The Logic of Six-Based Minor for Harmonic Analyses of Popular Music

Harmonic analyses of popular music typically take the minor tonic to be Roman numeral “one.” By nature, this “one-based” approach requires a new numbering scheme when songs shift between relative key centers. Recent scholarship has argued, however, that popular music often involves ambiguity between relative tonalities, as exemplified in the “Axis” progression, if not sometimes a tonal fusion of two relative keys. I thus argue for the utility of a “six-based” approach to the minor tonic, where the minor tonic is taken to be scale-degree 6. This six-based approach, common among practitioners of popular music as seen in the Nashville number system, avoids the forced choice of a single tonic, and it thus offers a consistent way to track chord function and behavior across shifts between relative key centers. After considering these shifts in a diatonic context on the levels of both phrase and song form, I posit that popular music involves three possible tonalities, together which form a “triple-tonic complex” akin to Stephenson’s three harmonic palettes: a major system, a parallel-minor system, and a relative-minor system. I conclude by considering how chromatic chords common in a major key, such as II and ".fn_flat('')."VII, correspond to their counterparts in the relative minor, IV and ".fn_flat('')."II, thereby collapsing the landscape of diatonic modes into three modal complexes. Overall, the paper serves to reveal the logic of six-based minor—why it is useful, what issues it resolves, and what insights it can afford us about harmonic syntax in popular music.

On the Colijn-Plazzotta numbering scheme for unlabeled binary rooted trees

Colijn & Plazzotta (Syst. Biol. 67:113-126, 2018) introduced a scheme for bijectively associating the unlabeled binary rooted trees with the positive integers. First, the rank 1 is associated with the 1-leaf tree. Proceeding recursively, ordered pair (k1, k2), k1 ⩾ k2 ⩾ 1, is then associated with the tree whose left subtree has rank k1 and whose right subtree has rank k2. Following dictionary order on ordered pairs, the tree whose left and right subtrees have the ordered pair of ranks (k1, k2) is assigned rank k1(k1 − 1)/2 + 1 + k2. With this ranking, given a number of leaves n, we determine recursions for an, the smallest rank assigned to some tree with n leaves, and bn, the largest rank assigned to some tree with n leaves. For n equal to a power of 2, the value of an is seen to increase exponentially with 2αn for a constant α ≈ 1.24602; more generally, we show it is bounded an < 1.5n. The value of bn is seen to increase with for a constant β ≈ 1.05653. The great difference in the rates of increase for an and bn indicates that as the index v is incremented, the number of leaves for the tree associated with rank v quickly traverses a wide range of values. We interpret the results in relation to applications in evolutionary biology.Mathematics subject classification05C05, 92B10, 92D15

A Unified Numbering Scheme for Class C β-Lactamases

ABSTRACT A standard numbering scheme has been proposed for class C β-lactamases. This will significantly enhance comparison of biochemical and biophysical studies performed on different members of this class of enzymes and facilitate communication in the field.

A Standard Numbering Scheme for Class C β-Lactamases

ABSTRACT Unlike for classes A and B, a standardized amino acid numbering scheme has not been proposed for the class C (AmpC) β-lactamases, which complicates communication in the field. Here, we propose a scheme developed through a collaborative approach that considers both sequence and structure, preserves traditional numbering of catalytically important residues (Ser64, Lys67, Tyr150, and Lys315), is adaptable to new variants or enzymes yet to be discovered and includes a variation for genetic and epidemiological applications.

