\\bf g}$ and ${\\bf g&#8217;}= (g&#8217;_L, g&#8217;_C)$ is ${\\bf gg&#8217;}= (g_Lg&#8217;_L, g_Cg&#8217;_C)$.<\/p>\n<p>Moreover, $G_L$ is the &#8220;dihedral&#8221; group $D_n$ of symmetries of an $n$-gon (see the illustration below), and $G_C$ is $D_{12}$.<\/p>\n<figure id=\"attachment_4358\" aria-describedby=\"caption-attachment-4358\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"wp-image-4358 size-full\" src=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/melody-2.png?resize=300%2C192&#038;ssl=1\" alt=\"Left: An octagon representing locations. The symmetry translation corresponds to a rotation of the octagon, and retrograde is a reflection. A melody maps the octagon to (right) a dodecagon representing the 12 pitch-classes. Transposition rotates the dodecagon, while inversion reflects it.\" width=\"300\" height=\"192\" \/><figcaption id=\"caption-attachment-4358\" class=\"wp-caption-text\">Illustration for a melody of length 8. Translation and retrograde are symmetries of the space of locations, while transposition and inversion are symmetries of the space of pitch-classes.<\/figcaption><\/figure>\n<p>Given a transformation $g$ in ${\\bf G}$, Ibragimova and Nehaniv construct a length-8 melody which is <i>symmetric<\/i> with respect to $g$. They choose $g = (R^2, r^3)$, where $R$ is translation forward by one position and $r$ is transposition up by one semitone. This means they want a melody which, when translated two steps to the right and transposed up three semitones, sounds the same as it did to start. They are free to choose the first two notes, which they take to be 1 (C) and 3 (D); their construction yields $[1~ 3~ 4~ 6~ 7~ 9~ 10~ 12]$.<\/p>\n<div style=\"margin-left: auto;margin-right: auto;width: 300px\">\n<audio class=\"wp-audio-shortcode\" id=\"audio-4276-6\" preload=\"none\" style=\"width: 100%;\" controls=\"controls\"><source type=\"audio\/mpeg\" src=\"https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/symmetric.mp3?_=6\" \/><a href=\"https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/symmetric.mp3\">https:\/\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/symmetric.mp3<\/a><\/audio>\n<\/div>\n<figure id=\"attachment_4326\" aria-describedby=\"caption-attachment-4326\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img data-recalc-dims=\"1\" loading=\"lazy\" decoding=\"async\" class=\"wp-image-4326 size-full\" src=\"https:\/\/i0.wp.com\/mathvoices.ams.org\/mathmedia\/wp-content\/uploads\/sites\/3\/2026\/03\/symmetric.png?resize=300%2C104&#038;ssl=1\" alt=\"\" width=\"300\" height=\"104\" \/><figcaption id=\"caption-attachment-4326\" class=\"wp-caption-text\">A symmetrical melody: C D D# F F# G# A B stays the same when translated two steps to the right and up three semitones.<\/figcaption><\/figure>\n<p style=\"text-align: right\"><em>\u2014Tony Phillips, Stony Brook University<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This column&#8217;s topics: Combinatorics in The New York Times The algebra of melody Combinatorics in The New York Times. Recent work by Neil J. A. Sloane and David Cutler addresses the &#8220;pancake-cutting&#8221; problem: If you cut a pancake $n$ times, how many pieces can you get? Of course, since this<span class=\"more-link\"><a href=\"https:\/\/mathvoices.ams.org\/mathmedia\/tonys-take-january-february-2026\/\">Read More &rarr;<\/a><\/span><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"advanced_seo_description":"","jetpack_seo_html_title":"","jetpack_seo_noindex":false,"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[3],"tags":[100,529,38,110,527,124,530,419,528,85,526,525],"class_list":["entry","author-tphillips","post-4276","post","type-post","status-publish","format-standard","category-tony-phillips-take","tag-combinatorics","tag-dihedral","tag-group-theory","tag-infinity","tag-inversion","tag-music","tag-pancake","tag-recreational-math","tag-retrograde","tag-symmetry","tag-translation","tag-transposition"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/4276","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/comments?post=4276"}],"version-history":[{"count":73,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/4276\/revisions"}],"predecessor-version":[{"id":4376,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/posts\/4276\/revisions\/4376"}],"wp:attachment":[{"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/media?parent=4276"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/categories?post=4276"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathvoices.ams.org\/mathmedia\/wp-json\/wp\/v2\/tags?post=4276"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}