Perpetuation of errors in illustrations of cranial nerve anatomy

For more than 230 years, anatomical illustrations have faithfully reproduced the German medical student Thomas Soemmerring's cranial nerve (CN) arrangement. Virtually all contemporary atlases show the abducens, facial, and vestibulocochlear nerves (CNs VI–VIII) all emerging from the pontomedullary groove, as originally depicted by Soemmerring in 1778. Direct observation at microsurgery of the cerebellopontine angle reveals that CN VII emerges caudal to the CN VIII root from the lower lateral pons rather than the pontomedullary groove. Additionally, the CN VI root lies in the pontomedullary groove caudal to both CN VII and VIII in the vast majority of cases. In this high-resolution 3D MRI study, the exit location of CN VI was caudal to the CN VII/VIII complex in 93% of the cases. Clearly, Soemmerring's rostrocaudal numbering system of CN VI-VII-VIII (abducens-facial-vestibulocochlear CNs) should instead be VIII-VII-VI (vestibulocochlear-facial-abducens CNs). While the inaccuracy of the CN numbering system is of note, what is remarkable is that generations of authors have almost universally chosen to perpetuate this ancient error. No doubt some did this through faithful copying of their predecessors. Others, it could be speculated, chose to depict the CN relationships incorrectly rather than run contrary to long-established dogma. This study is not advocating that a universally recognized numbering scheme be revised, as this would certainly create confusion. The authors do advocate that future depictions of the anatomical arrangements of the brainstem roots of CNs VI, VII, and VIII ought to reflect actual anatomy, rather than be contorted to conform with the classical CN numbering system.

Mode of binding of the antithyroid drug propylthiouracil to mammalian haem peroxidases

The mammalian haem peroxidase superfamily consists of myeloperoxidase (MPO), lactoperoxidase (LPO), eosinophil peroxidase (EPO) and thyroid peroxidase (TPO). These enzymes catalyze a number of oxidative reactions of inorganic substrates such as Cl−, Br−, I−and SCN−as well as of various organic aromatic compounds. To date, only structures of MPO and LPO are known. The substrate-binding sites in these enzymes are located on the distal haem side. Propylthiouracil (PTU) is a potent antithyroid drug that acts by inhibiting the function of TPO. It has also been shown to inhibit the action of LPO. However, its mode of binding to mammalian haem peroxidases is not yet known. In order to determine the mode of its binding to peroxidases, the structure of the complex of LPO with PTU has been determined. It showed that PTU binds to LPO in the substrate-binding site on the distal haem side. The IC50values for the inhibition of LPO and TPO by PTU are 47 and 30 µM, respectively. A comparision of the residues surrounding the substrate-binding site on the distal haem side in LPO with those in TPO showed that all of the residues were identical except for Ala114 (LPO numbering scheme), which is replaced by Thr205 (TPO numbering scheme) in TPO. A threonine residue in place of alanine in the substrate-binding site may affect the affinity of PTU for peroxidases.

How to specify the structure of substituted blade-like zigzag diamondoids

Abstract The dualist of an [n]diamondoid consists of vertices situated in the centers of each of the n adamantane units, and of edges connecting vertices corresponding to units sharing a chair-shaped hexagon of carbon atoms. Since the polycyclic structure of diamondoids is rather complex, so is their nomenclature. For specifying chemical constitution or isomerism of all diamondoids the Balaban-Schleyer graph-theoretical approach based on dualists has been generally adopted. However, when one needs to indicate the location of C and H atoms or of a substituent in a diamondoid or the stereochemical relationships between substituents, only the IUPAC polycycle nomenclature (von Baeyer nomenclature) provides the unique solution. This is so since each IUPAC name is associated with a unique atom numbering scheme. Diamondoids are classified into catamantanes (which can be regular or irregular), perimantanes, and coronamantanes. Regular catamantanes have molecular formulas C4n+6H4n+12. Among regular catamantanes, the rigid blade-shaped zigzag catamantanes (so called because their dualists consist of a zigzag line with a code of alternating digits 1 and 2) exhibit a simple pattern in their von Baeyer nomenclature. Their carbon atoms form a main ring with 4n + 4 atoms, and the remaining atoms form two 1-carbon bridges. All zigzag [n]catamantanes with n &gt; 2 have quaternary carbon atoms, and the first bridgehead in the main ring is such an atom. Their partitioned formula is Cn−2(CH)2n+4(CH2)n+4. As a function of their parity, IUPAC names based on the von Baeyer approach have been devised for all zigzag catamantanes, allowing the unique location for every C and H atom. The dualist of such a zigzag catamantane defines a plane bisecting the molecule, and the stereochemical features of hydrogens attached to secondary carbon atoms can be specified relatively to that plane. Graphical abstract